immiscible displacement of non-newtonian fluids in

Immiscible Displacement ofNon-Newtonian Fluids in Communicating

Stratified ReservoirsNoaman El-Khatib, SPE, Sudan U. of Science and Technology

SummaryThe displacement of non-Newtonian power-law fluids in commu-nicating stratified reservoirs with a log-normal permeability dis-tribution is studied. Equations are derived for fractional oil recov-ery, water cut, injectivity ratio, and pseudorelative permeabilityfunctions, and the performance is compared with that for Newto-nian fluids. Constant-injection-rate and constant-total-pressure-drop cases are studied.

The effects of the following factors on performance are inves-tigated: the flow-behavior indices, the apparent mobility ratio, theDykstra-Parsons variation coefficient, and the flow rate. It wasfound that fractional oil recovery increases for nw>no and de-creases for nw<no, as compared with Newtonian fluids. For thesame ratio of nw/no, oil recovery increases as the apparent mobilityratio decreases. The effect of reservoir heterogeneity in decreasingoil recovery is more apparent for the case of nw>no. Increasing thetotal injection rate increases the recovery for nw>no, and the op-posite is true for nw<no. It also was found that the fractional oilrecovery for the displacement at constant total pressure drop islower than that for the displacement at constant injection rate, withthe effect being more significant when nw<no.

IntroductionMany of the fluids injected into the reservoir in enhanced-oil-recovery (EOR)/improved-oil-recovery (IOR) processes such aspolymer, surfactant, and alkaline solutions may be non-Newtonian;in addition, some heavy oils exhibit non-Newtonian behavior.

Flow of non-Newtonian fluids in porous media has been stud-ied mainly for single-phase flow. Savins (1969) presented a com-prehensive review of the rheological behavior of non-Newtonianfluids and their flow behavior through porous media. van Poollenand Jargon (1969) presented a finite-difference solution for tran-sient-pressure behavior, while Odeh and Yang (1979) derived anapproximate closed-form analytical solution of the problem.Chakrabarty et al. (1993) presented Laplace-space solutions fortransient pressure in fractal reservoirs.

For multiphase flow of non-Newtonian fluids in porous media,the problem was considered only for single-layer cases. Salmanet al. (1990) presented the modifications for the Buckley-Leverettfrontal-advance method and for the JBN relative permeabilitymethod for non-Newtonian power-law fluid displacing a Newto-nian fluid. Wu et al. (1992) studied the displacement of a Binghamnon-Newtonian fluid (oil) by a Newtonian fluid (water). Wu andPruess (1998) introduced a numerical finite-difference solution fordisplacement of non-Newtonian fluids in linear systems and in afive-spot pattern. Yi (2004) developed a Buckley-Leverett modelfor displacement by a Newtonian fluid of a fracturing fluid havinga Herschel-Bulkley rheological behavior. An iterative procedurewas used to obtain a solution of the model.

The methods available in the literature to predict linear water-flooding performance in stratified reservoirs are grouped into twocategories depending on the assumption of communication or nocommunication between the different layers.

In the case of noncommunicating systems, no vertical cross-flow is permitted between the adjacent layers. The Dykstra-Parsons (1950) method is the basis for performance prediction innoncommunicating stratified reservoirs.

A model for communicating stratified reservoirs was presentedby Hiatt (1958). This model assumes complete crossflow betweenlayers to keep the pressure gradient the same in all layers (verticalequilibrium) at any distance. Warren and Cosgrove (1964) appliedthe Hiatt model to a system with log-normal permeability distri-bution and normal porosity distribution. El-Khatib (1999) pre-sented a closed-form analytical solution for communicating sys-tems with log-normal permeability distribution. Hearn (1971) usedthe same Hiatt model to develop expressions for pseudorelativepermeabilities that can be used to reduce a 3D model to a 2D arealmodel with average (pseudo) functions for the vertical direction.

To the best of the author’s knowledge, no analytical or numeri-cal models are available for the displacement of non-Newtonianfluids in multilayer stratified reservoirs. In this study, an analyticalmodel will be presented to study the performance of immisciblenon-Newtonian power-law fluids in stratified reservoirs. Althoughthe model is applicable for any permeability distribution, a strati-fied system with a log-normal permeability distribution is studiedbecause its behavior is well documented in the literature (Warrenand Cosgrove 1964; El-Khatib 1999) for Newtonian fluids.

Theoretical ConsiderationsRheological Model. Rheological models or equations of state forfluids describe the dependence of the shear stress T on the shearrate �̇. The most general rheological model is the Herschel-Bulkleymodel (1926), which is expressed in the following form:

T = T0 + H�̇ n, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where H is the consistency index and n is the flow-behavior index.Most of the familiar and commonly used rheological models canbe deduced from this model by assigning specific values for theparameters T0 and n. For n�1, we get the equation for the Bing-ham fluids, and for T0�0, we get the widely used power-lawmodel that is used in this study.

T = H�̇ n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

For T0�0 and n�1, Eq. 1 reduces to the Newtonian fluids case,with H being the viscosity �:

T = ��̇. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Capillary-Tube Model. The equation for the flow of a power-lawfluid in a bundle of capillary tubes is given by the equation (Teeuwand Hesselink 1980)

v = � 4n

3n + 1� �

Tn+1

2n

rn+1

n

22n+1

n H1

n

�−dp

dx�1

n. . . . . . . . . . . . . . . . . . . . . (4)

For a Newtonian fluid, n�1, Eq. 4 becomes

v =�

8T

r2

� �−dp

dx�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

Comparing with Darcy’s law, it follows that the permeabilityfor Newtonian-fluid flow is given by

Copyright © 2006 Society of Petroleum Engineers

This paper (SPE 93394) was first presented at the 2005 SPE Middle East Oil and Gas Showand Conference, Manama, Bahrain, 12–15 March, and revised for publication. Originalmanuscript received for review 5 January 2005. Revised manuscript received 6 April 2006.Paper peer approved 12 April 2006.

356 August 2006 SPE Reservoir Evaluation & Engineering

k =�

T

r2

8, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where T is the tortuosity of the porous medium.Using this expression, Eq. 4 can be expressed as

v = � 4n

3n + 1��

2�n−1

2n kn+1

2n

H1

n

�−dp

dx�1

n. . . . . . . . . . . . . . . . . . . . . . (7)

Darcy-Law Analogy. Different approaches were used to trans-form Eq. 7 into a form similar to Darcy’s law by introducing anapparent-viscosity term. Bird et al. (1960) used the form

v n =k

�eff�−

dp

dx�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

It is clear that the effective viscosity as defined in this equationwill not be a function of the rheological-model parameters H and nonly. It also will depend on the porosity and permeability of the rock.

Other investigators such as Gogarty et al. (1972) and Cannellaet al. (1988) suggested the form

v =k

�ap�−

dp

dx�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

As can be seen from Eq. 7, the apparent viscosity in this casealso will be dependent on the pressure gradient dp/dx in addition tothe rock and fluid properties.

In this work, we separate the effect of rock properties andpressure gradient from the definition of the apparent fluid viscosityand introduce the following definition for �ap, which depends onlyon the rheological-model parameters H and n.

�ap = 2n−1

2n �3n + 1

4n �H1

n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

With this definition, the Darcy-law analogy for power-law flu-ids may be written as

v = �k

�ap�−

dp

dx�1

n, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

where the non-Newtonian flow coefficient � is defined as

� = � k

��1−n

2n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

For multiphase flow, we may write

vi = �i

kkri

�iap�−

dp

dx�1

ni, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

where kri is the relative permeability to the specified phase i (oil,gas, or water).

The definition of the apparent viscosity in this formulation isdifferent from those of Bird et al. (1960), Gogarty et al. (1972),and Cannella et al. (1988) in that only the fluid parameters H andn are used in the definition. The rock parameters k and � and thepressure gradient dp/dx are not used. The effect of the rock prop-erties on the flow is introduced in the non-Newtonian-flow coef-ficient �. Although Eq. 7 is the same in all models, this formula-tion is more convenient in handling two-phase-flow problems.

Development of the Model EquationsThe following assumptions are made:

• The system is linear, horizontal, and of constant thickness.• The flow is isothermal, and rock and fluids are incompressible.• The initial fluid distribution is uniform with irreducible

water saturation.• The displacement is pistonlike, with only residual oil behind

and initial conditions ahead of the displacement front.• Capillary and gravity forces are negligible.

• The relative permeability characteristics (Swi , Sor , krw• , and

kro• ) are the same for all layers.

• The porosity is assumed the same in all layers.• Adsorption of fluids on the solid surface of the porous media

is negligible, so effects such as porosity and permeability alterationor inaccessible pore volume are not considered.

As in the conventional prediction methods for stratified reser-voirs, the layers are ordered in a decreasing order of absolutehorizontal permeability, with each layer i having a thickness �hi

and permeability ki. In the horizontal direction between injectionand production faces, the system is divided into N+1 zones sepa-rated by the displacement fronts. Zone 0 is at the production end,while Zone N is at the injection end, with N being the number oflayers. All layers in Zone 0 (zero zones flooded) are at initialconditions, while all layers in Zone N (N layers flooded) are at theresidual oil saturation. At Zone j (j layers flooded), Layers 1through j are at residual oil saturation, while Layers j+1 through Nare at initial conditions (see Fig. 1).

Fractional-Flow Formula. At the time of water breakthrough inthe jth layer, the fraction of water flowing at the outlet boundary(water cut) fwj, as derived in Appendix A, is given by

fwj =�Cwj

�Cwj + �Cot − Coj��fwj Qt �wap

Wkrw• Cwj

�nw

no− 1

. . . . . . . . . . . . . . . . (14)

It should be noted that Eq. 14 is implicit in fwj. It may be writtenin the following form:

fwj +�Cot − Coj�

�Cwj� Qt �wap

Wkrw• Cwj

�nw

no− 1

fwj

nw

no = 1. . . . . . . . . . . . . . . . . . (15)

Eq. 15 can be solved iteratively for fwj. This can be performed forj=1 . . . N−1. It is clear that fwo�0 and fwN�1.

Fractional Oil Recovery. The equations for oil recovery will bethe same as those for Newtonian fluids except for the expressionfor the fractional-flow formula.

Once the function fwj is evaluated, the equations for communi-cating stratified reservoirs as given in the literature can be used(Hiatt 1958; Warren and Cosgrove 1964; El-Khatib 1999):

Fig. 1—Stratified system showing zones and layers before wa-ter breakthrough in the first layer.

357August 2006 SPE Reservoir Evaluation & Engineering

The fractional oil recovery at time of breakthrough in the jthlayer is given by

Rj = hDj + �1 − fwj��j , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

where the dimensionless time � is the injected volume relative tothe ultimate oil recovery, and the fractional oil recovery is alsorelative to the ultimate oil recovery so that at j=N, R�1.

The dimensionless time at breakthrough in layer j is given by

�j =1

�fwj

�hDj

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

where

�fwj = fwj − fw�j−1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

with fw0�0.The dimensionless thickness �hDj and the cumulative dimen-

sionless thickness hDj are defined as

�hDj = �hj �ht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

and hDj = �i=1

j

�hDi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

where ht is the total thickness of the system.

Pseudorelative Permeability Functions. For non-Newtonian-fluid flow in a homogeneous system having a uniform permeabil-ity equal to the average permeability of the stratified system k andtotal thickness ht, Eq. 13 expressed for the water phase can bewritten as

Qwj = �w

khtWk̃rwj

�wap�−

dp

dx�1

nw, . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

where the average (pseudo) non-Newtonian-flow factor �w is de-fined as

�w =�i=1

j

�wiki�hi

�i=1

j

ki�hi

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

Comparing with Eq. A-2 of the stratified system, we get

k̃rwj = krw•

Cwj

Cwt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

Similarly for the oil phase using Eq. A-4,

k̃roj = kro•

Cot − Coj

Cot, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

where k̃rwj and k̃roj are the pseudorelative permeabilities for waterand oil, respectively.

The dimensionless pseudowater saturation at the time of waterbreakthrough in layer j is given by

SDj = hDj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

The dimensionless saturation SDj is defined as

SDj = �S̃w − Swi�/�Sw, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

S̃w = Swi + �Sw · hDj , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27)

�Sw = 1 − Swi − Sor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (28)

Unlike the case of pseudorelative permeabilities for Newtonian fluids,it can be seen from Eqs. 23 and 24 that for non-Newtonian fluids,

k̃rw

krw•

+k̃ro

kro•

� 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

Curves for oil and water pseudorelative permeabilities can be usedin reservoir simulation, collapsing the vertical direction into asingle block.

The pseudofractional-flow curve fw can be calculated frompseudorelative permeabilities by use of the relation

fwj =Qwj

Qwj + Qoj=

1

1 +�o

�w

k̃ro

k̃rw

�wap

�oap�−

dp

dx�1

no−

1

nw

. . . . . . . . . . . (30)

Injectivity Variation. As the displacement proceeds and more ofthe displacing fluid enters into the formation, either the injectionrate, the total pressure drop, or both will change. The variation isexpressed in terms of the injectivity ratio Ir or the resistivity ratio�, which is the inverse of the injectivity ratio and is defined as

�j =1

Ir=

��pt /Q tno�j

��pt /Q tno� in

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (31)

The expressions for the resistivity ratio � are derived in Ap-pendix B and given below.

Before water breakthrough in the first layer,

� = 1 − �� fw1

�hD1− Q t

nw−no

��wap

Wkrw• �nw

� �oap

Wkro• Cot

�no �j=1

N � fwj

Cwj�nw

�� fwj

�hDj��

. . . . . . . . . . . . . . . . . . . . . . . . . . (32)

At time of water breakthrough in layer j,

�j = �j Q tnw−no

� �wap

Wkrw• �nw

� �oap

Wkro• Cot

�no �i=j

N � fwi

Cwi�nw

�� fwi

�hDi�. . . . . . . . (33)

It is to be noticed from Eqs. 32, 33, and 14 that for the constant-injection-rate case, the calculation of the injectivity (resistivity)ratio is straightforward. In this case, the value of fwj needs to becalculated only once for each j (j�1, 2, . . . N). The resistivity ratiois linear with the dimensionless time � before water breakthroughin the first layer and linear with � between times of water break-through in the successive layers. On the other hand, for the case ofconstant total pressure drop �pt, the total injection rate Qt will varywith time. In this case, as shown by Eq. 14, the fractional flow fwwill be time dependent (i.e., it will depend on the location of thedisplacement front in the different layers). As realized from Eqs.32 and 33, the terms inside the summation involving fwj and �fwj

will change. At any given value of the dimensionless time �, therewill be a specific fractional-flow curve. The same also applies fortimes after water breakthrough in the first layer. An iterative pro-cedure must be used to estimate both Qt and fwj simultaneously atdifferent dimensionless times � before water breakthrough in thefirst layer and at the dimensionless times of water breakthrough inthe successive layers (j�1, 2, . . . , N). At the time of water break-through in Layer j, only values of fwi for i=j, j+1, . . . . . . , N−1need to be calculated (fwN�1). It is clear from the above discus-sion that for the case of constant total pressure drop, the resistivityratio is not linear with the dimensionless time �, neither beforewater breakthrough in the first layer nor between times of waterbreakthrough in the successive layers.

Computational ProcedureThe system parameters that are needed to perform the computa-tions include the number of layers N; the values of absolute hori-zontal permeability ki and thickness �hi for each layer; the poros-ity �; the rheological-model parameters Ho, no, Hw, and nw; andthe relative permeability endpoint characteristics Swi, Sor, krw

• , andkro

• . The displacement is specified either at constant injection rateQt or at constant total pressure drop �pt.


The layers are arranged in order of decreasing permeability,and the following terms are calculated for each layer:

• Oil and water non-Newtonian-flow coefficients �oj and �wj

are determined by use of Eq. 12.• For dimensionless thickness �hDj and dimensionless cumu-

lative thickness hDj, Eqs. 19 and 20 are used.• For non-Newtonian formation capacity Cwj and Coj, Eqs. A-3

and A-5 are used.• For pseudorelative permeabilities k̃rwj and k̃roj, Eqs. 23 and

24 are used; for dimensionless pseudowater saturation SDj, Eq. 25is used.

These values are used both for cases of constant injection rateand for cases of constant total pressure drop.

Constant Injection Rate Qt. The following procedure is performed:Calculate fwj for each layer (j�1, 2, . . . , N−1) from Eq. 15

using the following Newton-Raphson procedure:

g�fwj� = fwj +�Cot − Coj�

�Cwj� Qt �wap

Wkrw• Cwj

�nw

no− 1

fwj

nw

no − 1 = 0. . . . . . (34)

g��fwj� = 1 +nw

no

�Cot − Coj�

�Cwj� Qt �wap

Wkrw• Cwj

�nw

no−1

fwj wj

nw

no− 1

. . . . . . . . (35)

f wjk+1 = f wj

k −g�f wj

k �

g��f wjk �

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36)

The iteration is continued until Eq. 34 is satisfied within a speci-fied tolerance. Inspection of Eq. 35 shows that g� is always posi-tive (>1). This guarantees the convergence of the Newton-Raphsoniteration scheme because the function g does not have extremepoints. Furthermore, the procedure has a quadratic convergence.

Before Water Breakthrough in the First Layer. During thistime, R=� and fw�0. The following calculations are performed toobtain the total pressure drop �pt.

For X1�0.1, 0.2, . . . . . . . , 1.0, calculate• The dimensionless time � from Eq. B-6.• The total pressure drop �pt from Eq. B-7.After Water Breakthrough in the First Layer. At times of

water breakthrough in the successive layers j (j�1, 2, . . . , N),• The values of fwj, SDj, k̃rwj, and k̃roj are already calculated.• The dimensionless time of breakthrough in layer j is esti-

mated from Eq. 17.• The fractional oil recovery Rj is obtained from Eq. 16.• The total pressure drop �pt is calculated from Eq. B-9.

Constant Total Pressure Drop �pt. The same procedure as withconstant injection rate is followed, with the following modifications:

• The value from the previous time is used as an initial guessfor Qt at the start of the new time. At the first time, we choose Qtin

calculated from Eq. B-10.• The values of fwj are calculated for j�1, 2, . . . , N–1 before

water breakthrough in the first layer and for j, j+1, . . . , N–1 attimes of water breakthrough in layer j using the iterative procedureoutlined by Eqs. 34 through 36 using the assumed value of Qt.After all values of fwj are obtained, Qt is calculated from Eq. B-7before water breakthrough in the first layer and from Eq. B-9 at thetime of water breakthrough in layer j. Eq. B-9 is explicit in Qt andcan be solved directly. An iteration procedure is needed to solveEq. B-7 for Qt. Using a Newton-Raphson procedure, Eq. B-7 canbe written in the following form with y=Qt

no:

g�y� = ay + bynw

no − �pt = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . (37)

Inspection of Eq. 37 indicates that it does not have any maxima orminima (g��0), and, hence, the iteration procedure converges qua-dratically. After solving for y, the value of Qt is obtained (�y1/no).

A computer program was written to compute the performanceof the stratified system for a given set of model parameters.

Results and DiscussionThe developed method was applied to a hypothetical stratifiedreservoir of 20 layers with the permeability generated from a log-normal distribution with VDP�0.5. The porosity and endpointrelative permeabilities are assumed the same for all layers.

The log-normal distribution of permeability is given by

P�k� = 0.5 + 0.5 erf� ln�k/km�

�2k�, . . . . . . . . . . . . . . . . . . . . . . . (38)

where P(k) is taken as the relative cumulative thickness (fractionof the total thickness) with permeability less than k.

Noting that P(kj)�1−hDj, Eq. 38 may be rearranged as

k = km exp��2k erf −1�1 − 2hDj��. . . . . . . . . . . . . . . . . . . . . (39)

To investigate the effect of the different parameters that affectthe performances, 19 different combinations of the non-Newtonianflow-behavior indices nw and no, the Dykstra-Parsons variationcoefficient VDP, the total flow rate Qt, and the consistency indicesHw and Ho (as shown in Table 1) are considered. Values of theapparent mobility ratio for the different cases as given by Eq. A-8are calculated and listed in Table 1. The first case representing theNewtonian fluids is used for comparison.

Effect of Non-Newtonian Flow-Behavior Indices. The basicequations of the model are Eq. 14 for the water cut fw and Eq. 16for the fractional oil recovery R, with the formation capacity forwater and oil given by Eqs. A-3 and A-5, respectively. All theseequations include the ratio nw/no explicitly except Eq. 16, wherethe ratio is included implicitly in the terms fw and �. It is clear fromthese equations that for nw/no�1 (i.e., for Newtonian fluids ornon-Newtonian fluids with the value of the flow-behavior indicesfor the displacing and displaced fluids), the model equations re-duce to those for the case of displacement by Newtonian fluids.This case is well documented in the literature by Hiatt (1958),Hearn (1971), Warren and Cosgrove (1964), and El-Khatib (1999).These conditions are represented by Case 1 (Table 1) and areincluded for comparison.

To study the effect of the flow-behavior indices, Cases 1through 5 are considered. The reference Case 1 is for Newtonianfluids (nw/no�1). Cases 2 and 3 are for a Newtonian fluid (nw�1)displacing a non-Newtonian fluid (no�0.8, 1.2), while Cases 4


and 5 are for a non-Newtonian fluid (nw�0.8, 1.2) displacing aNewtonian fluid (no�1).

Fig. 2 shows the fractional oil recovery R as a function ofdimensionless time � for the five cases. For the case of a Newto-nian fluid displacing a non-Newtonian fluid, it is seen that the oilrecovery is higher for no�0.8 and lower for no�1.2 as comparedwith the Newtonian-fluids case. For the case of a non-Newtonianfluid (water) displacing a Newtonian fluid (oil), the oil recovery ishigher for nw�1.2 and lower for nw�0.8 as compared with theNewtonian-fluids case. It is clear that the recovery is high whennw>no and low when nw<no, as compared with the Newtonian-fluids case (nw/no�1).

Fig. 3 shows the water cut fw vs. the fractional oil recovery R.The results indicate delayed water breakthrough with lower watercut for nw>no and earlier water breakthrough with higher water cutfor nw<no as compared with the Newtonian-fluids case nw/no�1.This behavior can be explained by investigating Eq. 15, which canbe written as

fwj +�Cot − Coj�

�Cwj

� Qt �wap

Wkrw• Cwj

�nw

no

� Qt �wap

Wkrw• Cwj

� fwj

nw

no = 1. . . . . . . . . . . . . . . . . . . (40)

It is clear that the coefficient of the second term on the left side ofthe equation increases with the increase of nw/no. Hence, fw mustdecrease to keep the sum of the two terms on the left side of theequation constant at the value of unity. The decrease in fw is afavorable indication of reservoir performance.

Fig. 4 is a plot of oil and water pseudorelative permeabilitycurves. The results indicate that the water pseudorelative perme-ability is influenced only by nw and not by no, while the oil pseu-dorelative permeability is influenced only by no and not by nw. Thewater pseudorelative permeability decreases as nw increases, whilethe oil pseudorelative permeability increases as no decreases. Thiswill result in a decrease in fractional flow for increasing nw/no, ascan be realized from Eq. 30. This behavior can be seen in Fig. 5,which shows the fractional-flow curves vs. the dimensionless(pseudo) water saturation.

Fig. 6 is a plot of the total-pressure-drop ratio �P/�Pin. It canbe seen that the ratio drops from unity at the start of displacementto constant values at the time of breakthrough in the last layer. Theratio is, however, higher for nw>no and lower for nw<no as com-pared to the Newtonian-fluids case.

It is to be noted that at the time of breakthrough in the last layer,the pressure-drop ratio �Pfin/�Pin is not equal to the inverse of themodified mobility ratio, as would be the case for Newtonian fluids.As can be derived from Eqs. B-9 and B-10, this ratio is given by

�pt fin

�pt in=� Qt �wap

Wkrw• Cwt

�nw

� Qt �oap

Wkro• Cot

�no. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (41)

This ratio is equal to 1/� only if nw/no�1.

Effect of Apparent Mobility Ratio. The apparent mobility ratio �as given by Eq. A-8 is calculated for all cases and listed in Table 1.

Fig. 7 shows the performance in terms of water cut fw vs.fractional oil recovery R for Cases 1, 6, 7, 16, and 17. As statedpreviously, the fractional recovery increases and the water cutdecreases for nw>no. It may appear that the ratio nw/no controls the

Fig. 2—Effect of flow-behavior indices on performance (R vs. �). Fig. 3—Effect of flow-behavior indices on performance (fw vs. R).

Fig. 4—Effect of flow-behavior indices on pseudorelative per-meability curves.


performance. Three different cases with the same value of the rationw/no�1.5 but with different individual values for each nw and no

(1.2/0.8, 1.35/0.9, and 1.8/1.2) were investigated. As shown inFig. 7, the performance of the three cases is quite different. Thethree cases have largely different apparent mobility ratios (3.59,2.89, and 1.87, respectively). It is clear that the recovery increasesand the water cut decreases as the apparent mobility ratio de-creases. The results of Cases 1, 6, and 7 indicate that the effect ofthe apparent mobility ratio is secondary compared to the effect ofthe flow-behavior indices. Although the apparent mobility ratio forthe 0.8/1.2 case is 1.99 compared to the value of 2.5 for the 1/1case, the performance deteriorates because nw<no. On the otherhand, the performance improves for the 1.2/0.8 case over that ofthe 1/1 case despite the increase in the apparent mobility ratio from2.5 to 3.59. This is also attributed to nw being greater than no

despite the higher apparent mobility ratio.It can be seen from Eq. 15 or Eq. 40 that at the same value of

nw/no, a decrease in the value of � will cause the coefficient of thesecond term on the left side of the equation to increase, thusresulting in a decrease in fw to keep the sum of the two terms onthe left side of the equation constant at a value of unity.

As indicated by Eq. A-8 and Eq. 10, the value of the apparentmobility ratio is determined by the flow-behavior indices nw and no

and the consistency indices Hw and Ho. The above cases investi-gated the effect of apparent mobility ratio controlled by the non-Newtonian flow-behavior indices. The consistency indices weretaken as 1 and 5 for water and oil, respectively. For polymersolutions, the flow-behavior index as reported by many investiga-tors (Teeuw and Hesselink 1980; Cannella et al. 1988) is usuallyless than 1 (usually 0.3 to 0.7 and decreasing with increased poly-mer concentration). This will result in unfavorable displacementconditions (nw<no). The consistency factor for polymer solutions,however, is much higher than that of water (10 to 500 and increas-ing with increased polymer concentration), which will result in alower apparent mobility ratio and favorable displacement conditions.Gleasure and Phillips (1990) reported that some synthetic polymersolutions such as polyethylene oxide (PEO) and partially hydrolyzedpolyacracrylamides (PHPA) have a shear thickening behavior in coreflow (nw>1). This, combined with the high value of the consistencyindex, will result in very favorable displacement conditions.

The results reported by Gleasure (1990) are in agreement withthe results of the developed model. Two sets of polymers (XANand PEO) at polymer concentrations of 500, 1,500, and 2,500 ppmwere used to displace oil in unconsolidated porous media. Theconsistency index H increases while the flow-behavior index nw

decreases as polymer concentration increases. The increase in theconsistency index H, however, is much more than the decrease inthe flow-behavior index. This behavior causes a decrease in theapparent mobility ratio and, hence, an increase in the fractional oilrecovery with the increase of polymer concentrations for both setsof polymers. The consistency index H for the PEO polymers ismuch lower than that for the XAN polymer at all concentrations.This would result in a higher apparent mobility ratio and lowerfractional oil recovery for the PEO polymers as compared with theXAN polymers. However, because of the high values of the flow-behavior index, the fractional oil recovery for PEO polymers doesnot differ too much from that for the XAN polymers. These resultsare in agreement with the results of the developed model.

Effect of Reservoir Heterogeneity. The heterogeneity of the res-ervoir is described by means of the standard deviation of the per-meability distribution k or the Dykstra-Parsons coefficient of varia-tion VDP, which is related to k by the relation k=ln [1/(1 – VDP)].A value of VDP�0 represents a homogeneous reservoir (constantk), while a value of 1 represents a totally heterogeneous reservoir.

Fig. 6—Effect of flow-behavior indices on pressure-drop ratio.

Fig. 7—Effect of apparent mobility ratio on performance (fw vs. R).

Fig. 5—Effect of flow-behavior indices on fractional-flowcurves.


To investigate the effect of reservoir heterogeneity, permeabil-ity distributions were obtained from a log-normal distribution withVDP values of 0.25, 0.5, and 0.75 and nw/no values of 0.8/1, 1/1,and 1/0.8 (Cases 1, 2, 4, and 10 through 15). The results for thesecases are shown in Fig. 8 in terms of water cut fw vs. fractional oilrecovery R. As can be expected, heterogeneity (higher VDP) tendsto have a negative effect on the performance. For any pair of fluidswith a fixed value of nw/no, the value of fw increases and the valueof R decreases at higher values of VDP. This effect is more notice-able, however, when nw/no>1.

It also can be seen that for any value of VDP , water cut fwdecreases and the fractional oil recovery R increases for nw/no>1and vice versa, as was noticed before. This effect, however, ismore apparent in the less-heterogeneous reservoirs (VDP�0.25)than in the more-heterogeneous reservoirs (VDP�0.75).

Effect of Total Injection Rate. To investigate the effect of thetotal injection rate on the performance, six cases were considered.First, Cases 8, 2, and 9 (with values for Qt of 50, 100, and 200,respectively, and an nw/no value of 0.8/1), and then Cases 18, 4,and 19 (with values for Qt of 50, 100, and 200, respectively, andan nw/no value of 1/0.8) were investigated. All other parameters arethe same for all cases. Fig. 9 shows the fractional oil recovery R asa function of dimensionless time � for the three rates. It is clear thatthe increase in the total injection rate results in an increase in thefractional oil recovery R when nw>no and a decrease in the frac-tional oil recovery when nw<no. Again, this behavior can be ex-plained by Eq. 15. The total injection rate Qt is raised to the powerof nw/no−1. For nw/no>1, this power is positive, so the coefficientof the second term on the left side of the equation increases as Qt

increases. This causes fw to decrease to keep the sum at a value of1. On the other hand, for nw/no<1, Qt is raised to a negative power,and the coefficient of the second term decreases as Qt increases,resulting in an increase of fw.

Displacement at Constant Pressure Drop. The procedure out-lined before for injection at constant total pressure drop was ap-plied for nw/no values of 0.8/1.2 and 1.2/0.8. Fractional-flowcurves were calculated at values of dimensionless distance of thedisplacement front in the first (most-permeable) layer X1 of 0.2,0.4, 0.6, 0.8, and 1 and at the times of water breakthrough in thesuccessive 20 layers of the stratified system. Fig. 10 shows thefractional-flow curves as a function of the pseudo (average) watersaturation. It is seen that fw increases as X1 increases and as waterbreaks through the successive layers. This behavior is more effec-tive for the nw/no case of 0.8/1.2 than for the 1.2/0.8 case. For

Newtonian fluids, the fw/Sw curve remains the same during thedisplacement process.

Fig. 11 shows the fractional oil recovery vs. the dimensionlesstime, and Fig. 12 shows the water cut vs. the fractional oil recoveryfor displacement at constant total pressure drop in comparison withthose at a constant injection rate. The figures show a decrease infractional oil recovery and an increase in water cut for displace-ment at constant total pressure drop. This effect is more noticeablewhen nw<no than when nw>no. This behavior may be explained bythe large increase in the injection rate for nw<no as the displace-ment proceeds, as seen in Fig. 13.

Conclusions1. A mathematical model is developed for power-law non-

Newtonian-fluid displacement in communicating stratified res-ervoirs. Equations are derived for fractional oil recovery, watercut, injectivity ratio, and pseudorelative permeabilities. Cases ofdisplacement at constant injection rate and at constant total pres-

Fig. 8—Effect of reservoir heterogeneity on performance (fw vs. R).

Fig. 10—Fractional-flow curves for constant-pressure-dropdisplacement.

Fig. 9—Effect of total injection rate on performance (R vs. �).


sure drop are considered. An iterative procedure is applied tosolve for the implicit equations obtained.

2. The performance is controlled mainly by the relative values ofthe rheological-model indices nw and n0. The performance im-proves (higher recovery and lower water cut) over the Newto-nian case for nw>no, and the opposite occurs for nw<no. For thesame nw/no ratio, the performance is controlled mainly by theapparent mobility ratio. The recovery increases and the watercut decreases as the apparent mobility ratio decreases.

3. The recovery increases and the water cut decreases as the totalinjection rate is increased for nw>no; the opposite is true for nw<no.

4. Increased reservoir heterogeneity (high VDP) results in lower oilrecovery and higher water cut. This behavior becomes moreapparent for nw>no.

5. Fractional oil recovery is lower and water cut is higher fordisplacement at constant pressure drop as compared to those atconstant injection rate, with other parameters being the same.This effect is more noticeable for nw<no.

6. The displacement at constant pressure drop does not have asingle fw/Sw curve. The values are time dependent, and fw in-creases as displacement progresses. This effect is more notice-able for nw<no.

7. The best performance may be obtained if the displacing fluid isshear thickening (nw>1) and the displaced fluid is Newtonian orshear thinning (no1).

NomenclatureC � dimensionless formation capacityfw � water cut, dimensionless

hD � fraction of total thicknessht � total formation thickness, ft (m)H � consistency index in power-law modelIr � injectivity ratio, dimensionlessk � absolute horizontal permeability, md (�m2)

km � mean of log-normal permeability distribution, md (�m2)kro

• � oil relative permeability at irreducible water saturation,dimensionless

k̃ro � pseudorelative permeability for oil, dimensionlesskrw

• � water relative permeability at residual oil saturation,dimensionless

k̃rw � pseudorelative permeability for water, dimensionlessL � total length of flow system, ft (m)n � flow-behavior index in power-law model, dimensionlessN � total number of layersQ � flow rate, B/D (m3/s)r � radius of capillary tube, ft (m)R � fractional oil recovery, dimensionless

SD � dimensionless water saturationSor � residual oil saturation, fractionSw � water saturation, fractionS̃w � pseudowater saturation, fractionSwi � initial water saturation, fraction

T � tortuosity, dimensionlessv � average velocity, ft/D (m/s)

VDP � Dykstra-Parsons coefficient, dimensionlessW � width of layers, ft (m)X � dimensionless distance, dimensionless� � non-Newtonian-flow parameter� � resistivity ratio, dimensionless

Fig. 12—Performance (fw vs. R) for constant-pressure-dropdisplacement.

Fig. 13—Flow-rate ratio for constant-pressure-drop displacement.

Fig. 11—Performance (R vs. �) for constant-pressure-dropdisplacement.


�P � pressure drop, psi (kPa)�Sw � displaceable oil saturation, fraction

� � porosity, fraction� � apparent-mobility ratio, dimensionless�̇ � shear rate, s−1

� � mobility� � apparent viscosity, cp (Pa·s)

k � standard deviation of log-normal permeability� � dimensionless timeT � shear stress

Subscripts

ap � apparentD � dimensionless

eff � effectivefin � finalin � initialm � meano � oilr � relativet � total

w � water

Superscripts_ � average

ReferencesBird, R.B., Stewart, W.E., and Lightfoot, E.N. 1960. Transport Phenom-

ena, 206. New York City: John Wiley & Sons Inc.Cannella, W.J., Huh, C., and Seright, R.S. 1988. Prediction of Xanthan

Rheology in Porous Media. Paper SPE 18089 presented at the SPEAnnual Technical Conference and Exhibition, Houston, 2–5 October.

Chakrabarty, C., Tortike, W.S., and Farouq Ali, S.M. 1993. Complexitiesin the Analysis of Pressure-Transient Response for Non-NewtonianPower-Law Fluid Flow in Fractal Reservoirs. Paper SPE 26910 pre-sented at the SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania,2–4 November.

Dykstra, H. and Parsons, R.L. 1950. The Prediction of Oil Recovery byWaterflooding. In Secondary Recovery of Oil in the United States,second edition, 160–174. Washington, DC: API.

El-Khatib, N. 1999. Waterflooding Performance of Communicating Strati-fied Reservoirs With Log-Normal Permeability Distribution. SPEREE2 (6): 542–549. SPE-59071-PA.

Gleasure, R.W. 1990. An Experimental Study of Non-Newtonian PolymerRheology Effects on Oil Recovery and Injectivity. SPERE 5 (4): 481–486. SPE-17648-PA.

Gogarty, W.B., Levy, G.L., and Fox, V.G. 1972. Viscoelastic Effects inPolymer Flow Through Porous Media. Paper SPE 4025 presented at theSPE Annual Meeting, San Antonio, Texas, 8–11 October.

Hearn, C.L. 1971. Simulation of Stratified Waterflooding by Pseudo Rela-tive Permeability Curves. JPT 23 (7): 805–813. SPE-2929-PA.

Herschel, W. and Bulkley, R. 1926. Consistency Measurements on Rubber-Benzene Solutions. Koll. Zeit. 39: 291.

Hiatt, N.W. 1958. Injected-Fluid Coverage of Multi-Well Reservoirs WithPermeability Stratification. Drill. and Prod. Prac. 165: 165–194.Washington, DC: API.

Odeh, A.S. and Yang, H.T. 1979. Flow of Non-Newtonian Power-LawFluids Through Porous Media. SPEJ 19 (3): 155–163. SPE-7150-PA.

Salman, M., Baghdikian, S.Y., Handy, L.L., and Yortsos, Y.C. 1990.Modification of Buckley-Leverett and JBN Methods for Power-LawFluids. Paper SPE 20279 available from SPE, Richardson, Texas.

Savins, J.G. 1969. Non-Newtonian Flow Through Porous Media. Ind. Eng.Chem. 61 (10): 18–47.

Stiles, W.E. 1949. Use of Permeability Distribution in Water Flood Cal-culation. Trans., AIME 186: 9–13.

Teeuw, D. and Hesselink, F.T. 1980. Power-Law Flow and HydrodynamicBehaviour of Polymer Solutions in Porous Media. Paper SPE 8982presented at the SPE Oilfield and Geothermal Chemistry Symposium,Stanford, California, 28–30 May.

van Poollen, H.K. and Jargon, J.R. 1969. Steady-State and Unsteady-StateFlow of Non-Newtonian Fluids Through Porous Media. SPEJ 9 (1):80–88; Trans., AIME, 246.

Warren, J.E. and Cosgrove, J.J. 1964. Prediction of Waterflood Behavior ina Stratified System. SPEJ 4 (2): 149–157; Trans., AIME, 231. SPE-581-PA.

Wu, Y.-S. and Pruess, K. 1998. A Numerical Method for Simulating Non-Newtonian Fluid Flow and Displacement in Porous Media. Adv. inWater Res. 21 (5): 351–362.

Wu, Y.-S., Pruess, K., and Witherspoon, P.A. 1992. Flow and Displace-ment of Bingham Non-Newtonian Fluids in Porous Media. SPERE 7(4): 369–376. SPE-20051-PA.

Yi, X. 2004. Model for Displacement of Herschel-Bulkley Non-NewtonianFluid by Newtonian Fluid in Porous Media and Its Application inFracturing Fluid Cleanup. Paper SPE 86491 presented at the SPE In-ternational Symposium and Exhibition on Formation Damage Control,Lafayette, Louisiana, 18–20 February.

Appendix A—Derivation of the Fractional-FlowFormula for Non-Newtonian FluidsFor Layer i in Zone j (i�1, 2, . . . , j), from Eq. 13:

qwi = �wi

ki�hiWkrw•

�wap�−

dp

dx�1

nw, . . . . . . . . . . . . . . . . . . . . . . . (A-1)

where krw• and kro

• are the water and oil relative permeabilities atthe endpoints.The total flow rate of water in Zone j is

Qwj = �i=1

j

qwi =Wkrw

•

�wap�−

dp

dx�1

nwCwj , . . . . . . . . . . . . . . . . . . . (A-2)

where

Cwj = �i=1

j

�wi ki �hi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

Similarly for the oil phase,

Qoj = �i=j+1

N

qoi =Wkro

•

�oap�−

dp

dx�1

no�Cot − Coj�, . . . . . . . . . . . . . . (A-4)

where

Coj = �i=1

j

�oi ki�hi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

and

Cot = CoN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)

From Eqs. A-2 and A-4, we obtain

fwj =Qwj

Qwj + Qoj=

�Cwj

�Cwj + �Cot − Coj��−dp

dx�1

no−

1

nw

, . . . . . . . (A-7)

where � is the apparent-mobility ratio defined as

� =krw

0

kro0

�oap

�wap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-8)

From Eq. A-2, we can write

Qwj = fwjQt =Wkrw

•

�wap�−

dp

dx�1

nwCwj , . . . . . . . . . . . . . . . . . . . . . (A-9)

from which we get

�−dp

dx� = �fwjQt �wap

Wkrw• Cwj

�nw

. . . . . . . . . . . . . . . . . . . . . . . . . . (A-10)


Substituting in Eq. A-7,

fwj =�Cwj

�Cwj + �Cot − Coj��fwjQt�wap

Wkrw• Cwj

�nw

no− 1

, . . . . . . . . . . . . . (A-11)

which is the fractional-flow formula for non-Newtonian fluids. Itis clear that when nw/no�1, Eq. A-11 reduces to the familiar Stilesformula (1949) for communicating stratified reservoirs.

Appendix B—Derivation of theInjectivity-Variation FormulaAs the displacement proceeds and more of the displacing fluidenters into the formation, either the injection rate, the total pressuredrop, or both will change.

From Eq. A-10, we get, for j�1, . . . , N,

�pj = �fwjQt�wap

Wkrw• Cwj

�nw

L�Xj , . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

where

�Xj = Xj − Xj+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2)

and Xj = ��fwj

�hDj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)

Before water breakthrough in the first layer, for Zone 0,X0�1, and

�p0 = � Qt�oap

Wkro• Cot

�no

L�1 − X1�. . . . . . . . . . . . . . . . . . . . . . . . . (B-4)

From Eqs. B-1 and B-4, we obtain

�pt = � Qt�oap

Wkro• Cot

�no

L�1 − X1� + L�Qt�wap

Wkrw• �nw

�j=1

N � fwj

Cwj�nw

�Xj .

. . . . . . . . . . . . . . . . . . . . . . . . . (B-5)

Substituting for the values of Xj from Eq. B-3 and noting that

X1 = ��fw1

�hD1= �

fw1

�hD1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-6)

we obtain

�pt = L� Qt �oap

Wkro• Cot

�no�1 − �fw1

�hD1�

+ L��Qt�wap

Wkrw• �nw

�j=1

N � fwj

Cwj�nw

�� fwj

�hDj�. . . . . . . . . . . (B-7)

Eq. B-7 gives the change in total pressure drop �pt for constanttotal injection rate Qt up to the time of water breakthrough in thefirst layer �1, where

�1 =1

�fw1

�hD1

=1

fw1

�hD1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-8)

At the time of water breakthrough in Layer j (j�1, 2, . . . , N),from Eqs. B-1 through B-3 we obtain

�pt = L�j�Qt �wap

Wkrw• �nw

�i=j

N � fwi

Cwi�nw

�� fwi

�hDi�, . . . . . . . . . . (B-9)

where �j is given by Eq. 17.Initially, from Eq. B-7 at ��0,

�pt in = L�Qt in �oap

Wkro• Cot

�no

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-10)

Dividing Eqs. B-7 and B-9 by Eq. B-10 and rearranging, weobtain expressions for the inverse injectivity ratio 1/Ir or the re-sistivity ratio �.

Before water breakthrough in the first layer,

� =1

Ir=

�pt /Qtno

��pt /Qtno� in

= 1 − �� fw1

�hD1− Qt

nw−no

� �wap

Wkrw• �nw

� �oap

Wkro• Cot

�no �j=1

N � fwj

Cwj�nw

�� fwj

�hDj�� ,

. . . . . . . . . . . . . . . . . . . . . . . (B-11)

and at the time of water breakthrough in Layer j,

�j = �j

�Qtj�wap

Wkrw• �nw

�Qt in �oap

Wkro• Cot

�no �i=j

N � fwi

Cwi�nw

�� fwi

�hDi�. . . . . . . . . . . (B-12)

SI Metric Conversion Factorsbbl × 1.589 873 E–01 � m3

cp × 1.0* E–03 � Pa·sft × 3.048* E–01 � m

psi × 6.894 757 E+00 � kPa

*Conversion factor is exact.

Noaman El-Khatib is a professor of petroleum engineering atSudan U. for Science and Technology in Khartoum, Sudan.e-mail: [email protected]. Previously, he was a professorat King Saud U. in Riyadh, Saudi Arabia, where he had been onthe faculty for 25 years. He also taught at Cairo U. and the U.of Pittsburgh. His research interests include well-test analysis,reservoir simulation, enhanced oil recovery, reservoir charac-terization, and geostatistics. El-Khatib holds a BS degree fromCairo U. and MS and PhD degrees from the U. of Pittsburgh, allin petroleum engineering.


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