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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][2][3][4]
....................(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 (Rsi – Rs) 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 (Gp – GI).
- 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[1][4][5]
....................(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:
- Water influx (We)
- Initial free-gas-cap volume (which is proportional to Gfgi)
- Surface water injection (WI)
- Surface gas injection (by minimizing gas sales through Gps)
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 |
References
- ↑ 1.0 1.1 Walsh, M.P. 1995. A Generalized Approach to Reservoir Material Balance Calculations. J Can Pet Technol 34 (1). PETSOC-95-01-07. http://dx.doi.org/10.2118/95-01-07
- ↑ Walsh, M.P. 1994. New, Improved Equation Solves for Volatile Oil and Condensate Reserves. Oil & Gas J. (22 August): 72.
- ↑ Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 2 - Applications to Saturated and Non-Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27728-MS. http://dx.doi.org/10.2118/27728-MS
- ↑ 4.0 4.1 Walsh, M.P. and Lake, L.W. 2003. A Generalized Approach to Primary Hydrocarbon Recovery. Amsterdam: Elsevier.
- ↑ Walsh, M.P., Ansah, J., and Raghavan, R. 1994. The New, Generalized Material Balance as an Equation of a Straight Line: Part 1 - Applications to Undersaturated, Volumetric Reservoirs. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 16-18 March 1994. SPE-27684-MS. http://dx.doi.org/10.2118/27684-MS
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<a href="https://www.onepetro.org/conference-paper/SPE-62882-MS?sort=&start=0&q=material+balance+in+oil+reservoirs&from_year=&peer_reviewed=&published_between=&fromSearchResults=true&to_year=&rows=10#" alt="https://www.onepetro.org/conference-paper/SPE-62882-MS?sort=&start=0&q=material+balance+in+oil+reservoirs&from_year=&peer_reviewed=&published_between=&fromSearchResults=true&to_year=&rows=10#" title="https://www.onepetro.org/conference-paper/SPE-62882-MS?sort=&start=0&q=material+balance+in+oil+reservoirs&from_year=&peer_reviewed=&published_between=&fromSearchResults=true&to_year=&rows=10#">https://www.onepetro.org/conference-paper/SPE-62882-MS?sort=&start=0&q=material+balance+in+oil+reservoirs&from_year=&peer_reviewed=&published_between=&fromSearchResults=true&to_year=&rows=10#</a>
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