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

${\begin{aligned}&\underbrace {G_{fgi}E_{g}} _{\text{free-gas expansion}}+\underbrace {N_{foi}E_{o}} _{\text{free-oil expansion}}+\underbrace {WE_{w}} _{\text{free-water expansion}}+\underbrace {V_{pi}E_{f}} _{\text{pore-volume contraction}}+\underbrace {W_{e}} _{\text{water influx}}\\\\&=\underbrace {(G_{p}-G_{I})\left({\frac {B_{g}-B_{o}R_{v}}{1-R_{v}R_{s}}}\right)} _{\text{net gas withdrawal}}+\underbrace {N_{p}\left({\frac {B_{o}-B_{g}R_{s}}{1-R_{v}R_{s}}}\right)} _{\text{oil withdrawal}}+\underbrace {(W_{p}-W_{I})B_{w}} _{\text{net water withdrawal}},\end{aligned}}$

(1)

where:

• G_fgi, N_foi, and W are the initial free gas, oil, and water in place, respectively
• G_p, N_p, and W_p are the cumulative produced gas, oil, and water, respectively
• G_I and W_I are the cumulative injected gas and water respectively
• E_g, E_o, E_w, and E_f 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:

$N_{g}={\frac {2.63\times 10^{-3}k_{v}\Delta \rho r_{e}^{2}}{\mu q}},$

(2)

$t_{D}={\frac {0.03110k_{a}t}{\phi _{a}\mu _{w}c_{t}r_{o}^{2}}},$

(3)

$U=4.89\times 10^{-3}hfr_{o}^{2}\phi _{a}c_{t},$

(4)

$t_{D}={\frac {0.03110k_{a}t}{\phi _{a}\mu _{w}c_{t}L_{a}^{2}}},$

(5)

$L_{ac}={\sqrt {\frac {0.03110k_{a}t_{\text{max}}}{2\phi _{a}\mu _{w}c_{t}}}},$

(6)

$q_{c}={\frac {5.256\times 10^{-3}k_{o}h^{2}\Delta \rho }{\mu _{o}B_{o}}}q_{Dc},$

(7)

and

$q={\frac {5.35\times 10^{-4}k_{H}b(p_{e}-p_{w})}{\mu _{o}B_{o}\left(\ln {\frac {r_{e}}{r_{w}}}+s\right)}}\left[1+7{\sqrt {{\frac {r_{w}}{2b}}{\sqrt {\frac {k_{v}}{k_{H}}}}}}\cos \left({\frac {\pi b}{2h}}\right)\right],$

(8)

N_foi and G_fgi are related to the total original oil in place (OOIP) and original gas in place (OGIP), N and G, according to N = N_foi + G_fgi R_vi and G = G_fgi + N_foi R_si.

The expansivities are defined as

$E_{o}=B_{to}-B_{toi},$

(9)

$E_{g}=B_{tg}-B_{tgi},$

(10)

$E_{w}=B_{w}-B_{wi},$

(11)

and $E_{f}={\frac {V_{pi}-V_{p}(p)}{V_{pi}}}$, where B_to and B_tg are the two-phase formation volume factors (FVFs),

$B_{to}={\frac {B_{o}(1-R_{si}R_{v})+B_{g}(R_{si}-R_{s})}{1-R_{v}R_{s}}},$

(12)

and

$B_{tg}={\frac {B_{g}(1-R_{vi}R_{s})+B_{o}(R_{vi}-R_{v})}{1-R_{v}R_{s}}}.$

(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. B_to and B_tg 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, B_to = B_o + B_g (R_si – R_s) and B_tg = B_g. 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

$G_{fgi}E_{gwf}+N_{foi}E_{owf}+W_{e}=F,$

(14)

where:

• F = total net fluid withdrawal or production
• E_gwf = composite gas expansivity
• E_owf = composite oil expansivities

F, E_gwf, and E_owf are defined in

$F=G_{ps}\left({\frac {B_{g}-B_{o}R_{v}}{1-R_{v}R_{s}}}\right)+N_{p}\left({\frac {B_{o}-B_{g}R_{s}}{1-R_{v}R_{s}}}\right)+(W_{p}-W_{I})B_{w},$

(15)

$E_{owf}=E_{o}+{\frac {B_{oi}}{(1-S_{wi})}}\left({\frac {S_{wi}E_{w}}{B_{wi}}}+E_{f}\right),$

(16)

and

$E_{gwf}=E_{g}+{\frac {B_{gi}}{(1-S_{wi})}}\left({\frac {S_{wi}E_{w}}{B_{wi}}}+E_{f}\right).$

(17)

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

• F is expressed in reservoir volume units (e.g., RB or res m³)
• E_gwf is expressed in reservoir volume units per standard unit volume of gas (e.g., RB/scf)
• E_owf 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., G_fgi = 0) and the initial free oil in place is equal to the OOIP (i.e., N_foi = N) and Eqs.1 , 14, and 15 simplify, respectively, to

${\begin{aligned}&\underbrace {NE_{o}} _{\text{free-oil expansion}}+\underbrace {WE_{w}} _{\text{free-water expansion}}+\underbrace {V_{pi}E_{f}} _{\text{pore-volume contraction}}+\underbrace {W_{e}} _{\text{water influx}}\\\\&=\underbrace {N_{p}B_{o}} _{\text{oil withdrawal}}+\underbrace {(W_{p}-W_{I})B_{w}} _{\text{net water withdrawal}},\end{aligned}}$

(18)

$NE_{owf}+W_{e}=F,$

(19)

$F=N_{p}B_{o}+(W_{p}-W_{I})B_{w}.$

(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

Eq.21 succinctly shows that oil recovery increases with:

${\frac {N_{p}}{N_{foi}}}={\frac {{\frac {(W_{I}-W_{p})B_{w}+W_{e}}{N_{foi}}}+E_{owf}+{\frac {G_{fgi}E_{gwf}}{N_{foi}}}}{\frac {(B_{o}-B_{g}R_{s})+{\frac {G_{ps}}{N_{p}}}(B_{g}-B_{o}R_{v})}{(1-R_{v}R_{s})}}}.$

(21)

• Water influx (W_e)
• Initial free-gas-cap volume (which is proportional to G_fgi)
• Surface water injection (W_I)
• Surface gas injection (by minimizing gas sales through G_ps)

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

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

Noteworthy papers in OnePetro

• Ojo, K. P., and S. O. Osisanya. "Material Balance Revisited." Paper presented at the Nigeria Annual International Conference and Exhibition, Abuja, Nigeria, July 2006. SPE-105982-MS
• Peake, W. T.. "Steamflood Material-Balance Applications." SPE Res Eng 4 (1989): 357–362. SPE-17452-PA
• Kanu, Austin Ukwu, and Onyekonwu Mike Obi. "Advancement in Material Balance Analysis." Paper presented at the SPE Nigeria Annual International Conference and Exhibition, Lagos, Nigeria, August 2014. SPE-172415-MS
• Pletcher, J. L. "Improvements to Reservoir Material Balance Methods." Paper presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, October 2000. SPE-62882-MS

External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See also

Material balance in water drive reservoirs

Primary drive mechanisms

Oil fluid characteristics

Oil fluid properties

PEH:Oil_Reservoir_Primary_Drive_Mechanisms