Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

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International Journal of Non-Linear Mechanics ] (]]]]) ]]]–]]]

Contents lists available at SciVerse ScienceDirect

International Journal of Non-Linear Mechanics

0020-74

http://d

n Corr

E-m

ciriello.

Pleasporo

journal homepage: www.elsevier.com/locate/nlm

Analysis of a benchmark solution for non-Newtonian radial displacementin porous media

Valentina Ciriello n, Vittorio Di Federico

Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (D.I.C.A.M.), Universit�a di Bologna, Bologna 40100, Italy

a r t i c l e i n f o

Article history:

Received 23 October 2012

Received in revised form

16 January 2013

Accepted 16 January 2013

Keywords:

Porous media

Displacement

Non-Newtonian

Power-law

Compressible fluids

Similarity solution

Global sensitivity analysis

Polynomial chaos expansion

62/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.ijnonlinmec.2013.01.011

esponding author. Tel.: þ39 51 2093753; fax

ail addresses: [email protected],

[email protected] (V. Ciriello).

e cite this article as: V. Ciriello, V.us media, International Journal of N

a b s t r a c t

We present an analytical formulation useful to interpret the key phenomena involved in non-

Newtonian displacement in porous media and an analysis of the results obtained by considering the

uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider

the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing

and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and

velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a

modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into

account compressibility effects, yields a set of nonlinear second-order partial differential equations.

Considering two fluids with the same flow behavior index n allows transformation of the latter

equations via a self-similar variable; further transformation of the equations incorporating the

conditions at the interface shows for no1 the existence of a compression front ahead of the moving

interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface

and compression front, and the pressure distributions; the latter are derived in closed form for any

value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a

stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical

model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids

flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing

fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative

influence of input parameters on model outputs is evaluated by means of associated Sobol indices,

calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained

by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.

& 2013 Elsevier Ltd. All rights reserved.

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6768697071727374757677787980818283

1. Introduction

Displacement phenomena in porous media involving non-Newtonian fluid behavior are of considerable interest in severalareas of engineering and physics. In petroleum engineering, varioussubstances injected into underground reservoirs to enhance oilrecovery, by improving the overall sweeping efficiency and mini-mizing instability effects, reveal a nonlinear stress-shear raterelationship and other non-linear effects [1]: these include dilutepolymer solutions, emulsions of surfactants and foams [2,3]. On theother hand, heavy and waxy oils are often found to exhibit non-Newtonian characteristics at reservoir conditions [4,5]; therefore asituation may be envisaged in which a non-Newtonian fluidinjected into a reservoir displaces another non-Newtonian fluid

8485868788

ll rights reserved.

: þ39 51 6448346.

D. Federico, Analysis of aon-Linear Mechanics (2013

with different rheological characteristics. A similar situation mayarise in environmental remediation efforts geared towards in situtreatment, where injection of substances having nonlinear rheolo-gical properties such as colloidal or biopolymer suspensions isemployed to remove, or favor the removal of, liquid pollutants fromcontaminated soils; relevant examples include DNAPLs remediationby means of colloidal liquid aphrons [6], and the use of xhantangum to enhance mobility and stability of suspensions of nanoscaleiron employed in reactive barriers [7]. As in situ bioremediationmay create polymers with non-Newtonian characteristics [8], asubsequent injection may result in displacement of a non-Newtonian fluid by another. Similar situations may arise in indus-trial engineering, where non-Newtonian flows occur in filtration ofpolymer melts, food processing, and fermentation [9], and inorthopedic applications, where injectable cements used in a varietyof bone augmentation and bone reconstruction procedures alsodisplay a complex rheology [10].

The displacement phenomenon of a fluid by another in aporous domain has been extensively investigated in the literature

8990

benchmark solution for non-Newtonian radial displacement in), http://dx.doi.org/10.1016/j.ijnonlinmec.2013.01.011i

www.elsevier.com/locate/nlm

www.elsevier.com/locate/nlm

http://dx.doi.org/10.1016/j.ijnonlinmec.2013.01.011



mailto:[email protected]





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V. Ciriello, V.D. Federico / International Journal of Non-Linear Mechanics ] (]]]]) ]]]–]]]2

when either fluid, or both, exhibit non-Newtonian behavior.Pascal [11] adopted Muskat’s frontal advance model to studysteady-state immiscible displacement of a Bingham fluid byanother in plane/radial geometry. Steady-state displacement,and its stability, were analyzed in Ref. [12] for power-law fluidswith yield stress in plane geometry, and in Ref. [13] for power-law fluids in radial geometry; capillarity was added to the modelin Ref. [14]. In Ref. [15], transient plane displacement of a power-law compressible fluid by another was considered. In Refs. [16]and [17], transient plane/radial displacement of a power-law fluidby another was considered, allowing for two-phase flow behindthe displacement front but neglecting compressibility. An analy-tical solution for piston-like displacement of power-law dilatantfluids in plane and radial geometry was derived in Ref. [18]. In Wuet al. [19] an analytical solution of Buckley–Leverett type to two-phase flow determined by the displacement of a Newtonian fluidby a non-Newtonian power-law one was obtained and validatedby a numerical model. Wu and Pruess [20] developed a numericalsimulator for multiphase flow in porous media, including thepower-law and Bingham models. A novel two-phase numericalsimulator incorporating non-Newtonian behavior was proposedin Ref. [21]. Tsakiroglou [22] generalized the macroscopic equationsof the two-phase flow in porous media accounting for capillarity forthe case of a shear-thinning displacing fluid, and developed anumerical scheme of inverse modeling to estimate model para-meters from unsteady-state experiments. Other researchers inves-tigated the onset of instabilities in displacement of non-Newtonianfluids experimentally [22–25] or theoretically [26].

If a fingering instability does not develop at the interfacebetween displacing and displaced fluid, the frontal advancetheory may be considered an approximate yet acceptable descrip-tion of the displacement mechanism, with the advantage ofproviding analytical solutions, which in turn may prove usefulas benchmarks against which numerical solvers are tested.

An example of such solutions was provided by Ref. [15], whoderived a similarity solution for planar transient immiscibledisplacement of a power-law compressible fluid by another withthe same flow behavior index. The study of the radial case (e.g.,flow away from a wellbore), which represents a plausible simpli-fication of the geometry involved in several possible applications,is developed in this work. The assumption of identical flowbehavior index for displacing and displaced fluid is retained toderive a closed-form solution in the format of a system ofalgebraic nonlinear equations. As values on flow behavior indexin real applications, especially connected to reservoir engineering,tend to cluster around 0.6–0.8 [27], the proposed solution mayprovide a qualitative insight on relevant physical phenomena alsofor fluids whose flow behavior index differ to some extent. Theproblem is formulated in dimensionless form for different typesof boundary conditions in the origin of the flow domain (assignedpressure or flow rate), and novel closed-form expressions of thepressure field in the displacing and displaced fluids for a genericvalue of the flow law exponent are derived generalizing to twofluids the results of [28]; a discussion of deterministic results isthen provided.

Uncertainty plagues virtually every effort to predict thebehaviour of complex physical systems; in the problem underinvestigation, it affects to various degrees: (a) the properties ofthe porous medium, due to its inherent spatial heterogeneity andlack of complete characterization; (b) the descriptive parametersof the fluids involved, having a complex rheological behavior. Inthe first case, a random field description e.g., [29] represents themost complete methodology. In the sequel, to exemplify ourapproach and achieve easily interpretable indications, we modelkey problem parameters as independent random variables havingan assigned probability distribution.

Please cite this article as: V. Ciriello, V.D. Federico, Analysis of aporous media, International Journal of Non-Linear Mechanics (2013

Global Sensitivity Analysis (GSA), conducted by computingSobol indices as sensitivity measure (since no assumptions oflinearity or monotonic behavior on model equations are required)[30,31], is a powerful instrument to investigate the relativeinfluence of the different sources of uncertainty on the statevariables of interest and represents the basis for a rational designof a measurement strategy in contaminant transport in porousmedia [32]. A reliable technique for the evaluation of Sobolindices is constituted by the Polynomial Chaos Expansion (PCE)technique, introduced in engineering context inside the stochasticfinite elements analysis (SFEM) [33]. This method returns accu-rate results and drastically decreases the computational costassociated with GSA, otherwise unaffordable especially for com-plex numerical models [34].

In this work, we adopt GSA conducted by means of PCE tostudy how uncertainty affecting selected parameters: (a) thefluids flow behavior index, (b) the relative total compressibilityand mobility in the displaced and displacing fluid domains, and(c) the domain permeability and porosity, propagates to statevariables adopting the benchmark analytical model of non-Newtonian radial displacement derived earlier. The goodness ofthe results obtained by the PCE is then assessed by comparisonagainst a traditional Monte Carlo (MC) approach.

2. Analytical model of non-Newtonian displacement andsimilarity solution

2.1. Flow law for power-law fluid in a porous medium

Flow of Newtonian fluids in porous media is governed byDarcy’s law. Its extension to non-Newtonian fluids is complex,due to interactions between the microstructure of porous mediaand the rheology of the fluid, even in the creeping flow regime.The scientific literature of the past decades includes numerousworks dedicated to this problem: for exhaustive reviews seeRefs. [9,35,36]. A sizable part of them deals with power-lawfluids, described by the rheological Ostwald–DeWaele model,given for simple shear flow by

t¼m _g9 _g9n�1, ð1Þ

where t is the shear stress, _g the shear rate, m[ML�1Tn�2] and n

indices of fluid consistency and flow behavior respectively, withno1, ¼1 or 41 describing respectively pseudoplastic, New-tonian, or dilatant behavior. The power-law model, itself asimplification of more complex, and realistic, rheological beha-vior, is nevertheless often adopted in both porous media and free-surface flow modeling for its simplicity [37]. The correspondingmodified version of Darcy’s law takes in the literature the twoequivalent forms [35,38–44]

rP¼�mef

k9v9n�1

v¼�m

kn9v9n�1

v, ð2Þ

where P¼pþrgz is the generalized pressure, p the pressure, z thevertical coordinate, r the fluid density, g the specific gravity, v theDarcy flux, k the intrinsic permeability coefficient [L2], mef theeffective viscosity [ML�nTn�2], k* the generalized permeability[Lnþ1]; the ratio k/mef, termed mobility, is given by Ref. [45]

k

mef

¼1

2m

nf3nþ1

� �n 8k

f

� �ð1þnÞ=2

: ð3Þ

where f denotes the porosity. For n¼1, the effective viscosity mef

reduces to conventional viscosity m, and Eq. (2) reduces to Darcy’slaw rP¼�(m/k)v. Earlier literature reviews, e.g. Ref. [27] demon-strate that the bulk of applications to non-Newtonian flows in





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porous media involve pseudoplastic fluids with n mainly in therange 0.5–1, yet dilatant behavior is sometimes encountered.

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2.2. Problem formulation

We consider a well of radius rw located in the center of aporous domain of infinite horizontal extent, constant thickness h,and uniform properties, and analyze the dynamics of a movinginterface due to injection at the well of a non-Newtonian fluidinto the domain, initially saturated by another non-Newtonianfluid (Fig. 1); both fluids, displacing and displaced, are of power-law pseudo-plastic behavior with the same consistency index n.The interface between the fluids is considered to be stable andsharply defined, so that a piston-like displacement exists. Thepressure and velocity fields are assumed to be continuous at theinterface; the pressure is taken to be constant and equal to pe inthe domain occupied by the displaced fluid at time t¼0; thedisplacing fluid is injected at a constant pressure pw greater thanthe ambient pressure pe, or at a given injection rate Qw(t).

The flow and continuity equation for both fluids (i¼1 for thedisplacing, i¼2 for the displaced) are:

vi ¼ �k

mef i

@pi

@r

!1=n

, ð4Þ

@vi

@rþ

vi

r¼�c0if

@pi

@t, ð5Þ

where the Darcy velocities vi are the one-dimensional counter-parts of (2). In (4) and (5) r denotes the radial spatial coordinate, t

time, f and k the domain porosity and permeability, pi andc0i¼cfiþcp the pressures and total compressibility coefficients inthe two flow regions, with cfi being the fluid compressibilitycoefficient and cp the porous medium compressibility coefficient.The relative influence of fluid and medium behavior on the totalcompressibility coefficient may vary widely, depending on theirnature, and ranges from cases where one is negligible comparedto the other to instances where the two effects are of the sameorder. In the CO2 storage application presented by Ref. [46], brineand formation take the respective compressibilities cf¼3.5�10�10 Pa�1 and cp¼4.5�10�10 Pa�1. In enhanced oil recoveryapplications, the fluid compressibility coefficient typically lies inthe range 1–5�10�9 Pa�1 [47], while according to Ref. [15], thetotal compressibility coefficient c0 may vary between 1�10�8

and 5�10�8 Pa�1, implying a larger influence of medium com-pressibility. An example illustrating the differences betweencompressibilities for different fluids is the water–oil displacementcase study presented by Ref. [48], where water, oil and rockcompressibility are taken equal respectively to 4.5�10�10,1.3�10�9and 5�10�10 Pa�1. In general, oscillations betweentypical fluid compressibility values seem to be of one order ofmagnitude, while formation compressibility varies in a largerinterval [49].

121122123124125126127128129130131132Fig. 1. Domain schematic (either pw or Qw is assigned).


Substituting (4) in (5) one obtains for the two fluids (i¼1, 2):

@2pi

@r2þ

n

r

@pi

@r¼ nc0if

mef i

k

� �1=n

�@pi

@r

� �ðn�1Þ=n @pi

@t, ð6Þ

where pi(r,t)¼p1(r,t) for 0rrrz(t) and pi(r,t)¼p2(r,t) forz(t)rroN, with z(t) being the interface position; since theinjection starts at t¼0, z(0)¼0.

The initial condition for the displaced fluid is

p2ðr,0Þ ¼ pe: ð7Þ

Designated boundary conditions at the well r¼rw are eitherconstant pressure pw or flow rate Qw(t), indicated in the sequel asb.c. 1) and 2)),:

p1ðrw,tÞ ¼ pw, ð8Þ

v1 rw,tð Þ ¼ �k

mef 1

@p1

@r

!1=n

r ¼ rw

¼QwðtÞ

2phrw, ð9Þ

The expression of the injected flow rate for the second-typeboundary condition is taken to be

QwðtÞ ¼ Q0tc , ð10Þ

where Q0(40) is the injection intensity and c a real number.Lastly, the pressure within the displaced fluid at infinity equalsthe ambient pressure, i.e.,

limr-1

p2ðr,tÞ ¼ pe: ð11Þ

At the moving interface, the pressure and velocity fields arecontinuous; thus

p1½zðtÞ,t� ¼ p2½zðtÞ,t�, ð12Þ

�k

mef 1

@p1

@r

!1=n

r ¼ zðtÞ

¼ �k

mef 2

@p2

@r

!1=n

r ¼ zðtÞ

¼ V ¼fdzdt

, ð13Þ

where V is the common value of the Darcy velocity at theinterface.

We now define the following dimensionless variables (i¼1, 2):

r0,r0w,z0,h0,t0,p0i,v0i,V0,Q 0w,Q 00

� �¼

r

L,rw

L,zL

,h

L,

t

T,c01pi,

viT

L,VT

L,QT

L3,Q0T1�c

L3

!,

ð14Þ

where L is an arbitrary length scale of the order of the domain’sthickness h,

T ¼m1=n

1 c1=n01 Lðnþ1Þ=n

kð1þnÞ=2n0

ð15Þ

where T is a timescale, and k0 a reference permeability. WithLEhE10 m, c01E10�8 Pa�1, m1E1 Pa sn, k0E10�12 m2, onehas TE105 sE1 day for n¼0.5. The dimensionless form of(4) and (5) is therefore (primes are dropped for convenience)

v1 ¼fA�@p1

@r

� �1=n

, ð16Þ

v2 ¼fM1=n

A�@p2

@r

� �1=n

, ð17Þ

@2p1

@r2þ

n

r

@p1

@r¼ nA

@p1

@r

� �ðn�1Þ=n @p1

@t, ð18Þ

@2p2

@r2þ

n

r

@p2

@r¼ n

aA

M1=n

@p2

@r

� �ðn�1Þ=n @p2

@t, ð19Þ





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where the quantities

A¼ A n,f,Kð Þ ¼fð1þnÞ=2n

w1=nn K ð1þnÞ=2n

, wn ¼ 8ð1þnÞ=2nðn=ð3nþ1ÞÞn=2,

ð20a;bÞ

reduce for n¼1 to A¼f=K and wn¼1, and

M¼ ðk=mef 2Þ=ðk=mef 1Þ ¼m1=m2, a¼ c02=c01, K ¼ k=k0,

ð21a;b; cÞ

where M, a, K are respectively the mobility ratio, the compressi-bility ratio, and the dimensionless permeability.

Initial and boundary conditions (7), (8) and (11), and conditionat the interface (12) remain unchanged in dimensionless form.Boundary condition (9) becomes

v1 rw,tð Þ ¼fA�@p1

@r

� �1=n

r ¼ rw

¼QwðtÞ

2phrw: ð22Þ

Interface condition (13) reads in dimensionless form (primesomitted)

fA�@p1

@r

� �1=n

r ¼ BðtÞ¼

fM1=n

A�@p2

@r

� �1=n

r ¼ BðtÞ¼ V ¼f

dzdt

, ð23Þ

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2.3. Solution of the problem

Adopting the similarity variable

Z¼ r

tn=ðnþ1Þ, ð24Þ

Eqs. (16)–(19) take the form

v1 ¼fA

t�1=nþ1 �dp1

dZ

� �1=n

, ð25Þ

v2 ¼fM1=n

At�1=nþ1 �

dp2

dZ

� �1=n

, ð26Þ

d2p1

dZ2þ

n

Zdp1

dZ¼

n2

nþ1AZ �dp1

dZ

� �ð2n�1Þ=n

0rZrZ1

� �, ð27Þ

d2p2

dZ2þ

n

Zdp2

dZ¼

n2

nþ1

aA

M1=nZ �dp2

dZ

� �ð2n�1Þ=n

Z1rZo1� �

, ð28Þ

where Z1 is linked to the position of the moving interface by

zðtÞ ¼ Z1tn=ð1þnÞ: ð29Þ

The first-kind and second-kind boundary conditions at thewell (8) and (22) become, respectively

p1ðZwÞ ¼ pw, ð30Þ

�dp1

dZ

� �1=n

Z ¼ Zw

¼A

fQwðtÞ

2phrwt1=ð1þnÞ, ð31Þ

where Zw¼Z(rw,t). Initial and boundary conditions (7) and (11)expressed in terms of Z transform into

limZ-1

p2ðZÞ ¼ pe: ð32Þ

The conditions at the interface (12) and (23) become

p1ðZ1Þ ¼ p2ðZ1Þ, ð33Þ

dp1

dZ

� �Z ¼ Z1

¼Mdp2

dZ

� �Z ¼ Z1

: ð34Þ


The interface velocity takes the form

V ¼fdzdt¼

fA�

dp1

dZ

� �1=n

Z ¼ Z1

t�1=ð1þnÞ: ð35Þ

Integrating (35) with the initial condition z(0)¼0 yields

z tð Þ ¼1þn

nA�

dp1

dZ

� �1=n

Z ¼ Z1

tn=ð1þnÞ: ð36Þ

Coupling (29) and (36) leads to the following expression for Z1

Z1 ¼ z tð Þt�n=ð1þnÞ ¼1þn

nA�

dp1

dZ

� �1=n

Z ¼ Z1

, ð37Þ

and taking (34) and (37) into account yields

�dp2

dZ

� �Z ¼ Z1

¼1

M�

dp1

dZ

� �Z ¼ Z1

¼1

M

nA

1þnZ1

� �n

: ð38Þ

Eqs. (27) and (28) are Bernoulli differential equations; theirintegration with (38) yields, respectively

dp1

dZ ¼�Z�nða1�b1Z3�nÞ

n=ð1�nÞ ZwrZrZ1

� �, ð39Þ

dp2

dZ ¼�Z�nða2�b2Z3�nÞ

n=ð1�nÞ Z1rZo1� �

, ð40Þ

where

a1 ¼ a1 Z1,n,f,K� �

¼nA

1þn

� �1�n

Z2ð1�nÞ1 þ

nð1�nÞA

ð1þnÞð3�nÞZ3�n

1 , ð41Þ

b1 ¼ b1 n,f,Kð Þ ¼nð1�nÞA

ð1þnÞð3�nÞ, ð42Þ

a2 ¼ a2 Z1,n,f,K ,M,a� �

¼n

1þn

A

M1=n

� �1�n

Z2ð1�nÞ1

þnð1�nÞ

ð1þnÞð3�nÞ

aA

M1=nZ3�n

1 , ð43Þ

b2 ¼ b2 n,f,K ,M,að Þ ¼nð1�nÞ

ð1þnÞð3�nÞ

aA

M1=n: ð44Þ

From (39) it is evident that when no1, dp2/dZ¼0 for

Zn ¼ c1Z1; c1 ¼ c1 Z1,n,f,K ,M,a� �

¼ 1þð1þnÞnð3�nÞM

nnð1�nÞaAnZ1þn1

" #1=ð3�nÞ

41

ð45Þ

Eqs. (40) and (45) show for a pseudoplastic fluid (no1) theexistence of a compression front ahead of the moving interface,whose dimensionless position and velocity z* and V* (defined inanalogy to z and V) are given by

zn tð Þ ¼ Zntn=ð1þnÞ; Vn¼f

dzn

dt¼

nfnþ1

Znt�1=ð1þnÞ: ð46Þ

At and beyond the compression front, the displaced fluid Darcyvelocity v2 is null; hence, the fluid remains at the constantambient pressure pe for ZZZn. Therefore, (40) holds in theinterval Z1rZoZn, and the boundary condition (32) is replacedfor no1 by

p2ðZnÞ ¼ pe: ð47Þ

The velocities of displacing and displaced fluid can then bederived as

v1 t,Z� �

¼fA

t�1=ð1þnÞ

Zða1�b1Z3�nÞ

1=ð1�nÞ ZwrZrZ1

� �, ð48Þ

v2 t,Z� �

¼fM1=n

A

t�1=ð1þnÞ

Z ða2�b2Z3�nÞ1=ð1�nÞ Z1rZoZn

� �: ð49Þ





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The actual value of the front position Z1 in (41)–(49) may bederived by means of the boundary condition at the well (either(30) or (31)), the interface condition (33), and (47).

For b.c. 1) (assigned constant pressure at the well), taking (30)and (47) into account, the integration of (39) and (40) yields,respectively

p1ðZÞ ¼ pw�½Iða1,b1,n,ZÞ�Iða1,b1,n,ZwÞ�, ð50Þ

p2ðZÞ ¼ peþ½Iða2,b2,n,ZnÞ�Iða2,b2,n,ZÞ�, ð51Þ

where (i¼1, 2)

I ai,bi,n,tð Þ ¼

Zt�nðai�bit3�nÞ

n=ð1�nÞdt

¼an=ð1�nÞ

i t1�n

1�n� 2F1

1�n

3�n,�

n

1�n,2ð2�nÞ

3�n,bit3�n

ai

� �, ð52Þ

with 2F1 being the hypergeometric function. Appendix A reportssimpler expressions of (52), valid for certain special values of flowbehavior index n.

On the other hand, (50), (51) and the interface condition (33)give

Dp¼ pw�pe ¼ ½Iða1,b1,n,Z1Þ�Iða1,b1,n,ZwÞ�

þ½Iða2,b2,n,ZnÞ�Iða2,b2,n,Z1Þ�: ð53Þ

Note that in (50) and (53), I(a1,b1,n,Zw) may be set to zerogiven that Zwffi0 since ZwooZ. As the total pressure drop Dp

between well and reservoir is known, the only unknown in theimplicit algebraic Eq. (53) is Z1; once Z1 is determined, Z* is thencalculated through (45), and the pressure distributions behindand ahead the moving interface are evaluated via (50) and (51).

Finally, the injection flow necessary to maintain pw at the wellunder the approximation Zwffi0 is given by

Qw tð Þ ¼ 2phrwv1 Zw,t� �

¼2phf

Aða1�b1Z3�n

w Þ1=ð1�nÞt�ð1�nÞ=ð1þnÞ

ffi2phfa1=ð1�nÞ

1

At�ð1�nÞ=ð1þnÞ, ð54Þ

that is a decreasing function of time for a pseudoplastic fluid.For b.c. 2) (assigned time-variable flow rate at the well), using

(31) and (38) with Zwffi0 yields again (54). A self-similar solutionis possible in this case only when

QwðtÞ ¼Q0t�ð1�nÞ=ð1þnÞ, ð55Þ

with Q0 being the injection intensity defined in (10) and non-dimensionalized via (14). Taking (54) and (55) into account, thevalue of Z1 is determined solving the implicit algebraic equation

a1 Z1,n,f,K� �

¼Q0A

2phf

� �1�n

: ð56Þ

Once Z1 is known, the position of the compression front Z* isderived via (45), while the pressure in the displaced fluid p2(Z) isgiven again by (51), albeit with a different value of Z*. To derivethe pressure in the displacing fluid p1(Z), (39) is integratedbetween Z and Z1, yielding with the help of (34)

p1ðZÞ ¼ peþ½Iða1,b1,n,Z1Þ�Iða1,b1,n,ZÞ�þ½Iða2,b2,n,ZnÞ�Iða2,b2,n,Z1Þ�: ð57Þ

For Z¼Zw, (57) gives the pressure pw(Z) at the injection wellwhen the time variable injection rate is given by (55).

When n41, no pressure front is present and boundary condi-tion (32) holds; integrating (39) and (40) yields for assignedconstant pressure at the well (b.c. 1) an integral which isdivergent in the origin Zw¼0; therefore no similarity solutionexists in this case. For assigned flow rate at the well (b.c. 2), a self-similar solution is possible only for 1ono3 and when the


injection rate is given by (55), which for dilatant fluids is anincreasing function of time. Hence integrating (39) and (40) with(31)–(33) and (55) gives (note that in this case bio0)

p1ðZÞ ¼ peþ½Iða1,b1,n,Z1Þ�Iða1,b1,n,ZÞ�þ½Iða2,b2,n,1Þ�Iða2,b2,n,Z1Þ�, ð58Þ

p2ðZÞ ¼ peþ½Iða2,b2,n,1Þ�Iða2,b2,n,ZÞ�, ð59Þ

where I(a1,b1,n,Z), I(a1,b1,n,Z)1 are given by (52), and for x¼Z,Z1 [50]

I a2,b2,n,1ð Þ�I a2,b2,n,xð Þ ¼

Z 1x

t�nða2�b2t3�nÞ�n=ðn�1Þdt

¼ðn�1Þ

ðnþ1Þð�bÞn=ðn�1Þxðnþ1Þ=ðn�1Þ

�2F1

n

n�1,

nþ1

ðn�1Þð3�nÞ,

nþ1

ðn�1Þð3�nÞþ1,

a2

b2x3�n

!: ð60Þ

Again in (58) and (59) the position of the compression front Z1

is derived solving (56).When n¼1 (a Newtonian fluid displacing another one), the

situation is qualitatively analogous to the dilatant case, and asimilarity solution exists only for assigned constant injection rateQ0 at the well. The position of the interface Z1 can be derived,under the assumption Zwffi0, solving the implicit equation

Z21expðAZ2

1=4Þ ¼ ðQ0=ðpfhÞÞ, ð61Þ

and the pressure field is given by

p1 Z� �¼ pe�

A

4MZ2

1expaAZ2

1

4M

� �Ei �

aAZ2

4M

� �

�A

4Z2

1 Ei �AZ2

4

� ��Ei �

AZ21

4

� �� , ð62Þ

p2 Z� �¼ pe�

A

4MZ2

1expaAZ2

1

4M

� �Ei �

aAZ2

4M

� �, ð63Þ

where �Ei(� � ) is the exponential integral.

3. Results and discussion

In this section we discuss the behavior of the responses ofinterest (i.e., z(t), z*(t) and the pressure increment in the domainwith respect to the ambient value Dp(Z)) as functions ofthe dimensionless model parameters n, M, a, f and K, (a) bymeans of a deterministic analysis, and (b) modeling them asstochastic variables and considering the overall effect of theiruncertainty.

As far as the deterministic analysis is concerned, we startselecting f¼0.2 and K¼1 as a reference case; to grasp theinfluence of relative fluid mobility and compressibility, we thenconsider the following combinations for the mobility and com-pressibility ratios M and a: (I) M¼0.2, a¼0.2; (II) M¼0.2, a¼5;(III) M¼5, a¼0.2; (IV) M¼5, a¼5.

In Fig. 2a–c, the interface location z is depicted as a function oftime for the above combinations and b.c. 1) with pw¼1, pe¼0.1(Dp¼0.9) and n¼0.50,0.67,0.75, respectively; these values coverquite well the range of variation of n for pseudoplastic fluids infield cases [27]and references therein. Inspection of Fig. 2a–creveals that, for given value of flow behavior index n, the interfaceadvances slowly when the displaced fluid is less compressible andless mobile than the displacing one (case I), while it is fastestwhen the displaced fluid is more compressible and more mobile thanthe displacing one (case IV). Upon comparing results for differentvalues of n, it is seen that the interface position is an increasingfunction of flow behavior index. Differences between results at latelimes for different values of n are more pronounced for case I. Thisindicates, at least for the range of values of parameters examined





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Fig. 2. (a) Front position z(t) for injection at prescribed pressure versus time for Dp¼0.9, f¼0.2, K¼1, cases I–IV, n¼0.50; (b) as (a) but n¼0.67; (c) as (a) but n¼0.75; (d)

as (a) but compression front position z*(t); (e) as (d) but n¼0.67; (f) as (d) but n¼0.75.


here, that the maximum displacement for assigned well pressure isachieved with large values of the power law model exponent n, andof the compressibility and mobility ratios a and M. Fig. 2d–f showsthe compression front location z* as a function of time for the sameboundary conditions and cases I–IV listed above, respectively forn¼0.50,0.67,0.75. As expected, the compression front advancesfastest when the displaced fluid is more mobile, but less compres-sible than the displacing one (case III); the compression front isslowest for case II, when the displaced fluid is less mobile and morecompressible than the displacing one. The compression front location


is an increasing function of flow behavior index; in relative terms,this effect is compounded for cases I and III, when the displaced fluidis less compressible than the displacing one. Upon comparingFig. 2d–f with a–c, it is noted that the compression front location isfarther from the interface location when the displaced fluid is lesscompressible than the displacing one (cases I and III). The aboveconclusions hold true for all values of flow behavior index, withdifferences between the two fronts increasing with n.

When assigned flux in the origin is considered (i.e., b.c. 2)), thefront position at a given time is not a function of mobility and





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compressibility ratio, but only of flow behavior index; thus Fig. 3arepresents the front advancement over time for b.c. 2) with h¼1,Q0¼0.2 and n¼0.50, 0.67, 0.75; note that these values of flowbehavior index correspond to injection rates in the origin decreas-ing with time respectively as t�0.33,t�0.20,t�0.14; correspondingresults are thus not strictly comparable; the front advancesfurther for larger values of n at late times, while at small timesthe reverse is true. Fig. 3b–d shows the compression frontlocation z*(t) as a function of time for b.c.2) with the same valuesof Q0 in the cases I–IV listed above, respectively for n¼0.50, 0.67,0.75. As for b.c. 1), the compression front advances fastest in caseIII and slowest in case II; cases I and IV yield the same resultssince the location of the compression front is a function of theratio between mobility and compressibility ratios. As for b.c. 1),the relative distance between the compression front and theinterface location is greatest for case III and smallest for case II.In turn, the distance between the two fronts increases with thevalue of flow behavior index.

Fig. 4a and b shows for b.c. 1) with Dp¼0.9, n¼0.50 and caseII, the effect of a variation of K and f, respectively on the positionof the interface; Fig. 4c and d does the same for the location of thecompression front. It is seen that a permeability increase by afactor of 10 has a significant effect on the interface and compres-sion front position; less so a variation of porosity in the range0.15–0.30.

Finally, Fig. 5a and b shows the behavior of pressure in thedisplacing and displaced fluids, p1(Z) and p2(Z), as a function of Z

Fig. 3. (a) Front position z(t) for injection at prescribed rate versus time for Q0¼0.2,

position zn(t) and n¼0.50; (c) as (b) but n¼0.67; (d) as (b) but n¼0.75.


for selected cases with n¼0.50, M¼a¼5 (case IV), K¼1, f¼0.20;Fig. 5a does so for b.c. 1) with pw¼1 and pe¼0.1; Fig. 5b for b.c. 2)with h¼1, pe¼0.1 and Q0¼0.2. In both cases, note the disconti-nuity in the pressure derivative at the interface location Z1 andthe pressure asymptote at the compression front location Zn.

In the following we consider the stochastic nature of theparameters involved in the proposed model, representing themas independent random variables to exemplify our approach. Thisassumption makes the analysis consistent with the previousdeterministic one and enables to investigate the salient featuresof the proposed solution, not affecting the generality of theapproach. Furthermore, if the spatial variability of some of theparameters involved has to be investigated for specific character-ization purposes, the PCE-based approach can be combined withthe Karhunen–Loeve expansion to represent the stochastic pro-cesses in terms of uncorrelated random variables [51,52].

An hypothetical case study (i.e., two specific fluids and aporous domain) is simulated and the way in which the uncer-tainties associated with the values of the same five parametersinfluence the model responses is analyzed by means of GlobalSensitivity Analysis (GSA) performed through the PolynomialChaos Expansion (PCE) technique (Appendix B). We refer to theuniform distributions reported in Table 1 under a boundarycondition of assigned pressure at the well with pw¼1 andpe¼0.1 (Dp¼0.9).

Fig. 6a and b depicts the mean and associated standarddeviation (a) and the total sensitivity indices (b) of displacement

949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

f¼0.2, K¼1, cases I–IV, and n¼0.50, 0.67, 0.75; (b) as (a) but compression front





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100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

Fig. 4. (a) Front position z(t) for injection at prescribed pressure versus time for Dp¼0.9, case II, n¼0.50, f¼0.2 and K¼1, 2, 5, 10; (b) as (a) but compression front position

z*(t); (c) as (a) but K¼1 and f¼0.15, 0.20, 0.25, 0.30; (d) as (c) but compression front position z*(t).

Fig. 5. Pressure in the domain for injection at prescribed pressure versus similarity variable for n¼0.50, M¼a¼5 (case IV), K¼1, f¼0.2 and (a) pw¼1, pe¼0.1; (b) h¼1,

pe¼0.1, Q0¼0.2.


front position z(t) as a function of time. We find that uncertaintyin the front position increases, as expected, with time, doing solinearly except for very early times; the largest contribution to thetotal variance at any time is due to medium permeability and


porosity in almost equal fashion, while the flow behavior indexcontributes very little; the variance of flow behavior index,initially the highest, exhibits a non-monotonic behavior; com-pressibility and mobility ratios do not play a role. The total





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sensitivity index of permeability and porosity are almost equaland increase from 0% to 50% for dimensionless time around 5 andthen slightly decreases with increasing time. Correspondingly, thesensitivity to flow behavior index is initially close to 100%, thendecreases to almost zero, and again increases with time reaching10%. Fig. 6c and d does the same as Fig. 6a and b for the pressurefront position z*(t). We can observe that, while the variance ofpressure front again increases linearly with time, its value ismuch larger than that associated with the displacement front. Thelargest contribution to variance is here due by far to flow behaviorindex, then to porosity and permeability, and lastly to compres-sibility and mobility ratios. The total sensitivity index of flowbehavior index, initially largest, decreases to almost zero for very

808182838485868788899091

Table 1Intervals of variability of the selected uniformly

distributed random model parameters.

Random variable Distribution

N U(0.40–0.60)

a U(4–6)

M U(4–6)

K U(0.80–1.20)

f U(0.16–0.24)

Fig. 6. (a) Front position z(t) versus time and associated uncertainty calculated with the

(a) but calculated for compression front position z*(t); (d) as (b) but calculated for com


early times, then increases again reaching 60% at late times. Theinfluence of permeability and porosity is almost equal andincreases sharply for very early times, reaches a peak, then slowlydecreases to 10% for late times; the indices of the compressibilityand mobility ratios, almost identical between them, exhibit asimilar behavior but lower values.

An analogous analysis (not shown) for intervals of variabilityof random model parameters smaller (10%) and larger (30%) thanthose reported in Table 1 (20%) reveals a behavior over time oftotal and partial variances of the two fronts qualitatively similarto that shown in Fig. 6, with variance values increasing withincreasing variability. Consequently, the behavior of sensitivityindices is remarkably similar to that shown in Fig. 6.

When the sensitivity to uncertainty of the pressure incrementin the domain Dp(Z)¼p(Z)�pe is examined (not shown), thesensitivity indices exhibit a very irregular behavior, especiallynear the position of the displacement and pressure fronts; forsmall values of similarity variables (small radius/large times), theimpact of flow behavior index is the largest, while approachingthe displacement front the influence of porosity and permeabilityprevails; between the displacement and pressure fronts, theindices of porosity and permeability remain the highest, whilesensitivity to flow behavior index drops to almost zero; theinfluence of compressibility ratio, and, to a lesser extent, mobilityratio increases approaching the compression front.

9293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

PCE of order p¼2; (b) as (a) but total sensitivity indices (ST(O),O¼n,a,M,K,f); (c) as

pression front position z*(t).





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Table 2Comparison between the total variance (V) and partial variances

(V(O),O¼a,M,K,f) calculated for the front position z(t) at selected time instants,


Throughout all calculations first order sensitivity indicesexhibited insignificant differences from total ones, indicatingnegligible interaction among different inputs (Appendix B).

707172737475767778798081828384858687888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132

with the PCE of order p¼2 and with a different number of Monte Carlo iterations

(Nsim¼1000, 5000).

t 1 5 10 15

MC Nsim¼1000 V 4.32E–03 1.26E–02 2.05E–02 2.73E–02

V(f) 2.33E–03 5.52E–03 8.85E–03 1.25E–02

V(K) 2.14E–03 6.03E–03 8.98E–03 1.43E–02

V(M) 4.00E–05 3.00E–04 1.18E–03 3.90E–04

V(a) 4.00E–05 2.80E–04 1.22E–03 4.00E–04

MC Nsim¼5000 V 4.32E–03 1.29E–02 2.00E–02 2.63E–02

V(f) 2.04E–03 6.29E–03 1.03E–02 1.33E–02

V(K) 2.11E–03 6.29E–03 1.04E–02 1.22E–02

V(M) 5.00E–05 2.60E–04 5.80E–04 3.40E–04

V(a) 4.00E–05 2.80E–04 5.80E–04 3.70E–04

PCE p¼2 V 4.31E–03 1.26E–02 2.00E–02 2.62E–02

V(f) 2.16E–03 6.31E–03 1.00E–02 1.31E–02

V(K) 2.14E–03 6.26E–03 9.94E–03 1.30E–02

V(M) 1.00E–05 2.00E–05 2.00E–05 3.00E–05

V(a) 1.00E–05 2.00E–05 3.00E–05 4.00E–05

4. Accuracy and convenience of the PCE-based approach

The PCE-based approach (Appendix B) allows to obtain,through a simple analytical post-processing, all the results pre-sented in the previous section, i.e. when the PCE-surrogate modelis available all the information about the variability of the modelresponse is conserved in the set of expansion coefficients, result-ing in considerable savings in computational time.

In the selected case study, for each model response of interest,we calibrated surrogate models with the PCE at different orders,resorting to the Legendre Chaos space because the uncertaininput parameters are associated with uniform distributions[53,54]. Results obtained through the second-order PCE exhibitednegligible (or very minor) differences with higher order ones(generally 1–10%); thus only results for order 2 are reported.

The reliability of the results obtained through the PCE-basedsurrogate model is here analyzed by comparison against a tradi-tional approach in which the sensitivity indices are estimated in aMonte Carlo (MC) framework; this validation step, not shown inour previous work on non-Newtonian flows [28,49], can beperformed examining a considerable number of realizations, sincea benchmark analytical solution is available; when a complexnumerical model is investigated [34], the excessive computa-tional cost entails a limited amount of MC simulations. Validationis useful to assess: (a) the quality of the algorithm adopted toobtain the PCE approximation, (b) the applicability of the techni-que to this specific model, (c) the extent of computational saving.In particular we show the comparison between the total andpartial variances related to the front position z(t) in the selectedcase Dp¼0.9. Due to the non-negligible computational costassociated with Monte Carlo simulations (about 7 s for eachmodel run, i.e., about 2 h for 1000 iterations for each time instant,on a standard computer with a 2 GHz processor), we exemplifyour approach by considering only four time instants (t¼1,5,10,15)and a fixed value of flow behavior index, n¼0.50; this allowsusing the simpler expressions (A.1) of Appendix A for calculations.Note that this simplification does not affect the following valida-tion approach. The distributions of other random parameters areagain uniform with the same mean values of those reported inTable 1 and with a selected variability of 710% around the meanvalue for each one. Table 2 reports, for the considered timeinstants, the total variance of the model response, i.e., z(t), andthe partial variances due to the uncertainty on f, K, M and a,calculated with the PCE of order 2 and with a different number ofMonte Carlo iterations (Nsim¼1000, 5000). It is observed thatthere is a fine agreement between the variances evaluated viaMonte Carlo simulations and those predicted by the PCE, espe-cially when considering the total variance and the partial var-iances associated with f and K; furthermore the differencebetween the results of the two methods generally decreases asthe number of Monte Carlo iterations increases, even thoughconvergence of Monte Carlo results is not attained. The saving incomputational time is crucial as the calibration of the coefficientsof the surrogate model requires only 15 sampling points in thespace of the four selected uncertain parameters for each timeinstant. This advantage is even more important in the completeGSA discussed in the previous section, in which also n isconsidered uncertain. In that case the number of model runsnecessary for the calibration are 21 and 116 (respectively forsecond and third order PCE) and only the PCE method allows to


investigate the sensitivity of the presented similarity solutionquite continuously in time.

5. Conclusions

A novel analytical solution to non-Newtonian radial displace-ment of a power-law fluid by another in porous media has beenderived in self-similar format under the assumptions of thefrontal advance theory. Our analysis:

(i)

bench), http

extends to motion of two fluids the analytical approach andresults of [28] on flow of a single power-law fluid, takingcompressibility effects into account;

(ii)
may be used as a benchmark for complex numericalmodels;
(iii)
allows to investigate the key processes and dimensionlessparameters involved in non-Newtonian displacement inporous media.The PCE-based approach adopted allows to:
(iv)
perform a complete Global Sensitivity Analysis of thebenchmark solution by considering the uncertainty asso-ciated with key dimensionless parameters involved;
(v)
derive the variance associated with model outputs with noadditional computational cost;
(vi)
obtain accurate results when compared with traditionalsimulations conducted in a MC framework.
Appendix:A. Closed-form results

The hypergeometric function 2F1 in (57) reduces to simpleranalytical functions if n¼ l/(1þ l) where l is a positive integer. Forl¼1, 2, 3, corresponding to n¼1/2, 2/3, 3/4, these are respectively[50]:

Iðai,bi,1=2,tÞ ¼ 2ait1=2�bit3=3, ðA:1Þ

Iðai,bi,2=3,tÞ ¼ 3a2i t

1=3�3aibit8=3=4þbit5=5, ðA:2Þ

Iðai,bi,3=4,tÞ ¼ 4a3i t

1=4�6a2i bit5=2=5þ12aib

2i t

19=4=19�b3i t

7=7

ðA:3Þ

mark solution for non-Newtonian radial displacement in://dx.doi.org/10.1016/j.ijnonlinmec.2013.01.011i




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Appendix: B. Global Sensitivity Analysis (GSA) throughPolynomial Chaos Expansion (PCE)

Any square-integrable random function, y, can be approxi-mated through the Polynomial Chaos Expansion (PCE) techniqueas showed firstly by Wiener [55] for Gaussian processes. The PCEincludes multivariate orthogonal polynomials, Cj, made by anypossible combination of a given set of orthonormal randomvariables fziðyÞg

1

i ¼ 1 [33]:

y¼X1j ¼ 0

ajCjðfzng1

n ¼ 1Þ ðB:1Þ

Considering polynomials of degree not exceeding p and a finitenumber M of independent random variables collected in vector f,the expansion results truncated as follows:

yffi ~yðfÞ ¼XP�1

j ¼ 0

ajCjðfÞ ðB:2Þ

where P¼(Mþp)!/(M!p!) and aj in (B.1) and (B.2) are the expan-sion coefficient.

The Hermite-polynomial basis is suitable for Gaussian pro-cesses; on the contrary for, e.g., uniform processes, the LegendreChaos optimizes the convergence rate [53].

Now let y be the response of a stochastic model y¼ f(x). Here xrepresents the vector of the independent input parameters whosevalues are taken to be uncertain and dimðxÞ ¼M. If y is character-ized by a finite variance it is possible to approximate the modelresponse with the expansion (B.2) [33]. Note that the set ofindependent random variables f in (B.2) is related to x through anisoprobabilistic transform, x¼T(f), necessary if the distribution ofmodel parameters does not correspond to that required by theselected polynomial basis [53,54].

The definition of the PCE approximation of y is attainedthrough the computation of the expansion coefficients aj collectedin the vector 1. This can be achieved following differentapproaches, e.g., by adopting a regression-based method [34,55]:

ArgMinfE½ f ðxðfÞÞ� ~yðfÞð Þ2�,1g: ðB:3Þ

Here the variance of the difference between the PCE approx-imation ~y and the original model y is minimized with respect to 1in a certain number of regression points.

The variability of the original model is preserved in the set ofexpansion coefficients, rendering PCE a powerful tool for GlobalSensitivity Analysis (GSA) or Probabilistic Risk Assessment (PRA).The mean of the model response coincides, in fact, with thecoefficient of the zero-order term, a0, in (B.2), while the totalvariance of the response calculated through PCE is:

V ~y ¼VarXP�1

j ¼ 0

ajCjðfÞ

24

35¼XP�1

j ¼ 1

a2j E½C2

j ðfÞ�: ðB:4Þ

Adopting the Sobol indices as sensitivity measures [31,56,57],it is consequently possible to calculate the generic s-order indexanalytically from the expansion coefficients without additionalcomputational cost [54]:

Si1 ,...is ¼

PaAji1 ...is

a2aE½C2

a�

V ~y, ðB:5Þ

where the numerator represents the partial variance associatedwith ~y, due to the subset fxi1 ,. . .xis g of random model parametersdefined through indices i1,y,is, and j denotes a general termdepending only on the variables specified by the subscript.

Calculation of E½C2a� can be performed following, e.g., Abramo-

witz and Stegun [58].


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[1] Y.S. Wu, K. Pruess, Flow of non-Newtonian fluids in porous media, Advancesin Porous Media 3 (1996) 87–184.

[2] A.M. AlSofi, M.J. Blunt, Streamline-based simulation of non-Newtonianpolymer flooding, SPE Journal 15 (4) (2010) 901–911.

[3] W.R. Rossen, A. Venkatraman, R.-T. Johns, R. Kibodeaux, H. Lai, N. MoradiTehrani, Fractional flow theory applicable to non-Newtonian behavior in EORprocesses, Transport in Porous Media 89 (2011) 213–236.

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Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

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