Material Balance - Aziz

8/21/2019 Material Balance - Aziz

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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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0.0

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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0.2

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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0.0

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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Material Balance - Aziz

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