Models for ﬂow of non-Newtonian and complex fluids - porous media

J. Non-Newtonian Fluid Mech. 102 (2002) 447–473

Models for flow of non-Newtonian and complexfluids through porous media

J.R.A. Pearson∗, P.M.J. TardySchlumberger Cambridge Research, Cambridge CB3 0EL, UK

Abstract

This article first provides a brief and simple account of continuum models for transport in porous media, and ofthe role of length scales in passing from pore-scale phenomena to “Darcy” continuum scale representations usingaveraged variables. It then examines the influence of non-Newtonian rheology on the single- and multi-phase trans-port parameters, i.e. Darcy viscosity, dispersion lengths and relative permeabilities. The aim is to deduce functionalforms and values for these parameters given the rheological properties of the fluid or fluids in question, and theporosity, permeability, dispersion lengths and relative permeabilities (based on Newtonian fluids and equivalentcapillary pressures) of the porous medium. It is concluded that micro-models, typically composed of capillary net-works, applied at a sub-Darcy-scale, parameterised using data for flows of a well-characterised set of non-Newtonianfluids, are likely to provide the most reliable means. © 2002 Elsevier Science B.V. All rights reserved.

Keywords:Darcy viscosity; Newtonian fluids; Transport parameter

1. Scope and preface

This article does not seek to prove a new result or to justify a new technique. Nor is it intended as acomprehensive review. It is restricted to presenting in simple form selected parts of the most importantapproaches that have been used to predict flow behaviour in porous media, giving special attention to theeffects of non-Newtonian rheology. All of the seminal work reported here has been done by others andreference is made in the short bibliography below to a few of the textbooks and papers that are well-knownand have been found helpful in this work. The indulgence is craved of any other authors whose papers ortexts have not been referenced, particularly if they feel that their contributions deserve special mention.

The subject is important in a variety of contexts. This contribution has been prompted by efforts todesign treatment fluids and intervention processes to control the flow of gas, oil and water in hydrocarbonreservoirs. The difficulty of conducting realistic experiments to model every option, and hence the need to

∗ Corresponding author. Tel.:+44-1223-3509-27; fax:+44-1223-3530-99.E-mail address:[email protected] (J.R.A. Pearson).Dedicated to Professor Acrivos on the occasion of his retirement from the Levich Institute and the CCNY.

0377-0257/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0377-0257(01)00191-4

448 J.R.A. Pearson, P.M.J. Tardy / J. Non-Newtonian Fluid Mech. 102 (2002) 447–473

provide computational models for prediction, is overwhelming. Although even qualitative models basedon a basic understanding of the dominant mechanical, physical and chemical processes involved can beuseful, reliable quantitative models are found necessary for successful applications. This article concernscriteria for selecting such quantitative models.

2. Introduction

Many of the fluids used in the oil industry to treat and stimulate reservoirs are significantly non-Newtonian: in differing degrees they display shear-dependence of viscosity, thixotropy (or rheopexy)and elasticity. They are also multi-component and chemically complex, and can interact with the porousformations (sands, carbonates, clays or hard rock) through which they are pumped and with the otherfluids with which they come into contact within the pore structure. These structures are themselvescomplex, displaying inhomogeneities on a wide range of length scales, from 10−6 to 103 m. For quantitativepurposes, good mathematical models of the transport processes involved are necessary. This has in thepast been based on continuum representations.

2.1. Darcy’s law and permeability: Newtonian fluids

The starting point for such continuum models is typically Darcy’s law, relating gradients of pore fluidpressure to the pore fluid flux within a robust, essentially rigid, porous matrix. Symbolically, this is writtenas

u = −(k

µ

)(∇p − ρg) ≡

(− kµ

)∇p (2.1)

whereu, is the Darcy velocity, is a mean velocity of the pore fluid relative to the matrix,p the meanabsolute pressure of the pore fluid,ρ andµ are the density and viscosity of the fluid,g the gravity (orother field force) andk is the permeability of the matrix, with dimensions of length squared. Variousassumptions have been made in this much used model:

(i) a suitable averaging procedure has been prescribed to obtain the Darcy-scaleu andp from the actual‘point’ values of fluid velocityu∗ and pressurep∗ within the matrix;

(ii) the fluid is Newtonian;(iii) the matrix is isotropic;(iv) the flow is slow (in formal termsRep = ρ|u|k1/2/µ 1) and so the same linear law applies at each

instant.

For extended media, it is natural to suppose thatu, pandkcan become functions of positionx and timet.If both fluid and matrix are incompressible—an approximation that will be adopted for simplicity

throughout this account—then the mass conservation law becomes

∇ · u = 0 (2.2)

and the field equation for pressure is

∇ ln

(k

µ

)· ∇p + ∇2p = 0 (2.3)

J.R.A. Pearson, P.M.J. Tardy / J. Non-Newtonian Fluid Mech. 102 (2002) 447–473 449

If the porous medium and the pore fluid are homogeneous, and this was often assumed in early work, thenthe reduced pressurep obeys the harmonic equation. Boundary conditions on either pressure or velocitylead to well-posed problems, which can be solved relatively easily.

It is known that real porous media are not homogeneous, though in practice the spatial variation ofk(x) is at best inferred from a limited series of tests on small samples of the porous medium in question.It is, therefore, conventional to regardk(x) as a stochastic function of position, describable by a limitednumber of numerical parameters defining its statistics (those familiar with classical fluid turbulence will befamiliar with the use of one, two and three point correlations and covariances in this context). Solutionsnow become statistical (probabilistic) rather than exact, and the subtleties of predicting consequences(interpreting results) for given boundary conditions have led to a large and growing literature.

Reservoirs are normally highly stratified and so the scalark has to be replaced by a tensor permeabilityk. Darcy’s law becomes

u = −k · ∇pµ

(2.4)

with consequential complication of Eq. (2.3). This issue will not, again for simplicity, be consideredfurther here.

2.2. Dispersion: Newtonian fluid

If a tracer is present, at some averaged concentrationc(x, t), in the pore fluid, then it is well knownexperimentally that it is dispersed as well as advected by the mean flow fieldu. This dispersion isattributed to the pore-scale variations in the flow field, i.e. tou∗ − u, whereu is the mean fluid velocitywithin the pore space and differs from the Darcy velocityu by a factor 1/m, wherem is the porosity, theaverage fraction of total volume occupied by pores. In our case of a homogeneous isotropic medium, the‘advective–diffusive’ equation forc is conventionally written as

m∂c

∂t+ u · ∇c = ∇ · (D · ∇c) (2.5)

where

D ≈ Λ||uu|u| +Λ⊥

(I|u| − uu

|u|)

(2.6)

Λ||(x)andΛ⊥(x) being the dispersion lengths parallel and perpendicular to the Darcy velocity vector andwhere molecular diffusion is neglected. It is worth noting that the dispersion term, on the right-hand sideof Eq. (2.5), is linear inu as is the advection term on the left-hand side. This means that the change inthe tracer field depends not on the flow-rate, but only on the displacement. For Newtonian fluid flows,Λ|| andΛ⊥ are regarded as further parameters characterising the porous medium.

2.3. Multi-phase flow: Newtonian fluids

If two or more mutually immiscible fluids are present within the porous matrix, they are treated as‘interpenetrating’ fluids (simultaneously present atx in some average sense), and the phase fractionswithin the pore space are called theirI phase saturationss(I ). They are kept separate at the pore-scale by


internal interfaces across which interfacial tensions act. These lead, in some average sense, to capillarypressure differencesp(I,J )cap = p(I) − p(J) between phasesI andJ: clearly, these average quantities arefunctions not only of the porous medium in question, but also of the interfacial tensions between the phases,the wetting characteristics (usually described by contact angles for any given pair of fluids meeting alonga contact line on the pore surface) of the internal pore surfaces, the current saturations and, as has oftenbeen observed, the saturation history within the medium.

If one or more of the phases are driven by pressure gradients∇p(I ), then corresponding averagedvelocitiesu(I ) will arise. For slow flows, the relation between them is written formally, by analogy withEq. (2.1), as

u(I ) = −kk(I )rel

µ(I)(∇p(I) − ρ(I)g) (2.7)

where all the complexity of the situation is wrapped up in the relative permeabilitiesk(I)

rel which, inthe absence of any experimental or theoretical results, have to be regarded as functions of at least thesaturationss(I ), the capillary pressuresp(I,J )cap , the viscositiesµ(I ) and the flow-ratesu(I ), as well as of thepore geometry and nature of the formation.

The mass balance Eq. (2.2) is decomposed into phase relations

m∂s(I)

∂t+ ∇ · u(I ) = 0 (2.8)

It is not surprising that the general situation is so complex that severe approximations are always madewhen modelling multi-phase flows, and that these are chosen to suit the particular flow process involved.Thus, adjustments are made specifically to cover imbibition of water into oil-rich media (typically, cores),drainage of water from water-rich media, partially miscible phases, and flooding of depleted reservoirsby water or gas. Almost all calculations are based on locally one-dimensional flow; if capillary pressuregradients are neglected, as they often are at large-scales, the hyperbolic nature of equations (2.7) and(2.8) means that shocks ins(I ) can arise: these are typical of water floods and so lead to the tracking ofthe ‘fronts’ so defined; flow lines remain approximately perpendicular to pressure and saturation lines.

2.4. Flow of non-Newtonian fluids in porous media

We naturally wish to extend the basic flow Eq. (2.1) to cover fluids with complex rheology. The methodused for definingu and p in terms of any pore-scale flow will not be altered by the rheology of thefluid; it is now assumed, and all detailed pore-scale models confirm, that all non-Newtonian effects canbe subsumed in a suitable definition ofµ, which will continue to have the dimensions of Newtonianviscosity. Much of the rest of this article (Section 4, Appendix A) investigates the way bulk rheologicalbehaviour of the pore fluid will affect thisµeff ; put otherwise, how might we hope to derive the valueof µeff from a set of bulk rheological measurements, givenk and other readily available data? Next, weconsider in Section 5 extending the dispersion relations (2.5) and (2.6) with the hope of redefiningΛ||andΛ⊥ to cover the change in flow field associated with non-Newtonian rheology.

In Section 6, we discuss what changes have to be made to any set of relations found adequate formulti-phase flow of Newtonian fluids. In particular, is it sufficient to use the value ofµ

(I)

eff found relevantfor single-phase flow instead of the Newtonian viscosityµ(I ) in the phase flow (Eq. (2.7)), keepingk(I)rel


unaltered and estimatingp(I,J )cap in the same way as for Newtonian fluids? In Section 7, we consider othereffects, such as adsorption and desorption of viscosifying agents, boundary slip, adhesive layers and massinterchange between phases.

Few firm general conclusions are reached in Section 8. The best that can be hoped for at this stage isto set out some guidelines for those seeking to characterise flow behaviour of non-Newtonian systems inporous media and to analyse particular flow situations.

3. The role of length scales in characterising porous media and flow through them

3.1. Homogeneity in continuum models

How do we derive continuum models that do not require an exact knowledge of the detailed structureof any given porous medium? What do we mean when we say that a porous medium is homogeneous?Can we provide a simple and reasonable criterion for homogeneity?

To answer these questions, we start by noting that any such medium when dry (containing no fluid inits pores) will consist of one continuous region of solid, the matrix, and regions of void, the pores (forsimplicity, here we take the pore region to be continuously connected); these two regions will be separatedby a surface whose configuration (geometry) defines both matrix and pore spaces. We now take a sampleof characteristic dimensionlsample(e.g. the diameter of a spherical sample, the side length of a cube ormore generally the cube root of its volume) and refer pointsx within the sample to an origin0 fixed withinthe sample. We then associate with each pointx a presence function,I1(x), which has the value 1 ifx isin a pore and 0 ifx is in the matrix. The average porosity, a primary characteristic mentioned in Section2.2, is then given by

msample≡ C1 sample=∫

sampleI1(x)dx

volumesample(3.1)

We can also define a joint presence function

I2(x1, x2) = I1(x1)I1(x2) (3.2)

and hence, a mean correlation function

C2 sample(r) =∫

sample∗I2(x1 + r, x1)dx

volumesample∗msample(3.3)

The medium is isotropic and so we can takeC2 to be a function of |r|. The difficulty that arises in definingI2 whenx1 + r lies outside the sample will be addressed further later, and is accommodated formally hereby using the notation sample∗. It is relatively easy to see that higher-order correlationsCn can be definedsimilarly, involving (n− 1) separationsrj . All the Cn are dimensionless.

There is no unique way of defining a characteristic size for the pore space (pores). The simplest toenvisage is to consider the distances between intersections of straight lines passing through the samplewith the pore surfaces, and to take the average length of those segments corresponding to pores; thisdefines one pore size,lpore. A distribution of pore lengths is correspondingly obtained, which would be


formally related to theCn. If

ε0 = lpore

lsample 1, (3.4)

we have two widely separated length scales. The variable

xpore = xlpore

(3.5)

describes rapid variations ofI1 on the pore-scale, while

xsample= xlsample

(3.6)

defines variations on a sample scale. By definitionC1 is a constant, butC2 is a variable in terms of theargumentrpore. We expectC2 to asymptote rapidly tomsamplefor rpore � 1, and so we might expect

lcorr =∫ ∞

0

∣∣∣∣ C2(r)

msample− 1

∣∣∣∣ dr (3.7)

to be of orderlpore (this lcorr could be used as an alternative definition oflpore).We now consider the sample to have been taken from a larger physical block of an extended porous

medium, whose own length scale islblock. If

ε1 = lsample

lblock 1, (3.8)

we are able to take many separate samples from the block. Iflpore, lcorr (orC2(r)), and all higher correlationsprove to be independent1 of sample position in the block, then our block is said to be homogeneous (andevidently isotropic) and vice versa.

Now clearly, as the ratioε0 tends to unity by taking smaller and smaller samples, the sample will appearless and less homogeneous, while from experience in large sedimentary rock masses, by takinglblock

progressively larger, the block will appear to become inhomogeneous. The largest amount of informationabout the porous medium, treated as a continuum, will be obtained by using the widest range of lengthscales, given by the pair{l(min)

sample, l(max)block } 2 , that leads to local homogeneity. On the other hand, for many

problems the extent of the medium involved is so great that little information is available on the scaleof l(max)

block and so for calculation purposes the relevant continuum properties are required on a much largerscale. The issue of how to obtain formal relations between continuum parameters on different lengthscales is an important one, generally called the problem of up-scaling, but will not be addressed here.

3.2. Averaging in porous media

We now return to the question of averaging in porous media to obtain continuum parameters andvariables. In a formal sense, any sample can be regarded as a particular, geometrically defined, region of

1 Strictly speaking, we must expect some statistical variation, but we can accept a variance that is at most of the order of thelarger ofε0 or ε1, or even of a fractional power of these small quantities.

2 l(min)sampleis sometimes referred to as the representative effective volume (REV).


the medium, which remains unaffected by any of the fluids flowing through it. A prime requirement of anycontinuum model is that the parameters and variables defining it should be continuous variables spatially.This is easily achieved formally if the volume (sample, REV) over which averages are taken is thoughtto move continuously through the medium. Thus, a generalised form of Eq. (3.1) can be written as

A(xsample) =∫I1(xpore)W(xpore)A(xpore)dxpore∫

I1(xpore)W(xpore)dxpore(3.9)

where the origin of the co-ordinate systemxpore is atxsampleandW(xpore) is a suitable weighting function.Here, the same symbol,A in this case, is used for volume-averaged and pore-scale variables, the asso-ciated argument determining its meaning. The difficulty mentioned earlier in connection with Eqs. (3.2)and (3.3), and by extension with Eq. (3.7), is now removed becausex2 is now defined outside the av-eraging volume. The situation is different if experimental evaluations are undertaken. Physical samplesare unique, and cannot be moved continuously, so at best a discretised representation (usually, sparse inpractice) can be obtained for real media; however, this form of discretisation happens to coincide withmost computational representations for specific media, e.g. reservoirs or geological formations generally,and so the passage from given experimental data to computational model can in principle be direct.

A difficulty arises at the continuum scale when interpreting boundary conditions, which formally andphysically are applied at pore-scale, i.e. toA(xpore) over a two-dimensional surface. One way out of thedifficulty is to apply them formally at the sample scale, i.e. by writing them in terms ofA(xsample) overthe same surface; it should not be presumed, however, that these boundary conditions would be identicalat the two scales.

An associated difficulty arises when seeking to define the permeabilityk, used in Eq. (2.1), formally;we have assumed, rather than proved, that it will be a material constant, at least over length scales largecompared withlsampleand small compared withlblock; formally, we treatk as a slowly-varying function ofxsample. If we have a physical sample, say cubic of sideLsample, we can apply pressure differences acrosstwo of its faces and measure the fluxQ. We can replace∇p by p/Lsampleandu byQ/L2

sample; knowingµ,k is obtained. If, however, we seek to derive it independently by solving the Stokes equations of motion fora geometrically fully-defined pore-scale model, we soon realise that the pore spaces outside the samplevolume have to be considered when solving for the flow within the sample, particularly, close to the sampleboundary; in order to carry out the averaging (Eq. (3.9)) foru andp, an extended solution is required.

Some authors have sought to avoid this difficulty by prescribing spatially periodic media, where thesample represents one period along each of three principal axes. This has the added advantage that themedium becomes by definition homogeneous; the constantk (or its more general counterpartk) in Darcy’slaw can be calculated for a flow fieldu(xsample),∇p(xsample) which is uniform with respect toxsample;equally all theCn(r1 pore, . . . , r(n−1)pore) will be independent of positionxsample. In moving to the largerscalexblock, it has to be supposed that the assumption of periodicity is only asymptotically valid asε1 → 0,though this can only be done formally, since in any real contextlpore andlblock are given and soε1 andε0

must range between finite limits.

4. The ‘Darcy’ viscosity of non-Newtonian fluids

In practice, the permeability of porous media is obtained by direct measurement of pressure-drop andflow-rate using cylindrical cores from blocks of coherent porous media or lightly compacted cylinders of


unconsolidated granular media. It has been verified that the permeability obtained by using a Newtonianfluid with one viscosity is the same as that obtained using another Newtonian fluid of different viscosity,and that the pressure-drop and flow-rate are linearly related, thus confirming that Stokes law applies atthe pore-scale. This measurement of permeability requires but a single steady-state measurement on afully-saturated cylindrical sample of known length and cross-sectional area, sealed along its sides withentry and exit on plane ends orthogonal to its axis. If the fluid flowing through the medium of knownpermeability is now a non-Newtonian fluid (of known bulk rheology), how do we derive an equivalent ofDarcy’s law?

4.1. Generalised Newtonian fluids

These fluids, often called pseudo-plastic because most are shear-thinning, are such that the (Newtonian)stress/rate-of-strain law

T = µE (4.1)

with T the deviatoric stress tensor and

E = (∇v + v∇T) (4.2)

(twice) the rate-of-strain tensor, can be retained, but where the viscosity

µGN = µ(|E|) (4.3)

becomes a function of strain rate. This representation (withµ properly regarded as a characteristicparametric function of the fluid concerned) is suitable for solution of a pore-scale flow, but it is by nomeans clear how|E| = Eeffshould be defined for use ofµGN in the Darcy-scale relation (2.1). On purelydimensional grounds |u|/k1/2 is seen to be a rate-of-strain, and many authors have supposed that

Eeff = α|u|k−1/2 (4.4)

with α a near-universal constant would be a suitable approximation. On general grounds, we can supposethat any relation (4.3) will yield a pair of characteristic valuesµc, Ec and that Darcy’s law would become

u = − k

µcµ∇p (4.5)

with µ a joint matrix/fluid constitutive function of the reduced shear rate

E = |u|k1/2Ec

(4.6)

Viewed as a material function,µwould depend upon the detailed nature of the pore geometry, to allow forthe possibility that two separate media having the same permeability would distinguish between differentgeneralised Newtonian fluids.

An obvious way to evaluateµ would be to measure its value for a given sample over a suitable rangeof Darcy velocities (or pressure gradients), but this is seen to be impractical for the range of media, fluidsand flow-rates involved in many design procedures.

We are led to consider a series of model media that will reproduce relevant features of real porous mediaas far as permeability is concerned. As we have restricted ourselves to isotropic and homogeneous media,


simple one-dimensional tubular models suggest themselves. In Appendix A (Section A.2), we considerbundles of cylindrical tubes. It immediately becomes clear that it is convenient to replace relation (4.1)by a flow-rateQ versus pressure-dropP relation for a tube. By considering only circular pipes of lengthL and diameterD such a relation is readily obtained by integration, givenµGN(E) (it is interesting to notethat theQ, P relation is obtained directly in capillary rheometry, and thatµGN is derived from it using theRabinowitz differential relation, so that the former could in that case be regarded as more basic). If wetake a Carreau model (see [13], index) for the viscosity function (and this covers a wide variety of elasticand inelastic fluids in uniform steady simple shear flow) we can show that the breadth of the transitionregion from low to high Newtonian asymptotes is wider for a pipe (Q, P relation) than for uniform shear(µGN relation) and wider still for a bundle of pipes having a distribution of diameters. Thus, in general,it can be shown that the apparent viscosityµcµ will depend on

(i) the parameters in the model forµGN;(ii) the geometrical parameters defining each tube;

(iii) the statistical parameters defining the distribution of tube sizes and shapes.

We also consider (Section A.3) bundles of identical tubes with diameters varying along their length.Not surprisingly, we can show that, for model systems having the same values form andk, significantvariations can arise in the predicted apparent viscosity. There is an advantage in this, as we can hope torepresent any given medium, as far as Darcy pressure-drop alone is concerned, by a very simple capillarymodel with very few parameters, for all non-Newtonian fluids. We shall see later that this is not necessarilysufficient to represent other characteristics of the medium, such as dispersion/residence-time distribution.

More elaborate network models, typically on a square or cubical grid, have been used by several authors,often in connection with detailed modelling of two- or three-phase flow. Some of them have nodes ofzero volume connected by circular pipes, and are thus obvious extensions of the tubular models describedabove; others have most of the porosity contained in pores connected by throats which account for all ofthe pressure-drop. We shall not attempt in this article to generate any general results for either of thesenetwork models, though we shall provide in Section A.4 some preliminary results for power-law fluids ina square grid network. It is worth noting one curiosity: a regular square or cubical grid of equal capillariesbehaves isotropically in the case of Newtonian fluids, but not for shear-thinning fluids; thus, when someauthors line up an edge of the square with the flow direction, and others line up a diagonal, they will obtaindifferent predictions for most non-Newtonian fluids. This “cautionary tale” suggests that the geometry ofany real porous medium plays a greater role in determining flow behaviour of complex non-Newtonianfluids than of linear fluids.

The most “realistic” model geometries are described topologically rather than metrically. Pores areconnected to one another by a variable number of links that can have different lengths; a subset of thesepores are associated with the inlet and outlet points or faces. For calculation purposes, they are no moreinconvenient than regular grids.

4.1.1. The power-law fluidMany fluids display a significant power-law region in their viscosity function (4.3), as do the Carreau

models. The approximate form used for analysis over this region can be written as

µGN = µc

(E

Ec

)n−1

, n > 0 (4.7)


We now show that for any porous medium the constitutive function defined in Eq. (4.5) is given by

µ = Cn(E

Ec

)n−1

(4.8)

whereCn is a constant characteristic of the porous medium and of the power-law indexn. The proofsupposes that the steady velocity fieldu(xpore) and stress fieldT(xpore) + p(xpore)I is known for someparticular value of (Q, P) in a cylindrical sample of the medium. We then note that the velocity fieldfuand associated stress fieldf n(T + pI) satisfies the equations of motion at pore-scale, this correspondingto (fQ, fnP) at the sample scale. Eq. (4.8) follows directly.

4.2. Thixotropic fluids

We use the term thixotropic to describe an extension of generalised Newtonian fluids in which theviscosity is determined by the history of the rate-of-strain (many real fluids display such character whensubject to non-constant shear rate in a shear viscometer). Generically and formally, this can be written as

µthix(t) = µGN(E(t))t

Ms=−∞

(E(s)) (4.9)

whereM is a dimensionless functional over past timess. The simplest evolutionary equation forM is

dM

dt= 1 −M

t1(4.10)

with t1 a relaxation time for the fluid in question. An alternative form, supposedly more physicallyintuitive, has been written as

dµthix(t)

dt= µ0

t1

(1 − µthix

µ0− g(E)

)(4.11)

whereµ0 is the zero-shear-rate viscosity andg(E) is a dimensionless destruction function. The corre-sponding steady-shear-flow value ofµthix, µGN, would be given by

µGN(E) = µ0{1 − g(E)} (4.12)

from which Eq. (4.10) is trivially re-derived.This rheological model can be used with the tubular models for the porous medium. It is at once

apparent that the (Q, P) relation in any physical sample will depend, even in steady-state flow, upon theinlet value ofµthix and the dimensionless residence-time

tres = mLsampleAsample

Qt1(4.13)

If tres 1, then the fluid will appear to be Newtonian (assuming that it has been perfectly mixed beforeentry to the tube), with the entering value of viscosity, while iftres � 1, it will behave as a generalisedNewtonian fluid in an infinitely long domain. For intermediate values, the ‘apparent’ sample viscositywill be dependent upon the flow-rate and the sample length. In terms of the discussion of Section 2, theresponse within the averaging volume characterised byl

(min)samplewill depend upon other averaging volumes


(upstream). For a medium that is homogeneous up to the scalel(max)block , a criterion for treating a thixotropic

fluid as a generalised Newtonian fluid is thus:

ml(max)block

uDarcyt1� 1 (4.14)

However, this is not as simple a statement as it seems, because it cannot be assumed that the viscosityrepresented byµGN in Eq. (4.9) will be directly relevant in Darcy flow, nor that the relevant Darcyform will be as simply obtained as Eq. (4.8) even when the steady-state viscosity is given by Eq. (4.7).The difficulties involved in predicting the Darcy behaviour are just as great as those mentioned later inconnection with dispersion.

For cases where Eq. (4.14) is not obeyed, one may expect thatµ in Eq. (4.5) will be given by a functional

µ =t

MDarcys=−∞

(|uDarcy(xsample)|), dxsample(s)

ds= uDarcy(xsample(s)) (4.15)

GivenM as defined by Eq. (4.9), and a sufficiently representative tubular model for the porous medium,MDarcy can in principle be calculated for non-uniform flow fields.

For the particular case of simple bundles of tubes which we can suppose to be of infinite length, it isrelatively simple to consider unsteady flows, so thatuDarcy becomes a function ofs and not ofxsample.MDarcy then follows fairly simply fromM(E), as explained in Section A.2. Whether such an approachwould bear any relation to reality is another matter.

4.3. Elastic fluids

These are fluids which can be represented neither in the form Eqs. (4.1)–(4.3), withµ a scalar, nor inthe form (4.9). Formally, we can write a functional relation for the rheological equation of state:

T(t, a) = t

Ts=−∞

{E(s, a)} (4.16)

whereE(s, a) represents the past history of the rate of strain of the particle that is ata at time t, for ageneral memory-fluid. Typical examples are the evolutionary models of Oldroyd, based on the Jaumannderivative or the upper- and lower-convected Oldroyd derivatives, or the BKZ and Wagner integral models(see, e.g. [7], p. 345 et seq. and 436 et seq. or [13], index). At the pore-scale, flow fields are stronglyaffected by elastic forces which cause the principal directions of stress and rate-of-strain not to be parallel,as they have to be for pseudo-plastic and thixotropic fluids. However, at the Darcy-scale, the relationship(4.15) is expected to remain relevant, because all complexities at the pore-scale will be subsumed in thescalar functionalMDarcy.

Most elastic fluids are characterised by spectra of relaxation (and retardation) time scales, which canbe used as in Eq. (4.13) to determine whether history dependence is important or not. If it is not, it istempting to use the viscometric function for viscosity as defining an equivalent generalised Newtonianfluid for obtaining the effective Darcy viscosity at any flow-rate. Experimental measurements have notbeen conclusive: on theoretical grounds, it can be argued that the very high differences-of-normal-stressand Trouton ratios associated with polymeric fluids will produce increasing values of apparentµDarcy

(based onMDarcy) when flow channels in the porous medium are of rapidly-changing cross-section; insome cases, such sharp rises at the highest flow-rates used have been observed and called shear-thickening


by analogy with the behaviour of fluids, such as suspensions, in which the steady-state-shear viscositysimilarly rises at high shear rates; however, in other cases, the apparentµDarcy appears to decrease aspredicted by use of a viscometricµ(Eeff ) to describe the elastico-viscous fluid.

These two apparently conflicting sets of observations have their counterpart in the way pressure-dropscan be calculated for various network models. If junctions between tubes of different cross-section,including entry and exit points to large pores, are approximated by discontinuities in cross-section, thenpotentially dominant entry/exit pressure losses are predicted; if however, all cross-sectional changes aretaken to be smooth, and the lubrication approximation is invoked, no elastic losses are predicted. Realityprobably lies between these two extremes. Little has been done so far to analyse this possibility.

5. The dispersion tensor for non-Newtonian fluids

5.1. Origins of dispersion

If we consider a uniform fluid flowing through a single long circular–cylindrical tube of radiusa (typicalof micro-models), then a simply-calculable velocity distributionu(r), r the radial co-ordinate, arises forany generalised Newtonian fluid. If at any time,t = 0 say, a tracer of concentrationc0 is added to theentering fluid, it will be advected by the fluid. At a later timet > 0, the tracer particles will have moveda distance

xtr = u(r)(t − te) (5.1)

down the tube, wherer represents the radial position at which they entered at timete. It may easily beshown for a Newtonian fluid that the cross-sectional average concentration in the tube will be

c = c0(1 − x/L), x ≤ L0, x > L

}(5.2)

where

L = 2ut (5.3)

andc0 is a constant.This is the result of pure advection and neglects the effect of molecular diffusivityD on the tracer,

whose magnitude relative to advection is often associated with the Peclet number (Pe) for mass transport

Pe(m) = ua

D(5.4)

This is usually thought to be large (but foru = 10−4 m/s, a = 10−5 m, D = 10−9 m2/s, values notirrelevant to possible micro-models,Pe(m) = 1). However, a more informative dimensionless parameteris the mass Graetz number (Gz)

Gz(m) = ua2

Dl(5.5)

wherel is the length of the tube; this recognises that the time taken to be advected along the tube isO(l/u) and the time to diffuse across the tube is O(a2/D); Gz(m) is thus a ratio of times, smallGz


denoting diffusion dominance, largeGzdenoting advective dominance. Taylor has shown that the effectof diffusion dominance, achieved by makingl � a, is to lead to an apparent tracer diffusivity

DTaylor = a2u2

48D(5.6)

in the x-direction (this can be rationalised in dimensional terms by considering the mean advectivelengthL/2 achieved at timea2/D using Eq. (5.3); we now equate the advective flux 1/2uc0 at x = L/2to an apparent diffusive flux based on the concentration gradient (5.2) achieved at this position,i.e. to DTaylorc0/(2ua2/D), so that they sum to the total flowuc0, giving DTaylor ∝ u2a2/D as inEq. (5.6)).

These arguments are based on pore-scale kinematics, which we believe will remain important if we areto predict dispersive effects at a Darcy-scale, as in Eq. (2.6) for a porous medium. ClearlyD in Eqs. (2.5)and (2.6) does not have the same form asDTaylor in Eq. (5.6), while an advective effect like Eq. (5.2) isnot apparent. Why is this?

The answer almost certainly lies in the fact that the continuous streamlines (fixed in space) that mustcharacterise steady flow in a porous medium “sample” (almost) the full range of actual point valuescharacterising the velocity field in an extended porous medium: for part of their length, they will passthrough regions of high velocity at the centre of throats and for other parts they will be arbitrarily close tostagnation points. Thus, the mean velocity in the direction of the pressure gradient along any streamlinewill differ from uDarcy/m by an amount that decreases with streamline length; equivalently the distancetravelled along any streamline will grow more and more linearly with time. The stochastic nature of thegeometry of most porous media ensures that these departures approximate to random perturbations and soa diffusive term arises in the Darcy-scale continuum relation for mass transport, while there is no purelyadvective variation, scaling linearly as in Eq. (5.3). This process is called mechanical dispersion and isdominant even whenGz(m)samplebased onl = lsampleanda2 = k is large. The addition of cross-streamlinediffusion (based onD) will tend to decrease the values ofΛ|| andΛ⊥ In Eq. (2.6), although as Aris hasshown a relatively small value ofPe(m) will mean that a termD has to be added toDTaylor andDI/T(m) toD, whereT(m) the tortuosity for mass diffusion is of order unity.

If the fluid is Newtonian, then unsteadiness in the overall flow will not affect mechanical dispersion;low Peeffects will scale accordingly.

The first detailed papers on this subject were published by Saffman [11], though they were largelyqualitative; because he effectively assumed complete mixing at the inlet to each capillary the workreported inevitably led to estimates for dispersion lengths scaled by the length of the capillaries and sorepresented a version of the chemical engineer’s notion of a residence-time distribution in a sequence ofmixers.

5.2. Effect of fluid rheology on dispersion

It is quite clear that the main effect will be associated with its effect on the pore-scale velocity field,and hence, on the steady streamline pattern. If we consider a long tube (of lengthL), then a result similarto Eq. (5.6):

DTaylor = Nn−N u2a2

D(5.7)


will arise, but whereNn−N will be a function of the rheology of the fluid and the mean flow-rate (Darcyvelocityu); if we can characterise the fluid rheology for simple shear flow by a set of characteristic times[tn], n = (1,2, . . . , N), tn > tn−1:

Nn−N = Nn−N(utn

a

)(5.8)

This scalar relation will be applicable provided

Pe(m) � utN

a, i.e. D a2

t(5.9)

which implies that

utN

L 1 (5.10)

which is effectively the criterion for thixotropic or elastic fluids to behave as generalised Newtonian fluidsin Darcy flow, whenL is identified withlsample.

However, this is quite different from the effect onΛ|| andΛ⊥, which in principle has to be separatelycalculated. There is very little information in the literature about this aspect. We have noticed in our workthat the effective persistence length of a streamline before it is “mixed” with neighbouring streamlinesincreases as the fluid becomes more shear-thinning.

5.3. Fingering in unstable displacement

This is an effect that is observed in all cases of displacement of high viscosity fluid by one of lowerviscosity. It is usually treated as a continuum phenomenon, i.e. one where the scale of the fingers oflow viscosity fluid advancing into the higher-viscosity fluid is at least of the order of the Darcy scale,and is often compared with the (more carefully investigated) fingering observed in Hele–Shaw cells(involving low-Reynolds number (Re) flow between two closely-spaced parallel flat plates). It can beobserved in both miscible and immiscible displacements, i.e. when the displacing fluid is miscible orimmiscible with the displaced fluid. For miscible fluids, ultimate mixing, i.e. mixing at the pore-scale,depends upon molecular diffusion of one fluid into the other, or of the viscosifying agent when the solventis the same in both fluids; for immiscible fluids, interfacial tension will act directly at the pore-scale, andthrough capillary pressure gradients at the Darcy-scale (interfacial tension is the crucial factor in mostHele–Shaw cell flows). Both mixing and interfacial tension will affect the detailed mechanics of fingering.We concentrate here on miscible fingering.

Detailed study of pore-scale models shows that fingering can occur at all scales (specifically, thosebelow the Darcy-scale) in the sense that the less viscous fluid must move faster than the more viscousfluid when subject to the same pressure gradient, and evidently so when diffusion can be neglected. Thus,advective extension of the region over which mean3 concentration or saturation varies, of the type givenby Eqs. (5.2) and (5.3), can be expected. Molecular diffusion will now play a significant role in convertingthis advective process into a Taylor-like diffusion process, wherea in Eq. (5.6) must be re-interpreted as

3 The mean is to be taken as an areal average over planes orthogonal to the direction of displacement, the area cutting manyfingers.


finger widthwf . The changeover between advected and diffusive behaviour will occur at lengths of theorderuw2

f /D at most, since lateral dispersion will add to the molecular contribution envisaged by Taylor,and so, as the distance covered by any front increases, the characteristic width of fingers can be expectedto increase, until lateral restrictions constrain them. We have not been able to discover any definitivetreatment of this phenomenon, even for the highly homogeneous case wherelblock � lsample, lf (lf beingthe length of the fingers) and where the fluids are both Newtonian; in real porous media, the fingeringpattern is often determined by heterogeneity, so thatwf tends to be linked to the correlation scale ofm(xblock) as well as to the distance travelled by the front,Xfront. The one parameter of clear importanceis the ratio of effective viscosities:

M12 = µ1

µ2(5.11)

Pseudo-plasticity will have its most direct effect in alteringM12 whereµ1 andµ2 are now to be interpretedasµGN1(Eeff1) andµGN2(Eeff2). A difficulty arises at once because fingering itself suggests thatE will belarger for the less viscous than the more viscous fluid; in the miscible case, where the absolute magnitude,e.g.Cn in Eq. (4.8), rather than the shear-rate index, e.g.n in Eqs. (4.7) and (4.8), is altered by concentrationof the viscosifying agent, fingering will be intensified by pseudo-plasticity, e.g.n < 1.

Thixotropy, allied to pseudo-plasticity withn < 1, might be expected to stabilise large well-establishedfingers against tip splitting or lateral fingering instabilities, though there is no evidence for this that weknow of. Elastic fluids, that show apparent shear thickening in the Darcy representation, e.g. of the typedescribed approximately by

µ(Eeff) = µ0(1 + cSTDe2) (5.12)

where

De = u l−1poretelas (5.13)

cST is a dimensionless parameter, andtelas the characteristic relaxation time, can be expected to berelatively stabilised against the fingering instability, particularly ifDe based on the mean displacementvelocity leads tocSTDe2 ∼ 1 for at least one of the fluids.

6. Multi-phase flow of non-Newtonian fluids

6.1. The Buckley–Leverett approximation

Capillarity acts to distribute two or more phases within the pore space at the pore-scale, so that at restthey attain equilibrium in a time of orderµk1/2/σ (for µ as large as 1 Pa s,k as large as 10−12 m2 andσas small as 10−3 Pa m, the equilibration time will be<1 s). For most oil and gas reservoirs containingresidual brine, the equilibration time for samples large enough to make Darcy’s law valid becomes oforderµlsample/σ (if lsampleis 10−2 m,σ is 10−2 Pa m andµ is 10−3 Pa s, then the equilibration time is still<1 s) and is such that continuum equations assuming local equilibrium can be used.

For large-scale flooding experiments aimed at making O(1) changes to the saturations in an ex-tensive porous medium, the distancel over which the saturation changes from its initial to its finalvalue is often large enough for the expected capillary pressure gradients caused by the saturation


gradients to be small compared to the viscous pressure gradients4 given by the modification (Eq. (2.7))to Darcy’s law, and so at this scale capillarity might be neglected in the sense that saturation—ratherthan the distribution of the phases at pore-scale and minimum Darcy-scale—is determined by masstransport governed by the relative mobilitiesk(I)rel /µ

(I) while all thep(I ) can be taken equal at any givenpoint.

The set of Eqs. (2.7) and (2.8) together with∑s(I) = 1 (6.1)

for thes(I ), u(I ) andp is known as the Buckley–Leverett approximation and is used in many reservoircalculations.

6.2. Relative mobilities for non-Newtonian fluids

Since the equilibrium distribution of the phases at any given saturations is determined by capillarityand wettability, it may be assumed that non-Newtonian rheology is not significant at that level. It will ofcourse affect the rate of attainment of equilibrium, and in cases where the equilibrium is itself known to bedependent on saturation history, it may change the final state. However, these variations are not specificto non-Newtonian fluids and are similarly affected by a varying viscosity ratio for purely Newtonianfluids: as they are not generally considered important in the latter case, they may be neglected in theformer.

The interesting question is whether a knowledge ofk(I)

rel (as a function of saturation) for Newtonianfluids and ofµcµ, defined in Eq. (4.5), or alternatively ofMDarcy, defined in Eq. (4.15) for simple shearflow in the same medium, is sufficient to provide effective values for the relative mobilitiesk

(I)

rel /µ(I),

now to be functions ofs(I ) and |u(I )| in the case of non-Newtonian fluids.To try answering this question, we note that the pore space available for flow of phaseI at any given

saturations should, according to the argument given in the first paragraph of Section 6.2, be that which isdetermined by capillary and wettability forces, and therefore not rheology dependent. These pore spaceswill not be the whole pore space, but their basic geometrical characteristics in terms of flow resistancecan be supposedly given byk(I)rel . To derive the relevant value ofµ(I ), we note that the effective flow-ratein the reduced pore space now available to phaseI can be taken as |u(I )|s(I ), and use this value as |u| inthe already knownµ, or MDarcy.

6.3. Dispersion and fingering in multi-phase flows involving non-Newtonian fluids

In practice, these are important issues, primarily because complex treatment fluids tend to be veryviscous and so back-flow of reservoir fluids involves a large adverse mobility ratio, and because pore-scalemixing of inhomogeneous fluids can lead to unexpected and rapid variations in fluid properties and hence,to sharp gradients in continuum parameters describing the flow. Almost all models used are specific toparticular observed phenomena; they tend to be empirical rather than rigorous.

4 A suitable ratio is given byσk1/2/µlu—for σ = 10−2, k1/2 = 10−6,µ = 10−3, l = 102, u = 10−6, all in SI units, this gives avalue of 0.1.


7. Other effects in flow of complex fluids through porous media

The model media introduced in Section 4and the Appendix A were described and analysed for simplicityin the context of homogeneous fluids with bulk properties subject to a no-slip boundary condition at allmatrix or tube surfaces.

In practice, many of the fluids having non-Newtonian rheology are multi-component fluids with variouschemical species in solution; these can be ionic, polymeric and/or surfactant in various combinations.They can interact with the matrix at the pore walls; when more than one phase is present, they canexchange between phases or adsorb at mobile interfaces. These mass transfers can have long or shorttime scales relative to those mentioned earlier. They can alter the interfacial tension between phases, thewettability and slip at the matrix surfaces, and the bulk rheology of the fluid phases.

All of these complex properties can, in principle, be studied outside the porous medium, providingthe required solid–liquid interfaces can be prepared at laboratory scale. Their effect on micro-scalemodels can be studied directly by physical measurements at a pore-scale, or by suitable algorithms insimulations. Experiments on samples of porous media can provide information on their Darcy-scaleeffects. It does not follow, however, that observed results on real porous media could be predictedfrom micro-models selected to match behaviour for fluids not exhibiting these complex properties; theseadditional properties and their effects may well lead to further conditions that the micro-model wouldhave to meet.

8. Discussion and conclusions

Despite many attempts to derive continuum models from pore-scale information, whether formaland theoretical or directly experimental, there is little agreement on how to predict the behaviourof rheologically-complex fluids when they flow through porous media. Since most applications ofcontinuum theory use Darcy’s law or its straightforward extensions, most procedures for predictionattempt to do so by prescribing an effective viscosity for single-phase flow and relative permeabil-ities in multi-phase flow. Data on the steady-state viscosity function as a function of shear rate,obtained from bulk experiments, shifts the choice of effective viscosity to the choice of an effectiveshear rate, which is assumed to depend on flow-rate, permeability and porosity in a simple and uniformfashion.

Given that the pore space in a porous medium is geometrically very complex with different media dis-playing very different structures, it seems unlikely that simple universal algorithms for deriving effectiveviscosities from bulk data will be successful. Introduction of a detailed stochastic description of real poregeometry might be a way of parametrising the porous medium, but the labour of doing this, particularlywhen the significance of any such parameters is still obscure, makes it an unattractive option, at thisstage. The alternative that has been used is to simplify the geometry by using micro-models, analoguesbased on interconnecting, easily parameterised, unit channels placed on a regular lattice. This digitisedrepresentation is ideally suited to numerical (computational) techniques, which allow direct averagingof micro-model quantities to yield values of continuum (Darcy-scale) variables. Simple “exact” flowsolutions based on “exact” fluid rheology as well as “exact” wall effects exist for unit channels and sothe passage from pore-scale to continuum-scale variables becomes purely algorithmic, and thus, feasibleon modern computers.


The problem then becomes one of choosing the right distribution of components for the analoguegeometry. The proposal that we make here is that this distribution should be constrained to satisfyas many observations as can be made on the same or equivalent samples of a given porous mediumas possible. These observations should involve a limited number of test fluids covering the requiredrange of different rheological behaviour, over such a range of steady flow-rates that the full rheologicalcomplexity of each fluid be sampled, using tracer tests to measure dispersion lengths and static overallmolecular-diffusion effects to yield one measure of tortuosity. Comparison of forward predictions forcontinuum properties using a candidate micro-model (for the sample porous medium concerned) shouldsuggest optimal ways of meeting the required constraints, namely the observed data. We have decidedthat the best procedure is to disregard dispersion in the first instance, and to restrict attention to obtainingmodels that successfully predict pressure-drops; this has the pragmatic advantage that pressure-drops andflow-rates are much easier to measure accurately than residence-time distributions, while the dependenceof the latter on sample length can be more significant than for pressure gradients. Extending the modelto cover dispersion depends crucially on mixing at nodes: it remains to be discovered whether a singlesimple network model can predict dispersion lengths for fluids with widely different rheologies usingsimple algorithmic rules for particle tracking. This is described briefly in Section A.4.

Once an optimal micro-model has been derived, it could then be tested on a new fluid in similar flowconditions, and then be used to predict unsteady displacement flows of miscible fluids. If predictionsmatched observations for both tests, then confidence in the model would be sufficient to use it for routinepredictions until it could be shown to fail significantly in some particular application involving a hithertoinsignificant physical or chemical process. It would then be timely to increase the number of constraintson the initial model to include the new physical or chemical process.

One of the intermediate steps that has been interposed successfully is the use of small-scale physicalembodiments of the micro-models to confirm experimentally that the correct mechanics, physics andchemistry was being used at the micro, i.e. unit, scale in the computational model. Comparisons can beundertaken both at the detailed single-unit scale, and at the continuum, i.e. model-averaged, scale. Thisis of particular importance when trying to model capillarity in two- and three-phase flows and in trackingfronts in displacement flows. A related approach is to use tomographic techniques to follow flows in beador sand packs of suitable materials at a sub-Darcy-scale. These partially-averaged measurements on realmedia provide an independent and potentially crucial test of micro-scale simulators, both computationaland physical. They are expected to provide crucial information about dispersion, and how it should bemodelled.

A further issue which has to be mentioned is that most macro-scale flows in porous media are affected byheterogeneity and anisotropy. So continuum models for prediction of such flows have to include reliablemethods for up-scaling when computer simulators are used. As far as the effects of non-Newtonianbehaviour are concerned, the arguments given in the previous sections can almost always be adapted toapply to up-scaling techniques developed for simple Newtonian or two-phase flows.

For further reading see [1–6,8–10,12].

Appendix A. Micro-models

In this appendix, we consider simple micro-models that can be used to make predictions for Darcy-scalebehaviour. For simplicity, at this stage, we shall consider micro-models based on networks of intercon-


necting capillary tubes of fixed length, but different diameter; without loss of generality they can be placedon a finite cubical (square, or linear) lattice taken to represent a minimal REV. Nodes, i.e. points whereup to 6 capillaries meet (4 for two-dimensional, 2 for one-dimensional models), will be assumed to haveno volume. A porosity can be calculated for such a system.

For single-phase flow of a Newtonian fluid, an instantaneous linear flow-rate/pressure gradientrelationship, for flow between opposite faces of a cubical REV held at constant pressures, can read-ily be calculated, given the diameters of all the capillary units within the REV, based on Poiseuille’s lawfor flow in a single capillary. This provides a computed permeability. Modifications to cover exit/entrypressure losses associated with nodes can also easily be made. Analogous calculations for generalisedNewtonian fluids will be more complicated (but perfectly feasible), because the analogue of Poiseuille’slaw is flow-rate dependent, leading to a flow-rate and rheology-dependent effective Darcy viscosity.

For thixotropic or elastic fluids the entry state (given by the past history of flow) at each capillary entryhas to be specified. This is a non-trivial matter if exact histories of fluid elements are to be calculatedaccording to Eqs (4.8) and (4.9). Some simplification can be achieved if (a) a dimensionless residence-timein a single capillary based on Eq. (4.13) using the capillary length instead of sample length is either verylarge or very small, and (b) fluid entering a node is assumed to be fully mixed at nodes. This giveseither a generalised Newtonian response in each capillary (dealt with in the previous paragraph), ora Newtonian response that is slowly-changing. By imposing periodicity on the micro-scale model, aneffective steady-state Darcy viscosity can be obtained by iteration using the exit state from one REV asthe entry state for the next. This yields a behaviour that is analogous to that represented by Eq. (4.15).For elastic fluids, exit/entry losses (at nodes) that increase much more rapidly with flow-rate than thesimple-shear capillary-flow losses can be included.

Algorithms to predict dispersion are more difficult to devise if they are to be realistic, particularly ifthe massPeandGzbased on a single capillary are neither very large nor very small. The results given inSection A.4 are effectively based on smallGz.

Simple capillary models are not thought to be sufficiently realistic when modelling two- or three-phaseflow in real porous media, for which various pore/throat models have been used. Polygonal cross-sectionsfor throats allow a range of steady saturations within them, thus, leading to direct predictions of relativepermeability.

A.1. Flow in a single capillary

For Newtonian fluid of viscosityµN, Poiseuille’s law can be written as

µN = πd4G

128q(d,G), G = ∇p (A.1)

whered is the capillary diameter,q is the flow-rate at a pressure gradientG, and where the distinctionbetweenp and p will no longer be made. For a generalised Newtonian fluid satisfying Eq. (4.3), thecorresponding relation is

µGN(sw) = τw

sw= πd4G/32

{3qGN(d,G)+G(dqGN/dG)} (A.2)

whereτw is the wall shear stress andsw the wall shear rate. Normally, we think ofµGN(sw) as thebasic rheological function. However, for our purposes, it is simpler to use the functionqGN(d, G) or its


inverseGGN(q, d) as the primary rheological function characterising our fluid (this is the case in capillaryviscometry anyway). For example, for a power-law fluid, the relation (4.7) is replaced by

qGN =(q0

G1/n0

)G1/n (A.3)

whereq0/G1/n0 is a single dimensional constant, provided by any pair of values (q, G) for the capillary

concerned.The advection and diffusion caused by flow through such a capillary has been analysed for a Newtonian

fluid in Section 5.1. Very similar calculations can be carried out for any GN fluid. The definitions of thePeandGzcan be retained.

A.2. Flow in a parallel bundle of capillaries (variable-bundle model)

We consider now a set of capillaries of variable length5 lI and diameterdI , (dI+1 ≥ dI , lI+1 ≥ lI ifdI+1 = dI ), I = 1, . . . , N . We take them to be representative, at some level, of unidirectional Darcy flowin a homogeneous isotropic medium. We, thus, suppose them to represent the pore space in a cubicalREV of lengthl ≤ lI (to allow for tortuosity) for allI, each one starting on one face and ending on theopposite face. The porosity of the REV will be

mVB =∑I

πd2I lI

4l3(A.4)

For a given pressure-dropP across the entry and exit faces, the total flow-rateQ for a GN fluid will be

QVB =∑I

qGN

(dI , P

lI

)(A.5)

and so the ratiok/µ in Eq. (2.1) can be readily obtained for any particular fluid characterised byqGN. Ifwe choose a Newtonian fluid, of arbitrary viscosityµN and specify the setdI , lI , l, thenmVB andkVB

can be obtained. Ifmandk are known for a particular sample, then these represent two scalar constraintson the set characterising the bundle. Noting thatk1/2 has dimensions of length, a suitable dimensionlessformulation would use the quantities

dI = dI

k1/2, lI = lI

l(A.6)

as model parameters and

uVB = kQVB

l2qGN(k1/2, PVB/l), ∇pVB = PVB

lGGN(QVB/l2m, k1/2)(A.7)

as the corresponding Darcy-scale variables, all expected to be O(1). Eq. (A.4) becomes

m =∑I

πd2I lI

(k

4l2

)(A.8)

5 The requirement that all capillaries should have the same length introduced above can be maintained by supposing that eachof the capillaries is composed of a large but variable number of equal basic capillaries.


Since we expectk/l2 = O(ε20), whereε0 is defined by Eq. (3.4), the sum

∑I will be over a large number

of capillaries. For formal calculations it is convenient to replace the sum by an integral over a distributionof capillary diametersd and lengthsl, which are then parameterised by a small number of dimensionlessconstants. A simple example is given by

prob(d, l) = N1 exp{(d − d0)2} exp{(l − l0)2}, l0 > 1 (A.9)

whereN1 is a number yet to be specified. Eq. (A.8) is then replaced by

4ml2

k= N1

∫ ∞

1l exp{(l − l0)2} dl

∫ ∞

0d2 exp{(d − d0)

2} dd (A.10)

which can be taken to defineN1 in terms of 4ml2/k and l0, d0. The dimensionless analogue of Eq. (A.5)provides a further constraint on the distribution given by Eq. (A.9). For a Newtonian fluid, linearityimplies that the integral will be computable and it will introduce a single relation betweenl0 andd0. Ifwe were then to carry out a permeability test on a known power-law fluid, this would provide a furtherconstraint using the integral form of Eq. (A.5), and sol0 andd0 would at best be uniquely defined, thealternative being that no physically-acceptable solution were possible.

What is clear is that in general no simply defined capillary bundle model would satisfy flow measure-ments made using a large number of different generalised Newtonian fluids in equivalent samples of thesame real porous medium. To do so would require careful choice of a large number of parameter pairsdI , lI .

It is tempting to try to match dispersion data also. The residence-time distribution for any given parallelbundle model can be calculated. If the flow is purely advective, then the arguments of Section 5.1 canbe used to give a result similar to Eq. (5.2) withc representing a cross-sectional average concentrationacross all the tubes at any 0< x < l. For any given tube we takexI = xlI ; c will be a function of1 − (x/umaxt) whereumax is the maximum value ofuI max/lI , uI max being the maximum velocity intubeI.

If the meanGz

Gz(m)VB = QVBk

l3D(A.11)

is very small, then Taylor diffusion is relevant in each tube, from which a single-tube and hence aparallel-bundle residence-time can be calculated. If the distribution ofdI and lI are sufficiently closeto somed0,l0, then the residence-time distribution would be dominated by some mean Taylor diffusioncoefficient

D(m)Taylor ∝ Q2

VBk

l4D(A.12)

which as we have argued in Section 5.1 would not be consistent with the observed dispersion tensorsDproportional to mean velocityQVB/l2 as in Eq. (2.6). Broadening the(dI , lI )distribution would accentuatemean advective spread, but it is unlikely that this would provide the result (2.6).

On the other hand, if the REV dimensionl of the micro-model were taken to be many times smaller thanthe lengthlsampleover whichD were estimated, then the required dispersive model could be attained bysupposing all the flow in the micro-model REV to be instantaneously mixed each time it passed throughthe exit plane of the next identical REV model. This would not make any alteration to permeability


calculations, but would determine a particular value ofl/k1/2 that best fit observations. This argumentwould be strongest if the relevantGz(m)VB , based on thel fitting the sample dispersion measurements,were large.

Thixotropy could be included equally coarsely by supposing that the rheology was uniform at anycross-section in a tube, and that it was slowly varying in any tubeI according to Eq. (4.10) with(d/dt)I =uI (d/dx)I andEI = 2uI /dI . The mixing leading to dispersion would provide a volume-averaged valuefor M, based on the exit valuesMI for one REV as the entry values for the next. The calculation ofpressure-drops and flow-rates for each tubeI in any given REV would be more complex than in the caseof a GN fluid.

Elasticity is difficult to introduce in parallel tube bundles because there is no change in cross-section ofany of the channels; this could be done by introducing, purely to account for elastic entry and exit losses,a specified number of large pores along the length of each capillary.

A.3. Flow in a bundle of equal capillary tubes of slowly-varying diameter (equal-bundle model)

This represents an alternative way of matching permeability, porosity and tortuosity measurements insamples. Each of the tubes is now defined by a total lengthl0 ≥ l, composed in the digitised form bya connected set ofN capillaries of diameterdI and lengthlI . By definition

∑I lI = l0. The porosity is

defined by

mEB = nπ

4l0

∑I

lI d2I (A.13)

wheren is the number of tubes/unit area. IflI � dI , lubrication theory as implied by Poiseuille’s lawcan be applied here to each of theN component capillaries.qEB is now constant for all tubes and so

PEB =∑

lIGGn(qEB, dI ), QEB = NqEB (A.14)

This relation obviously covers all GN fluids. The arguments given in Section A.2 concerning thixotropicfluids can be applied to this case without significant alteration.

Elastic pressure losses can be introduced directly at each change in cross-section within any tube, i.e.after each (lI , dI ) unit. If a known loss can be attributed to any given fluid in passing from one diameterdIto anotherdI+1at any givenqEB, then the additional pressure losses in a REV of lengthl can be calculated.These losses which we can write formally asPelas(d1, d2, qEB) would yield

PEB =∑I

{lIGGN(qEB, dI )+ Pelas(dI , dI+1, qEB)} (A.15)

instead of Eq. (A.14). This is the most obvious advantage of the equal-bundle (EB) over the variable-bundle(VB) model. Conversion into an integral form for the elastic case would be rather elaborate and unphysical,though it would be reasonable for inelastic fluids.

Clearly, a combination of the two models would provide extra flexibility in matching data and as weshall see might be a useful simplification, for calculation purposes, of the network model described inthe next section.


A.4. Flow in a network of capillaries

The capillary-bundle models described in Sections A.2 and A.3 are potentially perfectly adequate forone-dimensional Darcy flows of single-phase fluids in homogeneous isotropic porous media. As argued atthe end of Section A.2, even dispersion data could be matched whenl lsample; the mixing process given iseasily applied to giveD||. A more elaborate process involving the output from laterally-neighbouring REVsmixing in specified proportions with the central REV to form the input for the next axially-neighbouringREV would be needed to modelD⊥. The use of tortuous (lI > l; l0 > l) capillaries in the (VB, EB)models can in principle match pure molecular diffusion data for tortuosity. Given rheological data in the(q, G) form for any specified fluid, the computational demands for forward prediction (calculation of theP, Q relation for givendI , lI , l) and inversion (calculation of optimaldI , lI for givenP, Q) are as modestas they could be.

The case for using networks, whether two- or three-dimensional, is not clear at this point. For simplicity,regular networks with equally-spaced nodes suggest themselves. On simple geometrical grounds, it is themultiple connectivity of the nodes that seems best to fit with pore-scale flow: tortuosity, now defined interms of mean lengths of flow paths, is achieved by “paths of least resistance” having lateral as well asaxial steps; mixing at nodes provides a simple approach to dispersion based on largePecovering bothaxial and lateral components; it provides far greater flexibility in modelling two- or three-phase flow; itsgreatest advantage lies in the possibility of following fingering at a pseudo-pore-scale. The concomitantof greater flexibility is increased computational demands, particularly for forward prediction in the caseof highly-non-linear GN fluids, when repeated iteration of matrix solutions is involved to give a singleP, Q value; this imposes a corresponding penalty on the even more computationally-intensive inversionprocess.

The case for networks must, therefore, be based on using micro-models to represent multi-phase flowand small-scale fingering in porous media, the details of which are beyond the scope of this article. Somepreliminary results for power-law fluid flow in square networks are given in Section A.4.1.

A.4.1. Flow of a power-law fluid in a two-dimensional capillary networkWe consider a power-law fluid, as defined in Eq. (4.7), and a regular two-dimensional capillary

network mapped on a rectangular Cartesian grid (co-ordination number 4). The radius of each capil-lary of the network is randomly chosen from a capillary-radius probability distribution function (pdf).For this network to be an effective model of a particular natural porous medium, for which flow datafor the power-law fluid under consideration are assumed to be available, the first parameter toreproduce is the permeabilityk. Clearly, an infinite number of the pdf parameters can give rise to thesame value ofk. One particular choice would be to have all the capillaries with the same radiusr1

(model 1), while another particular choice would be to have them distributed betweenr2 min andr2 max

(model 2). As we will see in the following, we cannot expect both models to match experimental dataobtained with the power-law fluid. Therefore,k is not sufficient to describe the flow of the power-lawfluid.

We are still left with a large number of degrees of freedom to attempt matching of additional data.Parameters from the pdf, geometry of the network, co-ordination number, length of capillaries suggestthemselves. In order to gain insight into which parameters matter, it is useful to study the forwardproblem whereby a given power-law fluid flows into a given network. The two-dimensional networks andassociated boundary conditions, which have been studied here, are illustrated in Fig. 1. The methodology


Fig. 1. Illustration of two-dimensional networks considered in this study. Capillaries are mapped onto a rectangular and regulargrid of 256 (flow direction) by 32 capillaries. Each capillary has a length of 200�m. Capillary radii are picked randomly accordingto one of the five pdf plotted on the right. A uniform inlet pressure is applied at both inlet and outlet, flow-rate is then computedfor different values of the inlet pressure, giving rise to apparent rheograms in porous media when measuring viscosity withEq. (2.1). Distribution 4 corresponds to the case where all capillaries have the same radius equal to 10�m.

for solving flow in such networks in described in ([12], p. 195). In the following, we present some generalobservations and measurements made with the numerical networks:

1. It can easily be proved and observed numerically (Fig. 2), that a power-law fluid still behaves as apower-law fluid at the network scale, and the apparent power-law exponent is the same as the bulk one:

u ∝ (2p)1/n

Fig. 2. Example of apparent rheology of three different power-law fluids as determined by a network model (distribution 1).The value of the slopes for these three lines is exactly the value of the bulk power-law exponent of the corresponding fluid. Theapparent viscosity is the value of the viscosity determined by Darcy’s law (2.1) when everything else is known. Shear rates arelinearly approximated from Darcy velocities through Eq. (4.4) with a fixed value ofα.


Fig. 3. Evolution of flow paths with power exponentn, n = 1 (top, Newtonian case),n = 0.1 (middle),n = 0.01 (bottom),network based on distribution 2. These pictures are generated by ‘dropping’ particles at the inlet and following them according tothe mean velocity in each capillary. At a junction between capillaries, the probability of going into a given capillary is proportionalto the flow-rate in this capillary. Each of these pictures has been obtained by using 10,000 particles, the colour indicating thefrequency of passing through a given capillary. White indicates that a large number of particles have passed through this capillary(more than 1000) while black indicates 0 occurrences.

Fig. 4. Tortuosity distributions along of flow paths for different pdf, in the Newtonian case and forn = 0.01. Tortuosity ismeasured by following a particle as it flows in the network and by recording the length of its flow path. This graphs shows thatdifferent pdf give rise to different tortuosity distributions and that power-law fluids follow paths of lower tortuosity as illustratedin Fig. 3.


Fig. 5. Evolution of the correction factorα (as defined in Eq. (4.4)). This factor is used to correct the Darcy velocity in order tocalculate what the shear-rate would be for the measured Darcy viscosity. This shear rate is called the apparent shear-rate.

2. The tortuosity and number of flow paths decreases with decreasing power-law exponentn, as illustratedin Figs. 3 and 4.

3. The correction factorα, as defined in Eq. (4.4), varies with both pdf and power-law exponentn (Fig. 5).4. The higher the mean tortuosity of the network (i.e. the mean tortuosity of flow paths for any Newtonian

fluid), the higher the correction factor (Figs. 4 and 5).5. For a given network, the effective pdf, as determined by following particles along flow paths, does not

vary significantly with the power-law exponent (Fig. 6).

Fig. 6. By recording the radius of each capillary visited by a particle, an effective pdf can be determined illustrating whichradii have been the most frequented. This is such an example with distribution 3 showing that this effective pdf does not varysignificantly with the power-law exponent.


From these observations, we conclude that the paths followed by power-law fluids have a pathwisepermeability very close to that of the paths followed by Newtonian fluids. Power-law fluids seem to followpaths of least tortuosity and not of least resistance in the permeability sense. It is now clear that model1 (or any capillary bundle model without tortuosity) is not a good choice as the tortuosity of its flowpaths is always unity, whatever the fluid. The medium’s tortuosity appears to be an important propertythat all models have to satisfy if they are meant to simulate flow of a power-law fluid; this is in line withthe model (Eq. (A.2)). This preliminary study suggests that an understanding of the relationship betweenthe effective pdf, which does not vary significantly with power-exponent, the initial geometrical pdf andthe tortuosity distribution of a porous medium, will enable one to constrain network models capable ofmodelling flow of Newtonian, power-law and possibly all generalised Newtonian fluids.

References

[1] P.M. Adler, Porous Media, Butterworth-Heineman, London, 1992.[2] P.M. Adler, H. Brenner, Transport processes in spatially periodic capillary networks: Parts I–III, Phys. Chem. Hydrodyn. 5

(1984) 245–297.[3] R. Aris, Mathematical Modelling, Dispersion in Flow, Academic Press, London, 1999 (Chapter 7).[4] G.I. Barenblatt, V.M. Entov, V.M. Ryzhik, Theory of Fluid Flows Through Natural Rocks, Kluwer Academic Publishers,

Dordrecht, 1990.[5] J. Bear, Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, Kluwer Academic Publishers,

Dordrecht, 1990.[6] J. Bear, J.M. Buchlin, Modelling and Applications of Transport Phenomena in Porous Media, Kluwer Academic Publishers,

Dordrecht, 1991.[7] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, 2nd Edition, Wiley, New York, 1987.[8] F.A.L. Dullien, Porous Media, Academic Press, New York, 1992.[9] C.M. Marle, Multiphase Flow in Porous Media, Technip, Paris, 1981.

[10] G. de Marsily, Quantitative Hydrogeology, Academic Press, New York, 1986.[11] P.S. Saffman, A theory of dispersion in a porous medium, J. Fluid Mech. 6 (1959) 321–349;

P.S. Saffman, Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries, J.Fluid Mech. 7 (1959) 1 94–208.

[12] K.S. Sorbie, Polymer-improved Oil Recovery, Blackie, Glasgow, 1991.[13] R.I. Tanner, Engineering Rheology, 2nd Edition, Oxford, 1988.

Models for ﬂow of non-Newtonian and complex fluids - porous media

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