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# PEH:Fluid Flow Through Permeable Media

Publication Information

Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume V – Reservoir Engineering and Petrophysics

Edward D. Holstein, Editor

Chapter 8 – Fluid Flow Through Permeable Media

John Lee, Texas A&M U.

Pgs. 719-894

ISBN 978-1-55563-120-8
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This chapter discusses fluid flow in petroleum reservoirs. Basic concepts, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids, are discussed first. Various flow geometries are treated, including radial, linear, and spherical flow. The pseudosteady-state equations provide the basis for a brief discussion of oil well productivity, and the unsteady-state equations provide the basis for a lengthy discussion of pressure-transient test analysis. For pressure-transient test analysis, semilog techniques, type curves, damage and stimulation, modifications for gases and multiphase flow, the diagnostic plot, bounded reservoirs, average pressure in the drainage area, hydraulically fractured wells, and naturally fractured reservoirs are included. The chapter also discusses transient and stabilized flow in horizontal wells and gas-well deliverability tests. It concludes with considerations of coning in vertical and horizontal wells.

## Basic Concepts

### The Ideal Reservoir Model

Many important applications of fluid flow in permeable media involve 1D, radial flow. These applications are based on a model that includes many simplifying assumptions about the well and reservoir. These assumptions are introduced as needed to combine the law of conservation of mass, Darcy’s law, and equations of state to achieve our objectives.

Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy’s law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation, [1]

....................(8.1)

In the derivation of this equation, it is assumed that compressibility of the total system, ct, is small and independent of pressure; permeability, k , is constant and isotropic; viscosity, μ, is independent of pressure; porosity, ϕ, is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible. The grouping 0.0002637k/ϕμct is called the hydraulic diffusivity and is given the symbol η.

### Line-Source Solution to the Diffusivity Equation

Assume that a well produces at constant reservoir rate, qB; the well has zero radius; the reservoir is at uniform pressure, pi, before production begins; and the well drains an infinite area (i.e., that ppi as r → ∞). Under these conditions, the solution to Eq. 8.1 is[1]

....................(8.2)

where p is the pressure at distance r from the well at time t, and

....................(8.3)

theEi function or exponential integral.

The Ei-function solution is an accurate approximation to more exact solutions to the diffusivity equation (solutions with finite wellbore radius and finite drainage radius) for 3.79 × 105 ϕμctrw2/k < t < 948 ϕμctre2/k. For smaller times, the assumption of zero well size (line source or sink) limits the accuracy of the equation; for larger times, the reservoir’s boundaries affect the pressure distribution in the reservoir, so that the reservoir is no longer infinite acting.

For the argument, x, of theEi function less than 0.01, theEi function can be approximated with negligible error by

....................(8.4)

For x > 10, theEi function is zero for practical applications in flow through porous media. For 0.01 < x < 10,Ei functions are determined from tables or subroutines available in appropriate software. [2]

### Altered Zone and Skin Factor

In practice, most wells have reduced permeability (damage) near the wellbore resulting from drilling or completion operations. Many other wells are stimulated by acidization or hydraulic fracturing. Eq. 8.2 fails to model such wells properly. Its derivation includes the explicit assumption of uniform permeability throughout the drainage area of the well up to the wellbore. Hawkins[3] pointed out that if the damaged or stimulated zone is considered equivalent to an altered zone of uniform permeability. ks, and outer radius, rs, the additional pressure drop, Δps, across this zone can be modeled by the steady-state radial flow equation

....................(8.5)

Eq. 8.5 states that the pressure drop in the altered zone is inversely proportional to ks rather than to k and that a correction to the pressure drop in this region must be made. Combining Eqs. 8.2 and 8.5, we find that the total pressure drop at the wellbore is

....................(8.6)

For r = rw, the argument of theEi function is sufficiently small after a short time that we can use the logarithmic approximation; thus, the drawdown is

....................(8.7)

We can conveniently define a dimensionless skin factor, s, in terms of the properties of the equivalent altered zone:

....................(8.8)

Thus, the drawdown is

....................(8.9)

Eq. 8.9 provides some insight into the physical significance of the algebraic sign of the skin factor. If a well is damaged (ks < k), s will be positive, and the greater the contrast between ks and k and the deeper into the formation the damage extends, the larger the numerical value of s, which has no upper limit. Some newly drilled wells will not flow before stimulation; for these wells, ks = 0 and s → ∞. If a well is stimulated (ks > k), s will be negative, and the deeper the stimulation, the greater the numerical value of s. Rarely does a stimulated well have a skin less than –7, and such skin factors arise only for wells with deeply penetrating, highly conductive hydraulic fractures. If a well is neither damaged nor stimulated (k = ks), s = 0.

The altered zone near a well affects only the pressure near that well; that is, the pressure in the unaltered formation away from the well is not affected by the existence of the altered zone. Thus, use Eq. 8.9 to calculate pressures at the sandface of a well with an altered zone, and Eq. 8.2 to calculate pressures beyond the altered zone in the formation surrounding the well. See Sec. 8.4 for more information on damage and stimulation.

### Inertial-Turbulent Flow and Rate-Dependent Skin

The diffusivity equation, Eq. 8.1, assumes that Darcy’s law represents the relationship between flow velocity and pressure gradients in the reservoir, an assumption that is adequate for low-velocity or laminar flow. However, at higher flow velocities, deviations from Darcy’s law are observed as a result of inertial effects or even turbulent flow effects. In 1D radial flow, these inertial/turbulent effects (often called non-Darcy flow effects) are confined to the region near the wellbore in which flow velocities are largest. This results in an additional pressure drop similar to that caused by skin, but the additional pressure drop is proportional to flow rate. The apparent skin, s′, for a well with non-Darcy flow near the wellbore is given by[4]

....................(8.10)

where D is the non-Darcy flow factor for the system. D can be regarded as constant, although, in theory, it depends slightly on near-well pressure. In practice, non-Darcy flow is ordinarily important only for gas wells, which have high-flow velocities near the wellbore, but it can be important for oil wells with high-velocity flow in some situations.

### Radius of Investigation and Stabilization Time

Radius of investigation is the distance a pressure transient has moved into a formation following a rate change in a well. This distance is related to formation rock and fluid properties and time elapsed since a rate change in the well. Consider this concept by visualizing the pressure distributions at increasing times as Fig. 8.1 shows for a well producing at constant rate from a reservoir initially at uniform pressure. (These pressure distributions were calculated using the Ei-function solution to the diffusivity equation.)

• The pressure in the wellbore, at r = rw, decreases steadily as flow time increases; likewise, pressures at other fixed values of r also decrease as flow time increases.
• The pressure drawdown (or pressure transient) caused by producing the well moves further into the reservoir as flow time increases. For the range of flow times shown, there is always a point beyond which the drawdown in pressure from the original value is negligible. This time-dependent point of "negligible drawdown" can be considered to be a radius of investigation.

Analysis shows that the time, t, at which a pressure disturbance reaches a distance, ri, which is called the radius of investigation, is given by the equation[2]

....................(8.11)

Investigators differ on the numerical constant in Eq. 8.11, but this difference is of little practical importance if the radius of investigation is used as a semiquantitative indicator of the distance into the reservoir to which formation properties have influenced the response of a well in a pressure-transient test.

The radius of investigation has several applications in pressure-transient test analysis and design. A qualitative use is to help explain the shape of a pressure buildup or drawdown curve. For example, a buildup test plot may have a complex shape at early times when the radius of investigation is in the altered zone near the wellbore, where the permeability is different from formation permeability. Or a buildup test plot may change shape at long times when the radius of investigation reaches the general vicinity of a reservoir boundary.

The radius-of-investigation concept provides a guide for well-test design. For example, you may want to sample reservoir properties at least 1,000 ft from a test well. The radius of investigation concept allows you to estimate the time required to achieve the desired depth of investigation.

Eq. 8.11 also provides a means to estimate the time required to achieve "stabilized" flow; that is, the time required for a pressure transient to reach the boundaries of a tested reservoir. For example, if a well is centered in a cylindrical drainage area of radius re, then the time required for stabilization, ts, is

....................(8.12)

For other drainage shapes, the time to stabilization can be quite different, as discussed later.

The Ei-function solution to the radial diffusivity equation is valid only while a reservoir is infinite-acting; that is, until boundaries begin to affect the pressure drawdown at the well. For the constant rate flow of a well centered in its drainage area of radius, re, and modeled by the Ei-function solution, these effects begin at t = 948 ϕμctre2/k. Before these boundary effects, the regime is called unsteady-state flow. After boundary effects are felt fully, the solution to the radial diffusivity equation for a well centered in a cylindrical drainage area and producing at constant rate is[2]

....................(8.13)

This equation for calculating pressure in the wellbore becomes valid for t > 948 ϕμctre2/k at the same time at which the Ei-function solution becomes invalid.

Another form of Eq. 8.13 is useful for some applications. It involves replacing original reservoir pressure, pi, with average pressure, , within the drainage volume of the well. The volumetric average pressure within the drainage volume of the well can be found from material balance. The pressure decrease resulting from removal of qB RB/D of fluid for t hours (a total volume removed of 5.615qBt/24 ft3) is

....................(8.14)

Substituting in Eq. 8.13, the time-dependent terms cancel, and the result is

....................(8.15)

Eqs. 8.13 and 8.15 are more useful in practice if they include skin factors to account for damage or stimulation. In Eq. 8.15,

....................(8.16)

....................(8.17)

and ....................(8.18)

### Productivity Index

The productivity index, J, of an oil well is the ratio of the stabilized rate, q, to the pressure drawdown, , required to sustain that rate. For flow from a well centered in a circular drainage area, Eq. 8.17 allows us to relate productivity index to formation and fluid properties:

....................(8.19)

Thus, if a well is tested at several different stabilized rates and the stabilized flowing bottomhole pressure (BHP), pwf, is measured at each rate (that is, if pseudosteady-state is attained at each rate), Eq. 8.19 implies that a plot of test data should produce a straight line with slope J and intercepts q = 0 when pwf = and when pwf = 0. (See Fig. 8.2.) In practice, actual field data will fall below the theoretical straight line for pressures below the bubblepoint pressure of the oil because of increasing gas saturations and oil viscosities that increase the resistance to flow.

### Generalized Drainage Area Shapes

Eq. 8.17 is limited to a well centered in a circular drainage area. A similar equation models pseudosteady-state flow in more general reservoir shapes[2]:

....................(8.20)

where A is the drainage area in square feet, and CA is the dimensionless shape factor for a specific drainage-area shape and configuration. Table 8.A-1 (Appendix) gives values of CA.

The productivity index, J, can be expressed for general drainage-area geometry as

....................(8.21)

Other numerical constants tabulated in Table 8.A-1 allow us to calculate the maximum elapsed time during which a reservoir is infinite-acting (so that the Ei-function solution can be used), the time required for the for the pseudosteady-state solution to predict pressure drawdown within 1% accuracy, and time required for the pseudosteady-state solution to be exact. For a given reservoir geometry, the maximum time a reservoir is infinite acting can be determined using the entry in the column "Use Infinite System Solution With Less Than 1% Error for tDA <." This tDA is defined as 0.0002637kt/ϕμctA, so this means that the time in hours is calculated from

....................(8.22)

Time required for the pseudosteady-state equation to be accurate within 1% can be found from the entry in the column titled "Less Than 1% Error for t DA., " Finally, the time required for the pseudosteady-state equation to be exact is found in the entry in the column "Exact for tDA >."

Figs. 8.3 and 8.4 show the flow regimes that occur at various times. These figures show pwf in a well flowing at constant rate, plotted as a function of time on both logarithmic and linear scales. In the transient region, the reservoir is infinite acting and is modeled by Eq. 8.9, which implies that pwf is a linear function of log t. In the pseudosteady-state region, the reservoir is modeled by Eq. 8.20 in the general case or Eqs. 8.15 or 8.13 for the special case of a well centered in a cylindrical drainage area. Eq. 8.13 shows a linear relationship between pwf and t during pseudosteady-state flow. This linear relationship also exists in generalized reservoir geometries.

At times between the end of the transient region and the beginning of the pseudosteady-state region, there is a transition region, sometimes called the late-transient region. This region is, for practical purposes, nonexistent for wells centered in circular, square, or hexagonal drainage areas, as Table 8.A-1 indicates. However, for a well off-center in its drainage area, the late-transient region can span a significant time region, as Table 8.A-1 also indicates.

Pseudosteady-state flow describes production from a closed drainage area (one with no-flow outer boundaries, either permanent and caused by zero-permeability rock or "temporary" and caused by production from offset wells). In pseudosteady-state, reservoir pressure drops at the same rate with time at all points in the reservoir, including at the reservoir boundaries. Ideally, true steady-state flow can occur in the drainage area of a well, but only if pressure at the drainage boundaries of the well can be maintained constant while the well is producing at constant rate. While unlikely, steady-state flow is conceivable for wells with edgewater drive or in repeated flood patterns in a reservoir. The solution to the radial diffusivity equation is based on a constant-pressure outer boundary condition, instead of a no-flow outer boundary condition. The steady-state solution, applicable after boundary effects have been felt, is

....................(8.23)

### Constant Pressure in the Well

Both the steady-state solution (constant pressure at the outer boundaries) and the pseudosteady-state solution (no-flow at the outer boundaries) assume constant rate production in the well. A well is actually more likely to be produced at something close to constant flowing BHP than constant rate. When pressure transients reach no-flow drainage area boundaries, the flow regime is not pseudosteady state; instead, it is more correctly called boundary-dominated flow. If the drainage boundaries are maintained at constant pressure, however, steady-state flow is achieved when the pressure transient reaches the reservoir boundaries.

These different flow regimes are clarified with figures showing pressure distributions in the drainage area of wells with constant flow rate and constant-pressure outer boundaries (Fig. 8.5); constant BHP and constant-pressure outer boundaries (also Fig. 8.5); constant flow rate and no-flow outer boundaries (Fig. 8.6); and constant BHP and no-flow outer boundaries (also Fig. 8.6).

### Wellbore Storage

TheEi-function solution to the diffusivity equation assumes constant flow rate in the reservoir, starting at time zero. In practice, only the rate at the surface can be controlled. Under ideal conditions, a constant surface rate can be maintained, but the first fluid produced will be fluid that was stored in the wellbore, and, at first, the flow rate from the reservoir into the wellbore will be zero. As the wellbore is unloaded, the reservoir rate approaches the surface rate (Fig. 8.7). Only as the reservoir and surface rates become approximately equal does theEi-function solution become valid. This wellbore unloading during flow tests is a special case of a general phenomenon called wellbore storage.

For a pressure buildup test, the surface rate is zero starting at the instant of shut-in. However, fluid continues to flow into the wellbore from the reservoir because of existing pressure gradients. Idealized models of pressure buildup tests assume a reservoir rate of zero starting at the time of shut-in for the test. This assumption is obviously violated because of the afterflow into the wellbore. As the afterflow rate diminishes, the downhole rate approaches the surface rate (zero), and only as the afterflow rate approaches zero closely can the idealized models closely approximate actual well behavior (Fig. 8.8). Afterflow during buildup tests is another special case of wellbore storage.

The relationship between changes in bottomhole pressure and wellbore unloading or afterflow rates can be modeled with mass balances on the wellbore. There are two special cases of interest: a wellbore completely filled with a single-phase fluid (Fig. 8.9, usually gas in practice) and a wellbore with a rising or falling liquid/gas interface in the well (Fig. 8.10).

For the wellbore filled with a single-phase fluid, [2]

....................(8.24)

For a well with a rising or falling liquid/gas interface, [2]

....................(8.25)

In most applications, pt is assumed to be constant, a convenient but frequently inaccurate simplification. Both equations can be written in the general form

....................(8.26)

where, for a fluid-filled wellbore,

....................(8.27)

and, for a moving liquid/gas interface with unchanging surface pressure,

....................(8.28)

C is called the wellbore storage coefficient.

For special cases in which, at earliest times for a flowing well, all the production is coming from fluid stored in the wellbore and none is entering the wellbore from the formation (or, for a shut-in well, the rate of afterflow is equal to the rate before shut in), the integration of Eq. 8.26 yields

....................(8.29)

where Δp is the pressure change in the time because either the start of flow or shut in and Δt is the elapsed time. On a log-log plot of Δp vs. Δt during these early times, a straight line with a slope of unity will result. For any point on this unit slope line, the wellbore storage coefficient, C, can be found from any point on the line (Δt, Δp) and Eq. 8.29 (Fig. 8.11). Alternatively, the slope (qB/24C) of a plot of Δp vs. Δt on Cartesian coordinates also leads to an estimate of the wellbore-storage coefficient.

### Linear Flow

Linear flow occurs in some reservoirs with long, highly conductive vertical fractures; in relatively long, relatively narrow reservoirs (channels, such as ancient stream beds); and near horizontal wells during certain times. For unsteady-state linear flow in an unbounded (infinite-acting) reservoir, [2]

....................(8.30)

### Spherical Flow

Spherical flow occurs in wells with limited perforated intervals and into wireline formation test tools. The solution to the spherical/cylindrical, 1D form of the diffusivity equation, subject to the initial condition that pressure is uniform before production and the boundary conditions of constant flow rate and an infinitely large drainage area, is[5]

....................(8.31)

where ....................(8.32)

and rsp = the radius of the sphere into which flow converges.

### Superposition

The principle of superposition indicates that the total pressure at any point in a reservoir is the sum of the pressure drops at that point caused by flow in each of the wells in the reservoir. A simple illustration of this principle is the case of three wells in an infinite reservoir. Consider wells A, B, and C, that start to produce at times tA, tB, and tC in an infinite-acting reservoir (Fig. 8.12). Application of the principle of superposition shows that[2]

....................(8.33)

For an infinite-acting reservoir, use theEi-function solutions, including the logarithmic approximation at Well A:

....................(8.34)

where tA, tB, and tC are times at which wells A, B, and C will begin to produce. The skin factor for Well A is included in Eq. 8.29. The skin factors for other wells are not, because skin factors for individual wells affect only pressures measured inside altered zones for those wells.

Next, consider the use of superposition to model the effects of boundaries in bounded reservoirs. Consider the well in Fig. 8.13, a distance L from a single no-flow boundary (such as a sealing fault). Mathematically, this problem is identical to the problem of a well at distance 2L from an "image" well; that is, a well that has the same production history as the actual well. The reason that the two-well system simulates the behavior of a well near a boundary is that a line equidistant between the two wells can be shown to be a no-flow boundary. That is, along this line the pressure gradient is zero, which means that there can be no flow. Thus, this problem is a simple problem of two wells in an infinite reservoir:

....................(8.35)

The drawdown term of the image well does not include a skin factor.

As examples, extend the imaging technique to model wells between boundaries intersecting at 90° (Fig. 8.14); wells between two parallel boundaries (Fig. 8.15); wells near single constant-pressure boundaries (Fig. 8.16); and wells at various locations in closed reservoirs (Fig. 8.17).

One of the most frequently used applications of superposition is to model variable-rate production. Consider Fig. 8.18, in which a well in an infinite-acting reservoir produces at rate q1 from time 0 to time t1; q2 from t1 to t2, and q3 for times greater than t2. To model the total drawdown for t > t2, add three drawdowns: the drawdown because of a well producing at rate q1 starting at time zero and continuing to produce to time t; the drawdown because of a well producing at rate (q2q1), starting at time t1 and continuing to time t; and the drawdown because of a well producing at rate (q3q2) starting at time t2 and continuing to time t. The total drawdown is thus

....................(8.36)

Horner[6] proposed a convenient alternative to superposition to model the many changes in rate in the history of a typical well. With this approximation, the sequence ofEi functions reflecting rate changes can be replaced with a singleEi function that contains a single producing time and a single producing rate. The single rate is the most recent nonzero rate at which the well has produced, qn. The single producing time, called tp, is the ratio of cumulative production, Np, to qn.

....................(8.37)

This approximation preserves the material balance in the drainage area of the well and properly gives greatest weight to most recent rate (as opposed to average rate), which dominates the pressure distribution near a well out to the radius of investigation achieved while the well was produced at rate qn. The approximation is particularly useful for hand calculations. Given the widespread availability of computer software for analyzing flow and buildup tests on well, the use of more rigorous superposition to model variable-rate production histories is generally more appropriate.

### Semilog Methods for Flow Tests

The logarithmic approximation to theEi-function solution can be used as a basis for analysis of an ideal constant-rate flow test in a well. Written in terms of log10, this equation, which models the BHP for a well in a homogeneous-acting formation with an infinite-acting drainage area and, in absence of wellbore unloading, becomes

....................(8.38)

This expression has the same form as the equation of a straight line, y = mx + b, with the analogies

....................(8.39)

....................(8.40)

....................(8.41)

and ....................(8.42)

These analogies suggest a graphical method of analysis. Eq. 8.38 indicates that a plot of pwf vs. log10(t) should be a straight line with slope m that will allow an estimate of effective permeability to the single liquid phase flowing. (See Fig. 8.19.)

....................(8.43)

From the intercept, b, at t = 1 hr [log10(1) = 0], p1hr, calculate the skin factor.

....................(8.44)

In these equations, the slope, m, is given by

....................(8.45)

Eq. 8.45 indicates that m is most easily determined by choosing values of times t1 and t2 that differ by powers of 10 and is especially easy if t1 and t2 differ by one log cycle. The intercept, p1hr, is the pressure at a time of 1 hour on the best straight line through the data. It may be necessary to extrapolate the straight line to a time of one hour to read the intercept.

### Semilog Methods for Pressure Buildup Test

Consider the rate history for an idealized pressure test shown in Fig. 8.20. A well is produced at constant rate q for a time tp, and then the well shut in (q = 0) for a pressure buildup test. The rate history is modeled as the sum of two constant flow rate periods, one at rate q, beginning at t = 0, and the other at rate –q, beginning at t = tp, at which the time elapsed since shut-in, Δt, is zero. Use the log approximation to theEi-function solution to model the drawdown, and sum them as Fig. 8.21 shows. Represented mathematically, the superposition process is

....................(8.46)

This can be simplified to

....................(8.47)

Like the drawdown equation, Eq. 8.43 can be interpreted as the equation of a straight line. The analogies are

....................(8.48)

....................(8.49)

....................(8.41)

and ....................(8.50)

The group [(tp + Δt)/Δt] is called the Horner time ratio (HTR) or sometimes simply the Horner time. Our simple model, which describes a buildup test in a homogeneous, infinite-acting reservoir, a well with one constant rate before shut in and without afterflow (wellbore storage), indicates that a graph of pws vs. the HTR should fall on a straight line. From the slope, m, of this line, the permeability to the single-phase liquid flowing into the wellbore can be estimated. The intercept, b, at log 10 [(tp + Δt)/Δt] = 0 or [(tp + Δt)/Δt] = 1 provides an estimate of original drainage area pressure, pi.

Obtain the slope, m, from

....................(8.51)

where {[ (tpt)/Δt]1, pws1} and {[ (tpt)/Δt]2, pws2} are any two points on the straight-line (Fig. 8.22). Normally, choose [tp + Δt)/Δt]1 and [(tp + Δt)/Δt]2 to be powers of 10.

In Fig. 8.22, which is called a Horner plot, the HTR on the horizontal axis decreases from left to right, so that shut-in time increases from left to right. In some Horner plots, the HTR increases from left to right; in that case shut-in time increases from right to left.

Skin factor can be estimated from a pressure buildup test, even though the skin factor does not appear in the buildup equation, Eq. 8.47. Simultaneously solve the equation modeling the drawdown at the instant of shut in (at time tp) with Eq. 8.47, discard terms that are ordinarily negligible, and arrive at the result

....................(8.52)

The radius-of-investigation concept is also useful for pressure buildup tests, as Fig. 8.23 illustrates. The approximate position of the point at which the pressure has built up to a uniform level intersects the region in which the pressure is little affected by the shut-in is given by Eq. 8.11, with elapsed time, t, interpreted as shut-in time, Δt.

## Type Curves

Type curves provide a powerful method for analyzing pressure drawdown (flow) and buildup tests. Fundamentally, type curves are preplotted solutions to the flow equations, such as the diffusivity equation, for selected types of formations and selected initial and boundary conditions. Because of the way they are plotted (usually on logarithmic coordinates), it is convenient to compare actual field data plotted on the same coordinates to the type curves. The results of this comparison frequently include qualitative and quantitative descriptions of the formation and completion properties of the tested well.

### Dimensionless Variables

The solutions plotted on type curves are usually presented in terms of dimensionless variables. To review dimensionless variables, consider theEi-function solution to the flow equation, Eq. 8.2, presented in terms of dimensional variables:

....................(8.2)

Eq. 8.2 can be rewritten in terms of conventional definitions of dimensionless variables. (Variables that when the parameters are expressed in terms of the fundamental units of mass, length, and time, have no dimensions are sometimes said to have dimensions of zero.)

....................(8.53)

In Eq. 8.53, the definitions of the dimensionless variables are

....................(8.54)

....................(8.55)

and ....................(8.56)

The dimensionless form of Eq. 8.2 has the advantage that this solution, pD, to the diffusivity equation can be expressed in terms of a single variable, tD, and single parameter, rD. This leads to much simpler graphical or tabular presentation of the solution than would direct use of Eq. 8.2. Solutions to the diffusivity equation for more realistic reservoir models also include the dimensionless skin factor, s, and wellbore storage coefficient, CD, where

....................(8.57)

### Gringarten Type Curve

Gringarten et al.[7] presented a type curve, commonly called the Gringarten type curve, that achieved widespread use. It is based on a solution to the radial diffusivity equation and the following assumptions: vertical well with constant production rate; infinite-acting, homogeneous-acting reservoir; single-phase, slightly compressible liquid flowing; infinitesimal skin factor (thin "membrane" at production face); and constant wellbore-storage coefficient. These assumptions indicate that the type curve was developed specifically for drawdown tests in undersaturated oil reservoirs. The type curve is also useful to analyze pressure buildup tests and for gas wells.

In the Gringarten type curve, pD is plotted vs. the time function tD/CD, with a parameter CDe2s (Fig. 8.24). Each different value of CDe2s describes a pressure response with a shape different (in theory) from the responses for other values of the parameter. However, adjacent pairs of curves can be quite similar, and this fact can cause uncertainty when trying to match test data to the "uniquely correct" curve.

### Derivative Type Curve

The derivative type curve proposed by Bourdet et al.[8] eliminates the ambiguity in the Gringarten type curve. The "derivative" referred to in this type curve is the logarithmic derivative of the solution to the radial diffusivity equation presented on the Gringarten type curve. Two limiting forms of this solution help illustrate the nature of the derivative type curve. First, consider that part of a test response where the distorting effects of wellbore storage have vanished. This portion of the test is described by the logarithmic approximation toEi-function solution, Eq. 8.9:

....................(8.9)

The derivative of (pipwf) with respect to ln(t), expressed more simply as t∂Δp/∂t, is 70.6qBμ/kh, a constant. In terms of dimensionless variables, tD(∂pD/∂tD) = 0.5. Thus, when the distorting effects of wellbore storage have disappeared, the pressure derivative will become constant in an infinite-acting reservoir, and, in terms of dimensionless variables, will have a value of 0.5.

When wellbore storage completely dominates the pressure response (all produced fluid comes from the wellbore, none from the formation),

....................(8.29)

The derivative, t∂Δp/∂t, is qBt/24C, the same as the pressure change itself. In terms of dimensionless variables, the derivative becomes

....................(8.58)

The implication of Eq. 8.58 is that, on logarithmic coordinates, graphs of pD and tD(∂pD/∂tD) vs. tD/CD will coincide and will have slopes of unity.

For values of tD(∂pD/∂tD) between the end of complete wellbore storage distortion and the start of infinite-acting radial flow, no simple solutions are available to guide us, but Fig. 8.25 shows the derivatives, including those times. Note the unit slope lines at earliest times and the horizontal derivative at later times. The shapes of the derivative stems are much more distinctive than those for the pressure-change type curve.

For test analysis, we plot pressure change, pD, and pressure derivative [tD(∂pD/∂tD)] on the same graph (Fig. 8.26). On this graph, a specific value of the parameter CDe2s refers to a pair of curves—one pressure-change curve and one pressure-derivative curve. Time regions can be defined conveniently on the basis of the combined pressure (Fig. 8.27) and pressure derivative type curves.

The shape of the pressure- and pressure-derivative type curves provides a qualitative estimate of skin factor (Fig. 8.28). For a well with a large skin factor, the derivative rises to a maximum and then falls sharply before flattening out for the middle-time region (MTR). The pressure change curve rises along the unit-slope line and then flattens quickly. The pressure-change and pressure-derivative curves are separated by approximately two log cycles when wellbore storage (WBS) ends.

When the skin is near zero, the pressure derivative rises to a maximum and then falls only slightly before flattening for the MTR. The pressure change and pressure derivative are separated by approximately one log cycle when WBS ends. When the skin factor is negative, the pressure derivative approaches a horizontal line from below. The pressure change and pressure derivative curves leave the unit slope line at relatively early times and take a relatively long time to reach the MTR.

### Differences in Drawdown and Buildup Test Type Curves

The shapes of drawdown and buildup type curves are different, as Fig. 8.29 illustrates. In this simplified case, in which wellbore storage distortion is absent, a well has produced for a dimensionless producing time, tpD, of 105, before shut-in. In the figure, note that, on a plot of pD and pD′ (the derivative) vs. tD (dimensionless time since each test began), the shapes of the buildup and drawdown curves for infinite-acting radial flow coincide up to tD = 104 and then begin to deviate. The buildup pressure-change curve is "flatter" than the drawdown curve at later times in an infinite-acting reservoir, and thus the slope of the buildup curve (the derivative) tends to deviate from the drawdown derivative. For many years, test analysts used a rule of thumb that buildup tests could be analyzed on a drawdown type curve only up to a maximum time of one-tenth the producing time before shut-in. That rule of thumb is appropriate for the conditions in Fig. 8.29.

### Equivalent Drawdown Time

Agarwal[9] suggested a method of plotting pressure change data from a buildup test on a logarithmic graph that alters the shape so that it corresponds to that of a constant rate flow test during infinite-acting radial flow. The basis for Agarwal’s "equivalent time" is a combination of logarithmic approximations toEi-function solutions to the diffusivity equation. The equation modeling the drawdown at the instant of shut-in is

....................(8.59)

We model a buildup test with

....................(8.60)

Combining Eqs. 8.59 and 8.60 and simplifying,

....................(8.61)

which can be rewritten as

....................(8.62)

The forms of Eqs. 8.62 and 8.59 are the same; thus Eq. 8.62 is an "equivalent" drawdown equation, with the equivalent pressure change, (pwspwf), a function of equivalent time, Δte = tpΔt/(tp + Δt). The analogies between these equations suggest that, just as Δp = pipwf vs. t were plotted for drawdown tests, Δp = pwspwf vs. Δte can be plotted for buildup tests and achieve the same shapes on logarithmic graphs. However, the theoretical basis for this radial-equivalent time indicates that the equivalence exists only for infinite-acting radial flow and not for data influenced by wellbore storage or by effects of boundaries or other conditions that cause the flow pattern to deviate from radial. In practice, buildup test data for infinite-acting radial flow, including data distorted by wellbore storage, are transformed to the same shape as drawdown test data. However, data affected by boundaries or by linear flow (as in wells with hydraulic fractures) may not be transformed accurately.

Radial equivalent time has the properties

....................(8.63)

### Type-Curve Matching

The steps in type-curve matching for wells with infinite-acting radial flow are outlined here. Details vary for more complex reservoirs, but the general procedure is similar to that for infinite-acting reservoirs.

• Plot field data on log-log coordinates with the same size log cycles as the type curve.
• Align the horizontal sections of the field data and the type curve.
• Align unit slope regions on the field data and the type curve.
• Select the value of CDe2s that best matches the field data.
• Select pressure and time match points (corresponding values of real and dimensionless variables from field data and type curve plots) from anywhere on the plot.
• Calculate permeability from the pressure match-point ratio,
....................(8.64)
or ....................(8.65)
• Calculate CD from the time match-point ratio,
....................(8.66)
or ....................(8.67)
• Calculate s from the matching stem value, CDe2s:;....................(8.68)

Fig. 8.30 shows an example interpretation of match points. In practice, this matching and match-point interpretation procedure is done on the computer and monitor, and much of the process is transparent to the analyst.

## Damage and Stimulation

### Causes of Formation Damage

The causes of formation damage that lead to positive skin factors include damage caused by drilling-fluid invasion, production, or injection.

When mud filtrate invades the formation surrounding a borehole, it will generally remain in the formation even after the well is cased and perforated. This mud filtrate in the formation reduces the effective permeability to hydrocarbons near the wellbore. It may also cause clays in the formation to swell, reducing the absolute permeability of the formation. In addition, solid particles from the mud may enter the formation and reduce permeability at the formation face.

The production process may also reduce permeability and introduce a positive skin factor. For example, in an otherwise undersaturated oil reservoir, pressure near the well may be below the bubblepoint pressure, causing a free-gas saturation and reducing the effective permeability to oil. In a retrograde gas reservoir, the pressure near the wellbore may drop below the dewpoint and an immobile liquid phase may form and reduce the effective permeability to gas near the wellbore.

Injection can also cause damage. The water injected may be dirty; that is, it may contain fines that may plug the formation and reduce permeability. In other cases, the injected water may be incompatible with the formation water, causing solids to precipitate and plug the formation. In still other cases, the injected water may be incompatible with clays in the formation (e.g., fresh water can destabilize some clays, causing fines to migrate and plug the formation).

### Altered Zone and Skin Effect

A two-region reservoir model (Fig. 8.31) is a convenient representation of a damaged well (and some stimulated wells with radially symmetric permeability alteration around the wellbore). In this model, the altered zone around the wellbore is assumed to have uniform permeability ks out to a radius rs, beyond which the formation permeability, k, is unaltered.

For a damaged well, the reduced permeability in the altered zone causes an additional pressure drop, Δps (Fig. 8.32). The dimensionless skin factor, s, and the additional pressure drop across the altered zone are related by

....................(8.69)

For a well with a known skin factor, s, Eq. 8.69 provides a method of translating the somewhat abstract dimensionless skin factor into a more concrete characterization of the practical effect of damage or stimulation.

In a two-region reservoir model, the skin factor, s, is related to the properties of the altered zone:

....................(8.70)

Rearrange Eq. 8.70 and solve for the permeability of the altered zone:

....................(8.71)

Rearrangements of Eq. 8.70 provide a second method of translating skin into a more concrete characterization of a well with altered permeability near the wellbore. If the depth of damage can be estimated for a well with a known skin factor, s, the permeability of the altered zone can be estimated. Even if the depth of permeability alteration, rs, is estimated Eq. 8.71 can still provide a reasonable estimate of altered zone permeability because rs appears in a logarithmic term. Alternatively, an estimate of the permeability reduction ratio (for example, from laboratory tests on cores) can produce an estimate of the depth of damage from another rearrangement of Eq. 8.70,

....................(8.72)

A third method of translating skin to a more concrete characterization of near-well conditions is to calculate apparent or effective wellbore radius, rwa. Apparent wellbore radius is defined as

....................(8.73)

or ....................(8.74)

For a stimulated well, the pressure drawdown at the wellbore is the same as it would be in a formation with unaltered permeability but with wellbore radius equal to the apparent wellbore radius. This concept has value in some simulation applications. Note that rwa can be calculated from the actual wellbore radius and skin factor.

Eqs. 8.73 and 8.74 are also useful to illustrate the minimum (i.e., the most-negative possible) skin factor. This minimum skin, smin, occurs when the apparent wellbore radius is equal to the drainage radius of the well:

....................(8.75)

For a well with a circular drainage area of 40 acres for which re is 745 ft and a wellbore radius of 0.3 ft, the minimum skin (maximum stimulation) is smin = - ln(re/rw) = −(745/0.3) = −7.82. Such a skin implies increasing the permeability throughout the entire altered zone to infinity—clearly an idealistic "upper limit." More realistically, research[10] has shown that the half-length, Lf, of a highly conductive vertical fracture is related to rwa by

....................(8.76)

or ....................(8.77)

Thus, for Lf = re = 745 ft, s = −7.12 is a more realistic minimum (for the given drainage radius and wellbore radius).

### Flow Efficiency

A fourth way to characterize a well with nonzero skin is to calculate the flow efficiency of the well. Flow efficiency, Ef, is defined as the ratio of the actual productivity index of the well (including skin) to the ideal productivity index if the skin factor were zero. Because the productivity index is the ratio of stabilized flow rate to pressure drop required to sustain that stabilized rate,

....................(8.78)

....................(8.79)

and ....................(8.80)

For a well with neither damage nor stimulation, Ef = 1; for a damaged well, Ef < 1; and for a stimulated well, Ef > 1.

### Geometric Skin

When the area open to flow decreases, the pressure drop is greater than when the area is unchanged all the way to the formation face. Examples include flow converging to perforations (Fig. 8.33), partial penetration (Fig. 8.34), and an incompletely perforated interval (Fig. 8.35).

Fig. 8.33 illustrates flow converging into perforations. When the perforation spacing is too large, this converging flow results in a positive skin factor. The skin increases as vertical permeability decreases and increases as shot density decreases.

Partial Penetration. Fig. 8.34 illustrates flow converging into an interval that is only partly penetrated by perforations. When a well is completed in only a fraction of the productive interval, the flow must converge through a smaller area, increasing the pressure drop near the well (compared to a fully completed interval). The additional pressure drop near the well results in a more positive skin. It increases as the vertical permeability decreases and as the perforated interval as a fraction of the total interval decreases. Formation damage (reduced permeability) near the completion face can significantly increase the additional pressure drop and thus the calculated skin factor.

Incompletely Perforated Interval. Partial penetration is a special case of an incompletely perforated interval (Fig. 8.35). In the general case, the well is perforated starting at a distance h1 from the top of the productive interval and has perforations extending over a distance, hp, in an interval of total thickness, h. The total skin for the well in this general situation is

....................(8.81)

In Eq. 8.81, sd is the skin caused by formation damage, and s p is the skin resulting from an incompletely perforated interval. This equation is not valid for a stimulated well.

The skin factor for an incompletely perforated interval, sp, can be quantified by[11]

....................(8.82)

where ....................(8.83)

....................(8.84)

....................(8.85)

....................(8.86)

and ....................(8.87)

The most significant limitation in applying Eq. 8.82 in practice is the difficulty in estimating accurately the vertical-to-horizontal-permeability ratio, kv/kh. Fortunately, this ratio appears only in a logarithmic term in Eq. 8.82, so errors will not seriously distort the calculated value of sp.

Deviated Well. For a deviated well (Fig. 8.36), which penetrates the formation at an angle other than 90°, more surface is in contact with the formation. This introduces a negative skin factor, sθ, which makes the total skin factor, s, more negative.

....................(8.88)

The effect increases as the vertical permeability increases and increases as the angle from the vertical, θw, increases. The deviated well skin factor, sθ, is given by a correlation of simulated results[12] (valid for θw < 75°):

....................(8.89)

where ....................(8.90)

and ....................(8.91)

Gravel-Pack Skin. When a well is gravel packed (Fig. 8.37), there is a pressure drop through the gravel pack within the perforations, given by[13]

....................(8.92)

where sgp is the skin factor because of Darcy flow through the gravel pack; h, the net pay thickness, ft; kgp, the permeability of the gravel in the gravel pack, md; k, the reservoir permeability, md; Lg, the length of the flow path through the gravel pack, ft; n, the number of perforations open; and rp, the radius of the perforation tunnel, ft. Eq. 8.92 does not include the effects of non-Darcy flow, which may be extremely important in high-rate gas wells.

Completion Skin. For a perforated well, any reduced permeability, kdp, in the zone surrounding the perforations (Fig. 8.38) introduces an additional pressure drop. The additional skin is[14]

....................(8.93)

and ....................(8.94)

where sp is the geometric skin from flow converging to the perforations; sd, the damage skin; sdp, perforation damage skin; kd, permeability of the damaged zone around the wellbore, md; kdp, permeability of the damaged zone around perforation tunnels, md; k, reservoir permeability, md; Lp, length of perforation tunnel, ft; n, number of perforations; h, formation thickness, ft; rd, radius of the damaged zone around the wellbore, ft; rdp, radius of the damages zone around the perforation tunnel, ft; rp, radius of the perforation tunnel, ft; and rw, wellbore radius, ft. Eq. 8.94 does not include the effects of non-Darcy flow.

Hydraulically Fractured Wells. Wells are frequently fractured hydraulically to improve their productivity, especially in low-permeability formations where fractures increase the effective drained area and in high-permeability formations where they penetrate near-well damage or promote sand control. These fractures, almost always vertical (Fig. 8.39), are high-conductivity paths between the reservoir and the wellbore. If the fracture conductivity is large enough relative to the formation permeability and fracture length, the pressure drop within the fracture will be negligible. This distributes the pressure drop caused by fluid influx into the wellbore over a much larger area, resulting in a negative skin factor, which is interpreted as a geometric skin.

Dimensionless fracture conductivity, Cr, is defined by

....................(8.95)

where wf is the fracture length, ft; kf, the permeability of the proppant in the fracture; k, the formation permeability, md; and Lf, the fracture half-length, ft. Pressure drop in the fracture is negligible for Cr > 100.

## Modifications for Gases and Multiphase Flow

### Diffusitivity Equation for Gas Flow

The diffusivity equation for liquids, Eq. 8.1,

....................(8.1)

was derived from three principles: conservation of mass, the equation of state for slightly compressible liquids, and Darcy’s law. This form of the diffusivity equation is linear, which makes solutions (such as theEi-function solution) much easier to find and which allows us to use superposition in time and space to develop solutions for complex flow geometries and for variable rate histories from simple, single-well solutions.

### Pseudopressure

Other forms of the equation for flow of gases must be developed because the equation of state for a slightly compressible liquid will not be applicable. First, introducing the real gas law,

....................(8.96)

to replace the slightly compressible equation of state results in a more complex, nonlinear partial differential equation. This equation can be partially linearized by introducing the pseudopressure transformation, [15]

....................(8.97)

where p0 is an arbitrary "base" pressure, frequently chosen to be zero psia. The resulting form of the diffusivity equation is

....................(8.98)

Eq. 8.98 has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, pp. However, this equation is nonlinear because the product μct is a strong function of pressure. Fortunately, research has shown that the equation can be treated as linear, and theEi-function is valid for gases if μct is evaluated at the pressure at the beginning of a flow period until the time when boundaries begin to have a significant influence on the pressure drop at the well; that is, as long as the reservoir is infinite-acting.

### Pressure-Squared and Pressure Approximations

By assuming that the product μz is constant, then, from Eq. 8.97, pseudopressure becomes

....................(8.99)

and the diffusivity equation becomes

....................(8.100)

The independent variable has become p2, and, in terms of this variable, theEi-function solution is valid when the assumption that μz is constant is valid. This is true (based on empirical evidence) even though Eq. 8.100 is nonlinear (pressure-dependent μct), but it is valid only for an infinite-acting reservoir.

Fig. 8.40 shows the range of validity of this assumption for a reservoir temperature of 200°F and several different gas gravities. The μz product is fairly constant at pressures below approximately 2,000 psia (the shaded area in the figure). Conclusions are similar at other temperatures from 100 to 300°F.

By assuming that the group p/μz is constant, from Eq. 8.97, pseudopressure becomes

....................(8.101)

and the diffusivity equation becomes

....................(8.102)

The independent variable has become p, and, in terms of pressure, theEi-function is valid (from empirical evidence) when the assumption that p/μz is constant is valid. This is true even though Eq. 8.102 is nonlinear (pressure-dependent μct) , but is valid only for an infinite-acting reservoir.

Fig. 8.41 shows the range of validity of this assumption (shaded area in the figure) for a reservoir temperature of 200°F and several different gas gravities. The group p/μz is fairly constant at pressures above approximately 3,000 psia as it is at other temperatures from 100 to 300°F.

The implication of these results is that the choice of variable for gas well-flow equations depends on the situation. The pressure-squared approximation is valid only for low pressures (p < 2,000 psia), the pressure approximation is valid only for high pressures (p > 3,000 psia), and the pseudopressure transformation is valid for all pressure ranges. For pressure transient test analysis using software, the pseudopressure is almost always the optimal variable to use. For hand analysis, only pressure or pressure-squared approaches are feasible.

### Pseudotime

Although the diffusivity equation written for gas flow has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, it is a nonlinear equation because the product, μct, is strongly pressure dependent. In some cases, the remaining nonlinearity cannot be ignored. To solve this problem, Agarwal[16] introduced the pseudotime transformation to further linearize the diffusivity equation for gas. (The linearization is not rigorous, but is adequate for many practical purposes. [17]) The definition of pseudotime is

....................(8.103)

In terms of pseudotime, tap, the diffusivity equation becomes

....................(8.104)

Subsequent studies[18] have shown that the pseudotime transformation is particularly useful for analysis of flow and buildup tests distorted by wellbore storage when using type curves designed to model flow of slightly compressible liquids.

Because the pressure in the integrand of Eq. 8.103 is a function of position in the reservoir, it is not obvious where the pressure is to be evaluated. Empirical observations[18] indicate that the pressure should be evaluated at BHP during wellbore storage distortion for both buildup and flow tests. During the middle time region for buildup tests, it should be evaluated at BHP, and, for flow tests, at the average reservoir pressure at the start of the test. For flow tests in infinite-acting reservoirs, this is equivalent to using ordinary time as the independent variable.

### Normalized Transformed Variables

The pseudopressure and pseudotime transformations provide excellent results when used as part of the analysis procedure for gas well tests. However, they are inconvenient for two reasons: the values of both variables will often be in the range of 105 to 109, and they do not have units of actual pressure and time. Thus, the intuitive "feel" for the transformed variables is lost, and they may tend to be regarded as "black box" output—never helpful in test analysis. The use of pseudopressure and pseudotime require different test interpretation equations for oil wells than for gas wells.

These difficulties are overcome by normalizing pseudopressure and pseudotime by multiplying them by constants[19]:

....................(8.105)

and ....................(8.106)

This normalization, or multiplication by appropriate constants, gives the new variables the same units—and similar ranges—as pressure and time, respectively. With these transformations, the equations for analysis of gas wells in terms of normalized pseudopressure and pseudotime, which are called adjusted pressure and adjusted time, are obtained from the equations for analysis of oil well tests by simple substitution. Of course, the transformations require the computer. Commercial well-test analysis software often provides these transformations.

Table 8.1 summarizes plotting methods and interpretation equations for oil well tests. It also presents information for gas well tests analyzed with ordinary pressure and time, adjusted pressure and time, pressure squared and time, and, finally, pseudopressure and time. The table includes a definition of pDMBH, a dimensionless pressure defined by Matthews, Brons, and Hazebroek[20] that is useful in estimating current average drainage pressure. See this topic in Section 8.8.

In Table 8.1, the HTR for gas well buildup tests is best estimated to be simply (tp + Δta)/Δta. This conclusion is based on the findings of Spivey and Lee. [18] Thus, when using adjusted pressure and time, the HTR is calculated using the actual producing time,tp.

### Non-Darcy Flow

The flow equations shown to this point assume that Darcy’s law is an appropriate model for gas flow into wells. However, as the flow velocity and Reynolds number near the well increase, the result is a transition from laminar and turbulent flow and then to turbulent flow. This transitional (and possibly turbulent) flow is called non-Darcy (non-laminar) flow. The high velocities at which the flow is transitional occur in the immediate vicinity of the well, and the additional pressure drop caused by this transitional flow is similar to a zone of altered permeability that is characterized with a skin factor. In the case of non-Darcy flow, however, the additional "skin effect" caused by the deviations from Darcy’s law is rate dependent.

An adequate model for the apparent skin factor, s′, determined from a flow or buildup test is

....................(8.107)

In Eq. 8.107, s is the "true" skin because of damage or stimulation; D is a non-Darcy flow coefficient (assumed constant), with units of D/Mscf; and qg is the gas flow rate with units of Mscf/D. The absolute value of the gas rate is used because the contribution to the skin is positive regardless of whether the gas well is a producer or an injector.

The true skin for a gas well cannot be obtained from information in a single test conducted at constant rate (including a buildup test following constant-rate production). However, skin calculated from tests conducted at several different rates (for example, associated with a multipoint deliverability test on a well) can be used to determine the true skin and the non-Darcy flow coefficient. Fig. 8.42 illustrates the process for a well tested at three different rates, with an apparent skin factor determined at each rate.

The apparent skin factor extrapolated to zero rate is the true skin (in this case, 3.4), and the slope of the curve is the non-Darcy flow coefficient, D (in this case, 5.1×10–4 D/Mscf). When this method is used, take care to ensure that the permeabilities obtained from the different tests are the same; otherwise, the skin factors will be inconsistent and erroneous.

Often, only one test is available. In this case, the non-Darcy flow coefficient, D, can be estimated from[4]

....................(8.108)

The turbulence parameter, β, can be estimated from[21]

....................(8.109)

The correlation represented by Eq. 8.109 will provide only a crude estimate of the turbulence parameter, β. Further, the correlation assumes that the non-Darcy flow occurs in the formation near the wellbore rather than through the perforations. In a gravel-packed well, the most significant additional pressure drop caused by non-Darcy flow may occur in the perforation channels through the casing.

### Multiphase Flow

The equations modeling flow in reservoirs can be modified to include multiphase flow. Perrine[22] suggested simple and easily applied modifications and Martin[23] gave them a theoretical basis. These modifications are based on the simplifying assumption that the saturation gradients in the drainage area of the tested well are small. Thus, as examples, the modifications may lead to reasonable approximations for solution-gas drive reservoirs and are inappropriate for water-drive reservoirs with a water bank (and saturation discontinuity) in the drainage area of the tested well. The Perrine-Martin modification for constant-rate flow in an infinite-acting reservoir is

....................(8.110)

and the Horner equation modeling a buildup test in an infinite-acting reservoir becomes

....................(8.111)

In Eqs. 8.110 and 8.111, qRt represents the total reservoir flow rate (RB/D) and is given by

....................(8.112)

and λt represents the total mobility, given by

....................(8.113)

The total mobility, λt, can be determined from a pressure buildup test run on a well that produces two or three phases simultaneously. Because Eq. 8.111 implies that λt is related to the slope, m, of a Horner plot of pws vs. log(tp+ Δt)/Δt by

....................(8.114)

The slope, m, of a plot of pwf vs. log(t) data from a constant-rate flow test has the same interpretation. Perrine[22] also showed that the permeability to each phase flowing can be estimated from the relations

....................(8.115)

....................(8.116)

and ....................(8.117)

The quantity (qgqoRs/1,000)Bg in Eqs. 8.112 and 8.116 is the free-gas flow rate in the reservoir; that is, the difference in the total gas rate, qg, and the dissolved gas rate, qoRs/1,000. Skin factor for multiphase flow test analysis using semilog plots is calculated from

....................(8.118)

For analysis of tests using type curves, note that the pressure match point on a type curve is related to total and individual phase mobilities and rates by

....................(8.119)

and the time match point is related to the dimensionless storage coefficient by

....................(8.120)

The practical implication of Eqs. 8.119 and 8.120 is that total mobility and individual phase permeability are determined from the pressure-match point on a type-curve match. The dimensionless storage coefficient is determined from the time-match point resulting in the calculation of skin factor from

....................(8.121)

just as for single-phase flow. When the conditions for applicability of the Perrine-Martin approximations (small saturation gradients in the drainage area of the tested well) are not satisfied, use of a reservoir simulator for test analysis is an appropriate alternative.

## Diagnostic Plot

### Introduction

The diagnostic plot is a log-log plot of the pressure change and pressure derivative (vertical axis) from a pressure transient test vs. elapsed time (horizontal axis). Fig. 8.43 shows an example. The diagnostic plot can be divided into three time regions: early, middle, and late. At the earliest times on a plot (the early-time region), wellbore and near-wellbore effects dominate. These effects include wellbore storage, formation damage, partial penetration, phase redistribution, and stimulation (hydraulic fractures or acidization). At intermediate times (the middle-time region), a reservoir will ordinarily be infinite acting. For a homogeneous reservoir, the pressure derivative will be horizontal during this time region. Data in this region lead to the most accurate estimates of formation permeability. At the latest times in a test (the late-time region), boundary effects dominate curve shapes. The types of boundaries that may affect the pressure response include sealing faults, closed reservoirs, and gas/water, gas/oil, and oil/water contacts. Several common flow regimes and the diagnostic plots associated with these flow regimes are discussed in the remainder of Section 8.6.

### Volumetric Behavior

Volumetric behavior is defined as that pressure response time dominated by the wellbore, reservoir, or part of the reservoir acting like a uniform-pressure "tank" with fluid entering or leaving the tank. The most common example of volumetric behavior is wellbore storage, which dominates during the early-time region. The "tank" is the wellbore, in which the pressure is uniform. Fluid either leaves this tank (earliest times in a flow test, before the reservoir begins to respond) or enters the tank (earliest times in a buildup test). Another example is pseudosteady-state (boundary-dominated) flow in a closed reservoir during constant-rate production. In this case, the reservoir is the tank; pressure is changing at the same rate throughout (although it is not the same at all points), and fluid is leaving the reservoir through the producing well. As a final example, in a test the reservoir may behave like a tank with recharge (fluid influx) entering from a secondary source of pressure support, such as a large supply of hydrocarbons in a lower-permeability medium in pressure communication with the reservoir being tested.

The equation modeling wellbore storage (derived from a mass balance on the wellbore) is

....................(8.29)

The equation modeling pseudosteady-state flow in a cylindrical drainage area is

....................(8.18)

The general form is

....................(8.122)

The derivative of the general form is

....................(8.123)

The implication is that the derivative plot will have unit slope (up one log cycle as it moves over one log cycle) on log-log coordinates, and the pressure change plot will approach unity at long times when b v is not equal to zero (Fig. 8.44). In wellbore storage, bv is zero, and the derivative and pressure change plots will lie on top of one another. During pseudosteady-state flow or recharge, the pressure change and pressure derivative plots will not coincide.

Infinite-acting radial flow is common in reservoirs, and data in the radial flow regime can be used to estimate formation permeability and skin factor. Common situations in which radial flow occurs include flow into vertical wells after wellbore storage distortion has ceased and before boundary effects, hydraulically fractured wells after the transient has moved well beyond the tips of the fracture, horizontal wells before the transient has reached the top and bottom of the productive interval, and horizontal wells after the transient has moved beyond the ends of the wellbore.

The equation used to model radial flow for a well producing at constant rate is the familiar logarithmic approximation to the line-source solution,

....................(8.124)

Equations modeling radial flow have the general form

....................(8.125)

with derivative

....................(8.126)

On the diagnostic plot (Fig. 8.45), radial flow is indicated by a horizontal derivative.

### Linear Flow

Linear flow is also common and occurs in channel reservoirs, hydraulically fractured wells, and horizontal wells. Data from linear flow regimes can be used to estimate channel width or fracture half-length if an estimate of permeability is available. In horizontal wells, an estimate of permeability perpendicular to the well can be made if the productive well length open to flow is known.

An equation that models linear flow in a channel reservoir of width w is

....................(8.127)

For a hydraulically fractured well with fracture half-length Lf,

....................(8.128)

The general form is

....................(8.129)

The derivative is

....................(8.130)

Linear flow on the diagnostic plot is indicated when a derivative follows a half-slope line—that is, a line that moves up vertically by one log cycle for each two cycles of horizontal movement (Fig. 8.46). The pressure change may or may not also follow a half-slope line. In a hydraulically fractured well, the pressure change will follow a half-slope line unless the fracture is damaged. In a channel reservoir, a hydraulically fractured well with damage, or a horizontal well, the pressure change will approach the half-slope line from above.

### Bilinear Flow

Bilinear flow occurs primarily in wells with low-conductivity hydraulic fractures. Flow is linear within the fracture to the well, and also linear (normal to fracture flow) from the formation into the fracture. Estimates of fracture conductivity, wfkf, can be made with data from this flow regime when estimates of formation permeability are available.

For a hydraulically fractured well, an equation that models bilinear flow is

....................(8.131)

The general form is

....................(8.132)

The derivative is

....................(8.133)

Bilinear flow derivatives plot as a quarter-slope line on the diagnostic plot (Fig. 8.47). The quarter-slope line moves up one log cycle as it moves over four log cycles. The pressure change does not necessarily follow a quarter-slope line. In a damaged, hydraulically fractured well, the pressure change curve will approach the quarter-slope line from above; in an undamaged hydraulically fractured well (Δps = 0), the pressure change will typically follow the quarter-slope line when the effects of wellbore storage have ended.

### Spherical Flow

The flow pattern is spherical when the pressure transient can propagate freely in three dimensions and converge into a "point." This can occur for wells that penetrate only a short distance into the formation (actually hemispherical flow), wells that have only a limited number of perforations open to flow, horizontal wells with inflow over only short intervals, and during wireline formation tests. Data in the spherical-flow regime can be used to estimate the mean permeability,

....................(8.134)

An equation that models spherical flow is

....................(8.31)

where ....................(8.32)

and rsp is the radius of the sphere into which flow converges. The general form is

....................(8.135)

and the derivative is

....................(8.136)

Spherical flow on the diagnostic plot produces a derivative line with a slope of −1/2. The pressure change during spherical flow approaches a horizontal line from below, and never exhibits a straight line with the same slope as the derivative (Fig. 8.48). Spherical flow can occur during either buildup or drawdown tests.

### Flow Regimes on the Diagnostic Plot

A major application of the diagnostic plot is the potential that it provides in identifying the flow regimes that appear in a logical sequence during a buildup or flow test. For example, consider Fig. 8.49. At early times, the unit slope line on both derivative and pressure change, indicating wellbore storage. Later, a derivative with a slope of −1/2, indicating possible spherical flow, followed by a horizontal derivative, indicating infinite-acting radial flow. Boundary effects, including a unit-slope line, follow, indicating possible recharge of the reservoir pressure.

## Behavior of Bounded Reservoirs

### Introduction

Reservoir boundaries have significant influences on the shape of the diagnostic plot. The effects of boundaries appear following the middle-time region (infinite-acting radial flow) in a test. Recognizing the influence of boundaries can be as important as analyzing the test quantitatively. However, a problem in recognition is that many reservoir models may produce similar pressure responses. The model selected to interpret the test quantitatively must be consistent with geological and geophysical interpretations. Once the proper reservoir model has been determined, test analysis may be relatively straight-forward type-curve matching or regression analysis using modern well-test analysis software.

The shapes of the diagnostic plots for a buildup test and a drawdown test are essentially identical during the early- and middle-time regions for most tests. However, boundary effects can cause quite different shapes for a given reservoir model at late times in buildup and drawdown tests. This problem is augmented by the common use of "equivalent time" functions to analyze buildup tests on drawdown type curves. (There are different equivalent time functions for radial flow, linear flow, and bilinear flow, as discussed in more detail in the section on analysis of hydraulically fractured wells.)

Basically, equivalent time functions apply rigorously only to situations where either the producing time and the shut-in time both lie within the middle-time region or, as is commonly the case, the shut-in time is much less than the producing time before shut in.

To further complicate matters for buildup test analysis, the shape of the derivative curve depends on how the derivative is calculated and plotted. The derivative of pressure change may be taken with respect to the logarithm of either shut-in time or equivalent time. The derivative may then be plotted vs. either of these time functions, and the shape differs for each plotting function. Some pressure transient test analysis software allows the user a choice in the time function used in taking the derivative and another choice in time plotting function; for other software, the time functions used are "hard-wired." The results can be bewildering.

### Well in an Infinite-Acting Reservoir

Infinite-acting, radial flow reservoirs were described in the previous section. Figs. 8.50 and 8.51 show their diagnostic curves. For these plots, the derivative was taken with respect to shut-in time and derivative and pressure change curves are plotted vs. shut-in time. Both pressure and time are in terms of dimensionless variables. Wellbore storage distortion is not included in any of the diagnostic plots in this section.

Notice the significant difference in the shapes of both the derivative and pressure change curves for buildup and drawdown tests, with the pressure change curves flattening for buildup tests and the derivatives moving downward with an ultimate slope of –1. The time at which the flattening of the pressure change curve (and corresponding downward movement of the derivative) becomes apparent is a function of the producing time before shut-in. The longer the producing time, the longer the flattening is delayed and the longer the time the buildup diagnostic plot is essentially identical to the drawdown diagnostic plot.

Fig. 8.52 is the diagnostic plot that results when the derivative is taken with respect to radial equivalent time and the time-plotting function is radial equivalent time. The drawdown and buildup curves appear to be identical for all times. However, the radial equivalent time has a maximum value of the producing time before shut-in, so, for the buildup plots, the curves terminate at these maximum values of the time plotting function, and all points "stack up" at these limiting values of the plotting function. Our conclusion is that radial equivalent time is more satisfactory as a variable for taking the derivative and as a plotting function for an infinite-acting reservoir because the shape of the diagnostic plot is the same as for a constant-rate drawdown test.

### Linear No-Flow Boundary

When a well is near a single no-flow boundary (Fig. 8.53) or, as a practical matter, much closer to one boundary than to any other, and when sufficient time has elapsed for the boundary to have an influence on the pressure response during the test, the characteristic diagnostic plot, as Fig. 8.54 shows, results for a constant-rate drawdown test. (Wellbore storage may distort some of the earlier data on this diagnostic plot.) The derivative will double in value over approximately 1 2/3 log cycles (from 0.5 to 1.0 on a plot of dimensionless variables). Similar responses occur in naturally fractured reservoirs with transient flow from the matrix to the fractures.

Fig. 8.55 is the diagnostic plot for a buildup test with the derivative taken with respect to shut-in time and plotted vs. shut-in time. (Wellbore storage may distort some of the earlier data on this plot.) The longer the producing time before shut-in, the more nearly the shape of the diagnostic plot for a buildup test resembles the diagnostic plot for a drawdown test. The derivative has a slope of –1 for shut-in times much longer than producing time before shut-in.

Fig. 8.56 is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. Derivatives double over a small fraction of a log cycle for short producing times and, in general, the shapes of the diagnostic plots for buildup tests are similar to drawdown diagnostic plots only for longer producing times before shut-in.

Fig. 8.57 is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. In this case, the diagnostic plot is similar to the drawdown response, but the plots are not identical. Notice that the derivative doubles over approximately 1 2/3 log cycle. This procedure for taking the derivative and preparing the diagnostic plot is the most satisfactory of the alternatives considered.

### Linear Constant-Pressure Boundary

When a well is much nearer a single boundary (similar to Fig. 8.53) but with a constant-pressure at that boundary and boundary effects are encountered during the test, the diagnostic plot shown in Fig. 8.58 will result in a constant-rate drawdown test. (Wellbore storage effects could also occur early in the test.) The derivative has a slope of –1 at late times on the diagnostic plot.

Fig. 8.59 is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. This diagnostic plot is identical to the drawdown plot if steady state was achieved during the flow period preceding the buildup test. For other cases, with shorter producing times, the derivative has a slope steeper than the drawdown slope of –1.

Fig. 8.60 is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. For short producing times, the derivative falls precipitously.

Fig. 8.61 is the diagnostic plot for a buildup test, with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The shapes of the diagnostic plots are similar to, but not identical to, the drawdown diagnostic plot for all producing times before shut-in. The diagnostic plot prepared in this way is the most satisfactory of the alternatives considered.

### Well in a Channel

When a well is between two parallel no-flow boundaries and the pressure transient encounters both during a test long before the ends of the reservoir influence the test data, the diagnostic plot in Fig. 8.62 results for a constant-rate drawdown test. Before the boundary effects, with characteristic derivative slope of 1/2, wellbore storage, radial flow (or hemiradial flow if the well is much nearer one boundary than the other) will usually appear on the diagnostic plot. Diagnostic plots with similar shapes occur for a well between two sealing faults, a hydraulically fractured well with a high-conductivity fracture, and a horizontal well during early linear flow.

Fig. 8.63 is the diagnostic plot for a buildup test, with derivative taken with respect to shut-in time and plotted vs. shut-in time. The longer the producing time before shut-in, the more similar the curve shape is to the drawdown-test diagnostic plot. The derivative has a slope of –1/2 when shut-in time is much larger than producing time.

Fig. 8.64 is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. equivalent time. This plot is not particularly useful for test analysis. However, linear equivalent time produces a more useful diagnostic plot as long as channel ends do not affect the pressure response.

Fig. 8.65 is the diagnostic plot for a buildup test with derivative taken with respect to radial equivalent time and plotted vs. shut-in time. The derivative is similar to, but not identical to, the drawdown response. This method is the most useful for test analysis among the alternatives discussed.

## Estimating Average Reservoir Pressure

Two different method types, one using data from the middle-time region and the second using data from the late-time region (LTR), are commonly applied in estimating average reservoir pressure. The middle-time region methods are the Matthews-Brons-Hazebroek (MBH) method[20] and the Ramey-Cobb method. [24] The LTR methods are the modified Muskat method[25] and the Arps-Smith method. [26]

### Middle-Time Region Methods

The MTR methods are based on extrapolation of the middle-time region and the correction of the extrapolated pressure. The advantage of these methods is that they use pressure data only from the middle-time region, which means they require relatively short tests. The disadvantages are the need for accurate fluid property estimates, a known drainage area shape and size, and the location of the well within the drainage area.

Drainage Area Shapes. The MTR methods depend on the shape of the drainage area. Matthews-Brons-Hazebroek[20] developed a series of curves that model buildup tests in many shapes. As a matter of interest, these graphs were generated using image wells to simulate boundaries.

Figs. 8.66 through 8.68 illustrate representative dimensionless pressures as calculated by the MBH method. Fig. 8.66 is a plot of dimensionless pressure as defined by the MBH method plotted against dimensionless producing time calculated using the drainage area. Dimensionless pressure is defined as

....................(8.137)

and dimensionless time is

....................(8.138)

In Eq. 8.137, p* = the extrapolated pressure at a HTR of unity, = the current average drainage area pressure, and m = the slope of the MTR straight line on a Horner plot. In Eq. 8.138, tp = the producing time before shut-in, and A = the well’s drainage area expressed in square feet.

The four different curves in Fig. 8.66 represent four different locations of a well within a square drainage area. On this plot of dimensionless pressure on a linear scale vs. dimensionless time on a logarithmic scale, these curves eventually become straight lines. For example, for a well centered in a square drainage area, the line becomes straight at a dimensionless time of approximately 0.2. The time at which the line becomes straight is an indication that a well has reached pseudosteady-state flow at that dimensionless time.

Fig. 8.67 shows the Matthews-Brons-Hazebroek correlations for a well in the various positions in a 2×1 rectangle. The wells eventually reach pseudosteady state and the lines become straight, but, in general, the time to reach pseudosteady state is longer for the 2×1 rectangle than it was for the square rectangle. Furthermore, the farther the well is off center within the drainage area, the longer the time required to reach pseudosteady state. The difference is on the order of one full log cycle between the case in which the well is centered in the drainage area and that for a well most off-centered in the drainage area, which is the lowest curve on this plot. Fig. 8.68 shows the MBH pressures for wells in various positions in a 4×1 rectangle. Matthews-Brons-Hazebroek generated many similar graphs for other drainage-area shapes.

Example of the Matthews-Brons-Hazebroek Method. This method will be applied to a well in a reservoir with the following properties: t p = 482 hours, ϕ = 0.15, μ = 0.25 cp, ct = 1.615 × 10–5, and A = 1,500 × 3,000 ft (a 2 × 1 reservoir, well centered).

First, plot well shut-in pressure against the HTR on semilog coordinates. In Fig. 8.69, which is an ordinary HTR plot, the wellbore storage affects the data at large values of HTR, followed by the straight-line middle-time region, in turn followed by a deviation of the curve as it begins moving toward a fully built-up pressure.

The MTR straight line on this Horner graph is extrapolated to a HTR of 1 to determine p*. In this case, p* = 2,689.4 psi. From the slope of the semilog straight line, 26.7 psi/cycle, we calculate k = 7.5 md.

Next, calculate the dimensionless producing time, tpAD, with Eq. 8.138.

To calculate dimensionless production time, use the same producing time used in preparing the Horner graph. If the actual producing time is quite long, replace it with the time required to reach pseudosteady state, but remember to use the same producing time in the HTR and in calculating the dimensionless time for the MBH function. The time to reach pseudosteady state is determined by observing the appropriate MBH graph and finding when the dimensionless pressure vs. time becomes a straight line.

The next step is to select the appropriate MBH chart for the drainage area shape and well location being evaluated. Because the example well is centered in a 2×1 rectangle, choose Fig. 8.67. On this chart, enter the graph at a dimensionless producing time of 0.35, as illustrated in Fig. 8.70, and read across to find the dimensionless pressure, pMBHD, which has a value of 2.05.

The next step is to calculate the average reservoir pressure, . From rearrangement of Eq. 8.137,

....................(8.139)

In this case, the extrapolated p* = 2,689.4 psi, the slope of the MTR = 26.7, and the dimensionless pressure= 2.05. Thus,

Ramey-Cobb Method. The Ramey-Cobb method[24] also uses information from a Horner plot of buildup test data. After determining permeability from the Horner plot, dimensionless producing time, tpAD, can be calculated.

The third step differs from the MBH method in that the Dietz shape factors, CA, from Table 8.A-1 for the drainage-area shape and well location that best describes the tested well are used. (For the physical significance of the shape factor, see Ramey and Cobb. [24]) For the example well, the drainage area is a 2 × 1 rectangle, and the shape factor is 21.8369. Ramey and Cobb found a relationship between shape factor and the HTR at which the pressure on the MTR is current average drainage area pressure, . The relationship is

....................(8.140)

In the example test, the dimensionless producing time is 0.35, so the HTR that corresponds to the average reservoir pressure is 7.63.

Enter the Horner plot at a HTR of 7.63, read up to the extrapolated MTR straight line, then read across to the vertical axis. The resulting average reservoir pressure is 2,665.8 (Fig. 8.71). The result, for practical purposes, is identical to the result obtained using the MBH method.

The MBH and Ramey-Cobb methods use only data in the MTR. Once enough data is available to identify the MTR, the test can be terminated, which reduces test costs. The disadvantages of these methods are the need to know the drainage area size, shape, location of the well within that drainage area, and an accurate measurement of fluid properties. In the MBH method, the well can be in transient flow at the time of shut-in, but in the Ramey-Cobb method, the well must have reached pseudosteady state before shut-in. Results for the two methods should be identical, because they are based on the same theory. When it is applicable (pseudosteady state before shut-in), the Ramey-Cobb method is preferred because it is easier to apply.

### Late-Time Region Methods

Methods using LTR data are based on extrapolation of the post-middle-time region data trend. The advantages of these methods are that they require neither accurate fluid property estimates nor the drainage area size and shape. They do require that the well be reasonably centered within its drainage area. The disadvantage is that they require the post-middle-time region transient data. Thus, they require longer and more expensive shut-in tests to provide the data required for analysis.

Modified Muskat Method. The modified Muskat method is based on the theoretical observation first published by Larsen[25] that, for late-time data (after boundary effects have appeared), the difference between current average reservoir pressure, , and shut-in BHP, pws, declines exponentially. In equation form,

....................(8.141)

or ....................(8.142)

Eq. 8.142 leads to a procedure for estimating average drainage-area pressure, . This method requires a trial-and-error approach. To select data suitable for analysis with this method, use the diagnostic plot to determine the start of boundary effects. Then assume a value for , and plot log vs. time. If the curve is concave downward, the assumed pressure is too low; if the curve is concave upward, the assumed pressure is too high. Try different values for until the graph is a straight line, as predicted by theory.

On the example (Fig. 8.72), once the data begin to fall on a straight line, they tend to remain on that straight line. Shown are curves for assumed values of = 5,600; 5,575; and 5,560. On the first curve, for = 5,600, the final data points are trending above the straight line. For the lower curve, with = 5,560, the last few data points are trending below the straight line. For the assumed value = 5,575, all of the data points fall on a straight line making this assumption the right estimate of . The advantage to this method is that it is very easy to apply. It works best with a well reasonably centered within a drainage area.

The weaknesses of this method are that it is more sensitive to estimates that are too low rather than to estimates that are too high and that it is not easily automated and, therefore, not as widely incorporated into well-test analysis software as some other methods.

Arps-Smith Method. This is an alternative method for analyzing LTR data. [26] The theoretical basis for this originally empirical method is also Eq. 8.141. Differentiating Eq. 8.141 with respect to time,

....................(8.143)

To apply this method, plot the change in BHP with time, dpws/dt vs. pws, on Cartesian coordinates. On such a plot, data for the LTR should fall on a straight line, and extrapolation of that line to dpws/dt = 0 provides an estimate of the average drainage area pressure, .

In Fig. 8.73, the final points from an example test fall on a straight line. Extrapolating the straight line to the horizontal axis gives the average pressure at the intercept. For this example, which is the same test illustrated with the modified Muskat method, the average pressure is 5,575 psi, which is the same value found with the Muskat method.

The advantages of the Arps-Smith method are that it is simple to apply and easily automated (which means that it is easily implemented into well-test analysis software or into spreadsheets). The disadvantages are that it requires data in the LTR, which means that it requires longer, more expensive tests. It assumes that shut-in pressure approaches average pressure exponentially, which is most nearly true for wells centered in the drainage area, and it requires numerical differentiation of pressure with respect to time, which tends to magnify any noise that may be present in the data.

The modified Muskat and Arps-Smith methods actually apply for shut-in times in the range,

....................(8.144)

In Fig. 8.74, the data points with darker dots are on the type curve for the derivative. These are the data in the range for which the modified Muskat and Arps-Smith methods work.

Fig. 8.74 illustrates one of the disadvantages of these two methods. Many other reservoir models will exhibit similar diagnostic plots, but data like that shown with the dark dots in this figure will not extrapolate to the correct average reservoir drainage area pressure. Examples of these other cases include dual-porosity reservoirs during the early transition from fracture flow to total system flow, layered reservoirs, and composite reservoirs with an inner zone mobility much lower than the outer zone mobility.

## Hydraulically Fractured Wells

Many wells—particularly gas wells in low-permeability formations—require hydraulic fracturing to be commercially viable. Interpretation of pressure-transient data in hydraulically fractured wells is important for evaluating the success of fracture treatments and predicting the future performance of fractured wells. This section includes graphical techniques for analyzing post-fracture pressure transient tests after identifying several flow patterns that are characteristic of hydraulically fractured wells. Often, identification of specific flow patterns can aid in well test analysis.

### Flow Patterns in Hydraulically Fractured Wells

Five distinct flow patterns (Fig. 8.75) occur in the fracture and formation around a hydraulically fractured well. [27] Successive flow patterns, which often are separated by transition periods, include fracture linear, bilinear, formation linear, elliptical, and pseudoradial flow. Fracture linear flow (Fig. 8.75a) is very short-lived and may be masked by wellbore-storage effects. During this flow period, most of the fluid entering the wellbore comes from fluid expansion in the fracture, and the flow pattern is essentially linear.

Because of its extremely short duration, the fracture linear flow period often is of no practical use in well test analysis. The duration of the fracture linear flow period is estimated by[27]

....................(8.145)

where tLfD is dimensionless time in terms of fracture half-length,

....................(8.146)

The dimensionless fracture conductivity, Cr, is

....................(8.147)

and ηfD is dimensionless hydraulic diffusivity defined by

....................(8.148)

Bilinear flow (Fig. 8.75b) evolves only in finite-conductivity fractures as fluid in the surrounding formation flows linearly into the fracture and before fracture tip effects begin to influence well behavior. Fractures are considered to be finite conductivity when Cr < 100. Most of the fluid entering the wellbore during this flow period comes from the formation. During the bilinear flow period, BHP, pwf, is a linear function of t1/4 on Cartesian coordinates.

A log-log plot of (pipwf) as a function of time exhibits a slope of 1/4 unless the fracture is damaged. The pressure derivative also has a slope of 1/4 during this same time period. The duration of bilinear flow depends on dimensionless fracture conductivity and is given by Eqs. 8.149a through 8.149c[27] for a range of dimensionless times and fracture conductivities:

....................(8.149a)

....................(8.149b)

and ....................(8.149c)

Formation linear flow (Fig. 8.75c) occurs only in high-conductivity (Cr ≥ 100) fractures. This period continues to a dimensionless time of tLfD ≅ 0.016. The transition from fracture linear flow to formation linear flow is complete by a time of tLfD = 10–4. On Cartesian coordinates, pwf is a linear function of t1/2, and a log-log plot of (pipwf) has a slope of 1/2 unless the fracture is damaged. The pressure derivative plot exhibits a slope of 1/2. Elliptical flow (Fig. 8.75d) is a transitional flow period that occurs between a linear or near-linear flow pattern at early times and a radial or near—radial flow pattern at late times.

Pseudoradial flow (Fig. 8.75e) occurs with fractures of all conductivities. After a sufficiently long flow period, the fracture appears to the reservoir as an expanded wellbore (consistent with the effective wellbore radius concept suggested by Prats et al.[10]). At this time, the drainage pattern can be considered as a circle for practical purposes. (The larger the fracture conductivity, the later the development of an essentially radial drainage pattern.) If the fracture length is large relative to the drainage area, then boundary effects distort or entirely mask the pseudoradial flow regime. Pseudoradial flow begins at tLfD ≅ 3 for high-conductivity fractures (Cr ≥ 100) and at slightly smaller values of tLfD for lower values of Cr.

These flow patterns also appear in pressure-buildup tests and occur at approximately the same dimensionless times as in flow tests. The physical interpretation is that the pressure has built up to an essentially uniform value throughout a particular region at a given time during a buildup test. For example, at a given time during bilinear or formation linear flow, pressure has built up to a uniform level throughout an approximately rectangular region around the fracture. At a later time during elliptical flow, pressure has built up to a uniform level throughout an approximately elliptical region centered at the wellbore. At a given time during pseudoradial flow, pressure has built up to a uniform level throughout an approximately circular region centered at the wellbore. The area of the region and the pressure level within that area increase with increasing shut-in time. Example 8.1 illustrates how to estimate the duration of flow periods for hydraulically fractured wells.

Example 8.1: Estimating Duration of Flow Periods in a Hydraulically Fractured Well For each case, estimate the end of the linear flow period and the time at which pseudoradial flow period begins. Assume that pseudoradial flow begins when tLfD = 3. Table 8.2 gives the data for each case.

Solution. The end of the linear flow regime occurs at a dimensionless time of tLfD ≅ 0.016 or, using Eq. 8.146,

Similarly, the time to reach pseudoradial flow is tLfD ≅ 3, or

Table 8.2 summarizes the results.

### Flow Geometry and Depth of Investigation of a Vertically Fractured Well

Fluid flow in a vertically fractured well has been described using elliptical geometry. [28] The equation for an ellipse with its major axis along the x-axis and minor axis along the y-axis is

....................(8.150)

where the endpoints of the major and minor axes are (±af, 0) and (0, ±bf), respectively. The foci of the ellipse are ±cf where cf2 = af2bf2. In terms of a well with a single vertical fracture with two wings of equal length, Lf, the relation becomes Lf2 = af2bf2, where Lf is the focal length of the ellipse. Fig. 8.76 shows the elliptical geometry of a vertically fractured well.

Hale and Evers[28] defined a depth of investigation for a vertically fractured well. Their definition is based on a definition of dimensionless time at a distance bf, the length of the minor axis:

....................(8.151)

Solving for the length of the minor axis,

....................(8.152)

Assuming that pseudosteady-state flow exists out to distance, bf, at dimensionless time tbD = 1/π as in linear systems, Eq. 8.152 becomes

....................(8.153)

which represents the depth of investigation in a direction perpendicular to the fracture at time, t, for a vertically fractured well. In gas wells, the terms μ and ct should be and , evaluated at average drainage-area pressure, .

The elliptical pattern of the propagating pressure transient can be fully described in terms of the lengths of the major axis, af, the minor axis, bf, and the focus, Lf. Using the estimate of bf from Eq. 8.153 and an estimate of Lf obtained by one of the methods described in sections that follow, the length of the major axis can be estimated from

....................(8.154)

Given values of af and bf, the depth of investigation at a particular time, t, in any direction from the fracture can be calculated using Eq. 8.150. Furthermore, the area, A, enclosed by the ellipse at time, t (the area of the reservoir sampled by the pressure transient), is given by

....................(8.155)

The coefficient 0.0002878 in Eq. 8.153 is strictly correct only for highly conductive fractures (Cr ≥ 100). As Cr becomes smaller, the ratio af/bf also becomes smaller. The lower bound of af/bf is 1 (a circle) as Cr approaches 0.

### Fracture Damage

Two major types of fracture damage are frequent: choked fracture damage and fracture-face damage. The choked-fracture damage means that the fracture has a reduced permeability in the immediate vicinity of the wellbore (Fig. 8.77). In this case, kf is used for the permeability in the propped portion of the fracture farther along the wellbore, and kfs for reduced permeability near the wellbore, out to a length, Ls, in the fracture.

The choked-fracture skin factor, sf, is[29]

....................(8.156)

Fracture face damage in a hydraulically fractured well (Fig. 8.78) is a permeability reduction around the edges of the fracture, usually caused by invasion of the fracture fluid into the formation or an adverse reaction with the fracturing fluid. The equation for fracture face skin is[29]

....................(8.157)

### Specialized Methods for Post-Fracture Well-Test Analysis

Generally, the objectives of post-fracture pressure-transient test analysis are to assess the success of the fracture treatment and to estimate the fracture half-length, fracture conductivity, and formation permeability. Three specialized methods of analyzing these post-fracture transient tests are included in this section: pseudoradial flow, bilinear flow, and linear flow.

Bilinear Flow Method. The bilinear flow method[30] applies to test data obtained during the bilinear flow regime in wells with finite-conductivity vertical fractures. Bilinear flow is indicated by a quarter-slope line on a log-log graph of pressure derivative vs. t or Δte.

During bilinear flow,

....................(8.158)

and ....................(8.159)

The following procedure is recommended for analyzing test data obtained in the bilinear flow regime (that is, data in the time range with quarter slope on the diagnostic plot). In Step 1, note the use of "bilinear equivalent time," ΔtBe. Radial equivalent time is rigorously correct as a plotting function only for infinite-acting radial flow.

1. For a constant-rate flow test, plot pwf vs. t1/4 on Cartesian coordinates. For a buildup test, plot pws vs. ΔtBe1/4, where
....................(8.160)
2. Determine the slope, mB, of the straight line region of the plot.
3. Determine the pressure extrapolated to time zero, po, and the fracture skin, sf, from
....................(8.161)
for drawdown and buildup tests, respectively.
4. From independent knowledge of k (for example, from a prefracture well test), estimate the fracture conductivity, wfkf, using mB and the relationship
....................(8.162)
where and , evaluated at , are used for a gas well test.

Fig. 8.79 is an example of bilinear flow analysis. The bilinear flow analysis method has the following important limitations.

• No estimate of fracture half-length, Lf.
• In wells with low-conductivity fractures, wellbore storage frequently distorts early test data for a sufficient length of time so that the quarter-slope line characteristic of bilinear flow may not appear on a log-log plot of test data.
• An independent estimate of k is required. This suggests that prefracture well tests should be conducted before fracturing the well, thus obtaining independent estimates of formation properties.

Linear Flow Method. The linear flow method[30] applies to test data obtained during formation linear flow in wells with high-conductivity fractures (Cr ≥ 100). After wellbore storage effects have ended, formation linear flow occurs up to a dimensionless time of tLfD = 0.016, which means that a log-log plot of pressure derivative against time will have a slope of one-half. The plot of pressure change vs. time, however, will have a half-slope only if the fracture skin is zero. The pressure and pressure derivative are

....................(8.163)

and ....................(8.164)

so that

....................(8.165)

which indicates that a log-log plot of the derivative against time will have a slope of one-half. Radial equivalent time applies rigorously only for radial flow in an infinite-acting reservoir. When linear flow is the flow pattern occurring at both times (tp + Δt) and Δt, a more useful equivalent time function is the linear equivalent time, ΔteL.

....................(8.166)

Test conditions in which linear flow occurs at both (tp + Δt) and Δt are rare, and, consequently, Eq. 8.166 is not necessarily rigorously correct for well-test analysis. Fortunately, when tp >> Δtmax, ΔteL ≈ Δt. Fig. 8.80 is an example of a plot used in linear flow analysis.

The linear flow analysis method also has limitations.

• The method applies only for fractures with high conductivities. Strictly speaking, linear flow occurs for the condition of uniform flux into a fracture (same flow rate from the formation per unit cross-sectional area of the fracture at all points along the fracture) rather than for infinite fracture conductivity. Therefore, only very early test data (tLf D ≤ 0.016) exhibit linear flow in a high-conductivity fracture.
• Some or all of these early data may be distorted by wellbore storage, further limiting the amount of linear-flow data available for analysis.
• Estimating fracture half-length requires an independent estimate of permeability, k, which suggests the need for a prefracture well test.

Pseudoradial Flow Method. The pseudoradial flow method applies when a short, highly conductive fracture is created in a high-permeability formation, so that pseudoradial flow develops in a short time. The time required to achieve pseudoradial flow for an infinitely conductive fracture (Cr ≥ 100) in either a flow test or a pressure buildup test is estimated by

....................(8.167)

The beginning of pseudoradial flow is characterized by the flattening of the pressure derivative on a log-log plot and by the start of a straight line on a semilog plot. Hence, when the pseudoradial flow regime is reached, conventional semilog analysis can be used to calculate permeability and skin factor. For a highly conductive fracture, skin factor is related to fracture half-length by[10]

....................(8.168)

Fig. 8.81 shows an example.

A recommended procedure for analyzing test data from the pseudoradial flow regime is as follows.

• For a drawdown test, plot pwf vs. log t. For a buildup test, plot pws vs. the HTR.
• Determine the position and slope, m, of the semilog straight line and the intercept, p1hr on the line.
• Using m, calculate values of k and s (or s′ for a gas well).
• Calculate the fracture half-length, Lf, using Eq. 8.168.

The pseudoradial flow method has the following limitations that seldom make it applicable in practice. [30]

• The conditions that are most favorable for the occurrence of pseudoradial flow are short, highly conductive fractures in high-permeability formations. These formations, however, are rarely fractured. The most common application of hydraulic fractures—wells with long fractures in low-permeability formations—require impractically long test times to reach pseudoradial flow.
• For gas wells, the apparent skin factor, s′, calculated from test data is often affected by non-Darcy flow.
• The pseudoradial method applies only to highly conductive (Cr ≥ 100) fractures. For lower conductivity fractures, fracture lengths calculated using the skin factor (Eq. 8.168) will be too low.

### Using Type Curves for Hydraulically Fractured Wells

Type curves are the most common method of analyzing hydraulically fractured wells. The independent variable for most type curves for analyzing hydraulically fractured wells is the dimensionless time based on hydraulic fracture half-length, tLf D. The dependent variable is usually the dimensionless pressure, pD.

For type curves used for manual type-curve matching, most vary only one parameter. The Cinco type curve[27] is obtained for zero CLf D and sf ; the only parameter is dimensionless fracture conductivity, Cr or FcD (where FcD = πcr). The choked-fracture skin is analyzed by assuming CLf D and infinite Cr with single parameter sf. The wellbore-storage type curve[31] sets sf to 0 and Cr (FcD) to infinity and varies the coefficient CLf D.

When using type curves in commercial software, the computer can set any two of the three parameters to fixed values (other than their limiting values) and vary the third parameter to obtain the matching stems.

Procedures for Analyzing Fractured Wells With Type Curves. The following steps outline the procedure for analyzing fractured wells with type curves.

• Graph field data pressure change and pressure derivatives.
• Match field data to the appropriate type curve.
• Find the match point and matching stem.
• Calculate the formation permeability from the pressure match point.
• Calculate Lf from the time match point.
• Interpret the matching stem value appropriate for a given type curve. For one type curve, this can be wfkf, which will provide an estimate of fracture conductivity. For another, it can be sf, the choked-fracture skin, or, for a third, it can be C, the wellbore-storage coefficient.

To interpret the match points for a test with unknown permeability, use Eqs. 8.169 and 8.170. The formation permeability, k, is determined from the pressure match point; that is, the relationship between the pressure derivative and pressure change found at a match point given by

....................(8.169)

From the time match point, calculate the fracture half-length:

....................(8.170)

Matching can be ambiguous for hydraulically fractured wells; the data can appear to match equally well in several different positions. The ambiguity can be reduced or eliminated if a prefracture permeability is determined, and the post-fracture test data forced to match the permeability.

Type Curves Used for Analysis in Fractured Wells. The Cinco type curve (Fig. 8.82),[27] assumes that CLf D = 0 and sf = 0. The type-curve stems on this curve are obtained by varying values of Cr or FcD. With the Cinco type curve, the fracture conductivity, wfkf, can be determined from the matching parameter:

....................(8.171)

Choked-Fracture Type Curve. Fig. 8.83 shows the choked-fracture type curve. [29] The choked-fracture type curve is generated with wellbore-storage coefficient, CLf D, of zero and infinite fracture conductivity, Cr. On this type curve, the stems represent different values of the fracture skin, sf. The fracture skin, sf, can be used to find the additional pressure drop from

....................(8.172)

Wellbore-Storage Type Curve. The wellbore-storage type curve (Fig. 8.84) takes into account the possibility of wellbore storage. The wellbore-storage type assumes sf = 0 and Cr = ∞. To interpret a best-fitting stem for this type curve, use the following:

....................(8.173)

### Limitations of Type-Curve Analysis in Hydraulically Fractured Wells

Although it is the most common methodology for analyzing hydraulically fractured well, type-curve analysis still has some limitations.

First, type-curves for analysis of hydraulically fractured wells are usually based on solutions for constant-rate drawdown tests. For buildup tests, shut-in time itself may possibly be used as a plotting function in those cases in which producing time is much greater than the shut-in time. Equivalent time can be used in some cases, but equivalent time has different definitions depending on the flow regime: radial, linear, and bilinear flow. Another possibility is to use a "superposition" type curve, which depends on the specific durations of flow and buildup periods. Superposition type curves can be readily generated with computer software.

Another problem with type curves is that they may ignore important behavior. The type curve that takes into account wellbore storage does not consider a variable wellbore storage coefficient. This can be caused by phase redistribution in the wellbore, for example. The widely available type curves that have been discussed do not include boundary effects. With gas wells, the probability of non-Darcy flow is high, but available type curves don’t take this into account.

An independent estimate of permeability may also be needed. A number of different type curves or a variety of stems on a given type curve may seem to match test data equally well. To remove this ambiguity, the best solution is to have an independent estimate of permeability.

## Naturally Fractured Reservoirs

This section focuses on interpretation of well test data from wells completed in naturally fractured reservoirs. Because of the presence of two distinct types of porous media, the assumption of homogeneous behavior (discussed in previous sections) is no longer valid in naturally fractured reservoirs. This section includes two naturally fractured reservoir models, the physics governing fluid flow in these reservoirs and semilog and type-curve analysis techniques for well tests in these reservoirs.

### Naturally Fractured Reservoir Models

Naturally fractured reservoirs are characterized by the presence of two distinct types of porous media: matrix and fracture. Because of the different fluid storage and conductivity characteristics of the matrix and fractures, these reservoirs often are called dual-porosity reservoirs. Fig. 8.85 illustrates a naturally fractured reservoir composed of a rock matrix surrounded by an irregular system of vugs and natural fractures. Fortunately, it has been observed that a real, heterogeneous, naturally fractured reservoir has a characteristic behavior that can be interpreted using an equivalent, homogeneous dual-porosity model such as that shown in the idealized sketch.

Several models have been proposed to represent the pressure behavior in a naturally fractured reservoir. These models differ conceptually only in the assumptions made to describe fluid flow in the matrix. Most dual-porosity models assume that production from the naturally fractured system comes from the matrix, to the fracture, and then to the wellbore (i.e., that the matrix does not produce directly into the wellbore). Furthermore, the models assume that the matrix has low permeability but large storage capacity relative to the natural fracture system, while the fractures have high permeability but low storage capacity relative to the natural fracture system. Warren and Root[32] introduced two dual-porosity parameters, in addition to the usual single-porosity parameters, which can be used to describe dual-porosity reservoirs.

Interporosity flow is the fluid exchange between the two media (the matrix and fractures) constituting a dual-porosity system. Warren and Root[32] defined the interporosity flow coefficient, λ, as

....................(8.174)

where km is the permeability of the matrix, kf is the permeability of the natural fractures, and α is the parameter characteristic of the system geometry.

The interporosity flow coefficient is a measure of how easily fluid flows from the matrix to the fractures. The parameter α is defined by[33]

....................(8.175)

where L is a characteristic dimension of a matrix block and j is the number of normal sets of planes limiting the less-permeable medium (j = 1, 2, 3). For example, j = 3 in the idealized reservoir cube model in Fig. 8.85. On the other hand, for the multilayered or "slab" model shown in Fig. 8.86, [34] j = 1. For the slab model, letting L = hm (the thickness of an individual matrix block), λ becomes

....................(8.176)

The storativity ratio, [33] ω, is defined by

....................(8.177)

where V is the ratio of the total volume of one medium to the bulk volume of the total system and ϕ is the ratio of the pore volume of one medium to the total volume of that medium. Subscripts f and f + m refer to the fracture and to the total system (fractures plus matrix), respectively. Consequently, the storativity ratio is a measure of the relative fracture storage capacity in the reservoir.

Many models have been developed for naturally fractures reservoirs. Two common models, pseudosteady-state and transient flow, that describe flow in the less-permeable matrix are presented here. Pseudosteady-state flow was assumed by Warren and Root[32] and Barenblatt et al.[35]; others, notably deSwaan, [36] assumed transient flow in the matrix. Intuition suggests that, in a low-permeability matrix, very long times should be required to reach pseudosteady-state and that transient matrix flow should dominate; however, test analysis suggests that pseudosteady-state flow is quite common. A possible explanation of this apparent inconsistency is that matrix flow is almost always transient but can exhibit a behavior much like pseudosteady-state, if there is a significant impediment to flow from the less-permeable medium to the more-permeable one (such as low-permeability solution deposits on the faces of fractures).

The pseudosteady-state flow model assumes that, at a given time, the pressure in the matrix is decreasing at the same rate at all points and, thus, flow from the matrix to the fracture is proportional to the difference between matrix pressure and pressure in the adjacent fracture. Specifically, this model, which does not allow unsteady-state pressure gradients within the matrix, assumes that pseudosteady-state flow conditions are present from the beginning of flow.

Because it assumes a pressure distribution in the matrix that would be reached only after what could be a considerable flow period, the pseudosteady-state flow model obviously is oversimplified. Again, this model seems to match a surprising number of field tests. One possible reason is that damage to the face of the matrix could cause the flow from matrix to fracture to be controlled by a sort of choke (the thin, low-permeability, damaged zone) and, therefore, is proportional to pressure differences upstream and downstream of the choke. In the next two sections, semilog and type-curve analysis techniques are presented for well tests in naturally fractured reservoirs exhibiting pseudosteady-state flow characteristics.

Semilog Analysis Technique. The pseudosteady-state matrix flow solution developed by Warren and Root[32] predicts that, on a semilog graph of test data, two parallel straight lines will develop. Fig. 8.87 shows this characteristic pressure response.

The initial straight line reflects flow in the fracture system only. At this time, the formation is behaving like a homogeneous formation with fluid flow originating only from the fracture system with no contribution from the matrix. Consequently, the slope of the initial semilog straight line is proportional to the permeability-thickness product of the natural fracture system, just as it is for any homogeneous system. Following a discrete pressure drop in the fracture system, the fluid in the matrix begins to flow into the fracture, and a rather flat transition region appears.

Finally, the matrix and the fracture each reach an equilibrium condition, and a second straight line appears. At this time, the reservoir again is behaving like a homogeneous system, but now the system consists of both the matrix and the fractures. The slope of the second semilog straight line is proportional to the total permeability-thickness product of the matrix/fracture system. Because the permeability of the fractures is much greater than that of the matrix, the slope of the second line is almost identical to that of the initial line.

Similar shapes are predicted for pressure buildup tests (Fig. 8.88). The lower curve, A, represents the ideal buildup test plot predicted by Warren and Root. [32] The shape of a semilog plot of test data from a naturally fractured reservoir is almost never the same as that predicted by Warren and Root’s model. Wellbore storage almost always obscures the initial straight line and often obscures part of the transition region between the straight lines. The upper curve, B, in Fig. 8.88 shows a more common pressure response.

The reservoir permeability-thickness product, kh [actually the kh of the fractures, or (kh)f, because (kh) m is usually negligible], can be obtained from the slope, m, of the two semilog straight lines. Storativity, ω, can be determined from their vertical displacement, δp. The interporosity flow coefficient, λ, can be obtained from the time of intersection of a horizontal line, drawn through the middle of the transition curve, with either the first or second semilog straight line. [33]

When semilog analysis is possible (i.e., when the correct semilog straight line can be identified), the following procedure is recommended for semilog analysis of buildup or drawdown test data from wells completed in naturally fractured reservoirs. Although presented in variables for slightly compressible fluids (liquids), the same procedure is applicable to gas well tests when the appropriate variables are used.

• From the slope of the initial straight line (if present) or final straight line (more likely to be present), determine the permeability-thickness product, kh. In either case, the slope, m, is related to the total kh of the system, which is essentially all in the fractures. The permeability-thickness product is given by
....................(8.178)
where is equal to (kh)f/h. Strictly speaking, the slope of the second straight line is related to [(kh)f + (kh)m ], but (kh)m ordinarily is negligible compared to (kh)f.
• If both initial and final straight lines can be identified (or the position of the initial line can at least be approximated) and the pressure difference, δp, established, then the storativity ratio, ω is calculated from
....................(8.179)

If the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by tl and t2, respectively, the storativity ratio may also be calculated from

....................(8.180)

For a buildup test, where the times of intersection of a horizontal line drawn through the midpoint of the transition data with the first and second semilog straight lines are denoted by [(tp + Δt)/Δt]1 and [(tp + Δt)/Δt]2, respectively, the storativity ratio may be calculated from

....................(8.181)

The interporosity flow coefficient, λ, is calculated[33] for a drawdown test by

....................(8.182)

or for a buildup test by

....................(8.183)

where γ = 1.781.

The terms (ϕV)m and (ct)m in Eq. 8.183 are obtained by conventional methods. A porosity log usually reads only the matrix porosity (not the fracture porosity) and thus gives ϕm, while (ct)m is the sum of coSo, cgSg, cwSw, and cf. Vm usually can be assumed to be essentially 1.0. From the definition of ω in Eq. 8.177,

....................(8.184)

The second semilog straight line should be extrapolated to p1hr, and the skin factor is

....................(8.185)

where Δp1hr is equal to (pip1 hr) for a drawdown test or [p1 hr - pwft=0)] for a buildup test.

• The second semilog straight line should be extrapolated to p* (Fig. 8.89). From p*, can be found using conventional methods (such as the Matthew-Brons-Hazebroek p* method).

Type Curve Analysis Technique. Particularly because of wellbore-storage distortion, type curves are quite useful for identifying and analyzing dual-porosity systems. Fig. 8.90 shows an example of the Bourdet et al.[37] type curves developed for pseudosteady-state matrix flow. Initially, test data follow a curve for some value of CDe2s where CD is the dimensionless wellbore storage coefficient. In Fig. 8.90, the earliest data for the well follow the curve for CDe2s = 1. The data then deviate from the early fit and follow a transition curve characterized by the parameter λe-2s. In Fig. 8.90, the data follow the curve for λe–2s = 3×10–4. When equilibrium is reached between the matrix and fracture systems, the data then follow another CDe2s curve. In the example, the later data follow the CDe2s = 0.1 curve.

At earliest times, the reservoir is behaving like a homogeneous reservoir with all fluid originating from the fracture system. During intermediate times, there is a transition region as the matrix begins to produce into the fractures. At later times, the system again is behaving like a homogeneous system with both matrix and fractures contributing to fluid production.

Fig. 8.91 illustrates the derivative type curves for a formation with pseudosteady-state matrix flow. [37] The most notable feature, characteristic of naturally fractured reservoirs, is the dip below the homogeneous reservoir curve. The curves dipping downward are characterized by a parameter λCD/ω (1 − ω), while the curves returning to the homogeneous reservoir curves are characterized by the parameter λCD/ω (1 − ω). Test data that follow this pattern on the derivative type curve can reasonably be interpreted as identifying a dual-porosity reservoir with pseudosteady-state matrix flow (a theory that needs to be confirmed with geological information and reservoir performance). Pressure and pressure derivative type curves can be used together for analysis of a dual-porosity reservoir. The pressure derivative data are especially useful for identifying the dual-porosity behavior. Manual type-curve analysis for well in naturally fractured reservoirs is tedious, and the interpretation involved is difficult. Most current analysis uses commercial software.

Transient Matrix Flow Model. The more probable flow regime in the matrix is unsteady-state or transient flow; that is, flow in which an increasing pressure drawdown starts at the matrix/fracture interface and moves further into the matrix with increasing time. Only at late times should pseudosteady-state flow be achieved, although a matrix with a thin, low-permeability damaged zone at the fracture face may behave as predicted by the pseudosteady-state matrix flow model even though the flow in the matrix is actually unsteady-state.

A semilog graph of test data for a formation with transient matrix flow has a characteristic shape different from that for pseudosteady-state flow in the matrix. Three distinct flow regimes have been identified that are characteristic of dual-porosity reservoir behavior with transient matrix flow. Fig. 8.92 illustrates these flow regimes on a semilog graph as regimes 1, 2, and 3.

Flow regime 1 occurs at early times during which all production comes from the fractures. Flow regime 2 occurs when production from the matrix into the fracture begins and continues until the matrix-to-fracture transfer reaches equilibrium. This equilibrium point marks the beginning of flow regime 3, during which total system flow, from matrix to fracture to wellbore, is dominant. The same three flow regimes appear when there is pseudosteady-state matrix flow. The duration and shape of the transition flow regimes, however, is considerably different for the two matrix flow models.

Serra et al.[34] observed that pressures from each of these flow regimes will plot as straight lines on conventional semilog graphs. Flow regimes 1 and 3, which correspond to the classical early- and late-time semilog straight-line periods, respectively, have the same slope. Flow regime 2 is an intermediate transitional period between the first and third flow regimes. The semilog straight line of flow regime 2 has a slope of approximately one-half that of flow regimes 1 and 3. If all or any two of these regimes can be identified, then a complete analysis is possible using semilog methods alone. Certain nonideal conditions, however, may make this analysis difficult to apply.

Flow regime 1 often is distorted or obscured by wellbore storage, which often makes this flow regime difficult to identify. Flow regime 2, the transition, also may be obscured by wellbore storage. Flow regime 3 sometimes requires a long flow period followed by a long shut-in time to be observed, especially in formations with low permeability. Furthermore, boundary effects may appear before flow regime 3 is fully developed.

Semilog Analysis Techniques. Serra et al.[34] presented a semilog method for analyzing well test data in dual-porosity reservoirs exhibiting transient matrix flow (Fig. 8.92). They found that the existence of the transition region, flow regime 2, and either flow regime 1 or flow regime 3 is sufficient to obtain a complete analysis of drawdown or buildup test data. Further, they assumed unsteady-state flow in the matrix, no wellbore storage, and rectangular matrix-block geometry, as Fig. 8.86 shows. The rectangular matrix-block geometry is adequate, although different assumed geometries can lead to slightly different interpretation results.

The major weakness of the Serra et al. method is that it assumes no wellbore storage. In many cases, flow regimes 1 and 2 are partially or even totally obscured by wellbore storage, making analysis by the Serra et al. method impossible or difficult. Despite this limitation, the Serra et al. method has great practical value when used in conjunction with type-curve methods. These calculations of the Serra et al. method apply to both buildup and drawdown test data and are applicable for well test analysis of slightly compressible liquids and gas well tests.

Type Curve Analysis Technique. Bourdet et al.[37] presented type curves for analyzing well tests in dual-porosity reservoirs including the effects of wellbore storage and unsteady-state flow in the matrix. The type curves are useful supplements to the Serra et al. semilog analysis. Fig. 8.93 gives an example of the pressure and pressure derivative type curves for transient matrix flow. Early (fracture-dominated) data are fit by a CDe2s value indicative of homogeneous behavior. Data in the transition region are fit by curves characterized by a parameter β′. Finally, data in the homogeneous-acting, fracture-plus-matrix flow regime are fit by another CDe2s curve.

On the derivative type curve, early data also are fit by a derivative curve reflecting homogeneous behavior. Fig. 8.94 shows an actual example. If wellbore-storage distortion ceases before the transition region begins (which did not happen in the example but is possible in other cases), the derivative data will be horizontal and should be aligned with the (tD/CD)pD′ = 0.5 curve. However, if the transition region is present (recall that its semilog slope is half that of the middle-time straight line), the derivative curve will flatten and should be aligned with the (tD/CD)pD′ = 0.25 curve as shown in this example. The homogeneous (fracture-plus-matrix) data should, after wellbore distortion has ceased and before boundary effects have appeared, be horizontal on the derivative type curve and should be aligned with the (tD/CD)pD′ = 0.5 curve as this example shows.

Manual type-curve matching is tedious and difficult, especially with the interpolation involved. Analysis ordinarily uses commercially available software to analyze these kinds of tests after the reservoir model has been identified.

## Horizontal Well Analysis

Productivity estimates in horizontal wells are subject to more uncertainty than comparable estimates in vertical wells. Further, it is much more difficult to interpret well test data because of 3D flow geometry. The radial symmetry usually present in a vertical well does not exist. Several flow regimes can potentially occur and need to be considered in analyzing test data from horizontal wells. Wellbore storage effects can be much more significant and partial penetration and end effects commonly complicate interpretation.

In vertical wells, variables such as average permeability, net vertical thickness, and skin are used. Horizontal wells need more detail. Not only is vertical thickness important, but the horizontal dimensions of the reservoir, relative to the horizontal wellbore, need to be known.

### Steps in Evaluating Horizontal Well-Test Data

There are three basic steps in evaluating pressure-transient data from a horizontal well. First, identify the specific flow regimes in the test data. Second, apply the proper analytical and graphical procedures to the data. Finally, evaluate the uniqueness and sensitivity of the results to properties derived from analysis or simply assumed.

Identify Flow Regimes. As discussed in previous sections, evaluation of data from a vertical wellbore will generally center on a single flow regime, such as infinite-acting radial flow, known as the MTR. However, a pressure-transient test in a horizontal well can involve as many as five major and distinct regimes that need to be identified. These regimes may or may not occur in a given test and may or may not be obscured by wellbore storage effects.

Apply the Proper Procedures. Each flow regime can be modeled by an equation that can be used to estimate important reservoir properties. At best, only groups of analytical parameters can be determined directly from equations. It is imperative that the proper analytical and graphical procedures be applied to the data. In many cases, when solving for specific parameters, the application of these analytical expressions may involve a complex iterative procedure.

Evaluate Uniqueness and Sensitivity. Experience indicates that results of horizontal well test analysis are seldom unique, so it is important that the uniqueness and sensitivity of the results to assumed properties be evaluated. Simulation of the test using properties that have been determined from the test can confirm that at least the analysis is consistent with the test data. A simulator can also determine whether other sets of formation properties will also lead to a fit of the data.

### Horizontal Well Flow Regimes

Different formation properties can be calculated from the data in each of the five different flow regimes. Any flow regime may be absent from a plot of test data because of geometry, wellbore storage, or other factors. Nor does the fact that they can appear mean that they do appear. The five different flow regimes that can occur are early radial, hemi-radial, early linear, late pseudoradial, and late linear.

Fig. 8.95 shows a horizontal well with length, Lw, within a reservoir that is assumed to be a rectangular parallelepiped or a "box reservoir" drainage area. In this discussion, it is assumed that the axes of the coordinate system coincide with the direction of principal permeability and the well produces over its entire length, Lw.

The axes for this box are the usual x-, y-, and z-axes. Notice that the x-axis is measured along the bottom edge of the reservoir, going from left to right in the direction perpendicular to the well. The y-axis lies along the axis from front to back of the reservoir, parallel to the wellbore. The z-axis is oriented in the direction of reservoir thickness.

The total width of the reservoir perpendicular to the wellbore is aH, the total length in a direction parallel to the wellbore is bH, and the total height of the reservoir is the net pay thickness, h. Notice the parameters for the distance from the well to the various borders. Along the axis of the well, the shortest distance from the end of the well to a boundary is dy, and the longest distance from the other end of the well to the boundary is Dy. In the vertical direction, the shortest distance to a vertical boundary is dz, and the longest distance to a vertical boundary is Dz.

Characteristics of Flow Regimes. Consider a well producing at a constant rate. The early radial flow regime occurs before the area drained or the pressure transient caused by this production encounters either of the vertical boundaries of the reservoir. Fig. 8.96 shows a radial flow pattern penetrating out into the reservoir. Actually, however, this flow pattern is likely to be elliptical, moving further into the reservoir at a given time in the higher-permeability x-direction than in the lower-permeability z-direction. This phenomenon causes no significant complications in our analysis.

When the wellbore is much nearer one vertical boundary than the other, another flow regime, called half-radial or hemiradial flow (Fig. 8.97) may exist. Hemiradial flow can occur immediately following the early radial flow regime, if the well is much nearer one of the vertical boundaries than the other. Eventually, the area affected by the production will include the entire thickness of the reservoir. When that happens, a linear flow pattern may develop, as Fig. 8.98 shows.

Eventually, flow will begin to come into the wellbore from beyond the ends of the well. Until these end effects become important, early linear flow continues. Once end effects become important a transition period is followed by a later pseudoradial flow regime, as Fig. 8.99 illustrates.

This flow regime continues until the area affected by the production reaches one of the sides of the reservoir. Once the area affected is the entire width of the reservoir (that is, the pressure transient has reached both sides of the reservoir), then the late-linear flow regime begins (Fig. 8.100).

### Identifying Flow Regimes in Horizontal Wells

All of these flow regimes in a test can be identified on a diagnostic log-log plot of the pressure change, Δp, and pressure derivative, p′, against the logarithm of time (Fig. 8.101).

A unit-slope line appears during wellbore storage; a horizontal derivative during early radial flow, and then, later, in pseudoradial flow; and a half-slope line in early-linear flow and then in late-linear flow. (These half-slope lines appear on the derivative but not on the pressure-change curves.) This does not imply that all these flow regimes will appear in any given test; in fact, that would be rare. But these are the shapes that identify the flow regimes that may appear in the test being analyzed.

The shapes that may appear in a drawdown test (which is the basis of Fig. 8.101) may not appear in a buildup test because of the complex superposition of flow regimes. For example, a test would have to be in linear flow both at time (tp + Δt) and at time Δ t to ensure appearance of a derivative with half-slope; this is highly unlikely. The best way to solve the problem is to ensure that a buildup test on a horizontal well is run with a producing time, tp, much greater than the maximum shut-in time in the test (that is, tp > 10 Δt max ).

Table 8.A-2 (see Appendix) summarizes the working equations for permeability, skin, and start and end of each of the recognized flow regimes. Different investigators have found different equations for start and end of various flow regimes, especially linear flow regimes. This is partly because of a difference of assumptions about flow into the wellbore. Uniform flux or infinite conductivity models are common; neither is rigorously applicable in practice. [38] In this section, the equations for duration of flow regimes derived by Odeh and Babu[39] are used. This model assumes uniform flux into the wellbore.

Early-Radial Flow. Early-radial flow is similar to the radial flow period in a vertical well (Fig. 8.96). The governing equation for this flow regime is

....................(8.186)

Data for this period may be masked by wellbore storage effects, but, when present, they may be analyzed on a semilog plot.

The early-radial flow regime may in theory start at time zero, in absence of wellbore storage effects. The end of the early-radial flow regime may occur when the transient reaches a vertical boundary or when flow comes from beyond the end of the wellbore. The end of the period is the smaller of these two values.

Eq. 8.187[39] says that the period must end when the transient reaches the nearest boundary, dz, from the well. This equation includes the permeability in the vertical direction:

....................(8.187)

The radial flow regime may also end when flow from beyond the end of the wellbore becomes important. Eq. 8.188 gives the time by

....................(8.188)

Lw is the completed length of the well, and k y is the permeability in the direction parallel to the wellbore. The actual end is the lesser of the two times calculated from Eqs. 8.187 and 8.188. It is helpful to check the expected duration of the early-radial flow regime after estimating the parameters necessary to make these calculations.

Eq. 8.186 suggests that possible radial flow on the diagnostic plot be identified and then bottomhole flowing pressure be plotted against time during the appropriate time range on semilog coordinates. The slope of the straight line that results is

....................(8.189)

The group can be found from the slope, merf:

....................(8.190)

Effective completed length of the well must be known to make this calculation. This is not necessarily the same as the perforated or completed length of the well. Some sections of the well may not produce at all.

The equation for calculating the altered permeability skin, sd, for early-radial flow is

....................(8.191)

When analyzing a buildup test rather than a constant-rate flow test, plot the HTR or equivalent time on the horizontal axis of the semilog plot, and then plot shut-in or equivalent time on the vertical axis. Note that this plotting is correct only if (tp + Δt) and Δt appear in this time period simultaneously; that is, radial flow must exist at both time (tp + Δt) and time Δt. This is unlikely because radial-flow regime may exist at time Δt, but a different flow regime is likely at time (tp + Δt).

Example 8.2: Well Erf-1 For drawdown test data from Well Erf-1, [39] the diagnostic plot indicates the data from approximately 0.24 to 24 hours may be in early-radial flow. The following information is available for this well: q = 800 STB/D, μ = 1 cp, B = 1.25 RB/STB, rw = 0.25 ft, ϕ = 0.2, ct = 15×10 –6 psi–1, centered in box-shaped drainage area, h = 200 ft, bH = 4,000 ft, and aH = 2,000 ft, Lw =1,000 ft, and, from analysis of data in early linear flow regime, kx = 200 md. Table 8.3 shows the pressure change data for 0.24 to 24 hours.

Plot (pipwf ) = Δp vs. t on semilog coordinates (Fig. 8.102). The plot results in a straight line with a slope of 8 psi/cycle. In Fig. 8.102, at t = 2.4 hours, the points begin to deviate from the straight line, as expected from calculations for flow regime duration that follow. The pressure change at 1 hour is 39 psia. Using the slope of 8 and Eq. 8.190,

Thus, because kx = 200 md, kz = 2 md. Using the value of 39 for Δp1hr from Fig. 8.102, skin from Eq. 8.191 is

The start of the early-radial flow regime is controlled by wellbore storage, which appears to have vanished at times earlier than 0.24 hours in this example. The end of the early radial flow regime is expected at the lesser of the two values derived from Eqs. 8.187 and 8.188. For a centered well, dz = h/2 = 100 ft, and Eq. 8.187 gives

Assuming ky = kx = 200 md, Eq. 8.188 gives

Thus, expect the early-radial flow regime to end at approximately 1.875 hours, which is the smaller value and is consistent with observed test data.

The hemiradial flow period (Fig. 8.97) will occur only when the well is close to one of the vertical boundaries (either the upper or the lower boundaries) and is analogous to a vertical well near a fault. The governing equation is[38]

....................(8.192)

A horizontal derivative on the diagnostic plot identifies hemiradial flow. If data appear to fall into this flow regime, a straight line on a semilog plot would provide more confidence that radial flow has been identified. Consistency checks in the analysis coupled with a well survey will be required to distinguish hemiradial flow from early radial flow.

The time range in which the analysis for hemiradial flow is valid begins after the closest vertical boundary, dz, affects the data and before the farthest boundary, Dz, affects them. In the absence of wellbore storage, the start of hemiradial flow is given by

....................(8.193)

Note that the start of the hemiradial flow regime involves the shortest distance to a vertical boundary and the permeability in the vertical direction. However, wellbore storage will most likely determine the actual start of hemiradial flow.

The end of hemiradial flow occurs when pressure is affected by the farther vertical boundary or flow from beyond the ends of the wellbore, whichever occurs first. It is the smaller of the times calculated using Eqs. 8.194 and 8.195. If the hemiradial flow regime ends when pressure reaches the farthest vertical boundary, it depends on the distance, Dz, and the vertical permeability, kz:

....................(8.194)

When the appearance of end effects—flow from beyond the ends of the wellbore—causes the end of the hemiradial flow regime to appear, the end of the flow regime occurs when

....................(8.195)

The completed length of the well, Lw, and the permeability ky, parallel to the wellbore appear in this equation. These parameters determine when enough flow has come from beyond the ends of the wellbore to distort the radial flow pattern that appeared earlier.

....................(8.196)

gives the slope of the semilog straight line for semiradial flow, mhrf. The multiplier, 325.2, is twice the multiplier for early-radial flow. The equation to estimate the damage skin factor is also similar to that for radial flow but has a multiplier that differs by a factor of two.

....................(8.197)

The equations relating slope and permeability and the equation for skin are similar in a buildup test to those for a drawdown test. The pressure change in the equation for skin is [p1hrpwft = 0)]. Semilog plots of buildup test data from the hemiradial flow regime cannot be analyzed rigorously using data from a Horner plot unless the pressure data at (tp + Δt) and at time Δt are simultaneously in this flow regime. As a practical matter, the hemiradial flow regime is likely to appear clearly in the buildup test only when the producing time is much greater that the shut-in time.

### Early Linear Flow

The governing equation for early-linear flow is[38]

....................(8.198)

The "convergence skin," sc, is discussed later in this section. The start of the early-linear flow regime (Fig. 8.98) depends on the farthest distance to a vertical boundary, Dz, and the vertical permeability, kz.[39]

....................(8.199)

Not until flow reaches that farthest vertical boundary can a linear flow pattern begin toward the well. This flow period ends when fluids flow from beyond the ends of the wellbore. Thus,

....................(8.200)

Notice that the end depends on the effective completed length of the well, Lw, and on the permeability in the direction parallel to the well. This is the time in which end effects—flow beyond the ends of the well—begin to significantly distort the linear flow pattern.

The early-linear flow regime is identified in a drawdown test with a half-slope for the derivative. (Because of the skin effect, the pressure change curve on the diagnostic plot will only approach a half-slope asymptotically.) For data identified as being in this flow regime, plot pressure against the square root of time.

The slope of the straight line on such a plot, melf, can be used to estimate the square-root of kx, the horizontal permeability perpendicular to the well:

....................(8.201)

To calculate the damage skin,

....................(8.202)

This equation includes a convergence skin, sc, which is[39]

....................(8.203)

This convergence skin is an additional pressure drop that acts like a skin effect caused by flow moving from throughout the entire formation until it converges down to the small wellbore in the middle of the formation (Fig. 8.103). This convergence skin is defined in terms of the ratio of the permeability in the x-direction, which is perpendicular to the wellbore, to the vertical permeability. It also involves the distance to the nearest vertical boundary, dz, and the net pay thickness, h.

Kuchuk[40] derived a different equation for convergence skin. As a practical matter, the Odeh-Babu and Kuchuk equations lead to the same result. When there has been a single rate preceding shut-in during early-linear flow, the buildup pressure is plotted against on a Cartesian plot, which is sometimes called a tandem root plot. The permeability, kx, is calculated from the slope, melf, of the plot and Eq. 8.201. kx has the same relationship to the slope that existed in a drawdown test. Skin for this flow regime is calculated with Eqs. 8.202 and 8.203.

Plots of buildup data from the early-linear flow regime cannot be analyzed rigorously with a plot of pws vs. (that is, the tandem-root plot) unless data at (tp + Δt) and Δt are simultaneously in this flow regime—highly unlikely—or unless tp is much greater than Δt, in which case simply plot pws vs. . Little error results from ignoring the (tp + Δt) term, which is essentially constant.

Example 8.3: Well Elf-2 The diagnostic plot for a drawdown test from Well Elf-2[39] indicates data in the early-linear flow regime because the derivative has a half-slope. The following data apply to this well: q = 800 STB/D; μ = 1 cp; B = 1.25 RB/STB; rw = 0.25 ft; ϕ = 0.2; ct = 15×10–6 psi–1; centered in box-shaped drainage area 100 ft thick, 4,000 ft long, 4,000 ft wide; Lw = 2,500 ft; and, from early radial-flow regime data, kxkz = 8,000 md2. In addition, Table 8.4 shows the pressure-change data for this well.

A plot of pressure change vs. the square root of time (Fig. 8.104) indicates early linear flow. The final point on the straight line is at a time of approximately 24 hours, but there may have been some deviation from the straight line by this time. From the slope of the straight line, melf = 0.934 psi/hr1/2 and Eq. 8.201, calculate the permeability in the horizontal plane perpendicular to the wellbore.

or kx = (20.1)2 = 404 md.

Analysis of data from the early radial flow regime indicated that kxkz is 8,000 md2; thus, kz is approximately 19.8 md. To calculate sd, use Eqs. 8.202 and 8.203, noting that the value for (pipwf ) = 3.1 at t = 0.

Then, sd = 4.91 – 4.91 = 0.

Check the expected time range for the early-linear flow regime.

Using Dz = h/2 = 50 ft and kz = 20 md, calculate the beginning of linear flow with Eq. 8.199:

Assuming ky = kx at 400 md, use Eq. 8.200 to find the end of early linear flow.

These limits are reasonably consistent with the time range analyzed assuming early linear flow.

The governing equation for late-pseudoradial flow is[38]

....................(8.204)

The late-pseudoradial flow period occurs only if [39]

....................(8.205)

Here, bH is the dimension of the reservoir parallel to the wellbore. As long as the completed length of the well is relatively short compared with the length of the drainage area late-pseudoradial flow can occur.

The start of this flow period occurs when fluid flows from well beyond the ends of the wellbore (Fig. 8.99). It is approximated with[39]

....................(8.206)

This starting time depends on the completed length of the well, Lw, and on the permeability in the direction of the well, ky. The end of this period, like others in this section, is approximated by the minimum of the results of two calculations. The first,

....................(8.207)

depends on dy and the length of the wellbore along with ky, the permeability in the direction parallel to the wellbore. This is the time at which horizontal boundary effects first appear.

The other equation gives a time at which the radial-flow pattern begins to be distorted depending on the shortest distance, dx, from the well to a boundary perpendicular to the wellbore and on kx, the permeability in that direction.

....................(8.208)

Whenever boundary effects first appear, whether in a direction that is parallel to the well or perpendicular to the axis of the well, the late-pseudoradial flow period will end.

The diagnostic plot helps identify the late-pseudoradial flow regime with the characteristic horizontal derivative. For data in the appropriate time range, prepare a semilog plot of pressure against time for a drawdown test. The slope of this plot will be mprf and the relationship between that slope and the square root of kxky, or the permeabilities in the horizontal plane, is given by

....................(8.209)

The skin equation is similar in form to those seen before:

....................(8.210)

Again, the total skin depends on Δp1hr. The convergence skin (Eq. 8.203) is subtracted from the "total" skin to determine the damage skin.

For a buildup test preceded by production at a single rate, plot pressure against the HTR on a semilog graph. Permeability is calculated from Eq. 8.209, the same as for a drawdown test. The skin equation is basically the same as for a drawdown test, except that the Δp1hr is now p1hrpwf. To obtain p*, the extrapolated pressure, extrapolate the semilog straight line to a HTR of unity.

Semilog plots of buildup test data from the late-pseudoradial flow regime cannot be analyzed rigorously using a Horner plot unless pressures at (tp + Δt) and at time Δt are simultaneously in the pseudoradial flow regime, which is highly unlikely. However, little error appears if the producing time before shut-in is much greater than the maximum shut-in time achieved in the buildup test.

Example 8.4: Well Prf-3 The diagnostic plot suggests that a constant-rate drawdown test from Well Prf-3[39] includes data in the late-pseudoradial period. The following data are available from the test: q = 800 STB/D, μ = 1 cp, B = 1.25 RB/STB, rw = 0.25 ft, ϕ = 0.2, ct = 15×10–6 psi–1, h = 150 ft, Lw = 900 ft, aH = 5,280 ft, bH = 5,280 ft, well centered in drainage volume, kx = 100 md (from analysis of early linear flow), kxkz = 1,000 md2, and kz = 10 md (from analysis of early radial flow). Table 8.5 gives the pressure change, Δp = pipwf vs. time.

A plot of pressure change vs. the logarithm of time (Fig. 8.105) confirms pseudoradial flow. A straight line fits all the data from 192 to 432 hours; the slope of the line, mprf, is 15.3 psi/cycle, and Δp1 hr = 18.94 psi (extrapolated). Then, from Eq. 8.209,

Thus,

From Eqs. 8.210 and 8.203 ,

Here,

The pseudoradial flow regime should start at the time given by Eq. 8.206:

It should end at a time given by the lesser of values from Eqs. 8.207 and 8.208. From Eq. 8.207,

where dy = 1/2(5,200-900) = 2,190ft for this centered well. From Eq. 8.208,

The smaller of these two values is 344 hours, which is thus the expected end of pseudoradial flow. The data on the figure that lie on the straight line show the time range from 192 to 432 hours, which is generally consistent with the expected duration of the flow regime.

### Late-Linear Flow

The governing equation for late-linear flow is[38][39]

....................(8.211)

The late-linear flow regime starts after the pressure transient has reached the boundaries in the z- and y-directions, and the flow behavior with regard to these directions has become pseudosteady state, as Fig. 8.100 shows.

The start of this time period is the maximum of two equations. [39] The first depends on the time to reach the boundary, Dy, beyond the end of the horizontal well. It also depends on the permeability, ky, in the direction parallel to the wellbore.

....................(8.212)

Another requirement for the start of the late-linear flow regime is the time to reach the maximum vertical distance, Dz, divided by the vertical permeability:

....................(8.213)

Usually, the start of the late-linear flow regime is dictated by the time to reach the boundaries in the y-direction. The end of this period is given by the equation

....................(8.214)

The end of the late-linear flow regime depends on reaching the nearest boundary in the direction perpendicular to the wellbore, which is the distance, dx, away, and on the permeability kx in that direction.

Identify the late-linear flow regime by a half-slope on the derivative in the diagnostic plot of drawdown test data. (The pressure change may approach a half-slope asymptotically.) For data that appear to be in this flow regime, plot pressure against the square root of time. From the slope mllf of the plot, estimate permeability in the x-direction from

....................(8.215)

Alternatively, if kx is known from an early-linear flow regime, estimate bH, the length of the drainage area, from

....................(8.216)

This late-linear flow regime is the only period that provides the data to calculate the total skin, s, including the partial-penetration skin, sp, and the convergence skin, sc. To calculate the damage skin, sd, use

....................(8.217)

The total skin depends on Δpt = 0. Subtracting the partial penetration skin, sp, and the convergence skin, sc, from the total skin yields the damage skin.

The partial-penetration skin is a complex function that is calculated with Eqs. A-25 through A-35 in Table 8.A-2. For a buildup test, plot pressure against the HTR. From the slope, mllf, calculate kx with Eq. 8.215, exactly the same as for drawdown tests. Or, if kx is known, estimate the length, bH, of the drainage area with Eq. 8.216. Calculate the damage skin, sd, from a pressure buildup test from Eq. 8.217, where Δpt = 0 = (pt = 0)extpwf(t = 0).

Note that the same difficulty arises in using superposition to find plotting functions plots of buildup data from the late-linear flow regime as existed with the previous flow regimes. Pressures at both time (tp + Δt) and time Δt must be in the late linear flow regime for a tandem-root plot to be valid. However, if tp >> Δtmax, there is little error.

Example 8.5: Well Llf-4 The diagnostic plot for a drawdown test from Well Llf-4[39] appears to include data in the late-linear flow regime (derivative with half-slope). The following data applies to this well: q = 800 STB/D, μ = 1 cp, B = 1.25 RB/STB, rw = 0.25 ft, ϕ = 0.2, ct = 15 × 10–6 psi–1 , h = 150 ft, Lw = 1,000 ft, bH = 2,000 ft (well centered), aH = 6,968 ft (well centered), Dz = 85 ft, dz = 65 ft, kxkz = 1,000 md2 (from analysis of early-radial flow), and kxky = 5,000 md2 (from analysis of pseudoradial flow). Table 8.6 gives pressure change, Δp = pipwf , data vs. time.

Fig. 8.106 is a plot of pressure change vs. the square root of time. The straight line on this plot for the entire time range (60 to 240 hours) confirms late-linear flow for this range. The slope of the line is 1.56 psi/hr1/2, and the intercept is Δpt = 0 =28.4 psi.

From Eq. 8.215,

Then, kx = 100 md. Because kkkz = 100 md2, kz = 10 md. Also, because kxky = 5,000 md2, ky =50 md. From Eq. 8.217,

From Eq. 8.203,

Calculate the partial penetration skin, sp , using the appropriate equation from among Eqs. A-25 through A-35 in Table 8.A-2. First, calculate

to determine whether "Case 1" or "Case 2" applies:

Because

this is Case 1 (Eq. A-26). Use Eqs. A-25 through A-31 from the table. From Eq. A-27, sp = pxyz + pxy. From Eq. A-25,

From Eq. A-28,

Here, from Eq. A-29,

The well is centered, so dy = Dy = 500 ft.

From Eq. A-30,

Also,

From Eq. A-31,

and

Then,

Then,

Now check the expected duration of the late-linear flow regime. The start is the larger of values from Eqs. 8.212 and 8.213. From Eq. 8.212,

From Eq. 8.213,

Thus, the start is expected to be at approximately 162 hours. Eq. 8.214 gives the end of the flow regime.

The data in this example spanned the time range from 60 to 240 hours. Some of the data that fall on the straight line are, in theory, from times before the start of the late-linear flow regime, but they appear to cause no problem in determining the slope of the line.

### Field Examples[41]

The following field examples illustrate the procedures used in analyzing horizontal well-test data.

Field Example Well A. Table 8.7 summarizes the reservoir and completion properties for Well A. The target for Well A, a horizontal exploration well, was vertical tectonic fracture development in a low-permeability shale. Because of the fractures, the permeability is assumed to be isotropic (kh = kz) and a result of the fractures. Fig. 8.107 is a diagnostic plot for Well A and includes a history match using an analytical model.

During the early part of the test there is a unit-slope line representing wellbore storage. Following that wellbore storage, there is a transition to radial flow. The final few data points may be in radial flow. On the Horner plot (Fig. 8.108), the last few data points fall on a straight line. From the slope of the straight line, the apparent permeability is 0.011 md and the altered zone skin is 2.9. There is no evidence of boundary effects on this Horner plot. The existence of the semilog straight line is not assured, but the data are at least on the verge of reaching it.

Guided by the Horner analysis results, engineers simulated these data with an analytical horizontal well model. The initial match was poor. The match was improved considerably by introducing a no-flow boundary approximately 16 ft above the wellbore, which led to a permeability estimate of 0.027 md and an apparent skin of 11.5. It was concluded that the flow regime observed in the test was hemiradial flow.

The final match, shown on the type-curve plot in Fig. 8.107, is still not a good match at all times, but the author stated that the poor match in the transition region could be the result of phase-redistribution effects in the wellbore. The distance to the no-flow boundary that led to the best match compared favorably with well survey data, which indicated that the well was drilled approximately 20 ft below the upper limit of the productive horizon.

Well B is in a west Texas carbonate formation. It was expected to have isotropic permeability caused by fracturing and dissolution. Table 8.8 gives the field data for this well.

Fig. 8.109 is the diagnostic plot for this well. After wellbore storage, a short period of radial flow appears to be followed by the onset of linear flow, because p′ approaches a slope of 0.5. In the time region where the derivative is horizontal, a straight line on the Horner plot (Fig. 8.110) yields k = 0.14 md. Using these results in the analysis shows that the end of radial flow occurs at tErf = 165 hours.

A tandem-root plot (Fig. 8.111) indicates linear flow and also suggests a distance to the nearest boundary of 29 ft. This is in good agreement with geological observations and helps to verify the assumption of isotropic permeability. The history match with an analytical horizontal well model, shown in Fig. 8.109, confirms the results of the Horner and tandem-root plots.

Field Example Well C. Well C data are from a buildup test of a horizontal well in a high-permeability sandstone where a 54-ft oil column overlies an extensive aquifer estimated to be approximately 180 ft thick. Table 8.9 shows the available data.

The diagnostic plot (Fig. 8.112) shows essentially no wellbore storage and a constant derivative, indicating radial, hemiradial, or elliptical flow at early times. The rapid decline of the derivative at the end of the test is caused by the aquifer underlying the oil column, which is acting like a constant-pressure boundary. A history match with an analytical horizontal well model with one no-flow and one constant-pressure boundary (the lower boundary), yielded kh = 313 md, kz = 7.5 md, sa = 1.5, and Lw = 356 ft. The no-flow boundary was estimated to be approximately 112 ft below the wellbore.

The time at which the early radial flow regime ends—the time where the derivative ceases to be flat on the diagnostic plot—is approximately 1.5 hours. For a wellbore with a volume of 130 bbl filled with a fluid of compressibility of 3.5 × 10–6 psi, the duration of wellbore storage (the unit-slope line) is estimated to be 0.0005 hours. With the gauge sampling rate set at 0.017 hours, the wellbore-storage unit slope simply could not be detected and does not appear at all on this plot. Fig. 8.113 is the Horner plot for this test. A straight line appears in the same range as the flat derivative on a diagnostic plot. From the slope of the line, the permeability is estimated to be 53 md, close to the regression analysis match value of 48 md.

### Running Horizontal Well Tests

The measurements in horizontal wells are usually made above the wellbore with the pressure gauge still in the vertical section.

The test string may often be too rigid to pass through the wellbore. However, in most cases, conventional hardware can be used for horizontal well tests. With longer horizontal wellbores, wellbore storage is an inherent problem for testing, even for buildup tests with downhole shut-in. As mentioned previously, problems arise in conducting buildup tests with short-duration production periods because superposition is inappropriate; therefore, Horner plots and tandem-root plots, which depend on superposition being applicable, are often inappropriate.

Another problem in conducting buildup test following a short production period is that significant pressure gradients along the length of the wellbore may cause crossflow within the wellbore during shut-in, so fluid may flow from one region to another in the wellbore. These gradients can be removed and this crossflow eliminated by a longer-duration flow period preceding a buildup test.

Factors That Affect Transient Responses. A number of factors may affect the transient response of a horizontal well test: horizontal permeability (normal and parallel to well trajectory), vertical permeability, drilling damage, completion damage, producing interval that may be effectively much less than drilled length, and variations in standoff along length of well.

In summary, seven or more factors may affect interpretation for horizontal wells in homogeneous reservoirs before the effects from boundaries. The problem is complex, so test results are frequently inconclusive. Furthermore, wellbore storage inhibits determination of properties associated with early-time transient behavior such as vertical permeability and damage from drilling and completion. Middle- and late-time behavior may require several hours, days, or months to appear in transient data.

Some practical steps will help ensure interpretable test data. First, it is helpful to run tests in the pilot hole before kicking off to drill the horizontal borehole section. From a test in the vertical section, it is possible to get usable estimates of horizontal and vertical permeabilities using modern wireline test tools. Second, a good directional drilling survey can frequently provide an adequate estimate of standoff. A production-log flow survey conducted with coiled tubing can determine what part of the wellbore is actually producing and, therefore, help provide an estimate of effective productive length. Wells in developed reservoirs should be flowed long enough to bring pressures along the wellbore to equilibrium and thus minimize crossflow. For high-rate wells, continuous borehole pressure and flow-rate measurements acquired during production can be used to interpret the pressure-drawdown transient response. If the downhole rates are not measured, the buildup test should be conducted with downhole shut-in to minimize wellbore storage distortion of test data.

### Estimating Horizontal Well Productivity

Because of two fundamental problems, estimating the productivity of a horizontal well accurately is even more difficult than estimating the productivity of a vertical well. The theoretical models available have a number of simplifying assumptions and the data required for even these simplified models are not likely to be available. Still, we must make estimates and decisions based on those estimates. In this section, two productivity models that have proved useful in practice are discussed. The first, published by Babu and Odeh[42] in 1989, is limited to single-horizontal wells. The second, published by Economides, Brand, and Frick[43] in 1996, is more general and is useful for multilateral wells.

Babu-Odeh Method. Babu and Odeh[42] obtained a rigorous solution to the diffusivity equation for a well in a box-shaped reservoir, subject to certain limiting assumptions. The assumptions include the following:

• Fluid flows to the well uniformly at all points along the wellbore (uniform flux) and the well is completed uniformly.
• The sides of the drainage volume are aligned with the principal permeability direction.
• The wellbore is parallel to the sides of the drainage area and is oriented parallel to one direction of principal permeability and perpendicular to the other two.
• The boundaries of the reservoir are all no-flow boundaries and the well reaches stabilized, pseudosteady-state flow.
• The formation damage around the wellbore is uniform at all points along the wellbore.

Fig. 8.95 introduces the nomenclature in the Babu and Odeh solution. The solution is quite complex but is approximated accurately with an equation written in the same form as the pseudosteady-state flow equation for a vertical oil well producing a single-phase, slightly incompressible liquid.

....................(8.218)

....................(8.219)

Table 8.A-2 gives equations to estimate CH and sp. Two examples adapted from Babu and Odeh[42] illustrate the application of these equations.

Example 8.6 A horizontal well 1,000 ft long (Lw) is drilled in a box-shaped drainage volume 4,000 ft long (aH), 2,000 ft wide (bH), and 100 ft thick (h). The well is off-center in the y -direction (parallel to the well), so dy = 250 ft and Dy = 750 ft. The well is also off-center in the x-direction so that dx = 1,000 ft and Dx = 3,000 ft. Finally, the well is centered in the z-direction so that dz = Dz = 50 ft. The wellbore radius is 0.25 ft; kx = ky = 200 md and kz = 50 md. Fluid properties are Bo = 1.25 RB/STB and μ = 1 cp. Calculate the productivity index.

Solution.

From Eq. A-38,

and

Thus, use Case 1 equations (Eqs. A-26 through A-31) to calculate sp.

To calculate pxy, determine ym, Lw/2bH, (4ymLw)/2bH, and (4ym + Lw)/2bH.

Thus,

Then,

Then, sp = pxyz + pxy =4.50+6.54=11.0, and

Example 8.7 A horizontal well is drilled in a box-shaped reservoir with the following characteristics: Lw = 1,000 ft, aH = 2,000 ft long, bH = 4,000 ft wide, and hw = 2,000 ft thick. The well is off-center in the y-direction (dy = 1,000 ft; Dy = 2,000 ft), centered in the x-direction (dx = Dx = 1,000 ft), and off-center in the z-direction (dz = 50 ft; Dz = 150 ft). Permeabilities are kx = ky =100 md and kz = 20 md. Wellbore radius is 0.25 ft, Bo = 1.25 RB/STB, μ = 1 cp, and sd = 0. Find the productivity index, J.

Solution.

From Eq. A-38 (Table 8.A-2),

Note that

Thus, use Case 2 equations to calculate sp.

To calculate py, determine ym. From Eq. A-29 (Table 8.A-2),

From Eq. A-35 (Table 8.A-2),

Thus, sp = 16.79 + 7.90 + 7.02 = 31.7. Then, from Eq. A-37 (Table 8.A-2),

Economides et al. Method. Economides et al.[44] presented a more general method to estimate productivity index for a horizontal well. The method has the advantage that it is applicable to multilateral wells in the same plane and is not limited to wells aligned with principal permeabilities. It includes solutions for wells with no pressure drop in the wellbore (infinite conductivity, as opposed to wells with uniform flux). It has the disadvantage that it requires interpolation in a table in which only certain drainage area shapes are given.

The basic working equation for the productivity index in this method is

....................(8.220)

where Σs refers to damage skin, turbulence, and other pseudoskin factors. In Eq. 8.220,

....................(8.221)

where

....................(8.222)

and se, describing eccentricity effects in the vertical direction, is

....................(8.223)

se = 0 when a well is centered in the vertical plane. This convergence skin differs only slightly from that used by Babu and Odeh. The difference is 0.25 ln (kx / kz) + h / Lw[2dz / h - 1/2(2dz / h)2 - 2/3], which is usually small (< 0.5). Table 8.10 gives values of CH for several drainage areas and multilateral configurations. The equations as written are for isotropic reservoirs. Certain variable transformations are required before substituting into the working equation:

....................(8.224)

....................(8.225)

where

....................(8.226)

and ....................(8.227)

ϕ is the azimuth of the well trajectory (relative to the y-axis). Reservoir dimensions:

....................(8.228)

....................(8.229)

....................(8.230)

and ....................(8.231)

Two examples, one from an isotropic reservoir and one from an anisotropic reservoir, illustrate this method.

Example 8.8 Economides et al.[43] provide this example. Consider a horizontal well 1,500 ft long in a reservoir with bH = 2,000 ft, aH = 4,000 ft, h = 20 ft, rw = 0.4, kx = ky = kz = 10 md, Bo = 1.25 RB/STB, and μ = 1 cp. Assume that the well is centered vertically so that se = 0. Also, assume Σs = 0.

Solution.

From Eq. 8.223,

(As a matter of interest, the Babu and Odeh sc for this case is also 2.07.) From Table 8.10, for 2bH = aH and Lw/bH = 1,500/2,000 = 0.75, CH = 2.53. From Eq. 8.221,

Then, from Eq. 8.220,

Example 8.9

Rework the Babu-Odeh Example 8.7 using the Economides et al. method.

Solution.

First transform the variables. From Eqs. 8.226 and 8.227,

Because the well is parallel to the x-axis, ϕ = 0, and

From Eq. 8.231,

From Eq. 8.224,

From Eq. 8.225,

From Eqs. 8.228 through 8.230,

Thus, the equivalent system is a rectangular-shaped drainage area twice as long parallel to the wellbore (3,050 ft) as perpendicular, with L′/bH ′ = 765/3,058 = 0.25. In the original example, the well was off-center in the horizontal plane; here, assume that a centered well is an adequate approximation. From Table 8.10, CH = 3.19.

From Eq. 8.223,

Then,

(The Babu-Odeh sc is 5.60 for this case.) Then, from Eq. 8.221,

Finally, from Eq. 8.220,

The result is slightly larger than the result using the Babu-Odeh method (J = 25.6 STB/D/psi). At least part of the reason for the difference is that, in this example, it was necessary to assume that the well was centered in its drainage volume, which was not true in the original example. The optimal location of a horizontal well to maximize productivity is to center it within its drainage volume.

Comparison of Recent and Older Horizontal Well Models. Ozkan[45] compared "contemporary" (generally 1990s) and "conventional" horizontal well models in a paper published in 2001. He pointed out that the older models are used for both pressure-transient test analysis and for estimating well productivity. Ozkan stressed three limitations of the conventional models, which include the Babu-Odeh model and other pioneering work.

Conventional models usually assume that the horizontal well is parallel to one of the principal permeability directions (preferably the minimum permeability direction in the horizontal plane). In many cases, this is not true. In fact, in many cases the principal permeability directions are unknown. When the principal permeability directions are known, corrections to length are possible (as in the Economides et al. model); if they are not known, there is no way to correct the analysis. Contemporary models show that the error in permeability estimates approaches 50% when the deviation angle exceeds 50°. Unfortunately, the models also indicate that there is nothing in a well’s response that provides any indication that the assumption that the well is parallel to a principal permeability direction is incorrect.

Ozkan pointed out that the damaged region around a horizontal well probably is nonuniform with distance (perhaps with the greatest damage near the heel of the well and the least near the toe, because filtrate invasion is of much longer duration near the heel). If there is variable permeability along the path of the well, the situation is even more complicated. Some contemporary models can take this variation into account; however, most conventional models cannot. Conventional models usually assume (implicitly) uniform skin effect along the wellbore. However, the contemporary models will not be helpful if the skin distribution along the length is unknown.

Ozkan notes that it is a common practice to complete horizontal wells selectively. Also, in other cases, some segment of the well may not be open to flow of reservoir fluids because of relatively low permeabilities or relatively large local skin effects. The absolute amount of the well that is open to flow and the location of the open intervals affect the pressure response in the well. Some contemporary well models can take these effects into account, but, again, the capabilities of the newer models may be limited if the location and length of the open intervals is unknown.

Many models assume negligible pressure drop in the wellbore (infinite conductivity). Others assume the same flow rate per unit length at all points along the well bore (uniform flux). In fact, there is likely to be finite pressure drop in the wellbore, resulting in neither uniform flow nor infinite conductivity. Contemporary models in which a reservoir model is coupled to a wellbore model can take these effects into account.

Unfortunately, contemporary horizontal well models have not led to simple, easily applied methods of well-test analysis or of predicting well productivity. Further, their full utility depends on availability of detailed well and reservoir description data. At present, the major use of such models may be to quantify the possible errors that arise from uncertainty and to be used to history-match observed information when sufficient data are available.

## Deliverability Testing of Gas Wells

### Introduction

This section discusses the implementation and analysis of the four most common types of gas-well deliverability tests: flow-after-flow, single-point, isochronal, and modified isochronal tests. A summary of the fundamental gas-flow equations, both theoretical and empirical, used to analyze deliverability tests in terms of pseudopressure is followed by a focus on specific tests and testing procedures, advantages and disadvantages of each testing method, and common analysis techniques. Examples illustrating deliverability tests analyses are included.

### Types and Purposes of Deliverability Tests

Deliverability testing refers to the testing of a gas well to measure its production capabilities under specific conditions of reservoir and bottomhole flowing pressures (BHFPs). A common productivity indicator obtained from these tests is the absolute open flow (AOF) potential. The AOF is the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the sandface. Although in practice the well cannot produce at this rate, regulatory agencies sometimes use the AOF to allocate allowable production among wells or to set maximum production rates for individual wells.

Another application of deliverability testing is to generate a reservoir inflow performance relationship (IPR) or gas backpressure curve. The IPR curve describes the relationship between surface production rate and BHFP for a specific value of reservoir pressure (that is, either the original pressure or the current average value). The IPR curve can be used to evaluate gas-well current deliverability potential under a variety of surface conditions, such as production against a fixed backpressure. In addition, the IPR can be used to forecast future production at any stage in the reservoir’s life.

Several deliverability testing methods have been developed for gas wells. Flow-after-flow tests are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHP. Each flow rate is established in succession without an intermediate shut-in period. A single-point test is conducted by flowing the well at a single rate until the BHFP is stabilized. This type of test was developed to overcome the limitation of long testing times required to reach stabilization at each rate in the flow-after-flow test.

Isochronal and modified isochronal tests were developed to shorten tests times for wells that need long times to stabilize. An isochronal test consists of a series of single-point tests usually conducted by alternately producing at a slowly declining sandface rate without pressure stabilization and then shutting in and allowing the well to build to the average reservoir pressure before the next flow period. The modified isochronal test is conducted similarly, except the flow periods are of equal duration and the shut-in periods are of equal duration (but not necessarily the same as the flow periods).

### Theory of Deliverability Test Analysis

This section summarizes the theoretical and empirical gas-flow equations used to analyze deliverability tests. The theoretical equations developed by Houpeurt[44] are exact solutions to the generalized radial-flow diffusivity equation, while the Rawlins and Schellhardt[46] equation was developed empirically. All basic equations presented here assume radial flow in a homogeneous, isotropic reservoir and therefore may not be applicable to the analysis of deliverability tests from reservoirs with heterogeneities, such as natural fractures or layered pay zones. These equations should not be used to analyze tests from hydraulically fractured wells during the fracture-dominated linear or bilinear flow periods. Finally, these equations assume that wellbore-storage effects have ceased. Unfortunately, wellbore-storage distortion may affect the entire test period in short tests, especially those conducted in low-permeability reservoirs.

Theoretical Deliverability Equations. The early-time transient solution to the diffusivity equation for gases for constant-rate production from a well in a reservoir with closed outer boundaries, written in terms of pseudopressure, pp,[47] is

....................(8.232)

where ps is the stabilized shut-in BHP measured before the deliverability test. In new reservoirs with little or no pressure depletion, this shut-in pressure equals the initial reservoir pressure, ps = pi, while in developed reservoirs, ps < pi.

The late-time or pseudosteady-state solution is

....................(8.233)

where is current drainage-area pressure. Gas wells cannot reach true pseudosteady state because μg(p)ct(p) changes as decreases. Note that, unlike , which decreases during pseudosteady-state flow, ps is a constant.

Eqs. 8.232 and 8.233 are quadratic in terms of the gas flow rate, q. For convenience, Houpeurt[44] wrote the transient flow equation as

....................(8.234)

and the pseudosteady-state flow equation as

....................(8.235)

where

....................(8.236)

....................(8.237)

and ....................(8.238)

The coefficients of q (at for transient flow and a for pseudosteady-state flow) include the Darcy flow and skin effects and are measured in (psia2/cp)/(MMscf/D) when q is in MMscf/D. The coefficient of q2 represents the inertial and turbulent flow effects and is measured in (psia2/cp)/(MMscf/D)2 when q is in MMscf/D.

The Houpeurt equations also can be written in terms of pressure squared and are derived directly from the solutions to the gas-diffusivity equation, assuming that μgz is constant over the pressure range considered. For transient flow,

....................(8.239)

....................(8.240)

The flow coefficients are

....................(8.241)

....................(8.242)

and ....................(8.243)

When the Houpeurt equation is presented in terms of pressure squared, the coefficients of q are measured in psia2/(MMscf/D) when q is in MMscf/D, while the coefficient of q2 is measured in units of psia2/(MMscf/D)2 when q is in MMscf/D. For convenience, all equations and examples in this section are presented with q measured in MMScf/D.

The pressure-squared form of the equation should be used only for gas reservoirs at low pressures (less than 2,000 psia) and high temperatures. To eliminate doubt about which equations to choose, use of the pseudopressure equations, which are applicable at all pressures and temperatures, is recommended. Consequently, all the analysis procedures in this section are presented in terms of pseudopressure.

An advantage of the pseudopressure form of the theoretical deliverability equation is that the flow coefficients are independent of the average reservoir pressure and, therefore, do not change as decreases during a flow test conducted under pseudosteady-state flow unless s, k, or A changes. Because the non-Darcy flow coefficient is a function of μg(pwf ), the coefficient b will change slightly if the BHFP is changed. In contrast, because of the pressure dependency of the gas properties on average reservoir pressure, the flow coefficients for the pressure-squared form of the deliverability equation must be recalculated for every new value. When s, k, or A changes with time, the only way to update the deliverability curve is to retest the well.

Empirical Deliverability Equations. In 1935, Rawlins and Schellhardt[46] presented an empirical relationship that is used frequently in deliverability test analysis. The original form of their relation, given by Eq. 8.244 in terms of pressure squared, is applicable only at low pressures:

....................(8.244)

In terms of pseudopressure, Eq. 8.244 becomes

....................(8.245)

which is applicable over all pressure ranges. In Eqs. 8.244 and 8.245, C is the stabilized performance coefficient and n is the inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate. Depending on the flowing conditions, the theoretical value of n ranges from 0.5, indicating turbulent flow throughout a well’s drainage area, to 1.0, indicating laminar flow behavior modeled by Darcy’s law. The value of C changes depending on the units of flow rate and whether Eq. 8.244 or 8.245 is used. All equations and examples in this section are presented with q measured in MMscf/D.

Houpeurt proved that neither Eq. 8.244 nor Eq. 8.245 can be derived from the generalized diffusivity equation for radial flow of real gas through porous media. Although the Rawlins and Shellhardt equation is not theoretically rigorous, it is still widely used in deliverability analysis and has worked well over the years, especially when the test rates approach the AOF potential of the well and the extrapolation from test rates to AOF is minimal.

### Stabilization Time

Unlike pressure-transient tests, the analysis techniques for conventional flow-after-flow and single-point tests require data obtained under stabilized flowing conditions. Although isochronal and modified isochronal tests were developed to circumvent the requirement of stabilized flow, they may still require a single, stabilized flow period at the end of the test. Consequently, there is a need to understand the meaning of stabilization time and have a method to estimate its value.

Stabilization time is defined as the time when the flowing pressure is no longer changing or is no longer changing significantly. Physically, stabilized flow can be interpreted as the time when the pressure transient is affected by the no-flow boundaries, either natural reservoir boundaries or an artificial boundaries created by active wells surrounding the tested well. Consider a graph of pressure as a function of radius for constant-rate flow at various times since the beginning of flow. As Fig. 8.1 shows, the pressure in the wellbore continues to decrease as flow time increases. Simultaneously, the area from which fluid is drained increases, and the pressure transient moves farther out into the reservoir.

The radius of investigation, the point in the formation beyond which the pressure drawdown is negligible, is a measure of how far a transient has moved into a formation following any rate change in a well. The approximate position of the radius of investigation at any time for a gas well is estimated by Eq. 8.246[48]:

....................(8.246)

Stabilized flowing conditions occur when the calculated radius of investigation equals or exceeds the distance to the drainage boundaries of the well (i.e., rire). Substituting r e and rearranging Eq. 8.246, yields an equation for estimating the stabilization time, ts, for a gas well centered in a circular drainage area:

....................(8.247)

As long as the radius of investigation is less than the distance to the no-flow boundary, stabilization has not been attained and the pressure behavior is transient. To illustrate the importance of stabilization times in deliverability testing, stabilization times were calculated as a function of permeability and drainage area for a well producing a gas with a specific gravity of 0.6 from a formation at 210°F and an average pressure of 3,500 psia , with a porosity of 10%. Table 8.11 shows that, for wells completed in low-permeability reservoirs, several days—or even years—are required to reach stabilized flow, while wells completed in high-permeability reservoirs stabilize in a short time.

A more general equation for calculating stabilization time is

....................(8.248)

where tDA is dimensionless time for the beginning of pseudosteady-state flow. Values for tDA are given in Table 8.A-1 for various reservoir shapes and well locations. [49] The time required for the pseudosteady-state equation to be exact is found from the entry in the column "Exact for tDA >."

The Rawlins-Schellhardt and Houpeurt deliverability equations assume radial flow. If pseudoradial flow has been achieved, however, these analysis techniques can be used for hydraulically fractured wells. The time to reach the pseudoradial flow regime, tprf, occurs[30] at and is estimated with

....................(8.249)

To illustrate the importance of achieving pseudoradial flow during a deliverability test, values of tprf were calculated for a hydraulically fractured well completed in a reservoir with ϕ = 0.15, = 0.03 cp, and = 1 × 10−4 psia−1 and with the range of permeabilities and hydraulic fracture half-lengths in Table 8.12. The results illustrate that a well with a long fracture in a low-permeability formation will take far too long to stabilize for conventional deliverability testing.

### Analysis of Deliverability Tests

This section discusses the implementation and analysis of the flow-after-flow, single-point, isochronal, and modified isochronal tests. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.

Flow-After-Flow Tests. Flow-after-flow tests, sometimes called gas backpressure or four-point tests, are conducted by producing the well at a series of different stabilized flow rates and measuring the stabilized BHFP at the sandface. Each different flow rate is established in succession either with or without a very short intermediate shut-in period. Conventional flow-after-flow tests often are conducted with a sequence of increasing flow rates; however, if stabilized flow rates are attained, the rate sequence does not affect the test. [46] The requirement that the flowing periods be continued until stabilization is a major limitation of the flow-after-flow test, especially in low-permeability formations that take long times to reach stabilized flowing conditions. Fig 8.114 illustrates a flow-after-flow test.

Rawlins-Schellhardt Analysis Technique. Recall the empirical equation that forms the basis for the Rawlins-Schellhardt analysis technique:

....................(8.245)

Taking the logarithm of both sides of Eq. 8.245 yields the equation that forms the basis for the Rawlins-Schellhardt analysis technique:

....................(8.250)

The form of Eq. 8.250 suggests that a plot of log (Δpp) vs. log (q) will yield a straight line of slope 1/n and an intercept of {–1/n[log(C)]}. The AOF potential is estimated from the extrapolation of the straight line to Δpp evaluated at a pwf equal to atmospheric pressure (sometimes called base pressure). This analysis technique is illustrated with Example 8.10.

Houpert Analysis Technique. Flow-after-flow tests require stabilized data or data measured during pseudosteady-state flow. Houpeurt[44] gives the theoretical equation for pseudosteady-state flow, which was derived from the gas-diffusivity equation, as

....................(8.235)

The coefficients a and b have theoretical bases and can be estimated if reservoir properties are known or they can be determined from flow-after-flow test data. Dividing both sides of Eq. 8.235 by the flow rate, q, and rearranging yields the equation that is the basis for the Houpeurt analysis technique:

....................(8.251)

The form of Eq. 8.251 suggests that a plot Δpp/q vs. q will yield a straight line with a slope b and an intercept a. The AOF is estimated in the Houpeurt deliverability analysis by solving Eq. 8.235 for q = qAOF at pwf = pb.

Example 8.10: Analysis of a Flow-After-Flow Test

Estimate the initial stabilized AOF potential of a well[50] with the well and reservoir properties listed. Use both the Rawlins-Schellhardt and the Houpeurt analysis techniques. In addition, estimate the AOF potential 10 years later when the static drainage area pressure has decreased to 350 psia. Evaluate the AOF potential at pb = 14.65 psia. Table 8.13 summarizes the flow-after-flow test data. L = 3,050 ft, rw = 0.5 ft, Ma = 20.71 lbm/lbm-mole, T = 90°F = 555°R, A = 640 acres, ϕ = 0.25, CA = 30.8828, and h =200 ft.

Current = 407.6 psia, pp( = 407.6) = 1.617 × 107 psia2/cp. after 10 years = 350 psia, pp( = 350) = 1.2239 × 107 psia2/cp. pb = 14.65 psia, pp(pb) = 2,674.8 psia2/cp.

The pseudopressure in this example (and all others in this section) were calculated using the methods suggested by Al-Hussainy et al.[15] These methods, which involve numerical evaluation of the integral in Eq. 8.97 and which require computational routines to estimate gas viscosity, μ, and deviation factor, z, are widely available in basic reservoir fluid flow analysis software.

Solution.

Rawlins-Schellhardt Analysis. Plot Δpp vs. q on log-log graph paper (Fig. 8.115). Table 8.14 gives the plotting functions. Construct the best-fit line through the data points. All data points lie on the best-fit line and will be used for all subsequent calculations.

Next, determine the deliverability exponent using least-squares regression analysis. Alternatively, because Points 1 and 4 both lie on the perceived "best" straight line through the test data, the reciprocal slope is estimated to be

Now, calculate the AOF of the well. Because 0.5 ≤ n ≤ 1.0, calculate C using either regression analysis or a point from the best-fit straight line through the test data. Estimating C with regression analysis results in log(C) = α =-3.09 . Thus,

Estimating C using Point 4 from the best-fit line and Eq. 8.245:

Therefore, the AOF potential of this well is

To update the AOF to a new average reservoir pressure, recall that for pseudopressure analysis, neither C nor n changes as drainage area pressure decreases. The AOF for the new drainage area pressure becomes

Houpeurt Analysis. Plot Δpp/q vs. q on Cartesian graph paper (Fig. 8.116). Table 8.15 gives the plotting functions. Construct the best-fit line through the last three data points. The first point, corresponding to the lowest flow rate, does not follow the trend and will be ignored in subsequent analyses.

Determine the deliverability coefficients, a and b, from a least-squares regression analysis, excluding the first point. The result is

Alternatively, use Points 2 and 4 from the line drawn through the test data to calculate a and b:

Then,

To update the AOF, note that for pseudopressure analysis neither a nor b changes as drainage area pressure changes. Therefore, the AOF for the new drainage area pressure is

A comparison (Fig. 8.117) of the results from the two parts of Example 8.10 shows that the Rawlins-Schellhardt equation appears to be valid for this range of test data; however, the line representing the Houpeurt equation deviates from the Rawlins-Schellhardt equation as BHFP decreases. Although the Rawlins-Schellhardt method is valid under many testing conditions, this deviation suggests that extrapolating the empirical equation over a large interval of pressure may not predict well behavior correctly.

Single-Point Tests. A single-point test is an attempt to overcome the limitation of long test times. A single-point test is conducted by flowing the well at a single rate until the sandface pressure is stabilized. One limitation of this test is that it requires prior knowledge of the well’s deliverability behavior, either from previous well tests or possibly from correlations with other wells producing in the same field under similar conditions. Ensure that the well has flowed long enough to be out of wellbore storage and in the boundary-dominated or stabilized flow regime. Similarly, for hydraulically fractured wells, the well must be flowed long enough to be in the pseudoradial flow regime and then stabilized.

To analyze a single-point test with the Rawlins-Schellhardt method, n must be known or estimated. An estimate of n can be obtained either from a previous deliverability test on the well or from correlations with similar wells producing from the same formation under similar conditions. The calculation procedure is similar to that presented for flow-after-flow tests. The AOF can be estimated graphically by drawing a straight line through the single flow point with a slope of 1/n and extrapolating it to the flow rate at . The AOF can also be calculated with

where C is estimated with

To use the Houpeurt analysis technique, the slope, b, of the line on a plot of

must be known. If a value of b is unavailable, estimate b using Eq. 8.238. Note that estimates of the formation properties are necessary to use Eq. 8.238. The remaining analysis procedure is similar to that for flow-after-flow tests.

Isochronal Tests. The isochronal test[51] is a series of single-point tests developed to estimate stabilized deliverability characteristics without actually flowing the well for the time required to achieve stabilized conditions at each different rate. The isochronal test is conducted by alternately producing the well then shutting it in and allowing it to build to the average reservoir pressure before the beginning of the next production period. Pressures are measured at several time increments during each flow period. The times at which the pressures are measured should be the same relative to the beginning of each flow period. Because less time is required to build to essentially initial pressure after short flow periods than to reach stabilized flow at each rate in a flow-after-flow test, the isochronal test is more practical for low-permeability formations. A final stabilized flow point often is obtained at the end of the test. Fig. 8.118 illustrates an isochronal test.

The isochronal test is based on the principle that the radius of drainage established during each flow period depends only on the length of time for which the well is flowed and not the flow rate. Consequently, the pressures measured at the same time periods during each different rate correspond to the same transient radius of drainage. Under these conditions, isochronal test data can be analyzed using the same theory as a flow-after-flow test, even though stabilized flow is not attained. In theory, a stabilized deliverability curve can be obtained from transient data if a single, stabilized rate and the corresponding BHP have been measured and are available.

The transient flow regime is modeled by

....................(8.232)

where ps is the stabilized BHP measured before the test. The transient equation can be rewritten in a form similar to the stabilized equation for a circular drainage area. To start this process, write

....................(8.252)

Further, a transient radius of drainage is defined as

....................(8.253)

By substituting Eq. 8.253 into Eq. 8.252 and rearranging, the transient solution becomes

....................(8.254)

which is valid at any fixed time because rd is a function of time and not of flow rate. rd has no rigorous physical significance. It is simply the radius that forces the transient equation to resemble the pseudosteady-state equation. In addition, do not confuse rd with ri, which is the transient radius of investigation given by Eq. 8.246.

Similar to Houpeurt’s equations, rewrite Eq. 8.254 as

....................(8.255)

where

....................(8.256)

and ....................(8.238)

b is not a function of time and will remain constant. Similarly, the intercept at is constant for each fixed time line or isochron.

The theory of isochronal test analysis implies that the transient pressure drawdowns corresponding to the same elapsed time during each different flow period will plot as straight lines with the same slope b. The intercept a t for each line will increase with increasing time. Therefore, draw a line with the same slope, b, through the final, stabilized data point, and use the coordinates of the stabilized point and the slope to calculate a stabilized intercept, a, independent of time, where (for radial flow) the stabilized flow coefficient is defined by

....................(8.257)

Rawlins-Schellhardt Analysis. In logarithmic form, the empirical equation introduced by Rawlins and Schellhardt for analysis of flow-after-flow test data is

....................(8.250)

For isochronal tests, plot transient data measured at different flow rates but taken at the same time increments relative to the beginning of each flow period. The lines drawn through data points corresponding to the same fixed flow time prove to be parallel, so the value of n is constant and independent of time. However, the intercept, log (C), is a function of time, so a different intercept must be calculated for each isochronal line. This "transient" intercept is log (Ct). In terms of this transient intercept, Eq. 8.248 becomes

....................(8.258)

is replaced by ps in the modified equation.

The conventional Rawlins-Schellhardt method of isochronal test analysis is to plot

for each time, giving a straight line of slope 1/n and an intercept of

Houpeurt Analysis. Recall that the Houpeurt equation for analyzing flow-after-flow tests is

....................(8.251)

Eq. 8.251 assumes stabilized flow conditions; however, in isochronal testing, measured transient data are being recorded. Consequently, for each isochronal (or fixed time) line, the equation for transient flow conditions is

....................(8.259)

where

....................(8.256)

and ....................(8.238)

The form of Eq. 8.259 suggests that a plot of Δpp/q = [pp(ps) – pp(pwf,s)]/q vs. q will yield a straight line with slope b and intercept at. This theory can then be extended to the stabilized point and calculate a stabilized intercept, a, using the coordinates of the stabilized point. The slope b remains the same.

Example 8.11: Analysis of Isochronal Tests Estimate the AOF of this well[51] using both the Rawlins and Schellhardt and the Houpeurt analyses. Table 8.16 summarizes the isochronal test data. Assume pb = 14.65 psia.

Solution. Rawlins-Schellhardt Analysis Technique. First, plot Δpp = pp(ps) – pp(pwf ) vs. q on log-log coordinates (Fig 8.119) and include the single stabilized, extended flow point. Table 8.17 gives the plotting functions.

Calculate the deliverability exponent, n, for each line or isochron using least-squares regression analysis. Note that, because the first data point for each isochron does not align with the data points at the last three flow rates (Fig. 8.119), the first data point is ignored in all subsequent calculations.

Table 8.18 summarizes the deliverability exponents determined with a least-squares regression analysis for each isochron. The arithmetic average of the n values in Table 8.18 is 0.89.

Because 0.5 ≤ ≤ 1.0, AOF can be calculated or determined graphically using Fig. 8.120. AOF will be calculated in this example. First, determine the stabilized performance coefficient using the coordinates of the stabilized, extended flow point and n = :

Then calculate the AOF potential:

To determine the AOF graphically, first calculate the pseudopressure at pb and compute

Then, draw a line of slope 1/ through the stabilized flow point, extrapolate the line to the flow rate at Δpp = pp(ps) − pp(pb), and read the AOF directly from the graph. The result is qAOF = 4.04 MMscf/D.

Houpeurt Analysis Technique. Plot Δpp/q = [pp(ps) – pp(pwf )]/q vs. q on Cartesian graph paper (Fig. 8.121). Table 8.19 gives the plotting functions. Construct best-fit lines through the isochronal data points for each time. Note that, for each flow time, the point corresponding to the lowest rate does fit on the same straight line, so all four data points will be used for the analysis of each isochron.

Next, determine the slope b of each line or isochron. Values of b from least-squares regression analysis are summarized in Table 8.20. The arithmetic average value of the slopes in Table 8.20 is 2.074 × 104 psia2/cp/(MMscf/D)2.

Calculate the stabilized isochronal deliverability line intercept using Δpp/q = 2.113 × 106 psia2/cp/(MMscf/D) at the extended, stabilized point.

Calculate the AOF potential using the average value of b and the stabilized value of a.

Fig. 8.122 illustrates the results.

Modified Isochronal Tests. The time to build up to the average reservoir pressure before flowing for a certain period of time still may be impractical, even after short flow periods. Consequently, a modification of the isochronal test was developed[52] to shorten test times further. The objective of the modified isochronal test is to obtain the same data as in an isochronal test without using the sometimes lengthy shut-in periods required to reach the average reservoir pressure in the drainage area of the well.

The modified isochronal test (Fig. 8.123) is conducted like an isochronal test, except the shut-in periods are of equal duration. The shut-in periods should equal or exceed the length of the flow periods. Because the well does not build up to average reservoir pressure after each flow period, the shut-in sandface pressures recorded immediately before each flow period rather than the average reservoir pressure are used in the test analysis. As a result, the modified isochronal test is less accurate than the isochronal test. As the duration of the shut-in periods increases, the accuracy of the modified isochronal test also increases. Again, a final stabilized flow point usually is obtained at the end of the test but is not required for analyzing the test data.

The well does not build up to the average reservoir pressure during shut-in; the analysis techniques for the modified isochronal tests are derived intuitively. Recall the transient flow equation, expressed in terms of the reservoir pressure at the start of flow, on which isochronal testing is based:

....................(8.254)

In new reservoirs with little or no pressure depletion, p s equals the initial reservoir pressure (ps = pi); in developed reservoirs, ps < pi. In addition, the transient drainage radius, rd, in Eq. 8.254 is defined as

....................(8.253)

Because rd is a function of time and not of flow rate, Eq. 8.254 is valid at any fixed time. For modified isochronal tests, use Eq. 8.254, in which the stabilized shut-in BHP, ps, is replaced with shut-in BHP, pws, measured before each flow period, where pwsps,

....................(8.260)

Eq. 8.260 can be rewritten as

....................(8.261)

where ....................(8.256)

and ....................(8.238)

Eq. 8.238 indicates that b is independent of time and will remain constant during the test. Similarly, Eq. 8.256 indicates that at is constant for a fixed time. The similarity of Eqs. 8.254 and 8.260 for the isochronal and modified isochronal tests, respectively, suggests that the modified isochronal test data can be analyzed like those from an isochronal test.

The theory developed for the modified isochronal test implies that, if the intuitive approximation of using pws instead of ps is valid, the transient data will plot as straight line for each time with the same slope, b. The intercept, at, will increase with increasing time. By drawing a line with slope b through the stabilized data point and using the coordinates of the stabilized point and the slope, a stabilized intercept, a, that is independent of time can be calculated, where

....................(8.257)

To calculate the AOF of the well, use the average reservoir pressure, ps, measured before the test instead of the pws value, or

....................(8.262)

Two variations of the modified isochronal test are considered: tests with a stabilized flow point obtained at the end of the test and tests run without that final point.

Modified Isochronal Tests With a Stabilized Flow Point. Rawlins-Schellhardt Analysis. Recall the empirical Rawlins and Schellhardt equation in terms of transient isochronal test data:

....................(8.258)

As in the graphical analysis techniques for isochronal tests, plot several trends of data taken at different times during a modified isochronal test. The slope n of each line through points at equal time values will be constant. However, the intercept, log(Ct), is a function of time but not flow rate. Therefore, a different intercept should be calculated for each isochronal test. Use pp(pws) instead of pp(ps) in Eq. 8.258, which gives

....................(8.263)

The conventional analysis technique for modified isochronal test data is to plot log [pp(pws) − pp(pwf )] vs. log (q) for each time, giving a straight line of slope 1/n and an intercept of {−1/n [log(Ct)]}. The Rawlins-Schellhardt analysis procedure for modified isochronal tests with a stabilized flow point is similar to that presented for isochronal tests, except the plotting functions are developed in terms of the shut-in pressure measured immediately before the next flow period. Only the stabilized, extended flow point is plotted in terms of the average reservoir pressure measured before the test, ps. Example 8.12 illustrates the procedure.

Houpeurt Analysis. As shown previously, the Houpeurt deliverability equation in terms of transient isochronal test data is

....................(8.259)

For modified isochronal test data, Eq. 8.259 should be modified with the assumption that pp(pws) can be used instead of pp(ps). With this assumption, Eq. 8.259 becomes

....................(8.264)

where ....................(8.256)

and ....................(8.238)

The form of Eq. 8.264 suggests that a plot of

will be a straight line with a slope b and intercept at. This theory can be extended to the stabilized point, and we can calculate a stabilized intercept, a, using the coordinates of the stabilized point, or

....................(8.265)

The slope b of the line through the stabilized point should remain the same. In addition, the average reservoir pressure, which is measured before the test, must be used to evaluate the pseudopressure, pp(ps) in Eq. 8.265. Example 8.12 illustrates the Houpeurt analysis procedure for modified isochronal tests with a stabilized flow point, which is similar to that presented for isochronal tests.

Example 8.12: Analysis of a Modified Isochronal Test With a Stabilized Flow Point Using the following data taken from Well 4, [53] calculate the AOF using both Rawlins and Schellhardt and Houpeurt analysis techniques. Assume pb = 14.65 psia, where pp(pb) = 5.093 × 107 psia2/cp. Table 8.21 gives the test data. h = 6 ft, rw = 0.1875 ft, ϕ = 0.2714, T = 540°R (80°F), ps= 706.6psia, = 0.015cp, = 0.97, = 1.5×10−3 psia−1, γg = 0.75, Sw = 0.30, cf = 3 × 10–6 psia–1, and A = 640 acres (assume that the well is centered in a square drainage area).

Solution.

Rawlins-Schellhardt Analysis. Plot

on log-log graph paper (Fig. 8.124). Table 8.22 gives the plotting functions. In addition, plot on the same graph the values of Δpp that corresponds to the stabilized, extended flow point evaluated at ps.

For each time, construct the best-fit line through the data points. Because the first data points for each isochron do not follow the trend of the higher rate points, they will be ignored for all subsequent calculations.

Calculate the deliverability exponent, n, for each line or isochron. For this example, use least-squares regression analysis. For example, at t = 0.5 hours, n1 = 0.72. Table 8.23 summarizes the deliverability exponents.

The arithmetic average of the values in Table 8.23 is

Because 0.5 ≤ ≤ 1.0, determine the stabilized performance coefficient, C, using the coordinates of the stabilized, extended flow point and n = . Note that the pseudopressure used to calculate the stabilized C value is evaluated at ps measured at the beginning of the test, rather than pws. From Eq. 8.245,

Then,

To determine the AOF graphically draw a line of slope 1/ through the extended flow point, extrapolate the line to the flow rate at Δpp = pp(ps) - pp(pb), and read the AOF directly from the graph (Fig. 8.125).

Houpeurt Analysis. Plot

on Cartesian coordinates (Fig 8.126). In addition, plot the Δpp/q value that corresponds to the stabilized, extended flow point. Table 8.24 gives the plotting functions. Construct best-fit lines through the modified isochronal data points for each time. The first data point at the lowest rate for each isochron does not fit on the same straight line as the last three rate points and is ignored in subsequent calculations.

Determine the slopes of the lines, b, for each isochron by least-squares regression analysis of the best-fit lines through the data points. For example, at t = 0.5 hours, b1 = 9.654 × 105 psia2/cp/(MMscf/D)2. Table 8.25 summarizes the slopes of the isochrons. The average arithmetic values of the slopes in Table 12.15 is

Calculate the stabilized isochronal deliverability line intercept, a:

Calculate the AOF potential using and the stabilized a value:

Fig. 8.127 shows the data for this example.

Modified Isochronal Tests Without a Stabilized Flow Point. Because the well is not required to build up to the average reservoir pressure between the flow periods, the modified isochronal approximation shortens test times considerably. However, the test analysis relies on obtaining one stabilized flow point. Under some conditions, environmental or economic concerns prohibit flaring produced gas to the atmosphere during a long production period, thus preventing measurement of a stabilized flow point. These conditions often occur when new wells are tested before being connected to a pipeline.

Two methods have been developed to analyze modified isochronal tests without a stabilized flow point. The Brar and Aziz method[53] was developed for the Houpeurt analysis, while the stabilized C method[54] was developed for the Rawlins and Schellhardt analysis. The stabilized C method requires prior knowledge of permeability and skin factor or determination of these properties using the methods Brar and Aziz proposed for analyzing modified isochronal tests. Both methods require knowledge of the drainage area shape and size.

Brar and Aziz Method-Houpeurt Analysis. The Brar and Aziz method[53] is based on the transient Houpeurt deliverability Eqs. 8.234, 8.236, 8.238, and ps, the stabilized BHP measured before the deliverability test.

Rewriting Eq. 8.236 as

....................(8.266)

where ....................(8.267)

and ....................(8.268)

m′ and c′ can be calculated using regression analysis of Eq. 8.266. Alternatively, these variables can be computed directly from the slope and the intercept of a plot of at vs. log t. Then calculate the permeability from the slope,

....................(8.269)

Combining Eqs. 8.267 and 8.268 yields an equation for the skin factor,

....................(8.270)

Estimating the AOF potential of the well requires a stabilized value of a. If the drainage area size and shape are known, the gas permeability calculated from Eq. 8.269 and the skin factor from Eq. 8.270 can be used to calculate a:

....................(8.271)

Table 8.A-1 gives shape factors for various reservoir shapes and well locations. The stabilized value of a then is used in Eq. 8.262 to calculate the AOF of the well:

....................(8.262)

Stabilized C Method-Rawlins-Schellhardt Analysis. Although the Houpeurt equation has a theoretical basis and is rigorously correct, the more familiar but empirically based Rawlins and Schellhardt equation continues to be used and is indeed favored by many in the natural gas industry. The Houpeurt and Rawlins-Schellhardt analysis techniques are combined here to develop a version of the Rawlins-Schellhardt method for analyzing modified isochronal tests. This analysis technique, called the "Stabilized C" method, [54] is derived by equating the stabilized Rawlins and Schellhardt empirical backpressure equation with the stabilized theoretical Houpeurt equation to obtain equations for the deliverability exponent, n, and the stabilized flow coefficient, C, in terms of the Houpeurt flow coefficients, a and b.

To obtain an equation for the exponent n , take the logarithm of both sides of the stabilized Rawlins and Schellhardt empirical backpressure equation ( Eq. 8.245 ).

....................(8.272)

n is the slope of a plot of ln(q) vs. ln(Δpp). Alternatively, note that n can be expressed as the derivative of ln(q) with respect to ln(Δpp):

....................(8.273)

Similarly, take the logarithms of both sides of the Houpeurt Eq. 8.235

....................(8.274)

and, thus,

....................(8.275)

or ....................(8.276)

and ....................(8.277)

In Eq. 8.277, let q be the unique value qe at which the d ln(Δpp)/dq values from the Rawlins-Schellhardt and Houpeurt equations are identical. Solving Eq. 8.277 for this value of q = qe,

....................(8.278)

and ....................(8.279)

Substituting in the Rawlins-Schellhardt equation and noting that, from the Houpeurt equation (Δpp)e = aqe + bqe2,

....................(8.280)

Rearranging,

....................(8.281)

To apply the stabilized C method, it is necessary to assume that the slope, n, of the Rawlins-Schellhardt deliverability plot is constant. This assumption implies that if values of a and b can be calculated for given reservoir properties, a flow rate can be calculated from Eq. 8.279, at which the change in pseudopressures calculated by the Rawlins-Schellhardt equation is equal to the change in pseudopressure calculated by the Houpeurt equation. The substitution this flow rate into Eq. 8.281 allows calculation of a stabilized value of C and this value of C can be used to calculate a value of AOF:

....................(8.282)

The stabilized C method is limited by the need for values of reservoir properties determined separately from the deliverability test analysis. These properties can be estimated either from drawdown or buildup test analysis or from the Brar and Aziz method.

Example 8.13: Analysis of Modified Isochronal Test Without a Stabilized Data Point The purpose of this example is to compare results obtained from the analysis of a modified isochronal test (see Table 8.26) with and without an extended, stabilized data point. Calculate the AOF for the following modified isochronal test data without the extended flow point. Use both the Brar and Aziz and the stabilized C methods. Compare these results with the results obtained by using the extended flow point. This example is Well 8. [53] Only the last four flow points from the test are used in the analysis. Reservoir data are summarized here: h = 454 ft, rw = 0.2615 ft, ϕ = 0.0675, T = 718°R (258°F), ps ≅ 4,372.6 psia, μ = 0.023 cp, z = 0.87, cg = 1.69 × 10–4 psia–1, γg = 0.65, Sw = 0.3, A = 640 acres. CA = 30.8828 (assume that the well is centered in a square drainage area). In addition, the results from a drawdown test in this well indicate kg = 4.23 md and s = −5.2.

The Rawlins and Schellhardt analysis with extended flow point gave C = 2.426 × 10–3, n = 0.54 and qAOF = 180.1 MMscf/D. The Houpeurt analysis with extended flow point gave a = 1.455 × 106 psia2/cp/MMscf/D, b = 1.774 × 104 psia2/cp/(MMscf/D)2, and qAOF = 205.6 MMscf/D.

Solution. Brar and Aziz Method. Step 1—Plot

on Cartesian coordinates (Fig. 8.128). Table 8.27 gives the plotting functions. Construct best-fit lines through the modified isochronal data points for each time. Although the data are scattered, all flow rates were used for each isochron.

Step 2—Determine the slopes of the lines, b, for each time by least-squares regression analysis. For example, at t = 3.0 hours, b1 = 1.823 × 104 psia2/cp/(MMscf/D)2. Table 8.28 summarizes the slopes for all isochrons. The arithmetic average value of the b values in Table 8.28 is

Step 3—Using least-squares regression analysis, calculate the transient deliverability line intercepts for each isochronal line. For example, at t = 3.0 hours,

Table 8.29 gives the intercepts for each isochron.

Step 4—Prepare a graph of at vs. log t (Fig. 8.129) and draw the best-fit line through data. Using all four data points, calculate m′ and c′ of the best-fit line of the plot of at vs. log t using least-squares regression analysis. The result is m′ = 3.871 × 105 psia2/(cp-MMscf/D)/cycle and c′ = 3.909 × 105 psia2/(cp-MMscf/D).

Step 5—Calculate the formation permeability to gas using the slope of the semilog straight line.

which compares with kg = 4.23 md estimated from the drawdown test analysis.

Step 6—Calculate the skin factor with Eq. 8.270.

This value agrees with s = –5.2 estimated from the drawdown test analysis.

Step 7—Calculate the stabilized flow coefficient, a. Assume that the well is centered in a square drainage area with CA = 30.8828.

Now, calculate the AOF potential using from Step 2 and the stabilized a value calculated in Step 7.

Stabilized C Method. Step 1—Plot

vs. q on log-log coordinates (Fig. 8.130). Table 8.30 gives the plotting functions. Construct best-fit lines through the data.

Step 2—Calculate the deliverability exponent, n, for each line. For this example, use the least-squares regression analysis of all points for each isochron. For example, for t = 3.0 hours, n = 0.63. Table 8.31 summarizes values of the deliverability exponent for each isochron. The arithmetic average slope of the values in Table 8.31 is

Step 3—Calculate the theoretical value of the Houpeurt coefficient, a, using the permeability and skin factor values calculated previously with the Brar and Aziz analysis (i.e., kg = 6.6 md, s = –5.0).

Use the average value for the coefficient, b = 1.878 × 104 psia2/(cp-MMscf/D), obtained from the Brar and Aziz analysis.

Step 4—Calculate the rate at which the change in pseudopressure determined with Rawlins-Schellhardt equation equals the change in pseudopressure determined with the Houpeurt equation. Use the average value for the coefficient, b = 1.878 × 104 psia2/(cp-MMscf/D), obtained from the Brar and Aziz analysis, and the a coefficient from Step 3.

Step 5—Calculate the stabilized C value.

Step 6—Calculate the AOF potential of the well using from Step 2.

Table 8.32 compares the results of the analyses with and without the extended, stabilized flow points. In general, the results are combrble and illustrate the validity of the Brar and Aziz and the stabilized C methods for modified isochronal tests with no extended, stabilized flow point.

## Coning

Coning is the production of an (usually) unwanted second phase simultaneously with a desired hydrocarbon phase in reservoirs with fluid contacts near the wellbore throughout much of the drainage area of a well. The term coning is used because, in a vertical well, the shape of the interface when a well is producing the second fluid resembles an upright or inverted cone (Fig. 8.131). Important examples of coning include production of water in an oil well with bottomwater drive, production of gas in an oil well overlain by a gas cap, and production of bottom water in a gas well.

In a horizontal well, the cone becomes more of a crest (Fig. 8.132), but the phenomenon is still customarily called coning. In a given reservoir, the amount of undesired second fluid a horizontal well produces is usually less than for a vertical well under combrble conditions. This is a major motivation for drilling horizontal wells, for example, in thin oil columns underlain by water.

Coning is a problem because the second phase must be handled at the surface in addition to the desired hydrocarbon phase, and the production rate of the hydrocarbon flow is usually dramatically reduced after the cone breaks through into the producing well. Produced water must also be disposed of. Gas produced from coning in an oil well may have a market, but also may not. In any event, production of gas in an oil well after the cone breaks through can rapidly deplete reservoir pressure and, for that reason, may force shut in of the oil well.

Several strategies may apply to wells with a potential to cone. One is to try to predict the rate at which a well will cone and produce at a lower rate as long as possible. Or, optimal economics may result by producing at a much higher rate, causing the well to cone, but increasing the cumulative hydrocarbon volume produced (and present value) at any future date. A horizontal well may be preferred to a vertical well.

Most prediction methods for coning predict a "critical rate" at which a stable cone can exist from the fluid contact to the nearest perforations. The theory is that, at rates below the critical rate, the cone will not reach the perforations and the well will produce the desired single phase. At rates equal to or greater than the critical rate, the second fluid will eventually be produced and will increase in amount with time. However, these theories based on critical rates do not predict when breakthrough will occur nor do they predict water/oil ratio or gas/oil ratio (GOR) after breakthrough. Other theories predict these time behaviors, but their accuracy is limited because of simplifying assumptions.

The calculated critical rate is valid only for a certain fixed distance between the fluid contact and the perforations. With time, that distance usually decreases (for example, bottom water will usually tend to rise toward the perforations). Thus, the critical rate will tend to decrease with time, and the economics of a well with a tendency to cone will continue to deteriorate with time.

Whether a cone will move toward perforations depends on the relative significance of viscous and gravitational forces near a well. The pressure drawdown at the perforations tends to cause the undesired fluid to move toward the perforations. Gravitational forces tend to cause the undesired fluid to stay away from the perforations. Coning occurs when viscous forces dominate.

The variables that could affect coning are density differences between water and oil, gas and oil, or gas and water (gravitational forces); fluid viscosities and relative permeabilities; vertical and horizontal permeabilities; and distances from contacts to perforations. Coning tendency turns out to be quite dependent on some of these variables and insensitive to others.

A number of prediction methods have been published. There is no guarantee of great accuracy when using any of these methods because they all contain significant simplifying assumptions. In particular, areal and vertical variations in vertical permeability (because of flow barriers of varying extent) can cause the prediction methods to differ significantly from what actually happens in the field. Accordingly, the prediction methods are best used for quick approximations, screening, and comparison of alternatives. Reservoir simulations, based on accurate reservoir characterization, will ultimately be required.

The coning prediction method proposed by Chaperon[55] is of particular interest because of the variables it includes and because variations of the method are applicable to gas and water coning in both vertical and horizontal wells. For vertical wells, the Chaperon method calculates the critical rate for coning from the expression

....................(8.283)

where

....................(8.284)

....................(8.285)

....................(8.286)

and hc = distance from perforations to fluid contact, ft. For horizontal wells, the critical rate is given by

....................(8.287)

where

....................(8.288)

and

....................(8.289)

Example 8.14 Consider a square, 160-acre drilling unit in an oil reservoir overlain by a gas cap. To determine the critical production rate (at or above which coning is likely to occur) for both horizontal and vertical well alternatives, assume the following well and formation properties: kh = 80 md, aH = 2,640 ft, μo = 0.5 md, ρo = 0.8 g/cm3, ρg = 0.3 g/cm3, Lw = 1,500 ft, h = 90 ft, Bo = 1.2 RB/STB, rw = 0.33 ft, distance from top of perforations in vertical well to GOC = 80 ft, and distance from horizontal well to GOC = 80 ft. Consider two cases: Isotropic Reservoir A, where kv = kh = 80 md, and Anisotropic Reservoir B, where kv = 8 md and kh = 80 md.

Solutions.

Isotropic Reservoir A (kv = kh = 80 md). For a horizontal well,

Here,

and

Then,

For a vertical well, note that = 1,489 ft.

where

and

Then,

Anisotropic Reservoir B (kv = 8 md and kh = 80 md). For a horizontal well,

For a vertical well,

The important conclusions and lessons from this example are:

• Vertical permeability has only a modest influence on critical coning rate, at least for this situation.
• The advantage of a horizontal well over a vertical well is substantial for both isotropic and anisotropic reservoirs in this situation.
• The same sorts of calculations could be made for an oil well coning water or a gas well coning water.
• These calculations give us no information on time at which the cone will break through to the producing well nor on GOR and oil production rate following breakthrough.

While these types of simple calculations can provide some insight on the potential for coning, a finely grided simulator model could be used to more effectively predict coning behavior including timing and the benefits of a horizontal well over a vertical one.

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## SI Metric Conversion Factors

 acre × 4.046 873 E + 03 = m2 bbl × 1.589 873 E – 01 = m3 cp × 1.0* E – 03 = Pa•s ft × 3.048* E – 01 = m ft2 × 9.290 304* E – 02 = m2 ft3 × 2.831 685 E – 02 = m3 °F (°F − 32)/1.8 = °C in.3 × 1.638 706 E + 01 = cm3 lbf × 4.448 222 E + 00 = N lbm × 4.535 924 E – 01 = kg psi × 6.894 757 E + 00 = kPa

*

Conversion factor is exact.