miscible flow through porous media richard booth

8/9/2019 Miscible Flow Through Porous Media RICHARD BOOTH

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Miscible Flow Through PorousMedia

Richard Booth

Linacre College

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Trinity 2008


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Acknowledgements

I would like to begin by thanking my supervisors John Ockendon and ChrisFarmer for their continued advice and support. I would like to thank Schlum-berger and EPSRC for their nancial support. I would also like to thank theentire OCIAM community for making postgraduate life so enjoyable.

Finally I would like to thank my parents for their kindness and generosity.


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Abstract

This thesis is concerned with the modelling of miscible uid ow throughporous media, with the intended application being the displacement of oilfrom a reservoir by a solvent with which the oil is miscible. The primarydifficulty that we encounter with such modelling is the existence of a ngeringinstability that arises from the viscosity and the density differences betweenthe oil and solvent.

We take as our basic model the Peaceman model, which we derive from rstprinciples as the combination of Darcys law with the mass transport of solventby advection and hydrodynamic dispersion. In the oil industry, advection isusually dominant, so that the Peclet number, Pe, is large.

We begin by neglecting the effect of density differences between the two uidsand concentrate only on the viscous ngering instability. A stability analysis

and numerical simulations are used to show that the wavelength of the insta-bility is proportional to Pe 1/ 2, and hence that a large number of ngers willbe formed. We next apply homogenisation theory to investigate the evolutionof the average concentration of solvent when the mean ow is one-dimensional,and discuss the rationale behind the Koval model. We then attempt to explainwhy the mixing zone in which ngering is present grows at the observed rate,which is different from that predicted by a nave version of the Koval model.We associate the shocks that appear in our homogenised model with the tips

and roots of the ngers, the tip-regions being modelled by Saffman-Taylornger solutions.

We then extend our model to consider ow through porous media that areheterogeneous at the macroscopic scale, and where the mean ow is not one-dimensional. We compare our model with that of Todd & Longstaff and alsomodels for immiscible ow through porous media.

Finally, we extend our work to consider miscible displacements in which bothdensity and viscosity differences between the two uids are relevant.


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Contents

1 Introduction 11.1 Oil recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Primary and secondary oil recovery . . . . . . . . . . . . . . . . . . 21.1.2 Enhanced oil recovery . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Miscible displacements . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Single-phase ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Derivation of Darcys law from steady Stokes ow . . . . . . . . . . 51.2.2 The Hele-Shaw cell analogue . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Homogenisation of the permeability . . . . . . . . . . . . . . . . . . 10

1.3 Two-phase ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 The Muskat problem . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2 Stability analysis of a planar interface . . . . . . . . . . . . . . . . . 141.3.3 The Saffman-Taylor nger solution . . . . . . . . . . . . . . . . . . 161.3.4 Regularisation of the Muskat problem . . . . . . . . . . . . . . . . . 17

1.4 Other models used in the oil industry . . . . . . . . . . . . . . . . . . . . . 201.4.1 The Peaceman model . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.2 The Buckley-Leverett equation . . . . . . . . . . . . . . . . . . . . 211.4.3 The Koval model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.1 Molecular diffusion of solvent and oil . . . . . . . . . . . . . . . . . 241.5.2 Taylor diffusion in a capillary . . . . . . . . . . . . . . . . . . . . . 251.5.3 Saffman dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.6 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.7 Statement of originality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8 A short note on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 The Peaceman model 33

2.1 Derivation of the Peaceman model . . . . . . . . . . . . . . . . . . . . . . . 342.1.1 Physical properties of mixtures of uids . . . . . . . . . . . . . . . . 34

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2.1.2 Homogenisation of miscible Stokes ow . . . . . . . . . . . . . . . . 352.1.3 The Peaceman model . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1.4 The Muskat limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.2.1 The many-uid Muskat problem . . . . . . . . . . . . . . . . . . . . 552.2.2 The linearised problem . . . . . . . . . . . . . . . . . . . . . . . . . 592.2.3 Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . . 602.2.4 The work of Hickernell and Yortsos . . . . . . . . . . . . . . . . . . 64

2.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Unidirectional ngering 793.1 Background Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2 Homogenisation of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3 The Koval model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.3.1 Limitations of the nave Koval model . . . . . . . . . . . . . . . . . 873.4 The homogenised problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.4.1 The Wooding problem . . . . . . . . . . . . . . . . . . . . . . . . . 933.4.2 The Yortsos problem . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.5 Tip rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.5.1 Order-one aspect-ratio tip-rescaling . . . . . . . . . . . . . . . . . . 1013.5.2 Concluding remarks on tip-rescaling . . . . . . . . . . . . . . . . . . 111

3.6 Dynamic scaling of nger-width . . . . . . . . . . . . . . . . . . . . . . . . 1113.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4 Multidirectional ngering 1154.1 The two-dimensional Koval model . . . . . . . . . . . . . . . . . . . . . . . 1164.2 Determination of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.3 Comparative studies for miscible and immiscible ow . . . . . . . . . . . . 1284.3.1 The Todd & Longstaff model . . . . . . . . . . . . . . . . . . . . . 130

4.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Gravitational effects 1385.1 The Muskat model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2 The Peaceman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.2.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.2.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 151


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5.3 The unidirectional nger model . . . . . . . . . . . . . . . . . . . . . . . . 1525.3.1 Scaling of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.3.2 Homogenisation of unidirectional ngers . . . . . . . . . . . . . . . 154

5.3.3 Closure of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 1565.4 Oblique viscous/gravity ngering . . . . . . . . . . . . . . . . . . . . . . . 162

5.4.1 Ellipticity of pressure problem . . . . . . . . . . . . . . . . . . . . . 1675.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6 Conclusions 1716.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

A Numerical algorithms 176A.1 Elliptic problem for the pressure eld . . . . . . . . . . . . . . . . . . . . . 176A.2 Advection of solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178A.3 Diffusion of solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

B Brinkman porous media 182

Bibliography 186


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List of Figures

1.1 A pad of solvent between the water and oil. . . . . . . . . . . . . . . . . . 41.2 Denition of V in a periodic medium. . . . . . . . . . . . . . . . . . . . . . 61.3 Cross-sectional view of two-phase ow in a Hele-Shaw cell. . . . . . . . . . 14

1.4 The Saffman-Taylor nger. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Oil and water are both present within the pores. One phase may close off

some of the pores, affecting the ow of the other phase. . . . . . . . . . . . 211.6 Relative permeability curves as used in the Buckley-Leverett equations. . . 221.7 A schematic of Saffmans capillary network. Particles move determinis-

tically through capillaries before randomly following a new streamline ateach junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1 Viscosity of a mixture of solvent, with a mobility ratio of 10 (the viscosityis nondimensionalised with the viscosity of the oil). Note that the depen-dence of the viscosity on the fraction of solvent is nonlinear, and a smallconcentration of solvent mixed with the oil leads to a large reduction inviscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2 The unit cell, V , of a periodic medium. . . . . . . . . . . . . . . . . . . . . 372.3 Streamlines for ow past a periodic array of cylinders. In Figure 2.3a the

mean ow is aligned with the array of cylinders so that the streamlines areperiodic. In Figure 2.3b the mean ow is slightly misaligned to the array of cylinders and the streamlines are not periodic. For large Pe l the effectivelongitudinal diffusion for 2.3a is much larger than the effective diffusion for2.3b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4 Viscosity prole in the inner region, where R = O(1). . . . . . . . . . . . . 532.5 Muskat problem with many uids . . . . . . . . . . . . . . . . . . . . . . . 552.6 Changing dispersion relation over time, as given by Woodings stability

analysis (2.56). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.7 A typical form of the function Q(X ). The dashed line shows 20 and the

dotted line shows dodX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.8 Turning points of Q(X ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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2.9 Sketch of the dispersion relation as given by (2.71). . . . . . . . . . . . . . 722.10 Schematic of the computational domain. . . . . . . . . . . . . . . . . . . . 732.11 Initial condition for the concentration of solvent, with red representing the

solvent (c = 1) and blue representing the oil ( c = 0). The (barely visible)changes to the concentration at the interface initiate the ngering instability. 74

2.12 Numerical simulation of the Peaceman model for an unstable miscible dis-placement. Here M = 5 and Pe = 2000, with red representing the lessviscous solvent (c = 1) and blue representing the more viscous, oil ( c = 0). 74

2.13 Numerical simulation of an unstable miscible displacement for various val-ues of the Peclet number. Here M = 5 and t = 0.5, with red representingthe less viscous solvent (c = 1) and blue representing the more viscous, oil

(c = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.14 Log-log plot of the number of ngers as a function of the Peclet number,

M = 5, t = 0.5. In these simulations we have assumed that the diffusionis isotropic and constant. The dashed, red line represents the line of bestt to the numerical results, and has a gradient of 0.48. . . . . . . . . . . . 76

2.15 Numerical simulations of an unstable miscible displacement with slightlydifferent initial conditions. Here M = 10, Pe = 2500 and t = 0.5, withred representing the less viscous solvent ( c = 1) and blue representing the

more viscous, oil (c = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.16 Transverse average of the numerical simulations presented in Figure 2.15,

with each different colour representing a different simulation . . . . . . . . 77

3.1 Idealised morphology of numerical simulations of the Peaceman model. Thered line represents a contour for a high concentration of solvent and theblue line represents a contour for a low concentration of solvent. . . . . . . 80

3.2 A cartoon depicting an idealised nger, with the red line representing acontour for a high concentration of solvent and the blue line representinga contour for a low concentration of solvent. The large aspect-ratio modelis only valid in region II. In the root region (I) and the tip region (III) themodel breaks down and further analysis is required. . . . . . . . . . . . . . 81


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3.3 Transversely-averaged concentration proles of two-dimensional simulationsof the Peaceman model. The red and blue dotted lines represent the trans-verse averages of two simulations, with small random differences in the

initial data. In the simulations we have taken M = 4, so that M e = 1.417,and Pe = 2000. The black dashed line represents the (incorrect) predictedconcentration prole when the nave Koval model is applied. The blackunbroken line represents the predicted concentration prole of the modi-ed Koval model in which Kovals effective mobility ratio is applied. Thelarge red dots represent the ends of the mushy region, as predicted by themodied Koval model. The ends of the mushy region as predicted by thenave Koval model are outside the plotted range. The transverse average

of the two numerical simulations appear to approximately coincide witheach other, but the nave Koval model vastly overpredicts the spread of themixing zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.4 Transversely-averaged concentration proles of two-dimensional simulationsof the Peaceman model. In the simulations we have taken M = 20, so thatM e = 2.404, and Pe = 2000. The representation of each curve followsthose of Figure 3.3. The large red dots represent the ends of the mixingregion, as predicted by the modied Koval model. The large red asterisks

represent the boundary between the solvent region and mushy region, aspredicted by the nave Koval model, with the boundary between the oilregion and mushy region far outside the plotted area (at ( x t)/t = 19).As in Figure 3.3 the nave Koval model vastly overpredicts the spread of the mixing region, but the modied Koval model is reasonably effective atcapturing both the size of the mixing region and the average concentrationprole within. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5 Numerical simulations of solutions of (3.23) in the moving frame = x t.Red represents the less viscous solvent ( c = 1) and blue represents the moreviscous oil (c = 0). The dashed, black lines represent the locations of theshocks representing the root and tip regions. We have taken M = 2 andPe = 1/ 100 for this simulation. . . . . . . . . . . . . . . . . . . . . . . . . 92

3.6 Characteristic diagram for the Wooding solution with the initial conditionc1(x, 0) = 0, even at x = 0. The dashed lines represent characteristics, andalong each line there are two coincident characteristics. A discontinuity atx = 0 persists for all t, but characteristics do not cross and no shock is

formed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95


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3.7 Characteristic diagram for the Wooding solution with initial conditionc1 = 0 for x 0 and c1(0, 0) = 0. The dashed lines represent characteristicsand the double, unbroken lines represent shocks. Along the vertical dashed

lines there are two coincident characteristics. . . . . . . . . . . . . . . . . . 953.8 Prole of c0 and c1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.9 The concentration of solvent in solution to (3.24). The solution is obtained

from (3.35) and (3.36) with = 0.5, = 1, = 1 and t = 3. Redrepresents the less viscous solvent ( c = 1) and blue represents the moreviscous oil (c = 0). The dashed, black lines represent the locations of theshocks representing the root and tip regions. Compare with Figure 3.5. . . 98

3.10 Plot of v and M e, the predictions for the speed of the leading nger-tip of

[63] and [33] respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.11 Illustrations of the three possible scenarios for the structure of the ngering

near to the leading nger-tip/s. Some leading nger-tips are shown insideblack circles and some interior nger-tips are shown inside red circles. Inthe second gure, depicting periodic leading nger-tips, the period betweenthe nger tips is shown as W . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.12 Numerical simulation of (3.23) in a moving frame = x t, with a small,articial longitudinal diffusion term. Red represents the solvent ( c = 1) and

blue represents the oil ( c = 0), and we have taken M = 2, and Pe = 1/ 200for this simulation. The initial condition in 3.12a consists of a leading ngersurrounded by two trailing ngers, and we observe that, as time progresses,the tip of the leading nger is pinched-off, so that eventually these ngersare of approximately equal length. . . . . . . . . . . . . . . . . . . . . . . . 110

3.13 Numerical simulations of the Peaceman model with M = 20, Pe = 2000.Red represents the solvent ( c = 1) and blue represents the oil ( c = 0). Notethat in addition to the horizontal spreading of the mixing zone, the mean

nger-width is also increasing with time. Inside the circled regions one canobserve an example of the convective coalescence that is observed to beresponsible for the increase in the mean nger-width. . . . . . . . . . . . . 112

4.1 Schematic of a miscible displacement, with solvent introduced from an in- jector well (solid red dot) and recovered from producer wells (red asterisks).The oil and solvent are separated by a large ngered (or mushy) region inwhich there are many long ngers that point in the direction of ow. . . . 116


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4.2 The red dots are a plot of 1/ against c, as dened by (4.23), and de-termined by taking the transverse average of simulations of the Peace-man model, as found in section 2.3. The black line is given by 1/ (c) =

M e + ( M M e)c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3 Plot of the harmonic average of viscosity as a function of the average con-

centration, for a linear ood with M = 25, Pe = 1500 and t = 0.3. The reddotted curve represents the average of our 2-dimensional simulation of thePeaceman model, and the solid, black curve shows the expression given in(4.25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4 A plot of the effective relative permeabilities that may be identied withthe Koval model (4.32)-(4.33), with M = 10 so that M e = 1.8817. Note

that kro is neither monotonic decreasing, nor less than one! Compare withFigure 1 .6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.5 The red dotted line is the transverse average of a numerical simulation of the Peaceman model (see section 2.3) with Pe = 2500, M = 5 and t = 0.5.The solid black line is the prediction of the Koval model (3.21), and thesolid blue line is the prediction of the Todd & Longstaff model with = 2/ 3.131

4.6 Plot of the harmonic average of viscosity as a function of the average con-centration, for a linear ood with M = 25, Pe = 1500 and t = 0.3. The red

dotted curve represents the average of a 2-dimensional simulation of thePeaceman model, and the solid, black curve shows the expression given in(4.25). The blue solid curve shows the prediction of the Todd & Longstaff model when we take = 1 log(M e)/ log(M ) to ensure that the modelpredicts the correct value of c. . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.1 Gravity ngering in carbon sequestration. Liquid CO 2 is injected at thebottom of the oil reservoir. As the CO 2 is lighter than the surroundingwater it rises until it becomes trapped beneath an impermeable cap rock.The carbon dioxide begins to dissolve into the surrounding water producinga dense mixture of water with dissolved CO 2. Gravity ngering then occursbetween the dense, CO 2-rich water and the pure water lying beneath. . . . 139

5.2 Numerical simulation of the carbon sequestration problem described inFigure 5.1. The colour represents the density of any water present, withred representing the highest density where the water is saturated withcarbon dioxide and blue representing the lowest density where there isno carbon dioxide present. Image courtesy of Walter Sifuentes (ImperialCollege/Schlumberger), simulation uses ECLIPSE software. . . . . . . . . . 140


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5.3 Solvent displacing oil in a porous medium in the presence of gravity. . . . . 1415.4 Dispersion relation for an interface which is stable for small wavenumber

disturbances, but unstable for moderately large wavenumber disturbances,

e.g. = 1 and M = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.5 Schematic of the computational domain. . . . . . . . . . . . . . . . . . . . 1515.6 Numerical simulation of equations (5.10), (5.11), (5.15) where viscous n-

gering is being damped by gravitational effects. Here M = 10, = 0.9and Pe = 2000, with red representing the less viscous, denser uid ( c = 1)and blue representing the more viscous, lighter uid ( c = 0). . . . . . . . . 152

5.7 Numerical simulation of equations (5.10), (5.11), (5.15) where viscous n-gering is enhanced by gravitational effects. Here M = 3, = 0 .5 and

Pe = 2000, with red representing the less viscous, lighter uid ( c = 1) andblue representing the more viscous, denser uid ( c = 0). . . . . . . . . . . . 153

5.8 Geometry of unidirectional problem . . . . . . . . . . . . . . . . . . . . . . 1535.9 Condition for existence of rarefaction wave solutions . . . . . . . . . . . . . 1585.10 Numerical simulations of equations (5.10), (5.11), (5.15) for gravity-driven

ngering. Red represents the lighter uid ( c = 1) and blue represents theheavier uid (c = 0). We have taken Pe = 2000 for these simulations. . . . 161

5.11 The transversely-averaged concentration proles of the simulation shown

in Figure 5.10. The black, unbroken line represents the average of thetwo-dimensional numerical simulations and the dashed, red line shows thetheoretical solution of the averaged concentration as given by (5.44) withe = 0.223. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.12 The transversely-averaged concentration proles of the numerical simula-tions shown in Figure 5.7. The black unbroken line represent the average of the two-dimensional numerical simulation and the dashed, red line showsthe theoretical solution of the averaged concentration as given by (5.46). . 163

5.13 Region of ell ipticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.1 Denition of pressure and ux in numerical scheme. . . . . . . . . . . . . . 177

B.1 Dispersion relation of disturbances to a free surface moving through aBrinkman porous medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 185


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Chapter 1

Introduction

This thesis is primarily concerned with the modelling of the ow of miscible uids througha porous medium. The motivation behind studying these ows lies in the recovery of oilin a process known as miscible displacement, in which a solvent, such as a short-chainhydrocarbon or pressurised carbon-dioxide, is injected into the oil reservoir. We shall seethat such ows are subject to a ngering instability, which leads to difficulty in accuratecomputation of the miscible displacement process. Although throughout this thesis ourfocus will be on miscible ow, we will spend some time, primarily in this introductorychapter, discussing immiscible ow, so that we are able to make comparisons between the

two mechanisms.We begin this introduction by giving an overview of the oil recovery process, and a

more detailed explanation of the difference between miscible and immiscible ows. Weintroduce the standard model for ow of a single uid through a porous medium, andinvestigate some simple free boundary problems for the displacement of one uid byanother. Strictly speaking these free boundary problems are only applicable to immiscibledisplacement, but they will be found to be limiting cases of miscible ood ows that shedlight on the mechanism by which the ngering instability arises. We then give a brief

review of some of the models applied in the oil industry which we shall revisit throughthe remainder of the thesis. We conclude the introduction with an overview of diffusionin porous media, which will be an important physical mechanism in many ows involvingmiscible displacement.

1.1 Oil recovery

Crude oil is found trapped inside the pore-space of rock formations several kilometres

below ground. Various techniques have been developed to enable the recovery of oil tothe surface, which we now outline.

1


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1.1 Oil recovery 2

1.1.1 Primary and secondary oil recovery

The simplest form of oil recovery is primary oil recovery, in which a well is drilled intothe oil reservoir and the oil is pushed up the well due to the naturally high pressure inthe oil reservoir. The high pressure is the result of denser rock and water resting abovethe level of the lighter oil. Pockets of trapped compressed gas may also contribute to thehigher pressure. As oil is produced from the well, the natural reservoir pressure dropsand the ow of oil to the surface may become greatly reduced. Some techniques thatare used to maximise the amount of oil recovered during this primary phase include theuse of pumps to lift oil up the well, and the use of explosives and high pressure pumpsto fracture the rock formation. In a typical oil reservoir only around 10% of the totalamount of oil available can be recovered by primary oil recovery.

In many oil reservoirs, once primary oil recovery has ceased, more of the oil may stillbe recovered by secondary oil recovery. In secondary oil recovery, water is injected intothe oil reservoir through one well, displacing the oil so that it can be extracted from aneighbouring well. Since the water is less viscous than the oil and the permeability of the rock is often highly heterogeneous, the time before breakthrough - i.e. when thewater nds a path between the injection well and the production well - is often veryshort. When a large fraction of water is being extracted from the production well, thesecondary recovery process becomes uneconomical. The amount of oil that may thus beextracted depends heavily on the structure of the rock in which the oil is contained andthe properties of the oil to be extracted; however, even under optimal conditions morethan half of the total amount of oil available will usually be left behind after primary andsecondary recovery is complete.

1.1.2 Enhanced oil recovery

With large amounts of oil remaining unrecovered in mature oil reservoirs, the oil recovery

industry has developed numerous techniques to extract this oil, referred to under thecatchall phrase enhanced oil recovery. The one feature that most enhanced oil recoveryschemes share is that they attempt to alter the physical properties of the oil.

For some applications, thermal recovery is appropriate. Here the oil is heated either bythe injection of steam or by in-situ combustion, where a controlled combustion is startedunderground using the oil itself as a fuel source. Heating the oil reduces its viscosity thusaiding recovery. Thermal recovery is particularly suited to the extraction of extremelyviscous heavy oils [53], for which the amount recovered by primary and secondary oil

recovery is particularly small. When the temperature of heavy oil is increased, its viscositydramatically decreases, vastly improving the effectiveness of primary and secondary oil


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1.1 Oil recovery 3

recovery techniques.During secondary recovery, it is possible to add various chemicals to the injected

water. The chemicals added are chosen to either increase the viscosity of the water orto be surface active and reduce the surface-tension at the interface between the oil andwater.

An increasingly important method of enhanced oil recovery, and one which forms thebasis for this thesis, is miscible displacement. Miscible displacement involves the injectionof a uid which, unlike water, is miscible with the oil.

1.1.3 Miscible displacements

There are two advantages of using a uid which is miscible with oil. Firstly, there are nosurface-tension effects, and so the two uids may more freely displace each other withinthe porous medium. Secondly, the introduction of a miscible uid, less viscous than theoil, leads to a mixture with a viscosity less than that of the oil, thus reducing the appliedpressure gradient required to displace the oil, and aiding recovery. Before we continue withmore details of the industrial process, we make precise the difference between miscibleand immiscible uids.

1.1.3.1 What is meant by miscible and immiscible ?

Two uids are dened as miscible if the molecules of the one uid are free to mix with themolecules of the other uid. There is no interface between two miscible uids. A commonexample of two miscible uids is water and ethanol. In any proportions it is possible tomix the water and ethanol together to form a single homogeneous phase. When two gasesmeet, they are always miscible; for example oxygen and nitrogen readily mix in air.

Two uids are dened as immiscible if the two uids scarcely mix at all at the molecularlevel, and not at all at the macroscale. The two phases remain distinct and there is a

well-dened interface between the two uids. Common examples of two immiscible uidsare water and most vegetable oils.

It is possible for two uids to be neither completely miscible nor completely immiscible.The molecules of the rst uid may mix with the molecules of the second uid until theconcentration of the rst uid reaches a certain saturation concentration. The saturationconcentration may be affected by the temperature and pressure of the system. A commonexample of two partially miscible uids is water and carbon dioxide. Some of the carbondioxide may dissolve in the water but there is still a well-dened interface between the

carbon dioxide gas and the water containing dissolved CO 2. At room temperature andpressure, the amount of dissolved carbon dioxide will be reasonably small, but at the


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1.1 Oil recovery 4

Water Mixture of

solvent and oil Oil

Figure 1.1: A pad of solvent between the water and oil.

temperatures and pressures experienced deep in an oil reservoir, there may be a signicantquantity of carbon dioxide dissolved in the water. At sufficiently high pressures carbon

dioxide and water eventually become completely miscible.It should be emphasised that when we speak of the two uids mixing we are referring

to mixing at the molecular level and not at the pore scale. When two immiscible uidsow through a porous medium, it is possible that the two uids will be simultaneouslypresent in an individual pore; however, the two uids will not freely displace each otherwithin the pore and any modelling needs to account for this.

1.1.3.2 Introduction of solvents

The earliest miscible displacement processes used short-chain hydrocarbons, such as methaneor propane, as a solvent. These solvents are fully miscible with oil and are considerablyless viscous than oil; however, they may also be expensive to use. Due to the high cost of these solvents they are also sometimes used in small quantities to form a pad betweenwater and oil in a secondary oil recovery process. The solvents used are immiscible withthe water and the viscosity of the solvent is less than that of water. It is found that themixing region, where both the solvent and oil are present, grows quite quickly. In thisthesis we will be concerned with modelling this mixing region, and shall assume that no

water is present.

1.1.3.3 Introduction of carbon dioxide

A more economically viable alternative to injection of hydrocarbon solvents is the injectionof carbon dioxide. Carbon dioxide can be captured from industrial processes above groundand injected at high pressure into the oil reservoir. At typical reservoir conditions, carbondioxide is a supercritical uid, with the carbon dioxide fully miscible with the oil, and isless viscous than the oil, so the mixing region may spread faster than one might navelyexpect. Carbon dioxide is also less dense than oil and so it tends to move upwards throughthe reservoir.


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1.2 Single-phase ow 5

An additional advantage to the use of carbon dioxide in miscible displacements isthat once the oil recovery process is complete the carbon dioxide is left behind in the oilreservoir. This removal of carbon dioxide from the atmosphere has attracted attention asa possible means of combating global warming. There is now also considerable interest insequestering carbon dioxide in aquifers, where the carbon dioxide displaces water ratherthan oil.

1.2 Single-phase ow

For ow of one phase through a saturated porous medium the basic equation is Darcyslaw,

u = K ( pg) , (1.1)where u is the ux of uid per unit area. The ux or Darcy velocity u is related to v,the averaged velocity through the pores, by

u = v, (1.2)

where the porosity is the volume fraction of pore space. The permeability tensor K maybe empirically determined from a sample of the porous medium, and has the dimensionsof length squared, being proportional to the square of the microscopic pore scale. Darcyslaw was originally suggested as an empirical law [13], but can also be formally derived byhomogenisation of steady Stokes ow, as we present in the next section.

1.2.1 Derivation of Darcys law from steady Stokes ow

Conservation of mass of the uid inside the pores gives

t + (v) = 0 , (1.3)

where is the density of the uid and v is the actual velocity of the uid inside the pores.The pore-scale Reynolds number in all oil recovery processes will be small (e.g. waterowing through pores of radius 0.1mm at a rate of 5mday 1 gives Re = 5 104) and wetherefore apply Stokes ow,

p + 2v + (4 / 3 + b)( v) = g, (1.4)

where p is the pressure, is the viscosity of the uid, b is the bulk viscosity of theuid and g is the acceleration due to gravity. We must also assume an equation of state


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V V P

V D

Figure 1.2: Denition of V in a periodic medium.

p = p(). At the interface between the uid space and the rock matrix we apply a no-slipboundary condition for the uid.

The equations (1.3), (1.4) are only valid inside the pores, and are not valid at themacroscopic scale. We dene a microscopic length scale (i.e. the length of a pore) to be

L, and a macroscopic length scale, over which the pressure gradient is applied, to be l.1The macroscopic length scale of interest (e.g. between two wells, or between large het-erogeneities in the rock) is much larger than the pore length scale, so that = L/l 1.We nondimensionalise the pressure with p, a typical pressure drop, x with l, the densitywith a reference value , the velocity v with U = L2 p/l , and t with l/U . The nondi-mensionalisation for the velocity is motivated by the scaling of the pressure in Poiseuilleow [20].

To apply homogenisation theory, one must make some assumptions about the struc-

ture of the porous medium that the uid is owing through. In principle one can applyhomogenisation either by assuming that the medium is periodic at the microscale or byassuming that the medium is constructed by some random process with reasonable sta-tistical properties such as stationarity and ergodicity [8]. In practice there are manytechnical details that arise when one derives Darcys law for a random porous medium,particularly related to the connectedness of the uid space, although these can be over-come [3]. Since the derivation of Darcys law for periodic porous media is much simpler tofollow and applies the same basic physical principles, we shall present the theory only for

1 Throughout this thesis we shall use lower-case letters to represent macroscopic variables and upper-case letters to represent microscopic variables.


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ow through periodic media. At least for this problem, the assumption that the mediumis periodic is simply for analytical convenience, and we recognise that natural porousmedia will not be periodic.

We dene V as the volume of pore space over which the medium is -periodic in x.The boundary of V , V , is composed of two surfaces as depicted in Figure 1.2. Theinterface between the solid rock matrix and the pore space is denoted by V D and theperiodic surface between adjacent volumes of pore space is denoted by V P . Inside V our non-dimensionalised model is

t

+ (u ) = 0 , (1.5)

p + 2

2

v + ( v) = Ge3, (1.6)where = 4/ 3 + b/ and G = gl/ p are dimensionless constants that will be of orderone (compared with ) for most oil recovery applications, and e3 is the unit vector inthe vertical ( x3) direction. We still have the equation of state p = p(), and on V Dwe apply the no-slip condition that v = 0. We now apply the rst crucial step toany homogenisation argument: we allow the velocity, pressure and density to vary bothslowly over the macroscopic length scale and rapidly over the microscopic length scale of the pores. This is achieved by proposing multiple-scales expansions of the form

v v0(x, X , t) + v1(x, X , t) + . . . , p p0(x, X , t) + p1(x, X , t) + . . . , 0(x, X , t) + 1(x, X , t) + . . . ,

where the microscale variable X = x/. We are on the manifold x = X and so

= 1X

+ x ,

as a result of the chain rule. This is equivalent to treating x and X as independentvariables.

The next key step to the homogenisation argument is to ensure that the expansionremains valid as 0. Since our problem is 1-periodic in X, we shall require thatv0, v1, p0, p1, 0, 1, . . . are all 1-periodic in X since otherwise we would nd that terms inour expansion would grow unboundedly as 0 and |X | . We cannot allow termsin our expansion to grow unboundedly as this will lead to the expansion breaking down

when, for instance we no longer satisfy p1 p0. The analogue to periodicity in randomporous media is stationarity (i.e. invariance under translation), and to avoid unbounded


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growth of terms in the expansion, we require stationarity of the terms in the expansion.To leading order in (1.6) we nd X p0 = 0 and so p0 = p0(x, t ). Also, from the equationof state p = p(), we conclude that 0 = 0(x, t ). The leading order term in (1.5) and theO(1) terms in (1.6) give

X v0 = 0, (1.7)X p1 =

2X v0 x p0 + 0Ge3 , (1.8)

v0 = 0 on V D . (1.9)

We must consider local cell-problems in V . We need to nd functions w j (X), P j (X)which are 1-periodic in X and solve the cell problem,

X w j = 0 , (1.10)X P

j = 2X w

j + e j , (1.11)

w j = 0 on V D , (1.12)

where e j is the unit basis vector in the X j direction. Crucially equations (1.10)-(1.12) arelinear and so we nd that

v0 = p0x j

w j + 0Gw 3. (1.13)

The O(1) term of (1.5) is

0t

+ x (0v0) + X (0v1 + 1v0) = 0 . (1.14)

We now come to the nal important step of the homogenisation process, we integratethe above equation over V to obtain an integrability condition that gives an averagedequation. Integrating (1.14) over V and applying the divergence theorem gives

V 0

t + x (0v0) dX = V (0v1 + 1v0) n ds.We can apply the no-slip condition on V D and the periodicity condition on V P to seethat the right-hand side of the above equation vanishes. Applying summation convention,we are left with the homogenised problem:

0t

+ x i 0K ij

p0x j

+ 0GK i3 = 0 , (1.15)


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where , the porosity, is equal to the volume of V , and

K ij =

V

w ji dX , (1.16)

is the permeability tensor; for some porous media it is reasonable to assume that the porestructure is isotropic so that K ij = k ij . After redimensionalisation we obtain

0t

0K

( p0 0g) = 0 ,

where the dimensional permeability tensor is given by K = L2K, i.e. the permeabilitytensor is proportional to the square of the pore length scale. It is also convenient to work

with the ux of the uid,

u = V v0 dX = K ( p0 0g) , (1.17)which satises

0t

+ (u 0) = 0 .Note that throughout this thesis we will use u to represent the ux of uid per unit area,rather than the average velocity within the pores. The ux of uid per unit area, alsoknown as the supercial velocity or the Darcy velocity, is simply related to the averagevelocity within the pores, v0 by the expression

u = v0,

where is the porosity.Although compressibility of the uid may be important near to injection wells where

the pressure gradient may be large, it is usually possible to justify neglecting the com-

pressibility throughout most of the reservoir. For incompressible ow we simply havethat

u = 0,where u is the ux as dened in (1.17). For the remainder of this thesis we shall assumethat the uids are incompressible. It is possible to justify this assumption, a posteriori,by the absence of large pressure gradients in all of our future models. In saturated,one-phase ow with no free surfaces, we may also incorporate the gravity term into thepressure gradient. When two or more uids are present gravity may only be incorporated

into the pressure gradient when the uids are neutrally buoyant (i.e. they have equaldensities).


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In a homogeneous, isotropic porous medium, the effective pressure (actual pressureminus the hydrostatic pressure) in an incompressible uid satises Laplaces equation

2 p = 0. This allows many simple analytical solutions to be found for one-phase incom-pressible ow through homogeneous, isotropic media.

1.2.2 The Hele-Shaw cell analogue

A useful analogue of ow through a porous medium is the Hele-Shaw cell. A Hele-Shawcell consists of a viscous uid owing between two parallel plates separated by a smalldistance, h. It is easy to show (see for example [41]) that when the reduced Reynoldsnumber and the aspect ratio of the Hele-Shaw cell are both much less than 1, the velocity,

averaged between the two plates, is given by

u = h2

12 p.

The averaged velocity also satises

u = 0,

and so ow through a Hele-Shaw cell is analogous to incompressible ow through a two-

dimensional porous medium with a permeability of h2/ 12. It is often difficult to constructexperiments for ow through porous media, and so a Hele-Shaw cell is used instead.

1.2.3 Homogenisation of the permeability

Homogenisation is not only useful for the derivation of Darcys law from pore-scale models.The technique also allows us to determine the effective permeability of a porous mediumthat is heterogeneous below the length scale of interest. Rock formations typically exhibitheterogeneities over many different length scales. Suppose a porous medium exhibitsheterogeneities with a characteristic length scale L, and the length scale of interest, i.e.the length scale over which a pressure difference is applied or the size of the reservoir, is l.For it to be possible to obtain an effective homogeneous permeability, without variationsover the length scale L, we require that the two length scales are well-separated, i.e. wenow have a small parameter = l/L 1.

The permeability can be described by K = K ij (x, X), where X = x/. To be ableto obtain an effective homogenised permeability, we require that the permeability has asensibly dened local average on the microscopic scale. A well-dened local average of


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the permeability is ensured if, for instance, the permeability is a strictly-stationary 2 andergodic3 random variable, and it is then possible to obtain a homogenised permeability,as shown in [43]. A simpler, but more restrictive, assumption that leads to a well-denedlocal average, is to assume that the permeability is 1-periodic in X, which we now apply.The pressure must solve

(K (x, X) p) = 0 ,and so we introduce the multiple-scales expansion for the pressure,

p p0(x, X) + p1(x, X) + . . . ,

where x and X are treated as independent variables. The pressure must be 1-periodic inX , as otherwise the pressure will grow unboundedly as 0 and |X | .

The leading order pressure satises

X (K (x, X)X p0) = 0 ,

which only holds for periodic p0 when p0 = p0(x). At next order we nd that

X (K (x, X)X p1) + X (K (x, X)x p0) = 0 ,

which has a solution of the form

p1 = F k(x, X)p0x k

+ p1(x),

where summation convention has been applied and F k satises

X i

K ikF jX k

+ K ijX i

= 0 , (1.18)

and is 1-periodic in X . Once the cell problems (1.18) are solved, we may proceed to nextorder to nd that

X (K (x, X)X p2)+X (K (x, X)x p1)+x (K (x, X)X p1)+x (K (x, X)x p0) = 0 ,

and, on integrating with respect to X over the unit cell, and applying the condition of periodicity to p0 and p1 we nd that

x

K

x p0 = 0 ,

2 I.e. the random variable representing the permeability is unchanged under translations in X .3 I.e. the average of the permeability over many realisations is the same as the average over X.


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1.3 Two-phase ow 12

whereK ij =

1

0 1

0 1

0K ik

F jX k

+ K ij dX 1dX 2dX 3.

The solutions to the cell problems, (1.18) are only unique up to the addition of afunction of x; however, since it is only the gradient of F k with respect to X that appearsin the homogenised permeability this function may be chosen arbitrarily. In general,there is no analytical solution to the cell problems, (1.18). A simple and physically-relevant example for which an analytical solution exists is a layered porous medium, inwhich K ij = k(x, X 2) ij . We then nd that without loss of generality F 1 = 0 and that

F 2 = A(x)

X 2

0

1k

dX 2 X 2,

and so F 2 will only be periodic if

A = k = 1

0

1k

dX 21

,

the harmonic average of the permeability. The homogenised permeability is thereforegiven by

K 11 = k, K 22 = k,

where k is the arithmetic mean of the permeability and k is the harmonic mean of thepermeability. Since the harmonic mean is always smaller than the arithmetic mean, it isalways more difficult for uid to ow across layers than along them.

The appearance of the arithmetic mean for ow along layers and the harmonic meanfor ow across layers may be familiar to the reader from studies of resistors in electricalcircuits. The mean resistance of resistors connected in series is given by the arithmeticmean of their resistance, while the mean resistance of resistors connected in parallel isgiven by their harmonic mean. Equivalently the mean conductance (reciprocal of resis-

tance) of resistors connected in series is given by the harmonic mean of their conductance,while the mean conductance of resistors connected in parallel is given by their arithmeticmean.

1.3 Two-phase ow

1.3.1 The Muskat problem

One of the simplest models for two-phase ow in a porous medium or a Hele-Shaw cellis the Muskat problem. Two uids are separated by an interface that is modelled by a


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1.3 Two-phase ow 13

moving boundary. We use subscript 1 to represent the properties of the displacing uidand subscript 2 to represent the properties of the displaced uid. In each uid we mustsatisfy

u i = ki pi , u i = 0 for i = 1, 2. (1.19)At the moving boundary we have a dynamic boundary condition,

p1 = p2, (1.20)

and a kinematic boundary condition

u 1

n = u 2

n = vn , (1.21)

where n is the normal to the boundary and vn is the normal velocity of the boundary.The single-phase Muskat problem, often referred to as the Hele-Shaw problem, con-

siders a single uid with a moving boundary separating a saturated region where the uidis present from an unsaturated region where it is not. In the saturated region the uidvelocity satises

u = k

p,

the dynamic boundary condition is

p = constant ,

and the kinematic boundary condition is

u n = vn .

The Hele-Shaw problem may be recovered from the Muskat problem by taking the limit

as the viscosity of either the displacing uid or the displaced uid tends to zero.The Muskat model appears to follow immediately from Darcys law, but in fact several

assumptions have been implicitly made. The microscopic ow near the interface is difficultto analyse and so the true interface between the two uids may be complicated. Inparticular, if we do not include any additional physics, as we apply no-slip boundaryconditions at solid surfaces we should conclude that the interface should not move fromany point where it meets the solid surface. This assumption will quickly lead to theformation of a tortuous moving boundary and the averaged equations for the ow that

we have wrote down in section 1.2 will not be relevant. In reality there are often extraphysical phenomena that allow one to apply the Muskat model. In immiscible ow the


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1.3 Two-phase ow 14

U 1 2

Figure 1.3: Cross-sectional view of two-phase ow in a Hele-Shaw cell.

physical phenomena are surface tension and the wettability of the uids at the interface.For two-phase ow in a Hele-Shaw cell it is possible to derive a moving boundary problemas demonstrated in a paper by Park and Homsy [45]. For two-phase ow in a porousmedium the situation is more complicated and it is not clear when the Muskat problem

is applicable.For miscible ows there is no longer a strict interface between the two uids. Never-

theless we shall see in Chapter 2 that the Muskat problem can be recovered as a limitingcase of the so-called Peaceman model for miscible ow in porous media.

1.3.2 Stability analysis of a planar interface

The Muskat problems permits a planar solution with an interface given by x = U t withuid of viscosity 1 in x < Ut and uid of viscosity 2 in x > Ut . The pressure in thetwo uids is given by

pi = U ik

,

where = x Ut. We now seek a solution where the interface is given by

= f (y, t ),

where is again a small parameter, and we have an initial condition,

f (y, 0) = f 0(y).

We seek pi of the form pi U

ik

+ pi( )et +i y ,

where is real. In each uid the pressure must satisfy

2 pi 2

2 pi = 0,


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1.3 Two-phase ow 15

and so the disturbance to the pressure is given by

p1 = P 1e| | for < 0,

p2 = P 2e| | for > 0,

where we have ensured that the disturbance decays as | | . The dynamic boundarycondition yields

U 1k

+ P 1 = U 2k

+ P 2,

and the kinematic boundary condition gives

k

1 |

|P 1 =

k

2 |

|P 2 = .

We nd the dispersion relation

= U ||2 11 + 2

. (1.22)

When a less viscous uid displaces a more viscous uid the interface is unstable, and thesmallest wavelength disturbances grow at the fastest rate.

More generally the interface will be given by

= f (y, t ),

where f (y, t ) is a linear combination of the exponential solutions already obtained, andis given by

f (y, t ) = 12 f 0()e( )tiy d, (1.23)

where f 0() is the Fourier transform of f 0, dened by

f 0() = f 0(y)eiy dy,

and () is the growth rate given by (1.22). If, for example,

f 0(y) = 1

11 + y2

, f 0() = e| |;

then when () > 0, (1.23) will only be integrable for

t < |

|() = 1 + 2U (2 1) ,


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1.3 Two-phase ow 16

y = 1

y = 1

U 1 22 V

Figure 1.4: The Saffman-Taylor nger.

and so an initially smooth solution may blow up in nite time. If

f 0(y) = 12e|y|, f 0() = 11 + 2 ,

then when () > 0, (1.23) is not integrable for any t > 0, and the solution blows upimmediately. We see that the linearised Muskat problem is ill-posed, as the solution neednot always exist. Indeed, the full Muskat problem with an analytic initial condition hasbeen shown to develop, in nite time, a singularity in the curvature of the free surface [52],suggesting that it too is ill-posed. In practice there is always a previously-neglected phys-ical phenomenon that regularises the Muskat problem and ensures that the true problem

to be solved is well-posed. We shall review some of these regularisation mechanisms insection 1.3.4.

1.3.3 The Saffman-Taylor nger solution

There is a well-known solution to the Muskat problem, known as the Saffman-Taylornger. We consider a single nger of uid, with viscosity 1, penetrating a seconduid, with viscosity 2, lying in the innite channel, < x < , 1 < y < 1 (seeFigure 1.4). Saffman and Taylor [49] found a family of travelling wave solutions, whereas x the displacing uid occupies a fraction of the channel and as x the velocity of the displaced uid is V . The displacing uid moves as a rigid body withconstant velocity U . The interface between the two uids is given by

x Ut = 1

log

1 + cos y2

, (1.24)

and the velocity of the nger is given by

U = MV 1 + (M 1)

, (1.25)


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1.3 Two-phase ow 17

where the mobility ratio isM =

21

.

We have a different solution for each value of between 0 and 1. The speed of the ngermay take any value between V and MV .

Saffman-Taylor ngers can be obtained as the large time limit of ows which initiallypossess an almost planar interface [31, 48, 28]. While Saffman-Taylor ngers are perhapsnot stable [38], their existence as the only travelling-wave, symmetric, nger solutions [28]and their appearance as the steady state of many particular solutions suggests they areat least important transient states.

It is worth noting that for the Hele-Shaw problem, one can nd a Saffman-Taylor

nger solution with arbitrarily large velocity by choosing to be arbitrarily small, andso one can nd solutions close to a planar interface that blow up almost instantaneously.It is perhaps, therefore, not surprising that the Hele-Shaw problem should be ill-posed.For the full Muskat problem, the velocity of the Saffman-Taylor nger solutions may notexceed MV , suggesting that instantaneous blow-up of solutions is less likely.

1.3.4 Regularisation of the Muskat problem

We have seen that the linearised Muskat problem is ill-posed, when M > 1, with probable

nite-time blow-up of solutions. The Hele-Shaw problem, in which the viscosity of thedisplacing uid is set to zero, is known to be ill-posed for a retreating interface [29], andcurvature singularities can appear in the full Muskat problem.

The physical problem cannot be ill-posed, and the ill-posedness of the mathematicalmodel is an indication that we have neglected an important physical effect. For the owof two immiscible uids in a Hele-Shaw cell, the missing physical effect is often surfacetension. A crude model for the effect of surface tension can be constructed [26] via achange in the dynamic boundary condition (1.20), so that

p2 p1 = 4

,

where is the surface tension between the two uids and is the curvature of the interface.The dispersion relation obtained is then

= 1

1 + 2U (1 2) ||

h2

12 ||3 ,

where we recall that h is the separation between the plates.The inclusion of surface tension stabilises large wavenumber disturbances, leading to


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1.3 Two-phase ow 18

the existence of a least-stable wavenumber, for which the growth rate is largest, and acutoff wavenumber, with all higher wavenumber disturbances being stable. The existenceof the least-stable and cutoff wavenumbers suggests a natural wavelength for the distur-bances, which was not present in the zero surface-tension problem. When the modiedcapillary number,

Ca = 122

(h/L )2,

is large (L is the macroscopic length scale), the least-stable wavenumber will be of orderCa 1/ 2, suggesting that disturbances with a wavelength of order Ca 1/ 2 will be observed,and this has been conrmed in experiments [59, 44]. The dispersion relation yields agrowth rate that is bounded, and so the linearised problem does not admit solutions

which blow-up in nite time such as those found in section 1.3.2. This result shows thatsurface tension can act to regularise the Muskat problem.In miscible displacements, there is no interface between the displacing and displaced

uids, and hence surface tension cannot act. Instead of surface tension we shall see inChapter 2 that transverse diffusion can regularise the Muskat problem, with the wave-length of disturbances observed depending on the magnitude of diffusion. The methodby which diffusion regularises the Muskat problem, is quite different theoretically to theregularisation by surface tension, since we cannot perturb about a known solution to amoving boundary problem.

1.3.4.1 Kinetic undercooling

While surface tension is the physically-relevant regularisation mechanism for immiscibleow of two uids, it is possible to associate the Muskat model with other physical mecha-nisms. One such physical problem is the Stefan problem, used to model a phase-change ina material, such as the freezing of water. The one-phase Stefan problem, with negligiblespecic heat (i.e. large Stefan number), is simply

(kT ) = 0 ,

on one side of a curve (t), and with boundary conditions on the curve ( t)

T = T 0, vn = kL

T n

,

where T represents the temperature, and may be associated with the pressure in the Hele-Shaw problem. The problem is exactly equivalent to the Hele-Shaw problem, and so the

problem may be ill-posed. An appropriate regularisation mechanism for this problem maybe kinetic undercooling, the rate at which the interface moves is too fast for the phase-


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1.3 Two-phase ow 19

change process to be in thermodynamic equilibrium at the free surface and so instead of applying T = T 0 at the free surface we apply, as in [16],

T = T 0 + vn ,

where vn is the normal velocity of the free surface and is the kinetic undercoolingparameter.

As in section 1.3.2 one can seek a travelling wave solution in which the interface travelsat speed U with the temperature given by

T = T 0 + U LU

k (x Ut).

If a small sinusoidal disturbance of wavenumber is made to this base-state solution thenwe nd that the growth rate of the disturbance is given by

= U ||

1 + kL ||.

A receding interface, corresponding to e.g. the freezing of a uid cooled below T 0, is unsta-ble. Even with kinetic undercooling included the process is unstable for all wavenumbers;however, the dispersion relation shows that for large wavenumbers the growth rate is con-stant, and so the solution will again not exhibit the nite-time blow-up that we observedin section 1.3.2, and so the linearised problem is well-posed. This observation suggeststhat while kinetic undercooling allows instability at all wavenumbers, it is still a possibleregularisation mechanism of the Hele-Shaw problem.

We shall see in Chapter 2 that the action of longitudinal diffusion, in which a sharpinterface is replaced by a region of non-zero width in which the fraction of solvent and oilgradually changes can be related to the kinetic undercooling regularisation.

1.3.4.2 Other possible regularisations

One can conceive of many different regularisation mechanisms for the Muskat problem,depending on the exact physical circumstances. One such regularisation, applicable forvery porous media and therefore unlikely to be appropriate for oil recovery applications,is presented in Appendix B.

In Chapter 2 we shall see that diffusion acts as a regularisation mechanism, althoughwe are unable to continue to use a free boundary model.


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1.4 Other models used in the oil industry 20

1.3.4.3 Shape selection of Saffman-Taylor ngers

The Saffman-Taylor nger solution, described in section 1.3.3, is valid for each value

of between 0 and 1. In experiments [49] it is often observed that only the solutioncorresponding to = 1/ 2 is obtained. There do not appear to be any simple physicalarguments as to why this particular solution should be preferred. It has, however, beenshown that when a small regularisation mechanism is included, such as surface tension[12, 9] or kinetic undercooling [10], the only possible solution is close to that of theSaffman-Taylor nger with = 1/ 2. In both cases the selection is a consequence of terms that are exponentially small in the regularising parameter. The shape-selectionof the Saffman-Taylor nger demonstrates the important role that small regularisationmechanisms play in determining the solution of the Muskat problem.

1.4 Other models used in the oil industry

1.4.1 The Peaceman model

The rst mathematical model developed specically for the simulation of miscible dis-placements was that of Peaceman [46]. The Peaceman model assumes that the ow isincompressible, and described by Darcys law, with a viscosity that is dependent on theconcentration of solvent (dened to be the volume fraction of solvent in the oil-solventmixture). The Peaceman model is

u = 0, u = K(c)

( p(c)g) , (1.26)

ct

+ (u c) = (Dc) , (1.27)where c is the concentration of solvent, u is the total volume ux of uid per unit area, K

and D are tensors representing respectively the permeability of the rock and the effectivediffusion of the solvent, (c) is the viscosity of the mixture, (c) is the density of themixture, and is the porosity of the medium. The viscosity of the solvent and oil mixturemay be experimentally determined, and is found to be a function of c. There is no generalrelationship for the viscosity of a mixture of two uids as a function of the fraction of eachuid present; however, for the mixtures under consideration in miscible displacement acommonly used relationship is due to Koval [33]:

(c) = c1/ 4s+ 1 c1/ 4o

4, (1.28)


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Oil

Rock

Water

Figure 1.5: Oil and water are both present within the pores. One phase may close off some of the pores, affecting the ow of the other phase.

where s and o are the viscosities of the pure solvent and oil respectively. Similarly thereis no general relationship for the density of a mixture of two uids as a function of thefraction of each uid present, but since the volume change on mixing of the two uids issmall, one typically assumes that the density is a linear function of c. We will take thePeaceman model to be the fundamental model for miscible ow through porous media,and we shall review it in much more detail in Chapter 2.

1.4.2 The Buckley-Leverett equation

The Buckley-Leverett equation models immiscible ow through porous media such asthe displacement of oil by water. Although the uids are considered to be completelyimmiscible at the molecular level, we nevertheless assume that the ow causes them to be

mixed together at the scale of the pores as in Figure 1.5. We introduce saturations of oiland water, i.e. the fractions of pore space which are occupied by oil and water respectively,as S o and S w . The uxes of oil and water are denoted by uo and uw respectively andthe pressure in the oil and water is given by po and pw . In two-phase ow, the owof each phase through the porous medium will be affected by the presence of the otherphase. Neglecting the effect of gravity for simplicity, this effect can be modelled by theintroduction of relative permeabilities so that in each uid,

u o = kroo K p

o, uw = krww K p

w , (1.29)


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0 0.5 1

0

0.5

1

S w

krwkro

Figure 1.6: Relative permeability curves as used in the Buckley-Leverett equations.

where the relative permeabilities, kro and krw are empirically determined and depend on

the fraction of the water-phase present, S w . An example of relative permeability curvesis given in Figure 1.6. Conservation of mass applied to the oil and water gives

S wt

+ u w = 0, (1.30)

S ot

+ u o = 0, (1.31)

where is the porosity of the rock. The difference between the pressure in the waterphase and the pressure in the oil phase is given by the capillary pressure,

pc(S w) = po pw .

The capillary pressure is a consequence of surface tension, and for an individual pore,lled with water, the capillary pressure could be determined by

pc = 2 cos

r (S w) ,

where is the coefficient of surface tension, is the wetting angle at the point where theoil, water and rock meet, and r(S w) is a typical radius for a pore in which both waterand oil are present. One can rearrange equations (1.29)-(1.31) in terms of the total ux,u = u w + u o , to obtain:

u = 0, u = Kkrww

+ kroo

pw Kkroo

pc, (1.32)

S wt

+ (F (S w)u ) = (G(S w) pc) , (1.33)


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where

F (S w) = 1 + kro (S w)wkrw (S w)o

1,

andG(S w) =

kro (S w)krw (S w)kro (S w)w + krw (S w)o

.

The Buckley-Leverett equation is obtained by considering a one-dimensional problem andneglecting the capillary pressure term giving

S wt

+ U F (S w)

x = 0, (1.34)

where U is the (constant) total ux.

Even though the Buckley-Leverett equation models immiscible diffusionless ow of u-ids that coexist on the pore scale, it has been used to model certain miscible ows throughporous media [33, 58, 18]. If one interprets the saturation as the local concentration of solvent within the pores then, since the mixture of oil and solvent is homogeneous withinthe pores, the physically sensible relative permeabilities are

kro = o(c)

(1 c), and krs = s(c)

c,

leading us to recover the Peaceman model, although without diffusion.It is possible to apply the Buckley-Leverett equation to modelling miscible ows by

averaging the concentration over a larger scale than the pore scale, with the uid not beinghomogeneous at this larger scale. The models that have been developed rely heavily onmatching with empirical data and several, carefully considered, but occasionally ad hocassumptions. It is desirable to place these models within a more systematic framework,with a better understanding of why the uid is not homogeneous, and what the relativepermeabilities should be. One important model for miscible ow through porous media,

which was originally derived via the Buckley-Leverett equations, is the Koval model, whichwe shall now examine.

1.4.3 The Koval model

It has been observed in experiments [6] that, in a miscible displacement through a channel,the evolution of the concentration of solvent is very different to that predicted by the one-dimensional Peaceman model. The cause of this difference is ngering of the solvent,as we shall see in Chapter 2. Koval developed a simple one-dimensional model [33] to


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1.5 Diffusion 24

describe the observations of Blackwell [6]. The Koval model is given by

c

t

+ U F (c)

x

= 0, where F (c) = M ec

1 + ( M e 1)c, (1.35)

and the effective mobility ratio, M e is dened to be the ratio between the viscosity of the oiland a specic mixture of solvent and oil with a volume-fraction of solvent ce . By matchingwith experiments, Koval determined that ce = 0.22. The Koval model was derived fromthe Buckley-Leverett equations, but, as explained above, it is not immediately clear whyany model other than the Peaceman model should be appropriate. Indeed, it is not readilyapparent under what conditions the Koval model applies, and one of the main objectivesof this thesis is to obtain the Koval model as an averaged version of the Peaceman model.

1.5 Diffusion

Diffusion of a solvent through a porous medium is a complicated process, particularlywhen the transport is convection dominated. The magnitude of the diffusion may dependon the structure of the porous medium and the mean velocity of the uid. When modellingmiscible displacements, on the scale of an oil reservoir, the transport will be convectiondominated and the small diffusion will be velocity-dependent and anisotropic. In this

section we shall review some of the popular models of diffusion in porous media.

1.5.1 Molecular diffusion of solvent and oil

The mixing of solvent and oil is governed by molecular diffusion at the microscopic scale.Molecular diffusion is described by Ficks law,

F = Dc, (1.36)

where F is the diffusive ux, D is the coefficient of diffusion, and c is the concentrationof solvent i.e. the (volume) fraction of the uid that is composed of solvent. If the uidis moving with a ux u then conservation of volume 4 gives

ct

+ (uc) = (Dc) , (1.37)

in free space. For low concentrations of solvent the coefficient of diffusion is constant;however, in the displacement of oil by a solvent the concentration of solvent is not ev-

4 Strictly speaking we should conserve mass, rather than volume, however, for all the solvents and oilsused in miscible displacements, the volume change on mixing is very small and so conservation of volumeis equivalent to conservation of mass. This subject is dealt with in more detail in section 5.2.


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1.5 Diffusion 25

erywhere small. In general D will be a function of the concentration of solvent, and inparticular D should depend on the viscosity of the mixture [4]. If the oil and solvent havereasonably similar properties the diffusion will be nearly constant. In Chapter 2 we shallmodel the effective diffusion of solvent through a porous medium, but rst we give anoverview of the effective diffusion of a passive tracer, i.e. a substance that does not affectthe properties of the ow.

When a passive tracer is transported through a porous medium, the effective diffusioncoefficient may be very different from molecular diffusion in free space. The dispersionof a substance through a porous medium is enhanced by the rapidly varying velocity atthe pore scale. Molecular diffusion does not adequately describe the effective diffusion atthe macroscopic scale of a substance, and so we must investigate the interaction between

diffusion and a rapidly varying velocity eld.

1.5.2 Taylor diffusion in a capillary

The simplest example of velocity dependent diffusion is the famous Taylor dispersion[57]. Taylor dispersion describes the dispersal of a solute dissolved in a uid owingdown a long thin pipe. The uid ow is Poiseuille ow and we assume that the solutediffuses with a constant coefficient of diffusion. The shear in the velocity prole interactswith transverse diffusion across the pipe to yield an enhanced longitudinal diffusion. Theconcentration, to rst approximation, is independent of the radial coordinate, is advectedwith the mean velocity of the uid, and diffuses in the longitudinal direction with thecoefficient of diffusion given by the famous Taylor-Aris dispersion coefficient [57, 1],

DT = a2U 2

48D + D,

where a is the radius of the pipe, U is the mean velocity of the uid and D is the coefficientof molecular diffusion. Small molecular diffusion may lead to a much larger dispersion of the concentration than one would immediately expect.

1.5.3 Saffman dispersion

Taylor dispersion provides a simple model for local diffusion in porous media; however asTaylor dispersion describes an essentially one-dimensional problem it can not be used todescribe the effective transverse diffusion in a porous medium. Saffman developed a model[50] in which the solute undergoes a random walk through a network of capillary pores.

The ow of uid in each capillary is described by Poiseuille ow. The solute particlestravel deterministically through the capillary until they reach a junction, at which point


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1.5 Diffusion 27

which is divergent. While the expected transit time of a particle passing through acapillary is nite, the variance is innite, and so the central limit theorem does not applyafter many transits through capillaries, and we therefore do not obtain a traditionaldiffusive model. The innite variance is a consequence of a long transit time when theparticle is close to the wall of the capillary, and leads to large longitudinal dispersionof the solute. Molecular diffusion will reduce the transit time for particles close to thecapillary walls by allowing diffusion towards the faster central ow; however, when theeffect of molecular diffusion is small, the long hold-up time for such particles will still leadto a large longitudinal dispersion of the solute.

Hold-up of the solute near the capillary walls is not the only mechanism by which dis-persion develops, because there is also hold-up of the solute in capillaries that are aligned

transversely to the direction of mean ow. The ux of uid down a capillary makingan angle with the mean direction of ow (and hence the applied pressure gradient), isproportional to cos , i.e. u = cos , for some constant . The component of the ux ineach capillary, parallel to the mean direction of ow is u cos , and the fraction of poresfor which the angle subtended between the pore and the mean direction of ow is between and + d is proportional to sin d. The total ux through the porous medium istherefore given by

U =

2

0

cos2 sin d,

and so we nd that = 3U . If we simply model the transit time of a particle through acapillary as equal to L/u then we nd that for a capillary making a random angle withthe mean direction of ow the expected transit time is given by

E (T ) =

2

0

2L3U

sin d = 2L3U

,

and

E (T 2

) =

2

0

2L2

9U 2 tan d,which is also divergent. Here the long transit times are a consequence of particles travellingalong a capillary that is nearly perpendicular to the mean direction of ow so that the uxof uid through the capillary is small. Molecular diffusion will reduce these long transittimes by allowing the particle to diffuse straight through the capillary.

Saffman suggested a simple formula [50] to represent the transit time of a solvent


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1.5 Diffusion 28

through the capillary without neglecting molecular diffusion:

t =

L2u 1 r

2

a 21 , t < t 1,

t1 + Lu , t1 t < t 0,t0, otherwise.

, (1.38)

where t1 = a2/ 8D is the timescale for molecular diffusion towards the centre of thecapillary and t0 = L2/ 2D is the timescale for diffusion through the capillary. With thisrepresentation of the transit time one nds that the variance of the transit time is niteand it becomes possible to produce a theory of diffusion via the central limit theorem.The dispersion described in Saffmans paper [50] assumes that at the microscopic scale

convection dominates, so that the timescales t0 and t1 are large. Saffman nds that theeffective diffusion in the longitudinal direction is given by

D = U L13

log 3UL

2D +

112

log 3Ua2

4DL

2

14

log 3Ua2

4DL +

1924

,

and the transverse diffusion is given by

D

= 316

UL.

Writing Pe l = UL/D for the Peclet number at the scale of the capillary, and = L/l ,where l is the macroscopic length scale, we see that

1Pe

= O((log Pel)2) + O(log Pel) + O(), (1.39)

1Pe

= O(), (1.40)

where Pe = Ul/D and Pe

= Ul/D

. We see that, at the macroscopic length scale,

convection will always be dominant since is small. We also see that the effective longi-tudinal diffusion has a weak dependence on the capillary Peclet number, but the effectivetransverse diffusion does not.

The available experimental evidence [5, 50, 21] appears to give reasonable qualitativeagreement with Saffmans model, with the effective transverse diffusion simply propor-tional to UL and the ratio of the effective diffusion and UL appearing to show a weakdependence on Pe l. While the Saffman model seems to agree fairly well with experiments,the model has attracted some criticism [7, 47] for the ad hoc assumptions involved in its

derivation. In particular, the assumed geometry of the medium is not particularly realisticand, as noted by Saffman [50], the result for the transverse diffusion is somewhat dubious,


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1.6 Structure of the thesis 29

at least for two-dimensional ow. For periodic porous media it is possible to apply thetechnique of homogenisation to develop a model for the effective diffusion [7, 47], andunlike Saffmans model, it is not necessary to assume that the pore-scale Peclet numberis large. We will revisit this in Chapter 2; however, periodic media are not representativeof real-life porous media [32].

1.6 Structure of the thesis

In the next chapter we shall derive the Peaceman model by rst considering the problemat the pore-scale. We thereby gain an understanding of the transport of the solventthrough the porous medium, both by convection and dispersion, and show that, for manyproblems relevant to the oil industry, the ow will be dominated by convection, and thatdispersion is small.

We have already seen that instabilities may arise when a less viscous uid displaces amore viscous uid through a porous medium, and so in Chapter 2 we will also investigatethe nature of these instabilities when the two uids are miscible. We show that instabilitiespersist in the miscible problem, but that there is now a maximum possible growth rate,shown to be a consequence of the smooth transition between the solvent and oil. Alsolarge-wavenumber disturbances are shown to be stable as a consequence of transverse

diffusion. Although the ill-posedness of the linear problem observed in section 1.3.1 isremoved, instabilities persist in the Peaceman model, and we show that the Peclet numberis critical in determining the appropriate length scale of these instabilities.

To conclude Chapter 2, we conduct numerical simulations of the Peaceman model.These are used to conrm the results of our stability analysis, and in particular to showthe importance of the Peclet number in determining the length-scale of the ngers that areformed. For the applications under consideration, the length scale of the instabilities thatform is small, and so accurate simulation of the Peaceman model requires a ne grid-size.

Although we show that it is possible to numerically simulate the Peaceman model, it isa numerically intensive task, and it may become unfeasible in three-dimensions at the oilreservoir-scale.

Our numerical simulations predict small scale variations well below the reservoir scale,yet it is only the locally-averaged solvent concentration that is of practical interest. Hence,in Chapter 3, we develop an averaged model which does not capture the detailed nger-ing behaviour but does accurately capture the spread of the solvent through the porousmedium on the reservoir scale. It is an important feature of the averaged model that

it should not exhibit the same instability that existed prior to averaging. We begin byconsidering ow in which the applied pressure gradient is unidirectional and the medium


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1.6 Structure of the thesis 30

is entirely homogeneous. For large values of the Peclet number, the amplitude of thengering instabilities is shown to be an order one fraction of the distance displaced bythe solvent, but the wavelength of the ngering instabilities is proportional to Pe 1/ 2. Onthe basis of these observations, we make the important assumption that the ngers thatare formed are long, thin and parallel. This assumption allows us to apply homogenisa-tion theory to the problem, allowing us to explicitly solve for the velocity eld, given theconcentration eld.

We show that development of a model for the evolution of the averaged concentrationof solvent is possible if one can solve a closure problem, which is equivalent to determininghow the concentration of solvent varies across the nger. A simple solution to the closureproblem is obtained by assuming that the solvent and oil do not mix locally; however, it

is found that this model signicantly overpredicts the rate at which the solvent spreads.Nevertheless, with additional empirical assumptions, this solution leads to a derivationof the Koval model, which we conrm to be in very close agreement with our numericalsimulations, at least for large Pe where the aspect ratio of the ngers is large.

In an effort to be able to predict the rate at which the solvent spreads out, we thenstudy the behaviour of the homogenised Peaceman model. As a special case we supposethat t

miscible flow through porous media richard booth

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