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This page introduces how [[ | This page introduces how [[Fluid_flow_through_permeable_media|fluid flow]] in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references. | ||
Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This page briefly discusses the statement of the IBVP for transient fluid flow in porous media. | Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This page briefly discusses the statement of the IBVP for transient fluid flow in porous media. | ||
==Equations of transient fluid flow in porous media== | == Equations of transient fluid flow in porous media == | ||
[[File:Vol1 page 0077 | In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]], and the density of the fluid, ''ρ'', as a function of the position (''x'', ''y'', ''z'') and time, ''t''; that is, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]] = [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]] (''x'', ''y'', ''z'', ''t'') and ''ρ''= ''ρ'' (''x'', ''y'', ''z'', ''t''). Relative to the fixed Cartesian axes, the velocity vector can be written as | ||
where ''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, and ''v''<sub>''z''</sub> are the velocity components, and [[File:Vol1 page 0077 inline 002.png]], [[File:Vol1 page 0077 inline 003.png]], and [[File:Vol1 page 0077 inline 004.png]] are the unit vectors in the ''x'', ''y'', and ''z'' directions, respectively. | [[File:Vol1 page 0077 eq 001.png|RTENOTITLE]]....................(1) | ||
where ''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, and ''v''<sub>''z''</sub> are the velocity components, and [[File:Vol1 page 0077 inline 002.png|RTENOTITLE]], [[File:Vol1 page 0077 inline 003.png|RTENOTITLE]], and [[File:Vol1 page 0077 inline 004.png|RTENOTITLE]] are the unit vectors in the ''x'', ''y'', and ''z'' directions, respectively. | |||
The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity, ''Φ'', which is filled with a fluid of viscosity ''μ''. '''Fig. 1''' illustrates an arbitrary closed surface in porous medium. | The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity, ''Φ'', which is filled with a fluid of viscosity ''μ''. '''Fig. 1''' illustrates an arbitrary closed surface in porous medium. | ||
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The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by | The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by | ||
[[File:Vol1 page 0078 eq 001.png]]....................(2) | [[File:Vol1 page 0078 eq 001.png|RTENOTITLE]]....................(2) | ||
Then, the time rate of change of mass within Γ is | Then, the time rate of change of mass within Γ is | ||
[[File:Vol1 page 0078 eq 002.png]]....................(3) | [[File:Vol1 page 0078 eq 002.png|RTENOTITLE]]....................(3) | ||
which, by the conservation of mass law, is equal to the rate at which mass enters ''V'' through the surface. | which, by the conservation of mass law, is equal to the rate at which mass enters ''V'' through the surface. | ||
Consider the differential surface element, dΓ, shown in '''Fig. 1'''. The mass entering the volume through dΓ at the normal velocity, [[File:Vol1 page 0078 inline 001.png]], in a time increment, Δ''t'', is [[File:Vol1 page 0078 inline 002.png]], and the total mass of the fluid passing through Γ during Δ''t'' is | Consider the differential surface element, dΓ, shown in '''Fig. 1'''. The mass entering the volume through dΓ at the normal velocity, [[File:Vol1 page 0078 inline 001.png|RTENOTITLE]], in a time increment, Δ''t'', is [[File:Vol1 page 0078 inline 002.png|RTENOTITLE]], and the total mass of the fluid passing through Γ during Δ''t'' is | ||
[[File:Vol1 page 0078 eq 003.png]]....................(4) | [[File:Vol1 page 0078 eq 003.png|RTENOTITLE]]....................(4) | ||
The surface integral in '''Eq. 4''' accounts for both influx and outflux through the surface of the volume; that is, Δ''M''<sub>''g''</sub> is the difference between the masses entering and leaving the control volume during the time increment, Δ''t''. Then, the mass rate entering the volume, ''V'', through its surface, Γ, can be written as | The surface integral in '''Eq. 4''' accounts for both influx and outflux through the surface of the volume; that is, Δ''M''<sub>''g''</sub> is the difference between the masses entering and leaving the control volume during the time increment, Δ''t''. Then, the mass rate entering the volume, ''V'', through its surface, Γ, can be written as | ||
[[File:Vol1 page 0078 eq 004.png]]....................(5) | [[File:Vol1 page 0078 eq 004.png|RTENOTITLE]]....................(5) | ||
By the principle of conservation of mass, equating the right sides of '''Eqs. 3''' and '''5''' yields | By the principle of conservation of mass, equating the right sides of '''Eqs. 3''' and '''5''' yields | ||
[[File:Vol1 page 0078 eq 005.png]].....................(6) | [[File:Vol1 page 0078 eq 005.png|RTENOTITLE]].....................(6) | ||
A more useful relation is found with the divergence theorem, which states that the flux of [[File:Vol1 page 0079 inline 001.png]] through the closed surface, Γ, is identical to the volume integral of [[File:Vol1 page 0079 inline 002.png]] (the divergence of [[File:Vol1 page 0079 inline 001.png]]) taken throughout ''V''; that is, | A more useful relation is found with the divergence theorem, which states that the flux of [[File:Vol1 page 0079 inline 001.png|RTENOTITLE]] through the closed surface, Γ, is identical to the volume integral of [[File:Vol1 page 0079 inline 002.png|RTENOTITLE]] (the divergence of [[File:Vol1 page 0079 inline 001.png|RTENOTITLE]]) taken throughout ''V''; that is, | ||
[[File:Vol1 page 0079 eq 001.png]]....................(7) | [[File:Vol1 page 0079 eq 001.png|RTENOTITLE]]....................(7) | ||
Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by | Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by | ||
[[File:Vol1 page 0079 eq 002.png]]....................(8) | [[File:Vol1 page 0079 eq 002.png|RTENOTITLE]]....................(8) | ||
and | and | ||
[[File:Vol1 page 0079 eq 003.png]]....................(9) | [[File:Vol1 page 0079 eq 003.png|RTENOTITLE]]....................(9) | ||
With the relation in '''Eq. 7''', '''Eq. 6''' can be recast into | With the relation in '''Eq. 7''', '''Eq. 6''' can be recast into | ||
[[File:Vol1 page 0079 eq 004.png]]....................(10) | [[File:Vol1 page 0079 eq 004.png|RTENOTITLE]]....................(10) | ||
If the functions involved in the argument of the integral in '''Eq. 10''' are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in '''Eq. 10''' is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained. | If the functions involved in the argument of the integral in '''Eq. 10''' are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in '''Eq. 10''' is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained. | ||
[[File:Vol1 page 0079 eq 005.png]]....................(11) | [[File:Vol1 page 0079 eq 005.png|RTENOTITLE]]....................(11) | ||
'''Eq. 11''' is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, '''Eq. 11''' is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ''ρ'', and velocity, [[File:Vol1 page 0077 inline 001.png]], in terms of pressure, we use an [[ | '''Eq. 11''' is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, '''Eq. 11''' is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ''ρ'', and velocity, [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]], in terms of pressure, we use an [[Equations_of_state|equation of state]] and a flux law, known as Darcy’s law, respectively. | ||
The following definition of isothermal fluid compressibility, ''c'', is a useful equation of state that relates density to pressure. | The following definition of isothermal fluid compressibility, ''c'', is a useful equation of state that relates density to pressure. | ||
[[File:Vol1 page 0079 eq 006.png]]....................(12) | [[File:Vol1 page 0079 eq 006.png|RTENOTITLE]]....................(12) | ||
If ''c'' is a constant (the compressibility of many reservoir liquids may be considered as constant), then '''Eq. 12''' can be integrated to yield | If ''c'' is a constant (the compressibility of many reservoir liquids may be considered as constant), then '''Eq. 12''' can be integrated to yield | ||
[[File:Vol1 page 0079 eq 007.png]]....................(13) | [[File:Vol1 page 0079 eq 007.png|RTENOTITLE]]....................(13) | ||
where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, ''c''<sub>''f''</sub>, is defined by | where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, ''c''<sub>''f''</sub>, is defined by | ||
[[File:Vol1 page 0080 eq 001.png]]....................(14) | [[File:Vol1 page 0080 eq 001.png|RTENOTITLE]]....................(14) | ||
and the total system compressibility, ''c''<sub>''t''</sub>, is given by | and the total system compressibility, ''c''<sub>''t''</sub>, is given by | ||
[[File:Vol1 page 0080 eq 002.png]]....................(15) | [[File:Vol1 page 0080 eq 002.png|RTENOTITLE]]....................(15) | ||
These definitions of compressibility help recast '''Eq. 11''' in terms of pressure. | These definitions of compressibility help recast '''Eq. 11''' in terms of pressure. | ||
Line 80: | Line 81: | ||
Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by | Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by | ||
[[File:Vol1 page 0080 eq 003.png]]....................(16) | [[File:Vol1 page 0080 eq 003.png|RTENOTITLE]]....................(16) | ||
In '''Eq. 16''', ''μ'' is the viscosity of the fluid, and ''k'' is the permeability tensor of the formation given by | In '''Eq. 16''', ''μ'' is the viscosity of the fluid, and ''k'' is the permeability tensor of the formation given by | ||
[[File:Vol1 page 0080 eq 004.png]]....................(17) | [[File:Vol1 page 0080 eq 004.png|RTENOTITLE]]....................(17) | ||
where ''α'', ''β'', and ''γ'' are the directions, and ''k''<sub>''ij''</sub> is the permeability in the ''i'' direction as a result of the pressure gradient in the ''j'' direction. | where ''α'', ''β'', and ''γ'' are the directions, and ''k''<sub>''ij''</sub> is the permeability in the ''i'' direction as a result of the pressure gradient in the ''j'' direction. | ||
Line 90: | Line 91: | ||
If '''Eqs. 13''' through '''16''' are used in '''Eq. 11''', an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained: | If '''Eqs. 13''' through '''16''' are used in '''Eq. 11''', an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained: | ||
[[File:Vol1 page 0080 eq 005.png]]....................(18) | [[File:Vol1 page 0080 eq 005.png|RTENOTITLE]]....................(18) | ||
'''Eq. 18''' is the PDE that governs transient fluid flow in porous media. In the present form, '''Eq. 18''' is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, ''μ'', is constant and '''Eq. 18''' can be linearized by assuming that the pressure gradients, ∇''p'', are small in the reservoir and the compressibility of the reservoir liquids, ''c'', is on the order of 10<sup>−5</sup> or smaller. Then, the second term of the left side of '''Eq. 18''' may be neglected compared with the remaining terms and the following linear expression is obtained: | '''Eq. 18''' is the PDE that governs transient fluid flow in porous media. In the present form, '''Eq. 18''' is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, ''μ'', is constant and '''Eq. 18''' can be linearized by assuming that the pressure gradients, ∇''p'', are small in the reservoir and the compressibility of the reservoir liquids, ''c'', is on the order of 10<sup>−5</sup> or smaller. Then, the second term of the left side of '''Eq. 18''' may be neglected compared with the remaining terms and the following linear expression is obtained: | ||
[[File:Vol1 page 0080 eq 006.png]]....................(19) | [[File:Vol1 page 0080 eq 006.png|RTENOTITLE]]....................(19) | ||
'''Eq. 19''' (or '''Eq. 18''') is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, ''k'', in '''Eq. 19''', may be represented by the following diagonal tensor: | '''Eq. 19''' (or '''Eq. 18''') is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, ''k'', in '''Eq. 19''', may be represented by the following diagonal tensor: | ||
[[File:Vol1 page 0081 eq 001.png]]....................(20) | [[File:Vol1 page 0081 eq 001.png|RTENOTITLE]]....................(20) | ||
Then, '''Eq. 19''' may be written | Then, '''Eq. 19''' may be written | ||
[[File:Vol1 page 0081 eq 002.png]]....................(21) | [[File:Vol1 page 0081 eq 002.png|RTENOTITLE]]....................(21) | ||
If each coordinate, ''j'' = ''x'', ''y'', or ''z'', is multiplied by [[File:Vol1 page 0081 inline 001.png]], where ''k'' may be chosen arbitrarily (to preserve the material balance, ''k'' is usually chosen to be [[File:Vol1 page 0081 inline 002.png]]), '''Eq. 21''' may be transformed into the [[ | If each coordinate, ''j'' = ''x'', ''y'', or ''z'', is multiplied by [[File:Vol1 page 0081 inline 001.png|RTENOTITLE]], where ''k'' may be chosen arbitrarily (to preserve the material balance, ''k'' is usually chosen to be [[File:Vol1 page 0081 inline 002.png|RTENOTITLE]]), '''Eq. 21''' may be transformed into the [[Source_function_solutions_of_the_diffusion_equation|diffusion equation]] for an [[Glossary:Isotropic|isotropic]] domain: | ||
[[File:Vol1 page 0081 eq 003.png]]....................(22) | [[File:Vol1 page 0081 eq 003.png|RTENOTITLE]]....................(22) | ||
where ''η'' is the diffusivity constant defined by | where ''η'' is the diffusivity constant defined by | ||
[[File:Vol1 page 0081 eq 004.png]]....................(23) | [[File:Vol1 page 0081 eq 004.png|RTENOTITLE]]....................(23) | ||
If the same transformation is also applied to the boundary conditions (see the "Initial and boundary conditions" section below), the problems in [[Glossary:Anisotropy|anisotropic]] reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces. | If the same transformation is also applied to the boundary conditions (see the "Initial and boundary conditions" section below), the problems in [[Glossary:Anisotropy|anisotropic]] reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces. | ||
Line 116: | Line 117: | ||
For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the ''c''(∇''p'')<sup>2</sup> term in '''Eq. 18''' may not be negligible. In these cases, an expression similar to '''Eq. 21''' may be obtained from '''Eq. 18''' in terms of pseudopressure, ''m'', as | For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the ''c''(∇''p'')<sup>2</sup> term in '''Eq. 18''' may not be negligible. In these cases, an expression similar to '''Eq. 21''' may be obtained from '''Eq. 18''' in terms of pseudopressure, ''m'', as | ||
[[File:Vol1 page 0081 eq 005.png]]....................(24) | [[File:Vol1 page 0081 eq 005.png|RTENOTITLE]]....................(24) | ||
Here, the pseudopressure is defined by<ref name="r1" /> | Here, the pseudopressure is defined by<ref name="r1">Al-Hussainy, R., Ramey Jr., H.J., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA</ref> | ||
[[File:Vol1 page 0081 eq 006.png]]....................(25) | [[File:Vol1 page 0081 eq 006.png|RTENOTITLE]]....................(25) | ||
where ''Z'' is the compressibility factor. To define a complete physical problem, '''Eq. 21''' (or '''24''') should be subject to the initial and boundary conditions discussed below. | where ''Z'' is the compressibility factor. To define a complete physical problem, '''Eq. 21''' (or '''24''') should be subject to the initial and boundary conditions discussed below. | ||
===Initial and boundary conditions=== | === Initial and boundary conditions === | ||
The solution of the diffusivity equation ('''Eq. 19''') should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is, | The solution of the diffusivity equation ('''Eq. 19''') should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is, | ||
[[File:Vol1 page 0082 eq 001.png]]....................(26) | [[File:Vol1 page 0082 eq 001.png|RTENOTITLE]]....................(26) | ||
The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, ''f'' (''x'', ''y'', ''z'') = ''p''<sub>''i''</sub>. | The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, ''f'' (''x'', ''y'', ''z'') = ''p''<sub>''i''</sub>. | ||
Line 133: | Line 135: | ||
The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as | The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as | ||
[[File:Vol1 page 0082 eq 002.png]]....................(27) | [[File:Vol1 page 0082 eq 002.png|RTENOTITLE]]....................(27) | ||
where Γ is the surface of the boundary, and ''n'' indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, ''g''(''t''), is related to a constant production rate, ''q''. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [''g''(''t'')=0] and leads to a pseudosteady state under the influence of boundaries. | where Γ is the surface of the boundary, and ''n'' indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, ''g''(''t''), is related to a constant production rate, ''q''. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [''g''(''t'')=0] and leads to a pseudosteady state under the influence of boundaries. | ||
Line 139: | Line 141: | ||
For some applications, pressure may be specified at the inner and outer boundaries. In this case, | For some applications, pressure may be specified at the inner and outer boundaries. In this case, | ||
[[File:Vol1 page 0082 eq 003.png]]....................(28) | [[File:Vol1 page 0082 eq 003.png|RTENOTITLE]]....................(28) | ||
When used at the inner boundary, this condition represents production at a constant pressure, ''p''<sub>''wf''</sub>; that is, ''h''(''t'') = ''p''<sub>''wf''</sub>. At the outer boundary, specified pressure, ''p''<sub>''e''</sub>, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir. | When used at the inner boundary, this condition represents production at a constant pressure, ''p''<sub>''wf''</sub>; that is, ''h''(''t'') = ''p''<sub>''wf''</sub>. At the outer boundary, specified pressure, ''p''<sub>''e''</sub>, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir. | ||
It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan<ref name="r2" /> contains more details about the common boundary conditions for the diffusion equation. | It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan<ref name="r2">Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.</ref> contains more details about the common boundary conditions for the diffusion equation. | ||
=== Assumptions and limits === | |||
Some assumptions have been made in the derivation of the diffusivity equation given by '''Eq. 19'''. These assumptions determine the limits of applicability of the solutions obtained from '''Eq. 19'''. One of the most important assumptions involved is the continuity of the properties involved in '''Eq. 19'''. (This was required to obtain '''Eq. 19''' from the more general integral form in '''Eq. 10'''.) Therefore, sharp changes in the [[Rock_types|properties of the reservoir rock]] and [[Pore_fluid_properties|fluid]] (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of '''Eq. 19'''. | |||
= | The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.<ref name="r3">Fancher, G.H., Lewis, J.A., and Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.</ref> In this case, Forchheimer’s equation,<ref name="r4">Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.</ref> which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore. | ||
Some | |||
== Mathematical tools used for solving transient analysis problems == | |||
These tools include: | These tools include: | ||
==Nomenclature== | *[[Bessel_functions_in_transient_analysis|Bessel functions in transient analysis]] | ||
*[[Laplace_transformation_for_solving_transient_flow_problems|Laplace transformation for solving transient flow problems]] | |||
*[[Green’s_function_for_solving_transient_flow_problems|Green’s function for solving transient flow problems]] | |||
== Nomenclature == | |||
{| | {| | ||
|- | |- | ||
|''c'' | | ''c'' | ||
|= | | = | ||
| | | fluid compressibility, atm<sup>−1</sup> | ||
|- | |- | ||
|''c''<sub>'' | | ''c''<sub>''f''</sub> | ||
|= | | = | ||
| | | formation compressibility, atm<sup>−1</sup> | ||
|- | |- | ||
| | | ''c''<sub>''t''</sub> | ||
| = | |||
| total compressibility, atm<sup>−1</sup> | |||
|- | |- | ||
|'' | | [[File:Vol1 page 0168 inline 004.png|RTENOTITLE]] | ||
| = | |||
| unit normal vector in the ''ξ'' direction, ''ξ'' = ''x'', ''y'', ''z'', ''r'', ''θ'' | |||
|- | |- | ||
|'' | | ''k'' | ||
|= | | = | ||
| | | isotropic permeability, md | ||
|- | |- | ||
|'' | | ''m'' | ||
|= | | = | ||
| | | pseudopressure, atm<sup>2</sup>/cp | ||
|- | |- | ||
| | | ''M''<sub>''g''</sub> | ||
|= | | = | ||
| | | mass, g | ||
|- | |- | ||
| | | [[File:Vol1 page 0169 inline 001.png|RTENOTITLE]] | ||
|= | | = | ||
| | | normal vector | ||
|- | |- | ||
|'' | | ''n'' | ||
|= | | = | ||
| | | outward normal direction of the boundary surface | ||
|- | |- | ||
|''p'' | | ''p'' | ||
|= | | = | ||
| | | pressure, atm | ||
|- | |- | ||
|''p''<sub>'' | | ''p''<sub>''e''</sub> | ||
|= | | = | ||
| | | external boundary pressure, atm | ||
|- | |- | ||
|''p''<sub>'' | | ''p''<sub>''i''</sub> | ||
|= | | = | ||
| | | initial pressure, atm | ||
|- | |- | ||
|'' | | ''p''<sub>''w f''</sub> | ||
|= | | = | ||
| | | flowing wellbore pressure, atm | ||
|- | |- | ||
|'' | | ''q'' | ||
|= | | = | ||
| | | production rate, cm<sup>3</sup>/s | ||
|- | |- | ||
|'' | | ''r'' | ||
|= | | = | ||
| | | radial coordinate and distance, cm | ||
|- | |- | ||
| | | ''t'' | ||
|= | | = | ||
| | | time, s | ||
|- | |- | ||
| | | [[File:Vol1 page 0077 inline 001.png|RTENOTITLE]] | ||
|= | | = | ||
| | | velocity vector | ||
|- | |- | ||
|'' | | ''V'' | ||
|= | | = | ||
| | | volume, cm<sup>3</sup> | ||
|- | |- | ||
|'' | | ''x'' | ||
|= | | = | ||
|distance in '' | | distance in ''x''-direction, cm | ||
|- | |- | ||
|'' | | ''y'' | ||
|= | | = | ||
|distance in '' | | distance in ''y''-direction, cm | ||
|- | |- | ||
|'' | | ''z'' | ||
|= | | = | ||
| | | distance in ''z''-direction, cm | ||
|- | |- | ||
|'' | | ''Z'' | ||
|= | | = | ||
| | | compressibility factor | ||
|- | |- | ||
|'' | | ''α'' | ||
|= | | = | ||
|permeability direction | | permeability direction | ||
|- | |- | ||
| | | ''β'' | ||
|= | | = | ||
| | | permeability direction | ||
|- | |- | ||
| | | Γ | ||
|= | | = | ||
| | | boundary surface, cm<sup>2</sup> | ||
|- | |- | ||
| | | ''γ'' | ||
|= | | = | ||
| | | permeability direction | ||
|- | |- | ||
| | | Δ | ||
|= | | = | ||
| | | difference operator | ||
|- | |- | ||
|'' | | ''η'' | ||
|= | | = | ||
| | | diffusivity constant | ||
|- | |- | ||
|'' | | ''θ'' | ||
|= | | = | ||
| | | angle from positive x-direction, degrees radian | ||
|- | |- | ||
|'' | | ''θ′'' | ||
|= | | = | ||
| | | source coordinate in ''θ''-direction, degrees radian | ||
|- | |- | ||
|'' | | ''μ'' | ||
|= | | = | ||
| | | viscosity, cp | ||
|- | |- | ||
|'' | | ''ρ'' | ||
|= | | = | ||
| | | density, g/cm<sup>3</sup> | ||
|- | |- | ||
|'' | | ''τ'' | ||
|= | | = | ||
| | | time, s | ||
|- | |- | ||
|'' | | ''Φ'' | ||
|= | | = | ||
| | | porosity, fraction | ||
|- | |- | ||
| ''φ''(''M'') | |||
| = | |||
| any continuous function | |||
|} | |} | ||
==References== | == References == | ||
< | <references /> | ||
== Additional references == | |||
Matthews, C. S., & Russell. D. G. 1967. Pressure Buildup and Flow Tests in Wells, SPE Monograph Series Vol. 1, Society of Petroleum Engineers, Richardson, TX, 163 pp. | |||
Van Everdingen, A. F., & Hurst, W. (1949, December 1). The Application of the Laplace Transformation to Flow Problems in Reservoirs. Society of Petroleum Engineers. doi:10.2118/949305-G | |||
== External links == | |||
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro | Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro | ||
==See also== | == See also == | ||
[[Bessel functions in transient analysis]] | |||
[[Bessel_functions_in_transient_analysis|Bessel functions in transient analysis]] | |||
[[Laplace_transformation_for_solving_transient_flow_problems|Laplace transformation for solving transient flow problems]] | |||
[[ | [[Green’s_function_for_solving_transient_flow_problems|Green’s function for solving transient flow problems]] | ||
[[ | [[Source_function_solutions_of_the_diffusion_equation|Source function solutions of the diffusion equation]] | ||
[[ | [[Solving_unsteady_flow_problems_with_Green's_and_source_functions|Solving unsteady flow problems with Green's and source functions]] | ||
[[Solving unsteady flow problems with | [[Solving_unsteady_flow_problems_with_Laplace_transform_and_source_functions|Solving unsteady flow problems with Laplace transform and source functions]] | ||
[[ | [[Mathematics_of_fluid_flow|Mathematics of fluid flow]] | ||
[[ | [[Differential_calculus_refresher|Differential calculus refresher]] | ||
[[ | [[PEH:Mathematics_of_Transient_Analysis]] | ||
[[ | [[Category:5.3.1 Flow in porous media]] |
Latest revision as of 10:33, 8 June 2015
This page introduces how fluid flow in porous media can be translated into a mathematical statement and how mathematical analysis can be used to answer transient-flow problems. This broad area is common to many other disciplines, such as heat conduction in solids and groundwater hydrology. The objective is to introduce the fundamentals of transient analysis, present examples, and guide the interested reader to relevant references.
Most physical phenomena in the domain of transient fluid flow in porous media can be described generally by partial differential equations (PDEs). With appropriate boundary conditions and sometimes with simplifying assumptions, the PDE leads to an initial boundary value problem (IBVP) that is solved to find a mathematical statement of the resulting flow in the porous medium. This page briefly discusses the statement of the IBVP for transient fluid flow in porous media.
Equations of transient fluid flow in porous media
In essence, fluid motion in porous media can be specified by the knowledge of the velocity vector, , and the density of the fluid, ρ, as a function of the position (x, y, z) and time, t; that is, = (x, y, z, t) and ρ= ρ (x, y, z, t). Relative to the fixed Cartesian axes, the velocity vector can be written as
where vx, vy, and vz are the velocity components, and , , and are the unit vectors in the x, y, and z directions, respectively.
The physical law governing the macroscopic fluid-flow phenomena in porous media is the conservation of mass, which states that mass is neither created nor destroyed. The mathematical formula of this rule is developed by considering the flow through a fixed arbitrary closed surface, Γ, lying entirely within a porous medium of porosity, Φ, which is filled with a fluid of viscosity μ. Fig. 1 illustrates an arbitrary closed surface in porous medium.
The conservation of mass principle requires that the difference between the rates at which fluid enters and leaves the volume through its surface must equal the rate at which mass accumulates within the volume. The total mass within the volume at any time is given by
Then, the time rate of change of mass within Γ is
which, by the conservation of mass law, is equal to the rate at which mass enters V through the surface.
Consider the differential surface element, dΓ, shown in Fig. 1. The mass entering the volume through dΓ at the normal velocity, , in a time increment, Δt, is , and the total mass of the fluid passing through Γ during Δt is
The surface integral in Eq. 4 accounts for both influx and outflux through the surface of the volume; that is, ΔMg is the difference between the masses entering and leaving the control volume during the time increment, Δt. Then, the mass rate entering the volume, V, through its surface, Γ, can be written as
By the principle of conservation of mass, equating the right sides of Eqs. 3 and 5 yields
A more useful relation is found with the divergence theorem, which states that the flux of through the closed surface, Γ, is identical to the volume integral of (the divergence of ) taken throughout V; that is,
Here, ∇ is the gradient operator, which in 3D Cartesian and cylindrical coordinates is given, respectively, by
and
With the relation in Eq. 7, Eq. 6 can be recast into
If the functions involved in the argument of the integral in Eq. 10 are continuous, then the integral is identically zero if and only if its argument is zero (because the volume integral in Eq. 10 is identically zero for any arbitrarily chosen volume). Then, the following continuity equation can be obtained.
Eq. 11 is a PDE that is equivalent to the statement of the conservation of mass for fluid flow in porous media. For practical purposes, however, Eq. 11 is expressed in terms of pressure because density and velocity cannot be measured directly. To express density, ρ, and velocity, , in terms of pressure, we use an equation of state and a flux law, known as Darcy’s law, respectively.
The following definition of isothermal fluid compressibility, c, is a useful equation of state that relates density to pressure.
If c is a constant (the compressibility of many reservoir liquids may be considered as constant), then Eq. 12 can be integrated to yield
where subscript 0 indicates the conditions at the datum. Similarly, the compressibility of the porous rock, cf, is defined by
and the total system compressibility, ct, is given by
These definitions of compressibility help recast Eq. 11 in terms of pressure.
Darcy’s law for fluid flow in porous media is a flux law. Neglecting the gravity effect, it is expressed by
In Eq. 16, μ is the viscosity of the fluid, and k is the permeability tensor of the formation given by
where α, β, and γ are the directions, and kij is the permeability in the i direction as a result of the pressure gradient in the j direction.
If Eqs. 13 through 16 are used in Eq. 11, an alternative statement of the conservation of mass principle for fluid flow in porous media is obtained:
Eq. 18 is the PDE that governs transient fluid flow in porous media. In the present form, Eq. 18 is not very useful in obtaining practical solutions because of the nonlinearity displayed in the second term of the left side. For liquid flow, the viscosity, μ, is constant and Eq. 18 can be linearized by assuming that the pressure gradients, ∇p, are small in the reservoir and the compressibility of the reservoir liquids, c, is on the order of 10−5 or smaller. Then, the second term of the left side of Eq. 18 may be neglected compared with the remaining terms and the following linear expression is obtained:
Eq. 19 (or Eq. 18) is known as the diffusivity equation. As an example in Cartesian coordinates, assuming that the coordinate axes can be chosen in the directions of the principal permeabilities, k, in Eq. 19, may be represented by the following diagonal tensor:
Then, Eq. 19 may be written
If each coordinate, j = x, y, or z, is multiplied by , where k may be chosen arbitrarily (to preserve the material balance, k is usually chosen to be ), Eq. 21 may be transformed into the diffusion equation for an isotropic domain:
where η is the diffusivity constant defined by
If the same transformation is also applied to the boundary conditions (see the "Initial and boundary conditions" section below), the problems in anisotropic reservoirs may be transformed into those in isotropic reservoirs provided that the system is infinite or bounded by planes perpendicular to the principal axes of permeability. In all other cases, this transformation distorts the bounding surfaces.
For the flow of gases, the assumptions of small fluid compressibility and pressure gradient may not be appropriate and the c(∇p)2 term in Eq. 18 may not be negligible. In these cases, an expression similar to Eq. 21 may be obtained from Eq. 18 in terms of pseudopressure, m, as
Here, the pseudopressure is defined by[1]
where Z is the compressibility factor. To define a complete physical problem, Eq. 21 (or 24) should be subject to the initial and boundary conditions discussed below.
Initial and boundary conditions
The solution of the diffusivity equation (Eq. 19) should satisfy the initial condition in the porous medium. The initial condition is normally expressed in terms of a known pressure distribution at time zero; that is,
The most common initial condition is the uniform pressure distribution in the entire porous medium; that is, f (x, y, z) = pi.
The boundary conditions are specified at the inner (wellbore) and outer boundaries of the reservoir. These are usually in the form of prescribed flux or pressure at the boundaries. The condition of prescribed flux can be formulated as
where Γ is the surface of the boundary, and n indicates the outward normal direction of the boundary surface. The prescribed flux condition may be used at the inner and outer boundaries of the reservoir. The most common use of the prescribed flux condition at the inner boundary is for the production at a constant rate. In this case, the function, g(t), is related to a constant production rate, q. At the outer boundary, the prescribed flux condition is usually used to indicate impermeable boundaries [g(t)=0] and leads to a pseudosteady state under the influence of boundaries.
For some applications, pressure may be specified at the inner and outer boundaries. In this case,
When used at the inner boundary, this condition represents production at a constant pressure, pwf; that is, h(t) = pwf. At the outer boundary, specified pressure, pe, is usually a result of injection or influx from an adjacent aquifer, which usually leads to steady state in the reservoir.
It is also possible to have boundary conditions of mixed type. These usually correspond to interface conditions in porous media. Raghavan[2] contains more details about the common boundary conditions for the diffusion equation.
Assumptions and limits
Some assumptions have been made in the derivation of the diffusivity equation given by Eq. 19. These assumptions determine the limits of applicability of the solutions obtained from Eq. 19. One of the most important assumptions involved is the continuity of the properties involved in Eq. 19. (This was required to obtain Eq. 19 from the more general integral form in Eq. 10.) Therefore, sharp changes in the properties of the reservoir rock and fluid (such as faults and fluid banks) should be incorporated in the form of boundary or interface conditions in the solution of Eq. 19.
The second important assumption is that Darcy’s law describes the flux in porous media. This assumption is valid at relatively low fluid velocities that may be appropriate to describe liquid flow. At high velocities (when Reynolds number based on average sand grain diameter approaches unity) such as those observed in gas reservoirs, Darcy’s law is not valid.[3] In this case, Forchheimer’s equation,[4] which accounts for the inertial effects, should be used. In petroleum engineering, it is a common practice to consider the additional pressure drop as a result of non-Darcy flow in the form of a pseudoskin because it is usually effective in a small vicinity of the wellbore.
Mathematical tools used for solving transient analysis problems
These tools include:
- Bessel functions in transient analysis
- Laplace transformation for solving transient flow problems
- Green’s function for solving transient flow problems
Nomenclature
References
- ↑ Al-Hussainy, R., Ramey Jr., H.J., and Crawford, P.B. 1966. The Flow of Real Gases Through Porous Media. J Pet Technol 18 (5): 624–636. SPE-1243-A-PA. http://dx.doi.org/10.2118/1243-A-PA
- ↑ Raghavan, R. 1993. Well Test Analysis, 28–31, 336–435. Englewood Cliffs, New Jersey: Petroleum Engineering Series, Prentice-Hall.
- ↑ Fancher, G.H., Lewis, J.A., and Barnes, K.B. 1933. Some Physical Characteristics of Oil Sands. Bulletin 12, Mineral Industries Experimental Station, Pennsylvania State University, University Park, Pennsylvania, 65–167.
- ↑ Forchheimer, P.F. 1901. Wasserbewegung durch Boden. Zeitschrift des Vereines deutscher Ingenieure 45 (5): 1781–1788.
Additional references
Matthews, C. S., & Russell. D. G. 1967. Pressure Buildup and Flow Tests in Wells, SPE Monograph Series Vol. 1, Society of Petroleum Engineers, Richardson, TX, 163 pp.
Van Everdingen, A. F., & Hurst, W. (1949, December 1). The Application of the Laplace Transformation to Flow Problems in Reservoirs. Society of Petroleum Engineers. doi:10.2118/949305-G
External links
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
See also
Bessel functions in transient analysis
Laplace transformation for solving transient flow problems
Green’s function for solving transient flow problems
Source function solutions of the diffusion equation
Solving unsteady flow problems with Green's and source functions
Solving unsteady flow problems with Laplace transform and source functions