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# Material balance in oil reservoirs

The material-balance equation is the simplest expression of the conservation of mass in a reservoir. The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal.

## Material balance equation

The applicable equation for initially saturated volatile- and black-oil reservoirs is ....................(1)

where:

• Gfgi, Nfoi, and W are the initial free gas, oil, and water in place, respectively
• Gp, Np, and Wp are the cumulative produced gas, oil, and water, respectively
• GI and WI are the cumulative injected gas and water respectively
• Eg, Eo, Ew, and Ef are the gas, oil, water, and rock (formation) expansivities

Most of the equations regarding primary drive mechanisms for oil reservoirs apply to any consistent set of units. A few equations, however, are written assuming English or customary units. Those equations are expressed in SI units: ....................(2) ....................(3) ....................(4) ....................(5) ....................(6) ....................(7)

and ....................(8)

Nfoi and Gfgi are related to the total original oil in place (OOIP) and original gas in place (OGIP), N and G, according to N = Nfoi + Gfgi Rvi and G = Gfgi + Nfoi Rsi.

The expansivities are defined as ....................(9) ....................(10) ....................(11)

and , where B to and B tg are the two-phase formation volume factors (FVFs), ....................(12)

and ....................(13)

The rock expansivity is obtained from direct measurement. See compaction driving oil reservoir for a greater discussion.

Physically, the two-phase FVF is the total hydrocarbon volume per unit volume of oil or gas at standard conditions. The two-phase FVF mimics the observations noted during a constant-composition expansion test. For instance, the two-phase oil FVF is the total hydrocarbon (oil + gas) volume of a saturated oil sample per unit volume of oil at standard conditions. In contrast, the two-phase gas FVF is the total hydrocarbon volume of a saturated gas sample per unit volume of gas at standard conditions. Bto and Btg typically are expressed in units of RB/stock tank barrel (STB) and RB/Mscf, respectively.

• For undersaturated oils, the two-phase oil FVF is equal to the oil FVF
• For undersaturated gases, the two-phase gas FVF is equal to the gas FVF.

Eqs. 12 and 13 account for volatilized oil in the equilibrium gas phase. If volatilized oil is negligible, these equations are simplified. For instance, Bto = Bo + Bg (RsiRs) and Btg = Bg. These equations apply for black oils. Eq.11 ignores dissolved gas in the aqueous phase.

Eq.1 broadly states that net expansion equals net withdrawal. More specifically, it shows the different forms of expansion and withdrawal. The different forms of expansion such as gas expansion are responsible for the different producing mechanisms.

For the sake of simplicity, Eq.1 is often written in the abbreviated form of ....................(14)

where:

• F = total net fluid withdrawal or production
• Egwf = composite gas expansivity
• Eowf = composite oil expansivities

F, Egwf, and Eowf are defined in ....................(15) ....................(16)

and ....................(17)

The composite expansivities include the connate-water and rock expansivities. Eq.15 includes Gps, which is the cumulative produced sales gas and is defined as (GpGI).

• F is expressed in reservoir volume units (e.g., RB or res m3)
• Egwf is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf)
• Eowf is expressed in reservoir volume units per standard unit volume of oil (e.g., RB/STB)

For strictly undersaturated oil reservoirs, no free gas exists (i.e., Gfgi = 0) and the initial free oil in place is equal to the OOIP (i.e., Nfoi = N) and Eqs.1 , 14, and 15 simplify, respectively, to ....................(18) ....................(19) ....................(20)

Eqs.18 through 20 ignore gas reinjection.

The material balance equation also helps explain most oil-recovery strategies. If the material-balance equation is solved for the produced fraction of the original free oil in place, then ....................(21)

Eq.21 succinctly shows that oil recovery increases with:

It also shows that oil recovery increases by minimizing water production (Wp).

The material balance equation and its many different forms have many uses including:

• Confirming the producing mechanism
• Estimating the OOIP and OGIP
• Estimating gas cap sizes
• Estimating water influx volumes
• Identifying water influx model parameters
• Estimating producing indices

## Nomenclature

 Bg = gas FVF, RB/scf Bo = oil FVF, RB/STB Btg = two-phase gas FVF, RB/scf Bto = two-phase oil FVF, RB/STB Btw = two-phase water/gas FVF, RB/STB Bw = water FVF, RB/STB cf = rock compressibility, Lt2/m, 1/psi ct = total aquifer compressibility, Lt2/m, 1/psi Ef = rock (formation) expansivity Eg = gas expansivity, RB/scf Egw = expansivity for McEwen method, RB/scf Egwf = composite gas/water/rock FVF, RB/scf Eo = oil expansivity, RB/STB Eow = expansivity for McEwen method, RB/STB Eowf = composite oil/water/rock FVF, RB/STB Ew = water expansivity, RB/STB F = total fluid withdrawal, L3, RB G = total original gas in place, L3, scf Gfgi = initial free gas in place, L3, scf Gi = cumulative gas injected, L3, scf Gp = cumulative produced gas, L3, scf h = pay thickness, L, ft k = permeability, L2, md ka = aquifer permeability, L2, md kH = horizontal permeability, L2, md kt = time constant, 1/t, 1/years kv = vertical permeability, L2, md La = aquifer length, L, ft N = total original oil in place, L3, STB Nfoi = initial free oil in place, L3, STB Ng = dimensionless gravity number Np = cumulative produced oil, L3, STB p = pressure, m/Lt2, psi pe = pressure at drainage radius, m/Lt2, psi pw = wellbore pressure, m/Lt2, psi q = producing rate at reservoir conditions (RB/D) or surface conditions (STB/D),v L3/t qc = critical coning rate, STB/D, L3/t qDc = dimensionless critical coning rate re = reservoir drainage radius rw = wellbore radius, L, ft R = instantaneous producing GOR, scf/STB Rs = dissolved GOR, scf/STB Rsw = dissolved-gas/water ratio, scf/STB Rv = volatilized-oil/gas ratio, STB/MMscf Swi = initial water saturation, fraction t = time, t, years tmax = maximum time, t, years tD = dimensionless time tDmax = maximum dimensionless time U = aquifer constant, L4t2/m, RB/psi Vpi = initial reservoir PV, L3, RB w = reservoir width, L, ft W = initial water in place, L3, STB WD = dimensionless cumulative water influx We = cumulative water influx, L3, RB WI = cumulative injected water, L3, STB Wp = cumulative produced water, L3, STB Δp = difference of time-averaged pressure, m/Lt2, psi Δρ = density difference, m/L3, lbm/ft3 and g/cm3 μg = gas viscosity, m/Lt, cp μo = oil viscosity, m/Lt, cp μw = water viscosity, m/Lt, cp