Modelling Two-phase Flow in Porous Media at the Pore Scale Using

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Modelling two-phase flow in porous media at the pore scale using

the volume-of-fluid method

Ali Q. Raeini , Martin J. Blunt, Branko Bijeljic

Department of Earth Science and Engineering, Imperial College, Prince Consort Road, London SW7 2BP, UK

a r t i c l e i n f o

Article history:

Received 2 September 2011

Received in revised form 24 February 2012

Accepted 3 April 2012

Available online 17 April 2012

Keywords:

Two-phase flow

Porous media

Volume of fluid

Pore-scale modelling

a b s t r a c t

We present a stable numerical scheme for modelling multiphase flow in porous media,

where the characteristic size of the flow domain is of the order of microns to millimetres.

The numerical method is developed for efficient modelling of multiphase flow in porous

media with complex interface motion and irregular solid boundaries. The NavierStokes

equations are discretised using a finite volume approach, while the volume-of-fluid

method is used to capture the location of interfaces. Capillary forces are computed using

a semi-sharp surface force model, in which the transition area for capillary pressure is

effectively limited to one grid block. This new formulation along with two new filtering

methods, developed for correcting capillary forces, permits simulations at very low capil-

lary numbers and avoids non-physical velocities. Capillary forces are implemented using a

semi-implicit formulation, which allows larger time step sizes at low capillary numbers.

We verify the accuracy and stability of the numerical method on several test cases, which

indicate the potential of the method to predict multiphase flow processes.

2012 Elsevier Inc. All rights reserved.

1. Introduction

Understanding multiphase flow at the micro-scale is of the utmost importance in a wide range of applications such as

enhanced oil recovery, carbon dioxide storage in underground aquifers and proton exchange membrane (PEM) fuel cells.

However, modelling multiphase flow in porous media is a challenging task, as it concerns forces acting at different scales

in the flow domain. Viscous forces are responsible for the dissipation of energy of the fluid system at larger scales, in the

bulk of individual fluids. Interfacial tension governs the shape and movement of the phase boundaries. Finally, wall adhesion

forces are active at the nano-scale thickness of the contact lines and control the contact angle and contact line dynamics. In

many transport problems in porous media, capillary forces are more significant at the pore scale than viscous forces. For

example, typical capillary numbers in petroleum reservoirs are in the range Nc= lu

d/r= 1010 to 105; whereu

dis the Darcy

velocity,lis viscosity andris the interfacial tension[1]. Consequently, any small errors in the numerical calculation of cap-illary forces, if not handled carefully, will introduce instabilities in the numerical method up to a point where it has no pre-

dictive capability.

Different approaches have been used to solve multiphase flow problems at the pore scale. Pore-network models [28],

LatticeBoltzmann simulations [914], Lagrangian mesh-free methods such as smoothed particle hydrodynamics (SPH)

[1517]and grid-based computational fluid dynamics with fluidfluid interface tracking/capturing and velocity-dependent

contact angles[18]have recently been reviewed by Meakin and Tartakovsky [19]. Pore-network models are computationally

efficient, and they are a viable tool for understanding the multiphase flow at the pore scale. These methods however, are

0021-9991/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jcp.2012.04.011

Corresponding author.

E-mail address: [email protected](A.Q. Raeini).

Journal of Computational Physics 231 (2012) 56535668

Contents lists available at SciVerse ScienceDirect

Journal of Computational Physics

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based on simplified geometry and physics, which limits their predictive capabilities. SPH, which is a Lagrangian mesh free

(particle based) method, does not require explicit and complicated interface tracking algorithms, and maintains a sharp

boundary between different fluids. This advantage makes it an attractive tool for studying multiphase flow in complex geom-

etries and in the presence of large interface motion, in cases where it is computationally feasible. Moreover, in contrast to

grid-based methods, LBM and SPH methods do not normally solve for an implicit pressure equation, and consequently they

can be easily and efficiently parallelized. In both of these methods, the surface tension and the contact angle arise from inter-

action forces between the particles of different fluids. The similarity of this approach to the molecular dynamics simulations

make it easier to relate the molecular scale interactions to the particle interactions in these methods.

Eulerian grid-based methods such as finite volume, finite difference or finite element, on the other hand, are traditionally

considered to be the preferable approach for solving the NavierStokes equations because of their superior numerical effi-

ciency and ability to simulate fluid flow with very large density and viscosity ratios[19]. Handling the motion of the interface

and contact line in these methods, however, is a challenging problem. Different methods have been developed to tackle this

problem: (i) the interface is marked and tracked, either with a height function [20]or a set of marker particles/segments

[21,22]; or (ii) the fluids on both sides of the interface are marked either by massless particles[23,24] or by an indicator func-

tion, which may be a volume fraction[20,25], a level-set function[26,27]or a phase-field variable[28,29].

The interfacial tension force has been applied to Eulerian grids using different approaches: (a) continuous surface stress

(CSS) method[30,31], (b) continuous surface force (CSF) method[32], and finally (c) sharp surface force (SSF)[33]and ghost

fluid models [34]. An often reported problem is the existence of the so-called spurious currents in the flow field of the

numerical simulations [32,31,35,36,25,22,33,34,37]. Lafaurie et al. [31] implemented a conserving form of the CSF and

showed that the magnitude of the largest spurious current (us, max) around a bubble/droplet can be given as:

us;max Krl: 1

Here,landr are the viscosity and surface tension, respectively. Kis a constant and is of order 102 in their method, leadingto a capillary number oflus,max/r= 10

2. This means that simulations at lower capillary numbers specifically for the range

of porous media problems we wish to study are rendered meaningless, as the spurious currents are larger than the physical

flow. Several attempts have been made to understand and solve this problem. Brackbill and Kothe[35]derived a CSF-like

formulation from an energy functional and suggested that spurious currents are best eliminated by controlling the interface

thickness. Tryggvason et al.[38]report that the problem is less pronounced in their marker method, whereKappears to be

around 105 [39]. Renardy and Renardy[33]implemented a parabolic reconstruction of surface tension (PROST) method that

reduced the spurious currents by two to three orders of magnitude. Despite these achievements, the problem of spurious

currents is still a main limitation when modelling multiphase flow in porous media: even with these corrections, the spu-

rious velocities can be larger than the actual velocities in the flow domain.

Another major problem in modelling multiphase flow at low capillary numbers is that it is difficult to maintain a zero net

capillary force on a closed interface in volume tracking methods [40]. When modelling movement of a micro-scale droplet at a

lowcapillary number, this defect in the numericalapproach canresultin the deviationof the velocity of droplets from the exact

solution by orders of magnitude. In Section3.5of this paper, we present a simple filtering method to resolve this problem.

The description of contact line motion is another challenge in numerical modelling of multiphase flow. The NavierStokes

equations with standard no-slip boundary conditions produce an infinite viscous dissipation [41]. On the other hand, solid

walls have small-scale heterogeneities that affect the contact line motion (see[19]for a review on the subject). Therefore,

the dynamics of the contact line can be predicted only in simulation methods where the heterogeneities are resolved down

to the molecular scale[42]. However, continuum numerical methods solve the flow to describe the micro-scale behaviour of

the interface; see Spelt[43]and Dupont and Legendre[37]and references therein, although the scale of the problem is dif-

ferent in these references from the pore scale that we are interested in. Therefore, proper boundary conditions should be

developed so that the numerical method can predict the large-scale behaviour of the contact line, without directly incorpo-

rating the molecular-scale interactions between the fluids and the solid wall.

Renardy et al.[44]investigated two different methods to implement the macroscopic effects of contact angle for smooth

solid boundaries in a volume-of-fluid (VOF) framework. Their first method modifies the interface normal vectors at the solid

boundaries, which are used in the calculation of curvature, so that they represent the contact angle. The second method

treats the problem as a three-phase situation and the contact angle is determined by the equilibrium of the capillary forces

on the three interfaces as in the Young equation. They report that the latter approach introduces an artificial localized flow,

and the first method is preferable. Huang et al. [18]used the VOF method to simulate two-phase flow in a 2-D rectilinear

finite difference grid. They conclude that the previous methods for implementing contact angle on smooth surfaces could

lead to large spurious velocities, when applied to complex solid surfaces (Fig. 1(a)). They used a staircase-like method for

representing the complex solid boundaries (Fig. 1(b)), which alleviates this and improves the stability of the numerical meth-

od. In this paper, we use a Gaussian kernel to smooth the solid wall normal vectors (Fig. 1(c)), which can be considered as a

generalization of the staircase-method. Moreover, we modify the capillary forces such that the net effect of the capillary

force and the capillary pressure gradient acts perpendicular to the interface in the close vicinity of the interface (see

Section3.4).

In this paper, we useinterFoam

code, a VOF-based interface capturing code developed by OpenFOAM[45], as the basis for

our numerical code and extend it to model two-phase flow at the micro-scale. The finite volume method is used for

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discretization of the governing differential equations. We develop a new sharp surface force formulation, which eliminates

the problem of spurious currents. A non-uniform scheme is introduced for smoothing the interface curvature away from the

interface. We present two new filtering schemes to eliminate the non-physical velocities, allowing stable simulation of two-

phase flow at low capillary numbers. We propose a semi-implicit formulation for stronger coupling between capillary forces

and the NavierStokes equations, which alleviates the capillary time step restriction for low capillary numbers. Finally, we

verify the numerical method on several test cases. We suggest that our method is a possible alternative to successful particle

based techniques[19].

2. Mathematical description of the problem

To avoid the difficulties in discretization of the capillary pressure jump, we write the momentum balance equation in

terms of a capillary pressure pcand a dynamic pressure pd=p pcfield as follows:

D

Dtqu r Trpd f

0; 2

where

f0

qg fcrpc; 3

uis the velocity vector, p is the total pressure, r T r lru ru rlis the viscous force[40],qgis the gravity forceand fcis the capillary force, as defined in Eq.(8). pcis calculated using the equation:

rrpcrfc; 4

with the boundary condition:

@pc@n

0; 5

wherenis the normal direction to the boundaries. Together(4) and (5)define a Neumann problem for pcwhich has a unique

solution up to an additive constant that can be obtained by fixing the value ofpcat a reference point (pc(xref) =pc,ref). This

approach for including the effect of capillary forces in the NavierStokes equations enables us to filter the numerical errors

related to inaccurate calculation offc. Finally,qandlare the average fluid density and viscosity, respectively, which are cal-culated using the indicator function (a):

q aq1

1 aq2

l al1 1 al2: 6

In the volume-of-fluid method, the indicator function (a) represents the volume fraction of one of the fluids in each gridcell. If the cell is completely filled with the first fluid thena = 1 and if it is filled with the second fluid a = 0. At the interfacethe value ofa lies between 0 and 1.a is evolved with an advection equation of the form:

@a@t

r au 0: 7

Once the indicator function is known, the capillary force (fc) can be computed as a body force[32],

fc rknsds; 8

wherek =r (ns) is the interface curvature and ns is the normal to the interface:

ns ra

jraj: 9

(a) (b) (c)

Fig. 1. Different approaches for representing solid wall normal vectors (nw) (a) no smoothing, (b) the staircase-like method by Huang et al.[18]and (c)

using a Gaussian filter to smooth the wall normal vector. The last approach is used in this paper.

A.Q. Raeini et al. / Journal of Computational Physics 231 (2012) 56535668 5655


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Finally,ds is a delta function concentrated on the interface that is discussed in Section 3.3 in more detail.

3. Numerical method

In this section, we present our formulation for computing capillary forces and applying them in the NavierStokes equa-

tions. First, we present our implementation of the discretization of the interface curvature,k, interface delta function,ds, and

consequently the capillary forces, fc. Then, we propose two new filtering schemes for correcting the computed capillary

forces to avoid numerical instabilities. Finally, we present our semi-implicit approach in coupling capillary forces and theNavierStokes equations. Further details of the numerical method, which are not covered here, can be found in [46,47,40,48].

3.1. Advection of the indicator function

The indicator function has the form of a step function in the continuum limit, while the numerical algorithm tends to

smear the interface sharpness. To deal with this problem, an extra artificial compression term, as proposed in [40], is used

to control the thickness of the interface. The discretised form of advection ofa, after implementing the artificial compressionterm, reads:

@ta Xf2Si

1

vi

haif/Cahaifh1 aif/r 0; 10

where for each grid blocki, viis its volume andfcounts for all the faces surrounding it.hifis used to denote the face-centred

fields that are calculated by interpolating the corresponding cell-centred field. /=uf Sfis the volumetric flux calculated

using Eq.(28), whereSfis the outward vector area of face f. Finally, /r(=ur Sf) is calculated as follows:

/r j/jhnsif nf: 11

3.2. Discretization of interface curvature

Solving Eq.(10)updates the indicator function,a, at the cell centres. Afterwards,ais obtained at solid boundaries using alinear extrapolation from the cell centres. For more accurate calculation of the interface normal vectors in the cells near the

interface, we smooth the indicator function by interpolating it from cell centres to face centres and then back to the cell cen-

tres recursively:

as;i1 CSKhhas;iic!fif

!c 1 CSKas;i; as;0 a: 12

Our simulations show that the coefficient CSKshould be less than one in order to prevent decoupling of the indicator function

from the smoothed indicator function; a value ofCSK= 0.5 and i = 2 is used in our simulations unless stated otherwise. The

smoothed indicator function (as) is then used in Eq. (9) to obtain the interface normal vectors at the centre of cells. At thesolid boundaries, however, the contact angle is used to define the direction of the normal vector to the interface at the con-

tact line (njw):

njw nw cos h sw sinh; 13

wherenwis the normal vector to solid walls andswis the tangent vector to the walls in the normal direction to the contact

line. The normal vectors to the solid walls (nw) are smoothed by interpolating them from the centre of faces (Fig. 1(a)) to the

corner points between the faces located at solid boundaries (Fig. 1(b)) and then interpolating them back to the face centres

(Fig. 1(c)), recursively (five times in this paper).njwis also used to extrapolate the indicator function to the face centres at the

solid boundaries; the magnitude of the gradient of indicator function used in the extrapolation is obtained from those cal-

culated at cell centres while its direction is obtained from Eq.(13). Once the interface normal vectors are computed, interfacecurvature at the cell centres can be obtained using the Gauss scheme:

k r ns Xf2Si

1

vi

Sf hnsif: 14

To model the motion of fluid interfaces more accurately, we smooth the calculated interface curvature in the direction

normal to the interface, recursively for two iterations (i= 0,1):

ks;i1 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffia1 a

p k 1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffia1 a

p k

s ; ks;0 k; 15

where,

k

s

hhks;iwic!fif!c

hhwic!fif!c; w ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffia1 a 106q : 16

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This smoothing method diffuses the variable kaway from the interface, 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1 a

p


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fc;f;filtk dsf

jdsfj e foldc;f;filtkCfc;filtkhrpc rpc nsnsif nf

; 22

wherefoldc;filtkrepresents the values offc, filtkat the previous time-step. The term dsf

jdsfjerestricts this correction term to the inter-

face. Eq.(22)gradually dampens those components offc rpcthat are parallel to the interface, so that they finally converge

to zero. This filtering may introduce small errors in the calculations, however the effect of this filtering was observed to be

negligible in the absence of non-smooth solid boundaries.Cfc,filtkis a coefficient determining how fast the non-physical veloc-

ities are filtered; a value ofCfc, filtk= 0.1 is used in our simulations.

3.5. Filtering capillary fluxes

Due to numerical errors in the calculation of interface curvature, it is difficult to maintain the zero net capillary force con-

straint (H

fc Ss= 0, where Ssis the interface vector area) in modelling the movement of a closed interface. This behaviour was

observed in modelling the steady movement of a micro-scale droplet in a capillary tube, in both the CSF and SSF formula-

tions. Since it is practically impossible to decrease the errors in the calculation of capillary forces to zero, it is difficult to

model the movement of fluid interfaces accurately at low capillary numbers, where the capillary forces are significantly lar-

ger than the viscous forces and therefore the violation of zero-net capillary force plays an important role. Therefore, filtering

the non-physical fluxes generated due to the inconsistent calculation of capillary forces is necessary to maintain the zero-net

capillary force constraint. We use a simple thresholding scheme to filter the capillary fluxes /c jSfjfc;f r?

fpc

and elim-

inate the problems related to the violation of the zero net capillary force constraint on a closed interface. This filtering will

explicitly set the capillary fluxes to zero when their magnitude is of the order of the numerical errors. The filtered capillary

flux reads:

/c;filtered /c maxmin/c;/c;threshold; /c;threshold; 23

where /c,threshold is a threshold value below which capillary fluxes are set to zero. The threshold value is chosen as

/c,threshold=C/c,filtjfc,fjavgjSfj, where jfc,fjavgis the average value of capillary forces (see Eq. (18)) over all faces where they are

non-zero. The filtering coefficient (C/c,filt) should be chosen so that it eliminates the capillary fluxes only when they are in

the range of numerical errors. In our simulations, we use C/c,filt= 0.01, which implies that the capillary fluxes are set to zero

if their magnitude are less than 1% of the average of the capillary forces. This filtering prevents numerical errors in capillary

forces causing instabilities or introducing large errors in the velocity field, by eliminating only a small fraction of the capil-

lary fluxes. Another effect of using this type of filtering is that it reduces the stiffness of the problem by eliminating the high

frequency capillary waves when the capillary forces are close to equilibrium with capillary pressure, allowing larger time

steps to be used when modelling interfacial motion at low capillary numbers. The effect of this filtering on the accuracy

and efficiency of the simulation results are presented in Section 4.2.

3.6. Momentum and pressure equations

The momentum equation, after discretization and linearization can be written as follows:

un u 1

Arpd; 24

where

u Hu f

0

A ; 25

Ais a vector field composed of the diagonal entries in the discretised form of the momentum equation, Eq.(2), andH(u) ac-

counts for all other terms except body forces and pressure gradients. The pressure equation is derived from the discretization

of the incompressibility condition and momentum conservation,24:

r 1

hAifr?fpd

!r uf

; 26

whereufis predicted as follows:

uf hHuif f

0f

hAif27

using the last known velocity field. Once the pressure field is computed, the face centred flux field is calculated as follows:

/ uf SfjSfj

hAifr?fpd; 28

which is used in Eq. (10)for advancing the indicator function.

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3.7. Pressurevelocity coupling

Several algorithms have been developed for the iterative solution of the pressure and velocity equations[49,50]. The pres-

surevelocity coupling algorithm used in this study is based on the pressure implicit with splitting of operators (PISO) algo-

rithm of Issa[50]. In previous studies, an explicit formulation is used for the capillary forces, which are computed from the

most recent values of the indicator function and used in the momentum equation as a source term[32,34]. This imposes a

limit on the time step by introducing a numerical capillary time scale; the numerical method is stable when the time step

resolves the propagation of capillary waves[32]:

dt< hqisdx

3

2pr

1=2; 29

where dxis the size of grid blocks and hqis is the average density of the two fluids.In this paper, a semi-implicit formulation is proposed for capillary forces, along with a CrankNicholson scheme for

advection of the indicator function to ensure the accuracy and stability of the numerical method when larger time steps

are used. In this formulation, first the indicator function is advected for half of the time step using the fluxes at the start

of the time step. Then the equations for the advection of indicator function for the second half of the time step, capillary

pressure, momentum and dynamic pressure are solved iteratively in two loops, as summarized below:

1. Update interface locations fortn1 +dt/2: Solve Eq.(10)for a.2. Outer-corrector loop (nouterCorriterations):

(a) Update interface locations fortn

: Solve Eq.(10).(b) Update viscosity and density (Eq. (6)), interface curvature (Eqs.(12)(17)) and capillary force (Eqs. (18)(21)).

(c) Relax the capillary force fc;f 0:7fn1

c;f 0:3fn

c;f

except for the last iteration.

(d) Solve Eq.(4) for capillary pressure.

(e) PISO loop (nPISO iterations):

i. Predict velocity field at cell centres (Eq.(25)), relax it (u= 0.7un1 + 0.3un) except for the last iteration, and then use

it to obtain the intermediate velocity ufat the face centres (Eq. (27)).

ii. Solve the pressure equation (Eq.(26)).

iii. Correct the velocity field (Eq.(24)) and flux field (Eq.(28)) using the new pressure field.

(f) End of PISO loop

3. End of outer-corrector loop.

4. Proceed to the next time step.

The outer-corrector loop is proposed for a semi-implicit coupling between the velocity and the capillary pressure, while

the PISO loop is responsible for coupling between the velocity and the dynamic pressure. We use two iterations in the PISO

loop and the effect of number of iterations in the outer-corrector loop is presented in Section 4.2.1 for three values of

nouterCorr= 1, 2 and 3.

4. Validation

We present numerical results for several test cases. First we model a stationary droplet (in the absence of gravity), where

we demonstrate the convergence of velocity and capillary pressure to the theoretical solution. Then we study the movement

of a droplet in a uniform velocity field and compare the numerical results for the three formulations: CSF, SSF and our pro-

posed FSF formulation. Next, we model resting of a droplet on a flat solid surface, in order to validate the implementation of

contact angle. In addition, in this test case, we present the effect of filtering capillary forces parallel to the interface on theconvergence of the velocity field, when the mesh is not aligned with the solid boundaries. Finally, we simulate two-phase

flow through capillary tubes, capturing the contact line dynamics.

4.1. Stationary droplet

We study a droplet of one fluid (either gas with viscosity of 10 5 Pa s and density of 1 kg/m3, or liquid with viscosity of

103 Pa s and density of 1000 kg/m3) immersed in a liquid with viscosity of 103 Pa s and density of 1000 kg/m3 in the ab-

sence of gravity, approaching its equilibrium state - a stationary spherical droplet. The interfacial tension is 0.07 N/m. The

aim of this test case is to show the convergence of the numerical results for the velocity field and capillary pressure to

the theoretical solution:

Dpc

2r

R

2 0:07N=m

24:8 106 m 5642

Pa

; umax

0;

30

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whereumaxis the maximum of the magnitude of velocity field in the entire flow domain. A visualization of the solution do-

main and the shape of droplet at the start and the end of the simulations is shown in Fig. 3.

4.1.1. Convergence of the velocity field

A plot of the maximum velocity for the gasliquid and the liquidliquid systems as a function of time is given in Fig. 4for

the CSF, SSF and FSF formulations. It is clear that the velocity field in the SSF and FSF formulations converges to the theoret-

ical solution, while it does not converge in the original CSF formulation. In the SSF formulation, this convergence is sensitive

to the discretization scheme for the fluxes; the self-filtered central differencing (SFCD) scheme was observed to produce

good results in terms of eliminating the spurious velocities and is used in our simulations.

In this case, we are only interested in the final static solution and we do not study the accuracy in the time evolution of

the velocity field and capillary waves. The results in Fig. 4, however, suggest that the SSF formulation can predict the cap-

illary waves that exponentially converge to zero. The FSF formulation, on the other hand, eliminates the capillary waveswhen their magnitude is small compared to the maximum velocity in the time and space domain, which is due to the filter-

ing of capillary fluxes presented in Section3.5. This is a desirable property for long term modelling of interface motion as it

allows larger time steps to be used without affecting the stability of the numerical method, and therefore it alleviates the

capillary time-step constraint (Eq.(29)). Moreover, the filtering of capillary forces parallel to the interface in the FSF formu-

lation (presented in Section3.4) eliminates the spurious velocities. This is essential if we want to use the numerical method

for prediction of multiphase flow at low capillary numbers.

Fig. 3. Solution domain for modelling the static droplet for a mesh resolution ofR/dx= 4; (a) initial condition a cube of size 40 lm, and (b) static shape of

droplet a sphere with radiusR = 24.8 lm (t= 0.001 s).

(a) (b)

Fig. 4. Plots of maximum velocity (umax) as a function of time: (a) gasliquid system and (b) liquidliquid system.



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4.1.2. Convergence of the capillary pressure

Obtaining the interface normal vectors from a smoothed indicator function can be used to ensure the convergence of the

calculated capillary forces and hence the capillary pressure to the theoretical solution (see [36]). The convergence results for

the original CSF formulation and the FSF formulation, with and without the smoothing approaches discussed in Section3.2,

are given inFig. 5, for the liquidliquid system presented previously. The results are presented in terms of the errors in pre-

dicted capillary pressure, E(Pc), defined as follows:

EPc

PcPc;exact

Pc;exact : 31

These results demonstrate that the calculated capillary pressure converges to the theoretical solution by refining the

mesh, when the interface curvature is obtained from a smoothed indicator function. It is clear that when the indicator func-

tion and interface normal vectors are not smoothed, the convergence rate is less than linear. The FSF formulation with

smoothing gives approximately second order convergence with grid refinement.

4.2. Moving droplet at a low capillary number

In this test case, we study the effect of violation of the constraint of zero net capillary force on a closed interface. For this

purpose, we model the steady movement of a micro-scale droplet of radius R= 12.5 lm in a uniform velocity field of

u= 0.001 m/s, corresponding to a capillary number of lur 2 105. The boundary condition is a constant value of

u= 0.001 m/s for the velocity field and zero-gradient for pressure, for all sides of the solution domain. The density and vis-

cosity of both fluids are the same as water at room temperature and the interfacial tension is r= 0.05 N/m. A visualization ofthe solution domain used in our simulations is given inFig. 6(a) for a mesh resolution ofR/dx= 5. We have presented snap-

shots of the velocity field in Fig. 6(b), the indicator function inFig. 6(c), and the capillary pressure inFig. 6(d) for the FSF

formulation.

Fig. 7compares the predicted velocity fields obtained for the CSF, SSF and FSF formulations. Clearly, the results of simu-

lations without filtering (CSF and SSF formulations) are not acceptable for modelling multiphase flow at the micro-scale. In

this case, the average velocity of the droplet deviates from the theoretical velocity by one and two orders of magnitude in the

SSF and CSF formulations, respectively, when no filtering is used to eliminate the numerical errors from the capillary forces.

This deviation will be more pronounced at lower capillary numbers. On the other hand, the FSF formulation in which the

capillary pressure is effectively limited to one grid block along with the two filtering methods presented in this study is able

to predict the movement of the droplet accurately. Aside from the filtering, our simulations show that the errors in the aver-

age velocity of the droplet are reduced by using the weighted interpolation of interface curvature from cell centres to face

centres (Eq.(17)) and by smoothing the curvature away from the interface (Eq. (15)).

Table 1presents the total error for the average velocity of the droplet,E(Udroplet), maximum of velocity magnitude in thesolution domain, E(umax), and the capillary pressure, E(Pc), averaged over a time interval of 0.02 s from the start of the

simulation.

These results show that the numerical method can predict the movement of the droplet accurately, without being af-

fected by the spurious currents or violation of the zero net capillary force constraint on a closed interface.Table 1also reports

the errors in solving the advection of the indicator function, Eadv, as the difference between displacement of the droplet pre-

dicted by solving Eq.(10)from the displacement calculated using the average velocity of the droplet R0:01 s

t0 Udropletdt

, which

shows that the interface compression scheme used in this study does not introduce any significant error in the predicted

displacement of the fluid interfaces.

Fig. 5. Convergence of capillary pressure for the CSF and FSF formulations and the effect of smoothing in improving the accuracy of computed capillarypressure. For very coarse mesh files dx

R> 0:2

, the CSF formulation was not stable. The dashed line indicates second order convergence.

http://-/?-http://-/?-


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Fig. 6. A 3D visualization of the solution domain for the case of moving droplet in a uniform velocity field (a), and the predicted results at time t= 0.02 s,

using the FSF formulation, for velocity field (b), the indicator function (c) and the capillary pressure (d). The interface compression coefficient is Ca= 1 and

the capillary compression coefficient is Cpc= 0.5.

Fig. 7. Average velocity of a droplet moving in a uniform velocity field of 1 mm/s inside an Eulerian mesh and violation of the zero net capillary forceconstraint in the CSF and SSF formulations due to numerical errors in calculation of capillary forces.



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4.2.1. Time step size

As it is clear fromFig. 4, capillary forces produce high-frequency waves that are gradually dampened by viscous forces.

When an explicit formulation is used for capillary forces, this imposes a limit on the time step size by introducing a numer-

ical capillary time scale given by Eq. (29). In contrast to the Courant-Friedrichs-Lewy (CFL) condition, umaxdt/dx< 1[51], Eq.

(29)implies that at low velocities we cannot use larger time step sizes. Consequently, the overall simulation time will be

higher for long-term prediction of flow at lower velocities, typical of fluid flow in porous media.

We performed simulations for the test case in the previous section using different time step sizes (dt= 106 to 103 s) and

different mesh resolutions (dx/R= 0.5 0.1). We repeat the simulations with different iterations in the outer-corrector loop

(nouterCorr), as presented in Section3.7. If the maximum local velocity during the simulation time deviated by more than 10%

from the theoretical velocity, the simulations were considered unstable. The following constraint on the time step size sum-

marizes the results of our simulations for the FSF formulation:

dt< Cdtldxr ; 32

whereCdt= 40 for the explicit implementation of capillary forces (nouterCorr= 1) and Cdt= 400 for the semi-implicit formula-

tion with nouterCorr= 2 and Cdt= 800 for nouterCorr= 3. As an example, for the test case discussed in this section with a mesh

resolution ofR/dx= 5 the capillary time-step constraint presented in Eq. (29)limits the time step to be less than dt= 0.2 ls.

In the FSF formulation presented in this paper, however, the simulations were stable for time step sizes up todt= 1 ls for the

explicit formulation (nouterCorr= 1) and up to dt= 20, 40 ls for the semi-implicit formulation withnouterCorr= 2, 3 iterations,

respectively. This observation implies that the semi-implicit formulation presented in Section 3.7can improve the efficiency

of the simulations by more than one order of magnitude for low capillary numbers.

4.3. Stationary droplet on a flat plate

In this section, we present the numerical results for modelling a droplet resting on a flat plate with different contact an-

gles. Viscosity and density of both fluids are the same as water at room temperature and the interfacial tension is 0.05 N/m.

The solution domain is restricted from the top to a hemisphere of radius L/2 = 50 lm, to save the computational time. The

droplet is in contact with the plate but not with the hemisphere. The equilibrium shape of the droplet will be a spherical cap,

as shown inFig. 8(a). It can be shown that the radius of the spherical cap corresponding to a contact angle ofh is:

R 3Va

p1 cos h22 cos h

!1=3; 33

Table 1

Errors in modelling a droplet moving in a uniform velocity field.

R/dx E(Pc) E(umax) E(Udroplet) Eadv

2.5 0.00475 0.0360 0.00837 0.0333

5 0.00238 0.0104 0.00671 0.00167

7.5 0.00171 0.00712 0.00214 0.000248

10 0.00433 0.0143 0.00367 7.9 105

Fig. 8. Two representations of a flat solid wall in a uniform Cartesian mesh. (a) The wall is aligned with the mesh. (b) The wall equation is y =x.



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whereVais the volume of the droplet calculated by numerically integratinga over the flow domain. Consequently, the the-oretical value of capillary pressure will be:

Pc;theoretical2rR : 34

The convergence of capillary pressure by mesh resolution is given in Table 2, for different mesh resolutions and different

contact angles. The errors in capillary pressure are small, although they do not appear to converge to zero by refining the

mesh.

4.3.1. Treatment of jagged solid walls

We now model the resting of a droplet of liquid on flat plates described mathematically by the equationy =awx+cwz, in

the absence of gravity. Two examples of the solution domain for a grid resolution ofdx= 5 lm are shown inFig. 8.

We performed simulations for different representations of a flat surface in a Cartesian mesh (i.e. different values ofawand

cw). When no filtering is used for correcting the computed capillary force near the jagged solid walls (i.e. when the mesh is

not aligned with the solid wall), the simulations produce large non-physical velocities. Visualization of the velocity field (u)

and effective capillary force (fc rpc) shows that the non-physical velocities are generated due to errors in capillary forcesthat are parallel to the interface (seeFig. 9). The non-physical velocities would be higher by nearly one order of magnitude if

the solid wall normal vectors were not smoothed (see Fig. 1). Nevertheless, we found that filtering capillary forces is inev-

itable to eliminate these velocities. The velocity field in presence of filtering converges to zero (umax< 1010 m/s) and the

errors in predicted capillary pressure (E(Pc)) are shown in Table 3.

Table 3

Fractional errors in predicted capillary pressure, E(Pc), for different values for the slopes of the solid walls (awand cw), different mesh resolutions (L/dx ) and a contact angle ofh = 60.

L/dx aw= 0, cw= 0 aw= 1, cw= 0 aw= 1, cw= 0.5

10 0.00996 0.0677 0.0366

20 0.00470 0.00217 0.00538

40 0.00952 0.00592 0.00503

Table 2

Fractional errors in predicted capillary pressure (E(Pc)) for modelling a droplet resting on a flat plate for a range

of mesh resolutions (L/dx) and contact angles (h).

L/dx h= 30 h= 60 h= 120

20 0.1300 0.0041 0.0063

30 0.0611 0.0142 0.0180

40 0.0270 0.0006 0.0001

50 0.0021 0.0032 0.0123

80 0.0091 0.0039 0.0036

100 0.0021 0.0027 0.0051

Fig. 9. A visualization of (a) the non-physical velocities, which are produced as a result of (b) the non-physical effective capillary force ( fc rpc) near

jagged solid boundaries, using SSF formulation (i.e. when no filtering is used to correct the computed capillary force). Only velocities on the interface are

shown. Similar behaviour was observed in the CSF formulation. In the FSF formulation, on the other hand, this non-physical behaviour is eliminated by

enforcing the effective capillary force to be perpendicular to the interface at the interface region (ds 0).



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These results show that the numerical method can predict the capillary pressure accurately. Moreover, the problem of

non-physical velocities when mesh is not aligned with the solid boundaries is solved by correcting the capillary force (fc)

so that the effective capillary force (fc rpc) remains perpendicular to the interface. The effect of different approaches

for applying the wall adhesion effects in dynamic situations is presented next.

4.4. Contact line dynamics

This section presents numerical results for contact line dynamics. The fluid properties are the same as in the previoussection. The accuracy of the contact line slip model is studied by modelling the steady movement of the interface between

two liquids in a capillary tube.

4.4.1. Introducing a slip length

One advantage of our method is that it can accommodate additional physical effects. For instance, experimental studies

have shown that even in single-phase flow the no-slip boundary condition may not hold at the micro scale, especially when

the fluids are not wetting the solid surfaces (see [52]for a review on the subject). Following the Navier slip law [53]the

velocity at solid walls is:

uw k@u

@n; 35

wherek is the so-called slip length. Since k depends on the fluid properties, in multiphase flow a constant value fork might

not be adequate. We assume thatk=ak1+ (1 a)k2+asks, wherek1is the slip length between phase 1 and the solid phase, k2is the slip length between phase 2 and the solid phase and asksis the additional slip length in the contact line region.as= 1 inthe neighbourhood of a contact line with radius rs and as= 0 outside this neighbourhood.

Although it is straightforward to modify the velocity field at the solid boundary according to Eq. (35)(see[37]), in this

study, we do not directly modify the velocities at the solid walls, as these velocities are not used in the advection of the indi-

cator function. It can be shown that we can reproduce the effect ofkby multiplying the viscosity of the fluids (l) at the faceslocated on solid boundaries by a factorCl

dx=2dx=2k

. The modified viscosity is then used in the discretization of the Laplacian

termr (lru) in the NavierStokes equations. From a physical point of view, this modification accounts for the differencein the momentum transfer to the solid wall when the Navier slip boundary condition is used instead of the no-slip boundary

condition.

To validate our implementation of slip velocity we performed simulations for single-phase flow, between two parallel

plates 2h= 25 lm apart, using different mesh resolutions and different slip lengths. The results show that, although the time

evolution of the velocity profile depends on time discretization scheme and time step size, when the steady state solution is

achieved the velocity profile matches the theoretical solution for steady state single phase flow exactly. Fig. 10shows theconvergence of the velocity field to the theoretical steady state solution [54]:

uxh

2

2l dpddx

1 y2

h2

2k

h

: 36

Fig. 10shows that the numerical method can match the parabolic profile of the velocity field exactly, even when the flow

domain is discretised using only two grid blocks across the parallel plates.

For the case of multiphase flow however, in addition to the slip length k presented in this section, the numerical imple-

mentation of the advection of the indicator function presented in Eq.(10), introduces an additional slip length proportional

to the grid size (similar to[44]). This additional slip length will prevent any leakage of the fluid in front of contact line, if the

Fig. 10. Convergence of the velocity field to the theoretical steady state solution for single-phase flow between parallel plates for different mesh resolutions(2h/dx). The plates are 25 lm apart. The same curves were obtained for three slip lengths ofk = 0, 0.5 and 1 lm.



14/16

thickness of leaked film is smaller than the grid block sizes. Nevertheless, the slip length, k, presented in this section can be

adjusted in such a way that the contact line velocity matches the observed behaviour in experimental measurements.

4.4.2. Capillary tube

The aim of this section is to present the accuracy of the numerical method in modelling contact line movement in a cap-

illary tube when using a uniform Cartesian mesh that is not necessarily aligned with the solid boundaries; this situation can

occur when modelling multiphase flow on micro-CT images of porous rocks. We apply a dynamic pressure difference of

4 104 Pa/m across the capillary tube Nc

luavg

r ffi105 . The boundary condition for velocity is zero-gradient along the

sides of the tube. A visualization of the solution domain for two sample mesh files and the corresponding velocity fields ob-

tained using the FSF formulation is presented inFig. 11(a)(d) for a mesh resolution ofdx/R= 0.1. For the capillary tube pre-

sented inFig. 11(b), which is not aligned with the grid blocks, the simulations using the CSF formulation were not stable and

the SSF formulation was not able to model the correct physics due to the presence of high non-physical velocities close to the

jagged solid boundaries in this case.

Table 4

Errors in predicted capillary pressure and the deviation of the flow rate from single-phase case when a slip

length of ks= 1 lm is used on a radius of 1 lm around the contact line. Other simulation parameters are:

k1= k2= 0, k s= 1 lm, rs= 1 lm, u = 0.001 m/s, L = 50 lm and R = 12.5 lm.

2R/dx y= 0 y= 0.5x

E(Pc) u usp E(Pc) u usp

5 0.2620 0.360 0.400 0.1334

10 0.0455 0.354 0.1329 0.382

20 0.0035 0.458 0.00344 0.42

30 0.0046 0.482 0.00542 0.482

40 0.0040 0.505 0.00163 0.495

Fig. 11. Two different representations of a capillary tube in a uniform Cartesian mesh: (a) the tube is parallel to the x axis, and (b) the tube makes an angle

of 45 to the x axis. (c) and (d) are the corresponding snapshots of the velocity field.



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Table 4presents the numerical results for the FSF formulation, for the two meshes presented inFig. 11. The numerical

results are similar which indicates that our implementation of contact angle can handle the solid boundaries accurately even

when the solid walls are not aligned with the mesh.

The decrease in the velocities presented inTable 4, (u usp), shows that the viscous forces active near the contact line

result in a decrease in the flow rate of the capillary tube for a given dynamic pressure drop, compared to single-phase flow.

Visualization of the interface shows that the interface is bent due to the presence of viscous forces, giving rise to an interface

curvature and a dynamic contact angle different from the static case. On the other hand, Table 4shows that the predicted

capillary pressure using Eq. (4) is nearly the same as in the static case, despite the fact that the dynamic contact angle is

different. This dynamic contact angle results in a high effective capillary force which is responsible for the contact line slip

velocity. The high viscous forces near the contact line are responsible for the decrease in the velocity which can also be

interpreted as an increase in the dynamic pressure drop for a given velocity of the contact line. This implies that in our for-

mulation the effect of the dynamic contact angle appears in an additional dynamic pressure drop and it does not affect the

capillary pressure. Our preliminary simulations show that this increase in the dynamic pressure drop is a function of the

applied slip length and is proportional to the contact line velocity. A more extensive study of the contact line dynamics is

beyond the scope of this study. Nevertheless, the results presented in this section show the importance of direct numerical

simulation in understanding the physics behind multiphase flow at the micro scale, where experimental studies alone

cannot reveal sufficient insight into the problem due to difficulties in measuring all of the flow parameters locally.

In summary, the simulation results presented in this section show that the FSF formulation presented can predict

two-phase flow without destabilisation due to numerical errors for the computation of capillary forces. Therefore, the FSF

formulation can be used to model fluid flow in complex geometries such as modelling two-phase flow on micro-CT images

of porous rocks[55,11,18,15,12,13]. These simulations are not possible using the original CSF formulation for low capillary

numbers due to presence of the spurious currents as well as non-physical velocities close to complex solid boundaries.

5. Conclusions

We have presented a stable and efficient method for modelling two-phase flow at low capillary numbers. We showed that

a sharp surface force model could eliminate the problem of spurious currents. We solve for the capillary pressure equation

separately from the dynamic pressure. This allows us to filter the capillary forces to avoid numerical errors and instabilities.

We presented a semi-implicit formulation for capillary forces, which alleviates the capillary time step-constraint and allows

larger time steps for long-term prediction of two-phase flow at low capillary numbers. Finally, we presented a series of

benchmark cases to demonstrate the stability and accuracy of the method. In future work, we will use this method to model

two-phase flow at low capillary numbers directly on micro-CT images of porous rocks to predict the macroscopic (Darcy

scale) properties of such systems.

Acknowledgements

This work is supported by the Imperial College consortium on pore-scale modelling (BG, BHP, BP, JOGMEC, Saudi Aramco,

Shell, Statoil, Total). We thank OpenFOAM developers and contributors for the use of their codes.

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