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Solution of thermodynamic equilibrium in black-oil
reservoirs: the classical material balance equations
Abstract
This work presents a derivation of the classical material balance equations
starting from black-oil equations in differential form. The existing link be-
tween both approaches is discussed and an interpretation for pressure in ma-
terial balance calculations is provided. It is shown that a condition stronger
than hydrostatic equilibrium is needed to reduce black-oil equations to the
classical approach. The classical equations are formulated for a variable
bubble-point pressure extending their applicability to reservoirs under pres-
sure maintenance. Sufficient conditions for solution uniqueness are presented
and a scheme for numerical solution is proposed.
Keywords: material balance, black-oil equations, reservoir engineering.
1. Introduction
The so called material balance equations are widely employed in reservoir
engineering and have become a classical tool in oil reservoir management.
The equations were originally conceived as a zero dimensional model for
estimating reservoir pressure from phase equilibrium and balance of fluid
production and expansion (Schilthuis, 1936). Although many derivations of
material balance equations have been reported, few explore their connection
with black-oil equations.
Preprint submitted to Elsevier February 1, 2013
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Black-oil equations arise from the generalization of the zero dimensional
concept to a multidimensional macroscopic model which incorporates hy-
drodynamics via Darcys law and entails exactly the same thermodynamic
approach for phase equilibrium (Aziz and Settari, 1979).
The termblack-oilrefers to specific hydrocarbon compositions whose vol-
ume and phase equilibrium can be well described by a two pseudo-component
system. The thermodynamic processes allowed for the system are restricted
to isothermal changes, and reservoir temperature must be sufficiently be-
low the mixture critical temperature. This restriction provides that different
fluid phases have clearly distinguishable properties. Details on the thermo-
dynamic behavior of hydrocarbons mixtures are found in Danesh (1998) and
Firoozabadi (1999).
The link between material balance and black-oil equations has been briefly
addressed in Ertekin et al. (2001), where the former approach was shown to
be a particular case of the latter under two basic assumptions: negligible
potential gradient and capillary pressure.
= p gz 0 , (1a)
p p , (1b)
for = o, g, w identifying by the initial letter oil, gas and water phases.
It is shown in the following sections of this work that a stronger assump-
tion is actually needed to integrate black-oil equations in the entire reservoir
volume and obtain material balance equations. Mechanical equilibrium in
the presence of gravity (p = gz) is not a sufficient condition: pressure
gradients must identically vanish instead, what means that gravity must be
neglected as well as capillary pressure.
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The restriction of a spatially constant pressure throughout the reservoir
does not require PVT properties to be averaged during the integration proce-
dure. For the same reason, spatial constancy is also imposed on bubble-point
pressure and it is shown that a constant bubble-point pressure is equivalent
to having constant fluid saturations.
The present derivation of material balance equations allows bubble-point
pressure to vary with time accordingly to the same concept widely employed
in reservoir simulation, which is found in Aziz and Settari (1979) and Ertekin
et al. (2001). The employed approach extends material balance equations to
saturated reservoirs under pressure maintenance (e.g. fluid injection).
After presenting the equations, sufficient conditions for solution unique-
ness are imposed on PVT properties. The conditions simply require fluids
to be compressible and that gas dissolution occurs with reduction of system
volume.
A simple scheme for numerical solution is proposed
2. Derivation of Equations
The derivation shall start presenting the more general black-oil equations.
It then proceeds restricting their validity until are met the specific conditions
for obtaining material balance equations.
2.1. Black-Oil Equations
As previously mentioned, black-oil is a designation to a simplified treat-
ment of hydrocarbon composition and phase equilibrium. Hydrocarbon species
are grouped into two pseudo-components called oil and gas, being a pseudo-
component one that consists of more than one chemical molecule.
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Oil component is composed by hydrocarbon species that remain in liquid
phase for standard surface pressure and temperature (stock tank oil). In con-
trast, gas component is composed by hydrocarbon species in gaseous phase
for the same surface conditions.
The mixture of reservoir fluids is then composed of the two pseudo-
components (oil and gas) plus the water component. The mixture can present
three fluid phases: oil, gas and water. Oil phase may be composed of oil and
gas components, gas phase may be composed uniquely by the gas component
and water phase uniquely by the water component. In other words, the only
allowed mass transfer is of gas component between oil and gas phases.
In addition to the three fluid components and phases, there is the solid
phase composed by the rock.
Mass balance equations for each fluid phase are:
t
(pP)SwBw(pw)
+
uw
Bw(pw)
= qws, (2)
t
(pP)SoBo(po, Pb)
+
uoBo(po, Pb)
= qos, (3)
t
(pP)
Sg
Bg(pg)+
SoRs(po, Pb)
Bo(po, Pb)
+
uoRs(po, Pb)
Bo(po, Pb) +
ug
Bg(pg)
= qgs,
(4)
where saturations must satisfy the constraint
So+Sw+Sg = 1 . (5)
Bubble-point pressure (Pb) is here defined as the minimum pressure for
which gas phase is absent, i.e., Sg = 0. Its local value is implicitly given by
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the solubility ratio (RS) satisfying the equation bellow:
Rs(po= Pb, Pb) = Bg(pg)
Bo(po, Pb)
SgSo
+Rs(po, Pb) . (6)
It must be remarked that bubble-point pressure is sometimes defined in a
different manner as the bubble-point pressure for the hydrocarbon composi-
tion in oil phase only. According to this definition, Pb = p for a saturated
reservoir.
The assumption of instantaneous mass transfer between oil and gas phases
is probably invalid far from phase interface, where a combined mechanism of
advection and diffusion exists. Black-oil equations, however, do not provide
a macroscopic diffusion of gas component in oil phase. Some heuristic models
have been proposed for problems where this assumption is critical, as for gas
injection (Utseth and MacDonald, 1981).
Momentum balance equations provide relationships between phase pres-
sure (p) and velocity (u) that comes from an extension of Darcys law to
multiphase flow. This extension consists in multiplying absolute (or single-
phase) permeability (k) by a scalar function called phase relative permeabil-
ity (kr,):
u = kr,k
(p gz) , = o, w , g . (7)
Since only isothermal process are considered, no energy balance is re-
quired to compute phase equilibrium.
Porosity () is assumed a function of pore pressure (pP). Since three
phases are present in general, such quantity is defined by some macroscopic
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relation of type (Kim et al., 2011):
pP =pP(po, pw, pg, So, Sw, Sg) . (8)
In order to close the system of equations, a connection among different
phase pressures must be provided. Such expressions are referred to as cap-
illary pressures and are usually employed in the following forms (Aziz and
Settari, 1979):
pc,ow(k, , S w) =po pw, (9)
pc,go(k, , S g) =pg po. (10)
2.2. Material Balance Equations
Derivation of the classical material balance equations consists in inte-
grating black-oil equations in the entire reservoir volume. The integration
process, however, requires that all quantities be either averaged or assumed
constant in space.
Defining averages of most quantities is intractable, though, as they appear
in equations forming products of up to three factors. Being the quantities self-
correlated, the averaging process would involve unclosed terms. Quantities
are thus required to be spatially constant whenever needed.
We start assuming mechanical equilibrium, i.e., no macroscopic fluid mo-
tion:
u= kr,k
(p gz) 0 . (11)
This condition implies that each phase pressure equals the gravity body force:
p= gz . (12)
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Black-oil equations then become:
t
SwBw(p)
= qws, (13a)
t
So
Bo(p, Pb)
= qos, (13b)
t
Sg
Bg(p)+
SoRso(p, Pb)
Bo(p, Pb)
= qgs, (13c)
So+Sw+Sg = 1 . (13d)
Eq. (12) gives an expression of how pressure, and hence PVT proper-
ties, depends on position. If gravity and capillarity are also neglected, fluid
pressure becomes independent of both position and phase:
po= pw =pg =pP =p , (14a)
p 0 . (14b)
Bubble-point pressure is also a function of position as it depends on local
values of oil and gas saturations. Assuming a spatially constant bubble-pointpressure is equivalent to assuming constant saturations, as the quantities are
related by Eqs. (5) and (6).
For spatially constant fluid and bubble-point pressures, Eqs. (13a), (13b)
and (13c) can be integrated over the entire reservoir volume (), which is
defined as the region where hydrocarbon saturation is greater than zero for
some arbitrary volume (provided that the restriction of macroscopic scale is
observed). This reservoir definition excludes any surrounding aquifer zone.
The interpretation of pressure computed from material balance equations
becomes clear at this point: it is the reservoir pressure in the absence of
pressure gradients, fluid segregation, capillary forces and gravity.
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Performing volume and then time integration of Eq. (13a) between times
t1 and t2, material balance for oil phase is obtained: t2t1
t
(p)SoBo(p, Pb)
dV dt=
t2t1
qosdV dt , (15)
1
Bo(p, Pb)
SodV
t2t1
= Np, (16)
Vp1So1Bo1
Vp2So2
Bo2=Np. (17)
Equations for gas and water phases are obtained following the same pro-
cedure:
Vp2
Sg2Bg2
+So2Rs2
Bo2
N
m
Bo1Bg1
+Rs1
= Gi Gp, (18)
Vp2Sw2Bw2
W =Wi+We Wp, (19)
So+Sw+Sg = 1 , (20)
where
N=Vp1So1
Bo1, (21)
W =Vp1Sw1
Bw1, (22)
m=Vp1Sg1
Bo1N . (23)
An equation for bubble-point pressure in integrated form can be obtained
by setting Sg = 0 in Eq. (18) and solving for the solubility ratio at bubble-
point pressure Rs(p= Pb):
Rs2(p= Pb) =
N(mB1/Bg1+Rs1)
+Gi Gp
/ (N Np) .(24)
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A single equation for pressure (without fluid saturations) can be obtained
by substituting Eqs. (17), (18) and (19) into (20). The resulting equation
allows pressure to be calculated for given volumes and fluid properties:
Fp Fi = N Eo+mNEg+W Ew total volume expansion E
+ Vp1 Vp2 volume contractionC
, (25)
where new terms are defined in Table 1. Eq. (25) is commonly referred to as
the Material Balance Equation (MBE).
3. Solution of Material Balance Equations
A numerical solution of Eq. (25) is presented here. We shall start defining
a residual function () for which the pressure in Eq. (25) is a root:
(p2) =E+Fi+C Fp
Vp1. (26)
We proceed showing that is monotonic for certain conditions on fluid prop-
erties, what implies that only one root exists at most.
3.1. Mathematical Aspects of the Residual Function
It is a known result of real analysis that a monotonic function f : D
R R can have one root at most in D. If f is continuous in interval
I= (a, b) D andf(a)f(b)
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Hence we conclude that has only one root in I= (a, b) if(a)(b)< 0. In
other words, only one value of pressure satisfies material balance equations
for any given volumes of production.
It can be shown that d/dp
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dBg
dp
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Bubble-point pressurePb2is only a function of the values in initial instant
t1 and produced and injected volumes.
Step 2. Compute fluid pressure for instant t2 finding the root of residual
function ().
Pressure in Eq. (25) is only a function of values in instant t1, produced and
injected values and bubble-point pressure Pb2, which was already computed
in step 1.
Step 3. Compute fluid saturationst2.
OnceSo2, Sg2, Sw2, p2 and Pb2 are known, solution for instant t2 is com-
plete.
4. Results
Material balance equations were applied to an oil reservoir initially pro-
ducing in depletion drive mechanism. Calculations show the steep fall of
pressure in the initial phases of production and the formation of a secondary
gas cap with the release of gas in solution (growing Sg).
Bubble-point pressure calculations reveal how far the current thermody-
namic state is from the undersaturation, allowing for planning of fluid injec-
tion projects to increase reservoir pressure and both redissolve and compress
the free gas.
5. Conclusions
1. Material balance equations can be obtained from black-oil equations under
the assumptions of:
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(a) mechanical equilibrium;
(b) no gravity and capillary forces;
(c) spatially constant bubble-point pressure or, equivalently, fluid satura-
tions.
2. One unique solution of pressure equation exists for a compressible system;
3. A residual function whose root solves pressure equation can be defined.
numerically found using nonlinear root-finding algorithms such as New-
tons and Bisection methods.
Nomenclature
Bg gas formation volume factor
Bo oil formation volume factor
Bw water formation volume factor
Rs solubility ratio of gas in oil phase
Sg gas saturation
So oil saturation
Sw water saturation
Gp produced volume of gas
m ratio of original free gas volume to original oil phase volume in reservoir conditio
N original volume of oil in place
Np produced volume of oil
Pb bubble-point pressure
Vp porous volume
Wi injected volume of water
Wp produced volume of water
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References
Allen III, M., Behie, A., Trangenstein, J., 1988. Multiphase flow in porous
media.
Aziz, K., Settari, A., 1979. Petroleum reservoir simulation. Vol. 476. Applied
Science Publishers London.
Danesh, A., 1998. PVT and phase behaviour of petroleum reservoir fluids.
Vol. 47. Elsevier Science.
Ertekin, T., Abou-Kassem, J., King, G., 2001. Basic applied reservoir simu-
lation. Richardson, TX: Society of Petroleum Engineers.
Firoozabadi, A., 1999. Thermodynamics of hydrocarbon reservoirs. Vol. 2.
McGraw-Hill New York.
Hamming, R., 1987. Numerical methods for scientists and engineers. Dover
Publications.
Herrera, I., Camacho, R., 1997. A consistent approach to variable bubble-
point systems. Numerical Methods for Partial Differential Equations 13 (1),
118.
Hilfer, R., Aug 1998. Macroscopic equations of motion for two-phase flow in
porous media. Phys. Rev. E 58, 20902096.
URL http://link.aps.org/doi/10.1103/PhysRevE.58.2090
Ismael, H., 1996. Shocks and bifurcations in black-oil models. SPE Journal
1 (1), 5158.
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Kim, J., Tchelepi, H., Juanes, R., 2011. Rigorous coupling of geomechanics
and multiphase flow with strong capillarity. In: SPE Reservoir Simulation
Symposium.
Schilthuis, R., 1936. Active oil and reservoir energy. Trans. AIME 118, 33.
Steffensen, R., Sheffield, M., 1973. Reservoir simulation of a collapsing gas
saturation requiring areal variation in bubble-point pressure. In: SPE Sym-
posium on Numerical Simulation of Reservoir Performance.
Thomas, L., Lumpkin, W., Reheis, G., 1976. Reservoir simulation of variable
bubble-point problems. Old SPE Journal 16 (1), 1016.
Trangenstein, J., Bell, J., 1989a. Mathematical structure of compositional
reservoir simulation. SIAM journal on scientific and statistical computing
10 (5), 817845.
Trangenstein, J., Bell, J., 1989b. Mathematical structure of the black-oil
model for petroleum reservoir simulation. SIAM Journal on Applied Math-
ematics, 749783.
Utseth, R., MacDonald, R., 1981. Numerical simulation of gas injection in
oil reservoirs. In: SPE Annual Technical Conference and Exhibition.
Highlights
1. The existing link between black-oil and material balance equations is ex-
amined in detail.
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2. It is shown that gravity must be neglected and saturations must be con-
stant in space so one can integrate black-oil equations in reservoir volume.
3. Material balance equations are shown to have a unique solution for a
compressible system.
4. Material balance equations are formulated for a variable bubble-point and
they are employed to a reservoir under pressure maintenance.
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Recovery Factor (Np/N)
0.00
0.05
0.10
0.15
0.20
0.25
WaterVolume[unitsofN]
production -Wp
injected -Wi
aquifer influx -We
Figure 1: Volumes of water.
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Recovery Factor (Np/N)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
GasVolume[unitsof10
N]
production -Gp
Figure 2: Volumes of gas.
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Recovery Factor (Np/N)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Pressure
[unitsofPb1
]
fluid pressure
bubble-point pressure
Figure 3: Reservoir pressure and bubble-point pressure as functions of recovery factor.
Pressure is expressed in units of the original bubble-point pressure.
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Recovery Factor (Np/N)
0.0
0.2
0.4
0.6
0.8
1.0
Fluid
Saturation
oil
gas
water
Figure 4: Phase saturations.
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0 . 0 0 . 5 1 . 0 1 . 5
2 . 0 2 . 5 3 . 0 3 . 5 4 . 0
P r e s s u r e [ u n i t s o f P
b 1
]
0 . 2 0
0 . 1 5
0 . 1 0
0 . 0 5
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
0 . 2 0
R
e
s
i
d
u
a
l
F
u
n
c
t
i
o
n
s o l u t i o n
b u b b l e - p o i n t p r e s s u r e
Figure 5: Residual function () for different volumes of production and injection. The
function monotonically decreases and it has no derivative at bubble-point pressure.
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Material Balance Equations
oil So2 = (NNp)Bo2/Vp2
water Sw2= (W+Wi+ We Wp)Bw2/Vp2
gas Sg2= {N[(Rs1 Rs2) + mBo1/Bg1] + NpRs2+ Gi Gp}Bg2/Vp2
bubble-point pressure Rs2(Pb) = [N(mB1/Bg1+ Rs1) + Gi Gp] / (N Np)
pressure Fp Fi= NEo+ mNEg+ WEw+ C
Definition of terms in pressure equation
fluid production Fp= Np[Bo2+ (Gp/Np Rs2)Bg2] + WpBw2
fluid injection and influx Fi= (Wi+ We)Bw2+ GiBg2
oil phase expansion Eo= (Bo2 Bo1) (Rso2 Rso1)Bg2
gas phase expansion Eg = Bo1(Bg2/Bg1 1)
water volume expansion Ew = Bw2 Bw1
porous volume contraction C= Vp2 Vp1
Table 1: Complete set of material balance equations.
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