spe-123971-pa (streamline-based simulation of non-newtonian polymer flooding)

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December 2010 SPE Journal 901

Streamline-Based Simulationof Non-Newtonian Polymer Flooding

Abdulkareem M. AlSofi,SPE, and Martin J. Blunt,SPE, Imperial College London

Copyright 2010 Society of Petroleum Engineers

This paper (SPE 123971) was accepted for presentation at the SPE Annual Technical

Conference and Exhibition, New Orleans, 47 October 2009, and revised for publication.Original manuscript received for review 7 July 2009. Revised manuscript received forreview 23 February 2010. Paper peer approved 8 March 2010.

Summary

Current commercial simulators for polymer flooding often makephysical assumptions that are not consistent with available experi-mental data and pore-scale modeling predictions. This may lead tooverly optimistic recovery predictions for shear-thinning polymers,while the potential advantages of reducing flow rate or using shear-thickening agents are overlooked.

We develop a streamline-based simulator that overcomes theselimitations and demonstrate how it can be used to design polymer-flooding projects. The simulator implements an iterative approachto solve the pressure field because the pressure depends on theaqueous-phase viscosity, which, in turn for non-Newtonian fluids,depends on shear stress and, hence, the pressure gradients. Thisis in contrast to the common approach in commercial simulatorswhere this viscosity/pressure interdependence is ignored, leading

to overestimation of sweep efficiency. Furthermore, in the simu-lator, non-Newtonian viscosities are defined to be cell-centeredwhile current simulators use a face-centered approach, therebyoverpredicting viscosities and the stability of the displacing fronts.In addition, we use a physically based rheological model wherenon-Newtonian viscosities in two-phase flow are taken at actualeffective stresses instead of single-phase equivalents.

To validate the simulator, we construct 1D analytical solutionsfor waterflooding with a non-Newtonian fluid. We then compareour results to those from commercial simulators. We discuss thesignificance of current assumptions to demonstrate the effect ofnon-Newtonian behavior on sweep efficiency and recovery.

Introduction

Polymer flooding is one of the most mature enhanced-oil-recovery(EOR) techniques; it relies on viscosifying the injected water tostabilize the displacing fronts and reduce channeling in heteroge-neous media (Lake 1989; Littman 1988; Sorbie 1991). The mainpolymers used for this purpose, polysaccharide and polyacryl-amide, are shear-thinning (Lake 1989). Although such thinningbehavior is desirable from an injectivity standpoint, it is undesir-able in terms of sweep and recovery.

However, there have been few studies that have discussed theeffect of shear-thinning behavior on sweep efficiency and recovery.It has been suggested that shear thinning can be beneficialwhichis the opposite of what we demonstrate in this paper. For instance,Martel et al. (1998) state, The use of polymer solutions showingshear-thinning behavior can be greatly beneficial for sweep effi-ciency and mobility control.

To explore the effect of non-Newtonian rheology on field-scalepolymer-flood performance, we extend an in-house streamlinesimulator based on the work of Batycky et al. (1997). The codeovercomes some of the limitations and nonphysical assumptionsin current simulators. This paper extends the preliminary workpresented in AlSofi et al. (2009). We will also compare our resultswith those from a commercial streamline-based polymer modeldescribed by Thiele et al. (2010).

Non-Newtonian Fluids and Their Effects on

Sweep Efficiency

Most polymers used for EOR are shear thinning; that is, abovesome critical shear rate (i.e., velocity), they exhibit an apparentviscosity that falls progressively with higher shear rates/velocities(Sorbie 1991; Tanner 2000). Laboratory data (Cannellaet al. 1988;Seright et al. 2009) indicate that shear thinning has significanteffect for the range of Darcy velocities occurring in typical poly-mer-flooding applications (Fig. 1).

Shear thinning can reduce sweep and recovery for two reasons(Fig. 2):increasing the velocity contrast (Jones 1980) and induc-ing instability (Lee and Claridge 1968). The first factor can beregarded as an additional preferential penetration a shear-thinningfluid has in comparison to a Newtonian fluid (AlSofi et al. 2009),which could consequently result in additional channeling. On

the other hand, using a shear-thickening chemical could enhancesweep (Jones 1980). The second factor, inducing instability, iswhen the local shear rate is sufficiently high that the aqueous-phase viscosity drops below the stability criterion, leading to anunstable displacement with fingering and bypassing. However,the example shown in Fig. 2 ignores connate water and crossflow,which means that the effect of shear thinning may not be so strongin field applications.

Pore-Network Modeling. Shah and Yortsos (1995) and Lopezand Blunt (2004) showed that, for single-phase flow, the subset ofthe pore space contributing to flow is smaller for a shear-thinningfluid. This suggests that thinning behavior would result in greaterbypassing and poorer sweep.

Laboratory Flooding. Lee and Claridge (1968) performed flood-ing experiments in a Hele-Shaw cell. Their main conclusion wasthat areal sweep at breakthrough is poorer for a polymer solutionthan for a Newtonian fluid with a comparable viscosity. In a secondstudy, Gleasure (1990) conducted core-scale polymer floods at dif-ferent flow rates. Their main conclusion was that non-Newtonianbehavior does not affect the oil production; however, this is prob-ably because the variations in pressure gradient were insufficientlylarge to induce a significant variation in aqueous-phase viscosity.

Simulation Studies.Many chemical simulators have been developedin the last 40 years with the capability of simulating shear-thinningrheology (Bondor et al. 1972; Patton et al. 1971; Scott et al. 1987;Wang et al. 1981; Wu and Pruess 1998). Nevertheless, only Wang et al.

(1981) investigated shear-thinning effects, simulating polymer flood-ing in a homogeneous reservoir. They concluded that shear-thinningbehavior could reduce the lifetime of a project significantly. On thebasis of their results, a thinning flood would underrecover approxi-mately 11 and 7% of the original oil in place (OOIP) compared to aNewtonian flood for a line drive and five-spot, respectively.

Streamline Simulation of Polymer Flooding

The methodology implemented to extend a Newtonian-waterfloodcode to handle polymer flooding can be subdivided into threemajor steps: solving for polymer mass balance, modeling thepolymer viscosity, and implementing an iterative algorithm tosolve the pressure field.

Solving for Polymer Mass Balance. The advantage of stream-line-based over grid-based simulation is its ability to transform a



902 December 2010 SPE Journal

3D transport problem into a number of 1D independent transportproblems, thus overcoming grid-based timestep limitations. This

transformation is achieved by means of tracing a set of streamlinesbased on the pressure solution along each of which the componentmass balances are solved (Datta-Gupta and King 2007). The equa-tion for water saturation is

+

=

S

t

fw w

0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where S is saturation, f is the fractional flow, t is time, is thetime-of-flight coordinate, and the subscript wrepresents the waterphase. Eq. 1 is discretized along the streamlines using upstreamweighting as in Obi and Blunt (2006):

S S t f fw

i n

w

i n

w

i n

w

i n, , , ,+ = ( )1 1d

, . . . . . . . . . . . . . . . . . . . . . (2)

where i is a cell identifier along the discretized streamline and nis the time level.

Similarly, the polymer mass balance with adsorption is (Sorbie1991) given by

( ) +

( ) +

( )=

tS C

tC f C

w p a w p

0, . . . . . . . . . . . . . . . . . . . (3)

where Cpis the polymer concentration normalized by the injectionconcentration and Cais the normalized sorbed concentration, whichis given by a linear isotherm, C A S C

a W p= ( )with some constantA.

Eq. 3 is discretized along streamlines as

CA S C t C f C

p

i n w

i n

p

i n

p

i n

w

i n

p,

, , , ,

+ = +( )

11 d

ii n

w

i n

w

i n

f

A S

+

( )+( )

1 1

11

, ,

,.

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

Modeling Polymer Viscosity. We first define the concentrationdependence of the unsheared aqueous-phase viscosity:

zero C P Cp w p( = ( , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where wis the pure-water viscosity and Pis the viscosity multi-

plier. Any function can be chosen; in this work, we take a lineardependence on Cp:

P C Cp w p( = ( ) + zero 1 1, . . . . . . . . . . . . . . . . . . . . . . . (6)

where *zero represents the unsheared viscosity at the injectionconcentration.

Second, to account for non-Newtonian behavior, we define athinning multiplierM, representing dimensionless viscosity:

Mw

w

=

aq

zero

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where aqis the aqueous-phase viscosity at a given shear rate andpolymer concentration.From Eqs. 5 and 7, we define a multiplier that takes into

account both viscosifying and non-Newtonian effects such thataqueous-phase viscosity is written as

aq= = +( )w w PM M 1 . . . . . . . . . . . . . . . . . . . . . . . (8)

Note that this approach of defining two multipliers is similar tothat used in commercial grid-based (Schlumberger 2007) andstreamline-based simulators (Thiele et al. 2010).

In-Situ Rheology. We define the rheology (i.e., the shear mul-tiplier) in terms of the shear stress . The shear multiplierrep-resenting bulk rheology measured in a viscometer or the in-siturheology measured in a corefloodcan be defined either through

a power-law model (for shear-thinning fluids),

M= 1 1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

where is the consistency index and is the shear exponent, orthrough a doublet Ellis model (Sorbie 1991) for shear-thinning,shear-thickening, and viscoelastic fluids:

M M MTH VE m TH

VE

o

TH= + = + ( )

+

1

1

11

1

,

++ ( )

m VE

VE,

11

1

, . . . . . . . . . (10)

ApparentViscosity,mPas

Darcy Velocity, m/d

Fig. 1Apparent viscosity as a function of Darcy velocity fora 1,200 ppm polysaccharide in a Berea core (Cannella et al.1988), green, and a 1,000 ppm polysaccharide in a Berea core(Seright et al. 2009), blue. Shear thinning does occur for therange of Darcy velocities typically experienced during polymerflooding.

(a) (b)

Newtonian Flood

Shear-Thinning Flood f(position, time)

-injector

stability

criteria

Fig. 2Shear-thinning sweep impairment caused by (a) increasing the velocity contrast and (b) inducing instability. (a) Where

the velocity is higher in high-permeability strata, a shear-thinning fluid will exhibit lower viscosities, leading to more channeling.(b) Shear thinning may locally lead to an unstable displacement.




where MTH accounts for shear thinning or shear thickening andMVEaccounts for polymer viscoelasticity, if any. THand VEarethe non-Newtonian and viscoelastic exponents, respectively; m,THand m,VEare the midpoint shear stresses. VEis the infinite-shearviscosity for a viscoelastic polymer, which is an empirical valueset to fit rheology measured in situ.

In-Situ Stresses. The value of shear stress, , used in Eqs. 9and 10 is computed as follows. For each gridblock, we assume thatthe stress is given by an expression for an equivalent cylindricaltube (Bird et al. 2007):

ijkijk

ijk

R

P= 4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

where ijkis the grid identifier and Pis the cell-centered pressuregradient that would give the computed cell-centered water velocity.Ris the hydraulic radius. Ignoring two-phase effects, the radius isgiven by the bundle-of-tubes model:

Rk

ijk

ijk

ijk

=8

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

where k is permeability in the flow direction obtained through anellipsoidal representation of anisotropic permeability.

We compute the shear stress, and hence the viscosity, for each

gridblock (not grid face). The final expression is

ijk ijk ijk ijk

ijk

ijk

ijkK

kP=

2, . . . . . . . . . . . . . . . . . . (13)

where: X accounts for the shift in rheology in a porous medium

compared to that measured in a viscometer (Littman 1988; Sorbie1991).

, the tortousity factor, represents the ratio of the grid tor-tousity to the core tortousity for which the in-situ rheology wasmeasured.

,the convention factor, accounts for the way effective shear

was defined.

Z, the two-phase factor, accounts for two-phase effects onthe effective radius.

In this work, we will assumeX = K = = 1. We will assumethat Z= 1, except for the field-scale simulations in the final sec-tion.

Note that single-phase experimental coreflood data often plotapparent viscosity as a function of Darcy velocity, q. To use suchdata, we convert Darcy velocity to shear stress:

=

q

k2 core core. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Implementing an Iterative Pressure/Viscosity Solver. Oursimulator uses an implicit-pressure/explicit-saturation (IMPES)approach (i.e., solving pressure implicitly and masses explicitly).An iterative approach is required for the pressure solution because,for non-Newtonian flow, the viscosity field is a function of flowrate, thus pressure gradient, which, in turn, is a function of viscos-ity. Fig. 3illustrates the polymer streamline-simulator algorithm.

Validating the Code

The simulator was validated in two steps: Newtonian and non-Newtonian validation.

Newtonian Validation. The Newtonian part of the code was thor-oughly validated by comparison to 1D analytical solutions (Patton

et al. 1971; Pope 1980), 2D streamline-simulation results fromThiele et al. (2010), and 3D results obtained using a commercialgrid-based code.

1D Validation Against Analytical Solutions. The input param-eters are shown in Table 1.

First, horizontal flooding with and without adsorption wassimulated. The adsorption constant Awas set to 0.2. Second, tovalidate the success of the operator-splitting technique in handlingpolymer flooding with gravity, up- and downdip flooding weresimulated. For the vertical case, the density difference betweenpolymer solution and oil was taken as 500 kg/m3and the perme-ability was set to be 100 darcy. Fig. 4 shows a successful com-parison of analytical and numerical results for horizontal injectionwith and without adsorption and up- and downdip injection withoutadsorption.

Fig. 3The streamline-simulation algorithm based on Baker et al. (2002) modified to model polymer flooding. Bordered with red

are modifications to account for polymer flooding. The right-hand side handles non-Newtonian effects, and the left-hand sidehandles the polymers viscosifying effects.




2D Validation Against Commercial Simulators. The codewas validated in two dimensions by comparison with results in aheterogeneous domain from Thiele et al. (2010). The oil-produc-tion predictions shown in Fig. 5show good agreement with thoseobtained using a commercial streamline-based code. These results,in turn, were virtually identical to those obtained using a conven-tional commercial grid-based simulator. Further details of the testcases can be found in Thiele et al. (2010).

3D Validation Against a Commercial Finite-Difference Simu-lator. Finally, the Newtonian part of the code was validated in 3Dby comparison to a commercial grid-based finite-difference codefor a quarter-five-spot flood in a heterogeneous reservoir. The res-

ervoir, which is 200

200 m and 15 m thick, was represented bya 50505 grid model. The reservoir was assumed to be layeredwith an isotropic permeability field. The permeability increaseddownward and was assumed to be 10, 50, 200, 500, and 1000md for each layer, from top to bottom. The fluid properties weredefined as in the 1D case (refer to Table 1). As shown in Fig. 6,forboth secondary and tertiary flooding, the production and pressureresponses are similar to those of an industry-standard grid-basedcode. Note that, in this and all subsequent polymer-flooding cases,the grid-based code runs fully implicitly.

Non-Newtonian Flooding Validation. The non-Newtonian codewas validated by comparison to novel 1D semianalytical solutions.

Semianalytical Solutions for 1D Flooding With Non-Newto-nian Fluids. Wu et al. (1991) demonstrated that the Buckley-Lev-

erett solution for 1D displacement of two Newtonian fluids can beextended to displacements involving non-Newtonian flow through

indirectly defining non-Newtonian viscosity in terms of saturation.From this, semianalytical solutions can be derived. For simplicity,we assume the connate water has the same composition as injectedwater (i.e., both are non-Newtonian). Thus, the aqueous-phaseviscosity is a function of only shear stress. Then, to construct the

fractional flow curve for flooding with a non-Newtonian fluid,shear stresses at each water saturation need to be determined. Forconstant flow rate Qt, the pressure gradient as a function of watersaturation is

( )= + ( )

P SQ

kA kr kr w

t

o o w

1

aq. . . . . . . . . . . . . . . . . . (15)

Substituting Eq. 15 into Eq. 13, with all the prefactors set to 1, theshear stress at a given saturation is given by

Sq

k kr kr w

t

o o w

( )= + ( )2

1

aq

. . . . . . . . . . . . . . . . . (16)

Consequently, to construct the fractional-flow curve for flood-ing with a non-Newtonian fluid, a guess-and-check technique isneeded to determine a self-consistent aqueous-phase viscosity ateach water saturation.

As an example, we will construct the fractional-flow curvesfor a synthetic case at different injection rates. Two scenarios will

TABLE 1INPUT PARAMETERSFOR THE 1D NEWTONIAN VALIDATION RUNS

Oil viscosity, o 10 mPas

Water viscosity, w 1 mPas

Polymer viscosity, zero* 20 mPas

Corey water exponent, n 3

Corey oil exponent, m 2

Grid size, x 0.001 of total length

Timestep, t 0.0125 PV

Irreducible water, Swir 2.0

Residual oil, Sor 2.0

Initial water, Swi 2.0

Endpoint water relative permeability 0.4

Endpoint oil relative permeability 1.0

Injection rate, Q D/VP52100.0

Permeability, k dm0001

(a) (b)

Fig. 4Water-saturation profiles at 0.25 pore volumes (PV) injected from simulation (solid) against analytical solutions (dashed).(a) A good match is obtained for horizontal flooding with (black) and without adsorption (blue). (b) A good match is obtained forboth downdip (green) and updip (red) injection. The analytical horizontal case (blue) is also shown for comparison. The polymercodes operator-splitting technique successfully predicts gravitational effects.

TotalProduction,m3/d

0 1,000 2,000 3,000 4,000 5,000

Time, days

Fig. 5Code validation in two dimensions. Total oil produc-tion from our simulator (dots) against results from Thiele et al.

(2010) (line) obtained using a commercial streamline-basedcode for both water (blue) and polymer flooding (green).




be investigated: a thinning and a viscoelastic aqueous phase. Theabsolute and relative permeabilities are the same as in the 1DNewtonian case (Table 1). However, the oil viscosity is 20 mPas,and the displacing-fluid rheologies are defined as shown in Fig. 7,based on polyacrylamide rheology measured by Masuda et al.(1992). We constructed self-consistent fractional-flow curves forflooding at different injection rates (Fig. 8).

From Fig. 8, for displacement with a thinning fluid, thefractional-flow curves are bounded by the Newtonian water andNewtonian polymer fractional-flow curves. This is because, at lowrates, the viscosity is along the upper Newtonian plateau; hence,the fractional-flow curve is that of a Newtonian polymer. Then,the curves shift to the left with increasing rates until rates are highenough that the viscosity is along the lower Newtonian plateau;

hence, the fractional-flow curve would replicate that of water. Fora viscoelastic fluid, the fractional-flow curves first shift to the leftuntil rates are high enough to yield thickening when the fractional-flow curves will shift to the right with increasing rate.

Simulations of 1D displacement were performed at differentinjection rates (the input parameters are shown in Table 2). Compar-ing breakthrough times estimated from the simulation runs againstthose from the analytical solutions (Fig. 9),we can see that the simu-lator successfully predicts the effect of non-Newtonian rheology.

Our Method Compared to Current Simulators

There are three main differences between our implementation of non-Newtonian rheology and that in current commercial simulators.

Using a Physically Based Rheological Representation. Our meth-odology uses actual in-situ stresses to define the non-Newtonianrheology (Eqs. 11 and 13). Current simulators either use interstitialwater velocities, which is analogous to defining shear stress as

ijk rw ijk ijk

Rk P=

core

4, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

(a) (b)

AverageFieldPressure,kPa

Injected Pore Volumes, PVInjected Pore Volumes, PV

RecoveryFactor

Fig. 6Code validation in three dimensions. Predictions from our simulator (dots) compared to a commercial grid-based code(line) for both water (blue) and polymer flooding during secondary (orange) and tertiary (green) production.

Viscosity,mPas

Shear Stress, Pa

Fig. 7Aqueous-phase rheologies for the 1D non-Newtoniancase: thinning (red) and viscoelastic (blue).

(a) (b)

Fig. 8Fractional-flow curves for displacement with a non-Newtonian fluid at different Darcy velocities: (a) displacement witha thinning fluid and (b) displacement with a viscoelastic fluid. Black curves represent Newtonian flooding with water (solid)and polymer (dotted). Colored are fractional-flow curves for non-Newtonian flooding at different Darcy velocities: 0.00125 (darkblue), 0.125 (light blue), 1.25 (green), 12.5 (orange), and 125 m/d (red). (a) For displacement with a thinning fluid, at very lowrates, fractional-flow curves replicate that of a Newtonian polymer (dotted black) because shear stresses are along the upperNewtonian plateau. Increasing the flow rate gradually shifts the fractional-flow curves to the left until very high rates where shearstresses are along the lower Newtonian plateau; hence, the fractional-flow curve will replicate the water curve (solid black). (b)

For displacement with a viscoelastic fluid, the fractional-flow curves will initially shift to the left as flow rates increase; however,once the flow rate is high enough to yield dilatancy, the fractional-flow curves will shift to the right.




whereRcore= 8kcore core (Eq. 12),or use water shear rates, (i.e.,an equivalent shear stress):

ijk ijk rw ijk ijk R

k P= 4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

with Rijk given by Eq. 12. In addition, some simulators incorpo-rate an effective-porosity definition such that the correspondingequivalent stress becomes

ijkijk rw ijk

w ijk

ijk

R k

SP=

4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

Using an interstitial-velocity model, Eq. 17, ignores the differ-ence between the core and reservoir permeability. Thus, if the coreand average reservoir pore radii (Eq. 12) are different, the in-siturheology needs to be corrected. In addition, if the field is heteroge-

neous, this correction needs to be made on a block-by-block basis.Finally, for multiphase flow, both the water-interstitial-velocity andthe water-shear rate models (Eqs. 17 through 19) define rheologyat the same water-flow rate. We suggest that the non-Newtonianrheology should be taken as that occurring in single-phase flowat the same pressure gradient (Eq. 11).On the basis of pore-scalenetwork modeling of shear-thinning multiphase flow by Lopez andBlunt (2004), at low water saturation, when the wetting phase residesin the smaller pores, calculating the rheology on the basis of thesame flow rate greatly underestimates the local pressure gradient inan equivalent single-phase displacement; using the same pressuregradient is a better way to account for the changing pore-spaceconfigurations of the fluid.

Implementing an Iterative Pressure/Viscosity Solver. Becausethe non-Newtonian viscosity depends on the pressure gradient,which, in turn, depends on viscosity, an iterative pressure/viscositysolver is implemented. In contrast, current simulators use a lag-ging approach in which viscosities from the previous timestep areused to solve for pressures at the new timestep. These viscositiesare then updated but used only to compute pressure at the nexttimestepthey are not necessarily consistent with the current flowfield. The importance of this iterative algorithm will depend onthe timestep takenfor short timesteps, a lagging approach mightbe adequate. Similarly, a lagging approach might be adequate forfully implicit grid-based simulators because the mass-balanceconvergence will represent a greater constraint on the timestep sizecompared to the viscosity/pressure-field convergence. However,because our code uses an IMPES formulation and because one of

the main advantages of streamline-based simulators is their abilityto take large timesteps, an iterative approach is needed.

Defining Viscosities To Be Cell-Centered As Opposed to Face-Centered. We define viscosities to be cell-centered, while currentsimulators estimate non-Newtonian viscosities on the basis ofvelocities at the faces. At first sight, this latter approach seemsreasonable because velocity is defined at cell faces. However, aswe illustrate in Fig. 10,a face-based algorithm will underpredictshear-thinning effects: If a component of velocity across a face issmall, the algorithm will use this low value to compute viscosity,even if the overall velocity is high with a high effective stress. Thiswill also affect the computations of transport that are traditionallyperformed along grid directions.

The Effect of Current AssumptionsHere, we investigate the significance of the assumptions discussed

earlier on the simulation results. We simulate polymer flooding ina 2D heterogeneous quarter-five-spot pattern with a constant injec-tion rate of 90 m3/d. The permeability of the reservoir (Fig. 11)wasbased on a section of the first layer of the SPE 10 model (Christieand Blunt 2001). The oil viscosity is 50 mPas. The remaining fluidproperties are the same as listed in Table 1, and the shear-thinningrheology is based on xanthan rheology measured by Seright et al.(2009), as shown in Fig. 10. First, to validate our code, Fig. 12shows example results using Eq. 17 for effective stress in both ourcode and a grid-based finite-difference simulator. The good agree-ment indicates that, when our code uses the same (poor) assump-tions as current commercial simulators, the results are similar.

The Effect of the Rheological Representation. Fig. 13shows the

recovery predictions obtained using our streamline code with thevarious rheological representations. The model we propose givesthe lowest recovery. We suggest that using current assumptionsmay tend to overestimate the recovery obtained from shear-thin-ning-polymer injection.

The Effect of a Lagging Pressure/Viscosity Solver. We ran ourstreamline code using a lagging and an iterative approach to thepressure-field and viscosity update and compared the results. Asin the conventional finite-difference simulator, all runs used thesame rheological model based on interstitial water velocities (Eq.17). We performed two runs with two different timestep sizes:100 days, corresponding to 60 timesteps in total, and 25 days(240 timesteps).

The results shown in Fig. 14indicate that, for this case, the effect

of this assumption on the streamline-based simulation is modest:Although the effect on recovery is relatively small, a lagging

TABLE 2INPUT PARAMETERSFOR THE 1D NON-NEWTONIAN VALIDATION RUNS

Oil viscosity, o 20 mPas

Water viscosity, w 1 mPas

Polymer viscosity, zero* 9.1 mPas

Corey water exponent, n 3

Corey oil exponent, m 2

Grid size, x 0.001 of total length

Timestep, t 0.005 PV

Irreducible water, Swir 2.0

Residual oil, Sor 2.0

Initial water, Swi 2.0

Endpoint water relative permeability 0.4

Endpoint oil relative permeability 1.0

Injection rate, Q suoirav

Permeability, k dm0001

BreakthroughTime,PV

Injection Rate, PV/D

Fig. 9Effect of injection rate on breakthrough from simula-tion and semianalytical solutions for non-Newtonian 1D floods.The simulator breakthrough predictions for displacement witha thinning () and a viscoelastic (+) fluid match those calcu-lated from the semianalytical solutions for thinning (red) andviscoelastic (blue) flooding. The black lines represent break-through times for Newtonian displacement with water (solid)and polymer (dotted).




method with a large timestep predicts slightly higher recovery;however, with short timesteps, the lagging approach converges tothe iterative solution.

The Effect of Face-Centered Viscosities.Our results were alsocompared with those obtained using a conventional finite-differ-ence simulator. As Fig. 14 indicates, compared to our code, thefinite-difference code predicts slightly higher recovery. Lookingat the viscosity profile during a simulation, Fig. 15,we see that

the higher sweep efficiency predicted by the finite-difference codefor the non-Newtonian case might be explained by an overpredic-tion of viscosity. Note that the low-viscosity streak predicted bythe streamline code follows the high permeability (Fig. 11). Incontrast, the finite-difference code has a discontinuous low-vis-cosity streak, which is unexpected. In addition, in approximatelyone-third of the blocks, the finite-difference code predicts shearedviscosities that are higher than the effective Newtonian viscosities,which is unphysical. This occurs because the effective viscositybased on the grid-centered polymer concentration is lower thanthe computed face-centered viscosity values.

Moreover, additional runs were carried out using our code withface-centered algorithms: an averaged viscosity and an averagedvelocity. The first takes the cell-centered viscosity as the average ofthe face values, while the second takes it at the average velocitynote, however, that, because the streamline and grid-based codesuse different transport algorithms, neither of our approaches is an

exact representation of face-centered viscosities computed in agrid-based simulator. Nevertheless, both face-centered approachesshow a similar discontinuity in the low-viscosity streak (Fig. 15).In addition, compared to the cell-centered approach, both face-centered algorithms overpredict recovery (Fig. 16).

Evaluation and Design of Polymer

EOR Projects

We conclude this paper by performing 3D simulations todesign a tertiary polymer flood. The reservoir model used is acoarse representation of the SPE 10 model; the dimensions are365.76670.5651.816 m, represented by 205517 gridblocks

(a) (b)

Viscosity,mPas

Shear Stress, Pa

Water Velocity, m/d

0.01 0.1

0.1 1 10 100

A. Face Algorithm-grid = 11.5 cp

B. Cell-center Algorithm

-grid = 3 cp

10

1

4.5m/d

4.5m/d

0.1m/d0.1m/d

Fig. 10The effect of defining non-Newtonian viscosities at grid faces. In this example, with a face-centered approach, the aver-age grid viscosity is predicted to be approximately 11.5 mPas, while using the cell-centered velocity it should be approximately3 mPas. In addition, a face-centered approach will significantly overpredict viscosity across faces with a small normal velocitycomponent. For a shear-thinning system, this could lead to the prediction of an unrealistically stable displacement along low-flowdirections. The rheology shown in the figure is based on Seright et al. (2009) measurements of xanthan rheology in a Berea corewith permeability of approximately 551 md and porosity of approximately 0.2. This rheology will be used in our later simulations.We used Eq. 14 to plot viscosity as a function of shear stress (upper axis).

Fig. 11Permeability for the 2D comparison based on the firstlayer of the SPE 10 model.

(a) (b)Injected Pore Volumes, PV

RecoveryFactor

Vicscosity

Dimensionless Distance

Fig. 12Results for Newtonian polymer flooding (green), and non-Newtonian polymer flooding at two injection rates: low and

high (orange and red, respectively) obtained with a finite-difference simulator (solid lines) and our code (dots). Both models useEq. 17 for the effective stress. (a) Recovery predictions (b) viscosity profiles at 0.27 PV injected.




(Beraldo et al. 2009). The upscaled permeability field is shown inFig. 17,while the other properties are given in Table 1. Waterflood-

ing for 1,200 days (approximately 0.47 PV of water injected) is fol-lowed by polymer injection. In this section, we attempt to includefurther multiphase effects using a bundle-of-tubes approximationwhereZ k S

rw w= .

Flooding With a Shear-Thinning Polymer. The first scenario isflooding with a strictly shear-thinning polymer; the polymer rhe-ology was based on xanthan rheology measured by Seright et al.(2009), as shown in Fig. 10.

Compared to results with current assumptions, recovery predic-tions using the streamline codewith an iterative pressure solu-tion, face-centered viscosities, and a physically based rheologicalmodelare much less optimistic; see Fig. 18. We suggest thatcurrent commercial codes may be overestimating recovery.

Fig. 19illustrates the effect of using a shear-thinning polymer

at an injection rate of 1000 m3/d on recovery compared with New-tonian polymers. Note that, although the shear-thinning polymerused in this case has an upper plateau viscosity of 20 mPas, shear-thinning results in a recovery performance similar to a Newtonianpolymer with a much lower viscosity of 2 mPas.

Flooding With a Viscoelastic Polymer.The second scenario looksat flooding with a viscoelastic polymer. The polymer rheology is

RecoveryFactor

Injected Pore Volumes, PV

Fig. 13Tertiary-recovery predictions with the various possiblerheological representations: interstitial-water-velocity model,Eq. 17 (black); Eq. 17 corrected to account for the difference inaverage permeability between the core experiment and the field(orange); the equivalent stress models, Eq. 18 (blue) and Eq. 19(green); and the actual stress model, Eq. 13 (red).

Injected Pore Volumes, PV

RecoveryFactor

Fig. 14The effect of a lagging approach on recovery predic-tions for tertiary non-Newtonian polymer flooding. The resultsobtained with the finite-difference simulator (black) and ourstreamline code (colored): iterative and lagging with longtimesteps (100 days) in red and green, respectively, and laggingwith short timesteps (25 days) in dashed green. Note that thecommercial simulator predicts slightly higher recovery. Moreo-ver, running our code in a lagging mode with long timesteps(green) also results in slightly higher recovery predictions, butthe effect is small in this case. In addition, the results suggestthat, with sufficient timesteps, the lagging approach converges

to the iterative solution. Note that, as in the conventional finite-difference simulator, all runs used the same rheological modelbased on interstitial water velocities (Eq. 17).

)c()b()a(

Fig. 15The viscosity profile (values shown in mPas) predicted by streamlines with face-centered (a) and cell-centered viscosi-ties (b), and that predicted by a commercial grid-based simulator (c), for a secondary non-Newtonian polymer flood at 1.34 PVinjected. Similar results were observed for the tertiary case. Note that the low-viscosity region is discontinuous for the grid-based

code, despite a continuous high-permeability region; see Fig. 11. Running the streamline code with a face-centered approachpredicts a similar but less prominent discontinuity.

based on polyacrylamide rheology measured by Masuda et al.(1992), as shown in Fig. 7.

Compared with results with current rheological representations,the streamline codewith a physically based rheological modelsuggests a lower critical injection rate at which in-situ thickeningeffects are realized (Fig. 20).In addition, while current rheologicalmodels overpredict recovery below the critical thickening rate (i.e.,along the thinning regime), at rates above the critical thickeningrate (i.e., along the thickening regime), they underpredict recovery

compared with the model we propose.

Summary and Conclusions

We have extended a streamline simulator to model polymer flood-ing with non-Newtonian behavior, implementing a physically basedrheological model. We tested our model successfully against ana-lytical 1D solutions and with multidimensional comparisons withresults from commercial grid-based and streamline-based codes.




Our methodology differs from current commercial simulators inthree ways: (1) we use a rheological representation for multiphaseflow based on pore-network modeling results, (2) we take intoaccount the interdependence between pressure and non-Newtonianviscosities through the implementation of an iterative pressure/vis-cosity solver, and (3) we define viscosities to be cell-centered asopposed to face-centered.

All three effects lead to less-optimistic recovery predictionsfor shear-thinning polymers; we suggest that current models may

RecoveryFactor

Injected Pore Volume, PV

Fig. 16The effect of face-centered viscosities on recovery pre-dictions for tertiary non-Newtonian polymer flooding: our stream-line (colored) and the finite-difference-simulator results (black).Streamline runs were carried out in a lagging mode but withshort timesteps (25 days to ensure convergence; see Fig. 14).Predictions with face-centered viscositiesaveraged viscosities(blue) and averaged velocities (green)overestimate recoverycompared to the cell-centered approach (red). Note that, as in theconventional finite-difference simulator, all runs used the samerheological model based on interstitial water velocities (Eq. 17).

Fig. 17SPE 10 upscaled permeability used for the 3D simula-tion case. The values of permeability are shown in millidarcies.

CumulativeProduction,m3


0 1,000 2,000 3,000 4,000

Time, days

0 1,000 2,000 3,000 4,000

Time, days(a) (b)

Fig. 18Cumulative-oil-production predictions for a 3D tertiary shear-thinning polymer flood. (a) Predictions using streamlines(red) against that predicted also using our code but with assumptions contained in current simulators. In green are results withthe same rheological representation with cell-centered viscosities but using a lagging approach to compute the pressure. Dashedgreen shows the lagging results with smaller and smaller timesteps (solid: 200 days; dashed: 100 days; and dotted: 50 days)theresults converge to the iterative solution once a 50-day timestep is used. In blue are results using cell-centered viscosities butwith an equivalent-stress/effective-porosity model (Eq. 19) for the rheology and with the lagging approach. Finally, in black areresults using face-centered viscosities and an equivalent-stress/effective-porosity model (Eq. 19), for the rheology and the lag-ging approach. (b) Predictions with an interstitial-water-velocity model (Eq. 17) using a commercial grid-based simulator (dots)and our streamline simulator (solid lines) with cell-centered viscosities (red) and face-centered viscosities: averaged viscosities(blue) and averaged velocities (green). Those two streamline runs used a lagging approach with small timesteps (50 days).


0 1,000 2,000 3,000 4,000

Time, days

Fig. 19Cumulative oil production for tertiary Newtonian polymerflooding (lines), continuation of waterflooding (blue), and tertiaryshear-thinning-polymer flooding (dots). The Newtonian polymershave viscosities of 2 (red), 5 (orange), 10 (yellow), and 20 mPas

(green). The performance of the 20-mPas shear-thinning polymermatches that obtained with a 2-mPas Newtonian polymer.

Fig. 20Incremental oil recovery at approximately 0.9 PVwith different injection rates for a tertiary viscoelastic floodpredicted with the equivalent-stress models, Eq. 18 (blue) andEq. 19 (green), and the actual stress model, Eq. 13 (red). The

dashed black line plots the incremental recovery for a 9.1-mPas Newtonian polymer.

IncrementalRecovery,%OOIP

Injection Rate, m3/d


http:///reader/full/spe-123971-pa-streamline-based-simulation-of-non-newtonian-pol


significantly overestimate recovery from shear-thinning-polymerfloods. Nevertheless, we shall note that the importance of a lag-ging approach will likely depend on the complexity of the problemas well as the number of timesteps taken. On the other hand, forviscoelastic polymers, current rheological models might overlookthe benefit of viscoelasticity or the possibility of exploiting suchviscoelastic behavior by increasing the injection rate.

Nomenclature

A= adsorption level, dimensionless

c= volumetric concentration, L3/L3, ppm

C= normalized concentration, dimensionless f= fractional flow, dimensionless

k= permeability, L2, m2

K= convention factor, dimensionless

m= tortuosity exponent, dimensionless

M= shear multiplier, dimensionless

n= Corey exponent, dimensionless

P= viscosity multiplier, dimensionless

q= Darcy velocity, L/T, m/s

Q= injection rate, L3/T, m3/d

R= hydraulic radius, L, m

S= saturation, dimensionless

t= time, T, seconds

X= bulk-to-grid factor, dimensionless

Z= two-phase factor, dimensionless

= tortuosity factor, dimensionless

= polymer multiplier, dimensionless

= shear exponent, dimensionless

= consistency index, dimensionless

= viscosity, ML1T1, Pas

= shear stress, ML1T2, Pa

= time of flight, T, seconds

= porosity, dimensionless

P= pressure gradient, ML2T2, Pa/m

Subscripts

a = adsorbed

aq = aqueous phase i = initial

ijk= grid identifiers

inj = injection

ir = irreducible

m = midpoint

o = oil

p = polymer

r = residual

TH= thinning or thickening

VE = viscoelastic

w = water

zero = zero shear

Superscripts i= cell identifier in the discretization along a streamline

n= time level

* = injection condition

Acknowledgments

The authors thank Schlumberger and Streamsim Technologiesfor the use of their simulators. Abdulkareem AlSofi thanks BilalRashid for his help with reading Eclipse output files and SaudiAramco and EXPEC ARC for funding his post-graduate studies.

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Abdulkareem M. AlSofiis a PhD candidate in the Departmentof Earth Science and Engineering at Imperial CollegeLondon. emai: [email protected]. His PhD research ison the design and evaluation of augmented waterfloods.He holds a BS degree, with high honors, from The Universityof Texas at Austin. Before entering graduate school, AlSofiworked as a reservoir engineer with Saudi Aramco. MartinJ. Blunt is professor of petroleum engineering and headof the Department of Earth Science and Engineering atImperial College London. email: [email protected], he was an associate professor in the Departmentof Petroleum Engineering at Stanford University and workedat the BP Research Centre. He holds MA and PhD degrees

in physics from Cambridge University. Blunt, winner of the1996 Cedric K. Ferguson Medal, served as associate exec-utive editor of the SPE Journal from 1996 to 1998 and wason the editorial board from 1996 to 2005. He was a 2001Distinguished Lecturer.

spe-123971-pa (streamline-based simulation of non-newtonian polymer flooding)

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