Comput Geosci (2013) 17:151–166DOI 10.1007/s10596-012-9322-2

ORIGINAL PAPER

Computational modeling of moving interfaces betweenfluid and porous medium domains

Chongbin Zhao · Thomas Poulet ·Klaus Regenauer-Lieb · B. E. Hobbs

Received: 25 April 2012 / Accepted: 17 September 2012 / Published online: 1 November 2012© Springer Science+Business Media Dordrecht 2012

Abstract This paper presents a numerical procedurefor computational modeling of moving interfaces be-tween fluid and porous medium domains. To avoid thedirect description of the interface boundary conditionsat the interface between the fluid and porous mediumdomains, Darcy’s law is used to simulate fluid flowsin both the fluid and porous medium domains. Forthe purpose of effectively simulating the fluid flowusing Darcy’s law, the artificial permeability of thefluid domain is used to establish a permeability ratiobetween the artificial permeability in the fluid domainand the real permeability in the porous medium do-main. Using the proposed permeability ratio, the ratioof fluid pressure gradient in the porous medium domainto that in the fluid domain can be appropriately sim-ulated. To verify the proposed numerical procedure,analytical solutions have been derived for a benchmarkproblem, which can be simulated using the proposednumerical procedure. Comparison of the numerical so-lutions with the derived analytical solutions has demon-strated the correctness and accuracy of the proposed

C. Zhao (B)Computational Geosciences Research Centre,Central South University, Changsha 410083, Chinae-mail: chongbin.zhao@iinet.net.au

T. Poulet · K. Regenauer-LiebCSIRO, Division of Earth Science and ResourceEngineering,P. O. Box 1130, Bentley, WA 6102, Australia

K. Regenauer-Lieb · B. E. HobbsSchool of Earth and Environment,The University of Western Australia,Crawley, WA 6009, Australia

numerical procedure. Through applying the proposednumerical procedure to several examples associatedwith the fluid–porous medium interface propagationproblems, the related numerical solutions have demon-strated that: (1) the proposed numerical procedure iscapable of simulating the morphological instability ofthe fluid–porous medium interface in a coupled fluidflow–chemical dissolution system when the system is ina supercritical state; and (2) the permeability ratio canhave a considerable effect on the evolved morphologiesof the fluid–porous medium interface in the coupledfluid flow–chemical dissolution system.

Keywords Computational modeling · Fluid domain ·Porous medium domain · Moving interface ·Coupled system · Chemical dissolution

1 Introduction

When acidic fluids pass through carbonate rocks, theacid reacts with the carbonate rock through chemicaldissolution reactions. As a result, the chemical reactioninterface divides the whole problem domain into twodomains, namely a fluid domain where the carbonaterock is completely dissolved and a porous mediumdomain where the acid has not reached. Although thechemical dissolution reaction only takes place at thefluid–porous rock interface, the fluid flow must passthrough both the fluid and porous medium domains asa direct consequence of fluid mass conservation. Thismeans that the carbonate rock performs just like anacid filter, which can filter the acid when the acidicfluid passes through the carbonate rock. However, as

152 Comput Geosci (2013) 17:151–166

time goes on, the fluid–porous medium interface (i.e.,the acid filter) will propagate in the whole problemdomain. If the fluid flow is strong enough, then thefluid–porous medium interface can become unstablewhen it propagates in the whole problem domain. Insuch a case, the morphological shape of the fluid–porous medium interface can be changed with time.This is called the instability problem of the movingfluid–porous medium interface in the fluid-saturatedporous medium. This kind of instability problem hasbeen encountered in the related scientific and engineer-ing fields. For example, the instability of moving fluid–porous medium interfaces in carbonate rocks is animportant mechanism of the karst formation (i.e., largecavities in carbonate rocks) that is commonly observedin nature. This mechanism has been successfully usedto increase the secondary oil recovery in petroleumengineering through injecting hydrochloric acid intocarbonate rocks in the surroundings of drilling wells[7, 10–13, 17]. In order to describe the above processesappropriately, it is necessary to consider not only acoupled problem between porosity, fluid flow, masstransport, and chemical dissolution reactions but also amoving fluid–porous medium interface problem in thewhole problem domain.

From the continuum mechanics point of view, thefluid–porous medium interface propagation problemcan be simulated using the following three differentapproaches. In the first approach, fluid flows in thefluid and porous medium domains are simulated usingthe Stokes equations and Darcy’s law, respectively.This approach is only valid if the interface velocity issmall, which is currently acceptable for most chemicaldissolution problems. To use this approach, appropri-ate boundary conditions at the fluid–porous mediuminterface must be established beforehand. Althoughinterface boundary conditions can be established forsome planar interfaces [1], it is very difficult, if notimpossible, to establish such interface boundary con-ditions for the interfaces of complicated morphologies.In the second approach, fluid flows in both the fluidand porous medium domains are simulated using theDarcy–Brinkman equation [2], which was originallydeveloped as a correction to Darcy’s law in the caseof large porosities and was further justified on thetheoretical considerations [18, 20, 22]. If the gravityterm is neglected, the Darcy–Brinkman equation hasthe following form: ∇ p = − μ

K�V + μ

φ∇2 �V, where p is

the fluid pressure; �V is the fluid velocity vector; μ is thedynamic viscosity of the fluid; φ and K are the porosityand permeability of the porous medium, respectively.In the fluid domain where K is very large (theoreti-

cally infinite), the Stokes equation is recovered fromthe Darcy–Brinkman equation, while in the porousmedium domain where K is relatively small, the viscousdiffusive term becomes negligible so that the Darcy–Brinkman equation is degenerated to Darcy’s law.Since the Darcy–Brinkman equation is used to simulatethe fluid flows in both the fluid and porous mediumdomains simultaneously, it is avoidable to establish anyinterface boundary conditions between the fluid andporous medium domains. For this reason, the Darcy–Brinkman equation was used to develop a computa-tional model for capturing wormhole formation duringthe dissolution of a porous medium [11]. In the third ap-proach, through appropriately determining the valuesof permeability, Darcy’s law is used to simulate fluidflows in both the fluid and porous medium domainssimultaneously. In the porous medium domain, the realpermeability is used to simulate the fluid flow, while inthe fluid domain, an artificial permeability is assumedto simulate the fluid flow. By defining the ratio of fluidpressure gradient in the porous medium domain to thatin the fluid domain, the ratio of artificial permeabilityin the fluid domain to the real permeability in theporous medium domain can be determined. Since thisapproach is a natural extension of the well-developedapproach, which was widely used to simulate a mov-ing interface between two different porous mediumdomains [3–6, 15, 16, 23–25], it has attracted muchattention in recent years [7, 12, 13, 17]. Although thethird approach has been successfully used to simulatedifferent wormhole structures in the acidization of car-bonate rocks by assuming the fluid pressure-gradientratio to be 100 between the porous medium and fluiddomains [12, 13, 17], research on some numerical simu-lation issues associated with this approach, such as theverification of the numerical solution and the effect ofthe permeability ratio between the fluid and porousmedium domains, is still lacking. This will become themain purpose of this paper.

Keeping the above-mentioned considerations inmind, the forthcoming contents of this paper arearranged as follows. In Section 2, the mathematicaland numerical modeling of the fluid–porous mediuminterface propagation problem is described. In partic-ular, the concept of the critical porosity is presented torepresent the permeability ratio between the fluid andporous medium domains. In Section 3, the analyticalsolutions for a benchmark problem have been derivedso that the numerical solutions obtained from the pro-posed numerical procedure can be verified throughcomparing them with the derived analytical solutions.In Section 4, the proposed numerical procedure is

Comput Geosci (2013) 17:151–166 153

applied to several application examples related to thefluid–porous medium interface propagation problem.Finally, some conclusions are given in Section 5.

2 Mathematical and numerical modeling of the movinginterface problem between fluid and porousmedium domains

2.1 Mathematical model of the problem

To avoid any difficulty in tracking the moving interfacebetween a fluid domain and a porous medium domain,it is assumed that the governing equations for boththe fluid domain and porous medium domain are ofthe same form. Through the careful use of parame-ters in the governing equations, the fluid domain andthe porous medium domain can be automatically de-termined in the process of numerical simulation. Forexample, theoretically speaking, when the porosity ofa part of the porous medium is equal to unity, thispart of the porous medium can be regarded as a partof the fluid domain. Thus, the moving interface prob-lem between a fluid domain and a porous mediumdomain is, in essence, a problem of dealing with thepropagation of a porosity front, in which the porosityis equal to unity, in the fluid-saturated porous medium.Based on these considerations, the governing equationsand related formulas of the moving interface problembetween a fluid domain and a porous medium domaincan be written as follows:

∂φ

∂t+ ∇ • �u = 0 (1)

∂

∂t(φC) + ∇ • [

C�u] = ∇ • [φD∇C

] − α0(φ f − φ

)C

(2)

∂φ

∂t= χα0

ρs

(φ f − φ

)C (3)

�u = −k (φ)

μ∇ p (4)

α = α0(φ f − φ

)(5)

where �u is the Darcy velocity vector within the car-bonate rock; p and C are the fluid pressure and theacid concentration (with a unit of moles per fluid vol-ume); μ is the dynamic viscosity of the fluid; φ isthe porosity of the carbonate rock; D is the effective

diffusivity/dispersivity of the acid; k(φ) is the perme-ability of the porous medium; χ is the stoichiometriccoefficient of the carbonate rock; α is the mass ex-change coefficient between the acid and the carbonaterock; ρs is the molar density (i.e., moles per volume)of the carbonate rock; φ f is the final (i.e., maximum)porosity when the carbonate rock is completely dis-solved.

To distinguish the fluid domain from the porousmedium domain, it is necessary to consider the perme-ability change caused by a change in porosity using theCarman–Kozeny law [9, 14, 19].

k (φ) = k0 (1 − φ0)2 φ3

φ30 (1 − φ)2 (6)

where φ0 and k0 are the initial reference porosity andpermeability of the porous medium, respectively.

Equation 6 indicates that when the porosity of aporous medium approaches unity, the correspondingpermeability will tend to infinity. According to Darcy’slaw, as expressed in Eq. 4, the pressure gradient ofthe fluid must approach zero, so that the product ofthe Darcy velocity and dynamic viscosity of the fluidcan be maintained as a real number of finite value,from the mathematical point of view. This implies thatas long as the permeability of a porous medium islarge enough, the pressure gradient of the fluid can bekept small enough to represent the pressure change inthe fluid domain. From the computational simulationpoint of view, it is impossible to use the permeabilityof an infinite value in the numerical simulation. Tocircumstance this difficulty, it is assumed that if theratio of the pressure gradient in the porous mediumdomain to that in the fluid domain is large enough,then the pressure change in the fluid domain can bereasonably simulated in the numerical simulation. Thismeans that for a given pressure-gradient ratio, thereexists a critical porosity that can be used to distinctthe fluid domain from the porous medium domain inthe numerical simulation. Suppose the minimum andcritical porosities of the porous medium are φ0 and φC,respectively, the corresponding ratio of permeabilitycan be expressed as follows:

λ = k (φC)

k (φ0)= ∇ p (φ0)

∇ p (φC)= (1 − φ0)

2 φ3C

φ30 (1 − φC)2 . (7)

Equation 7 can result in the following equation for φC:

(1 − φ0)2

λφ30

φ3C − φ2

C + 2φC − 1 = 0. (8)

154 Comput Geosci (2013) 17:151–166

To find solutions for Eq. 8, it is convenient to define thefollowing function:

f (φC) = (1 − φ0)2

λφ30

φ3C − φ2

C + 2φC − 1. (9)

It can be demonstrated that the first differentiation offunction f (φC) in Eq. 9 with respect to φC is alwaysgreater than zero for any realistic value of the porosity.

d[

f (φC)]

dφC= 3 (1−φ0)

2

λφ30

φ2C−2φC+2>0 (for φC <1) .

(10)

This indicates that function f (φC) varies monotonicallywith φC within the whole possible variation range of φC.Thus, when

(q2

)2 + ( r3

)3> 0, the third power equation

that is expressed in Eq. 8 can only have one real numbersolution as follows:

φC = 1

3a+ 3

√

−q2

+√(q

2

)2 +( r

3

)3

+ 3

√

−q2

−√(q

2

)2 +( r

3

)3((q

2

)2 +( r

3

)3> 0

)

(11)

where

r = 6a − 1

3a2, q = −27a2 − 18a + 2

27a3, a = (1 − φ0)

2

λφ30

.

(12)

When(q

2

)2 + ( r3

)3< 0, the solution for φC can be also

derived analytically. For this purpose, substituting φC =φC1 + 1

3a , where φC1 can be viewed as an auxiliaryvariable, into Eq. 8 yields the following equation:

φ3C1 + rφC1 + q = 0 (13)

On the other hand, the following equity exists mathe-matically:

cos (3α) = 4 cos3 α − 3 cos α (14)

Letting φC2 = cos α, where φC2 can be viewed asanother auxiliary variable, leads to the followingequation:

φ3C2 − 3

4φC2 − 1

4cos (3α) = 0 (15)

Similarly, substituting φC1 = nφC2 into Eq. 13 yields thefollowing equation:

φ3C2 + r

n2φC2 + q

n3= 0 (16)

By comparing Eqs. 15 and 16, both n and α can bedetermined as follows:

n =√

−4r3

, α = 1

3arccos

(−4q

n3

)(17)

As a result, the following three possible solutions forφC2 in Eq. 16, namely φC2 = cosα, φC2 = cos

(α + 120

180π),

and φC2 = cos(α + 240

180π), can be obtained.

Consideration of the porosity variation range yieldsthe following solution for φC when

(q2

)2 + ( r3

)3< 0:

φC = 1

3a+ n cos

(α + 120

180π

) ((q2

)2

+(

r3

)3

< 0

)

(18)

Figure 1 shows the variation of φC with both φ0 andλ. In this figure, φC, φ0, and λ are represented byPhi_critical, Phi_0, and Lambda, respectively. It is ob-vious that for a given value of the initial porosity(i.e., Phi_0), the critical porosity (i.e., Phi_critical) in-creases with an increase in the value of the permeabilityratio (i.e., Lambda). Similarly, for a given value of thepermeability ratio (i.e., Lambda), the critical porosity(i.e., Phi_critical) increases with an increase in the valueof the initial porosity (i.e., Phi_0).

0.0

0.3

0.5

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5

Phi_0

Ph

i_cr

itica

l

Lambda=10

Lambda=100

Lambda=1000

Fig. 1 Variation of the critical porosity with the initial porositydue to different permeability ratios

Comput Geosci (2013) 17:151–166 155

From the mathematical point of view, the governingequations of the moving interface problem between thefluid and porous medium domains can be rewritten intothe following dimensionless form:

∂φ

∂ t̄− (

φ f − φ)

C = 0 (19)

∂

∂ t̄

(φC

)−∇ •

[D∗ (φ) ∇C + Cψ∗ (φ)∇ p

]+ ∂φ

ε∂t=0

(20)

∂φ

∂t− ∇ • [

ψ∗ (φ)∇ p] = 0 (21)

where

x = xL∗ , y = y

L∗ , C = CC0

,

p = pp∗ , �u = �u

u∗ , t = tt∗

(22)

t∗ = ρs

α0χC0, L∗ =

√φ f Dt∗, p∗ = φ f D

ψ(φ f

) ,

u∗ = φ f DL∗ , ε = χC0

ρs, D∗ (φ) = φD

φ f D,

ψ∗ (φ) = ψ (φ)

ψ(φ f

) , ψ (φ) = k (φ)

μ(23)

where C0 is the injected acid concentration in the car-bonate rock.

2.2 Numerical simulation of the problem

In this subsection, Eqs. 19–21 are solved using a com-bination of both the finite element method and thefinite difference method. The finite element methodis used to discretize the geometrical shape of theproblem domain, while the finite difference methodis used to discretize the dimensionless time. Since thesystem described by these equations is highly non-linear, the segregated algorithm, in which Eqs. 19–21are solved separately in a sequential manner, is usedto derive the formulation of the proposed numericalprocedure.

For a given dimensionless time-step, t + �t, theporosity can be denoted by φ t+�t = φt̄ + �φ t+�t, whereφ t is the porosity at the previous time-step and �φ t+�t

is the porosity increment at the current time-step. Using

the backward difference scheme, Eq. 19 can be writtenas follows:[ ε

�τ+ Ct+�t

]�φ t+�t = (

φ f − φ t)

Ct̄+�t̄ (24)

where Ct+�t is the dimensionless acid concentration atthe current time-step and �t is the dimensionless timeincrement at the current time-step.

Mathematically, there exist the following relation-ships in the finite difference sense:

∂(φC

)

∂t=

�(φ t+�tCt+�t

)

�t

= Ct+�t�φ t+�t

�t+ φ t+�t

�(

Ct+�t

)

�t(25)

∂φ

∂t= �

(φ t+�t

)

�t= Ct+�t

(φ f − φ t+�t

)(26)

∇ •[

D∗ (φ)∇C]

= ∇ •[

D∗ (φ t+�t

) ∇Ct+�t

](27)

∇ •[Cψ∗ (φ) ∇ p

]

= C∇ • [ψ∗ (φ) ∇ p

] + [ψ∗ (φ)∇ p

] •[∇C

]

= Ct+�t∇ • [ψ∗ (

φ t+�t) ∇ pt+�t

]

+ [ψ∗ (

φ t+�t) ∇ pt+�t

] •[∇Ct+�t

](28)

Substituting Eqs. 25–28 into Eq. 20 yields the followingfinite difference equation:[

1

�tφ t+�t + 1

ε

(φ f − φ t+�t

)]Ct+�t

− ∇ •[

D∗ (φ t+�t

) ∇Ct+�t

]

− [ψ∗ (

φ t+�t) ∇ p̄t+�t

] •[∇Ct+�t

]= 1

�tφ t+�tCt. (29)

Similarly, Eq. 21 can be rewritten in the followingdiscretized form:

∇ • [ψ∗ (φ)∇ p

] = ∇ • [ψ∗ (

φ t+�t) ∇ pt+�t0

]

= 1

�t

(φ t+�t − φ t

). (30)

Using the proposed segregated scheme and finite ele-ment method, Eqs. 24, 29, and 30 are solved separatelyand sequentially for the porosity, dimensionless acid

156 Comput Geosci (2013) 17:151–166

concentration and dimensionless fluid pressure at thecurrent time-step. To consider the coupling effect be-tween these three equations, an iteration scheme is alsoused in the process of finding the numerical solution.The following convergence criterion needs to be sat-isfied so that a convergent solution can be obtained.

E = Max

⎛

⎜⎝

√√√√

Nφ∑

i=1

(φk

i,t+�t − φk−1i,t+�t

)2,

√√√√

N C∑

i=1

(C

ki,t+�t − C

k−1i,t+�t

)2,

√√√√

N p∑

i=1

(pk

i,t+�t − pk−1i,t+�t

)2

⎞

⎟⎠ < E (31)

where E and E are the maximum error at the k−thiteration step and the allowable error limit; Nφ , NC,and N p are the total numbers of the degree-of-freedomfor the porosity, dimensionless acid concentration, anddimensionless fluid pressure, respectively; k is the indexnumber at the current iteration step and k − 1 is theindex number at the previous iteration step; φk

i,t+�t,

Cki,t+�t, and pk

i,t+�t are the porosity, dimensionless acidconcentration, and dimensionless fluid pressure of nodei at both the current time-step and the current iteration

step, respectively; φk−1i,t+�t, C

k−1i,t+�t, and pk−1

i,t+�t are theporosity, dimensionless acid concentration, and dimen-sionless fluid pressure of node i at the current time-stepbut at the previous iteration step, respectively.

3 Verification of the proposed numerical procedurefor simulating the propagation of fluid–porousmedium interfaces

3.1 Derivation of analytical solutionsfor a benchmark problem

When a planar interface between the fluid and porousmedium domains is considered to propagate in the fullspace, analytical solutions can be obtained for boththe propagation speed of the interface between thefluid and porous medium domains and the downstreampressure gradient of the fluid in the fluid-saturatedcarbonate rock. In this special case, a planar inter-face between the fluid and porous medium domains isassumed to propagate in the positive x direction, so

that all quantities are independent of the transversecoordinates y and z. For this reason, Eqs. 19–21 can berewritten as follows:

∂φ

∂ t̄− ∂

∂ x̄

[ψ∗ (φ)

∂ p∂x

]= 0 (32)

∂

∂t

(φC

)− ∂

∂x

[

D∗ (φ)∂C∂x

+ Cψ∗ (φ)∂ p∂x

]

+ 1

ε

(φ f − φ

)C = 0 (33)

∂φ

∂t− (

φ f − φ)

C = 0. (34)

If the acid is injected at the location of x approachingnegative infinity, then the boundary conditions of theproblem can be expressed as

limx→−∞

C = 1, limx→−∞

φ = φ f ,

limx→−∞

∂ p∂x

= p ′f x

(upstream boundary

)(35)

limx→∞

C = 0, limx→∞

φ = φ0,

limx→∞

∂ p∂x

= p ′0x

(downstream boundary

)(36)

where φ0 is the initial porosity of the carbonate rock;C is the injected dimensionless acid concentration inthe carbonate rock; p ′

f x is the fluid pressure gradientas x approaching negative infinity in the upstream ofthe fluid flow; and p ′

0x is the unknown fluid pressuregradient as x approaching positive infinity in the down-stream of the fluid flow. Because p ′

f x drives the fluidflow continuously along the positive x direction, it hasa negative algebraic value (i.e., p ′

f x < 0).To transform a moving interface problem between

the fluid and porous medium domains (in an x − t co-ordinate system) into a steady-state boundary problem(in a ξ − t coordinate system), the following coordinatemapping can be used:

ξ = x − vfrontt (37)

where vfront is the dimensionless propagation speedof the planar interface between the fluid and porousmedium domains in the fluid-saturated carbonate rock.

Comput Geosci (2013) 17:151–166 157

From the mathematical point of view, the followingrelationships exist between the partial derivatives withrespect to ξ and t and those with respect to x and t [21]:

(∂

∂ t̄

)

ξ

=(

∂

∂t

)

x+ ∂

∂x∂x∂t

=(

∂

∂t

)

x+ vfront

∂

∂x,

(∂

∂ξ

)

t=

(∂

∂x

)

t(38)

where derivatives are taken with x or t held constant asappropriate.

Since the transformed acid dissolution system in theξ − t coordinate system is in a steady state, the follow-ing equation can be derived from Eq. 38:

(∂

∂t

)

x= −vfront

∂

∂ξ,

(∂

∂ξ

)

t=

(∂

∂x

)

t(39)

Substituting Eq. 39 into Eqs. 32–34 yields the followingequations:

∂

∂ξ

[ψ∗ (φ)

∂ p∂ξ

+ vfrontφ

]= 0 (40)

∂

∂ξ

[

D∗ (φ)∂C∂ξ

+ Cψ∗ (φ)∂ p∂ξ

+ vfront

(C + 1

ε

)φ

]

= 0

(41)

vfront∂φ

∂ξ+ (

φ f − φ)

C = 0. (42)

Integrating Eqs. 40 and 41 from negative infinity topositive infinity and using the boundary conditions (i.e.,Eqs. 35 and 36) yield the following equations:

ψ∗ (φ0) p ′0x + vfrontφ0 − p ′

f x − vfrontφ f = 0 (43)

p ′f x + vfrontφ f

(1 + 1

ε

)− vfrontφ0

1

ε= 0 (44)

Note that according to the upstream boundary con-ditions (Eqs. 35 and 36), the dimensionless acid con-centration approaches unity and zero as x approachesnegative and positive infinity, respectively, indicatingthat ∂C

∂x = 0 at both the negative infinity and the positiveinfinity. From Eq. 37, it is obvious that ∂ξ = ∂x, so that∂C∂ξ

= 0 at both the negative infinity and the positiveinfinity. This is the reason why the first term of Eq. 41varnishes in Eq. 44 after integration.

Solving Eqs. 43 and 44 simultaneously results in thefollowing analytical solutions:

vfront = −p ′f x

φ f + (φ f − φ0

)1ε

(45)

p ′0x = p ′

f x + vfront(φ f − φ0

)

ψ∗ (φ0)(46)

Equations 45 and 46 indicate that since p ′f x is known in

the upstream of the fluid flow, both vfront and p ′0x can

be expressed explicitly, so that they can be used in theanalytical solutions hereafter. It needs to be pointed outthat according to Darcy’s law, the dimensionless Darcyvelocity can be expressed as u f x = −p ′

f x and u0x =−ψ∗ (φ0) p ′

0x when x approaches negative and positiveinfinity, respectively. As a result, Eq. 46 can be furtherrewritten as u0x = u f x − vfront

(φ f − φ0

). Since the term

of vfront(φ f − φ0

)represents the fluid consumption due

to the propagation of the fluid–porous medium inter-face, the meaning of u0x is exactly the dimensionlessDarcy velocity at the downstream boundary, indicatingthat Darcy’s law holds for both p ′

f x and p ′0x at the

upstream and downstream boundaries.The planar interface between the fluid and porous

medium domains divides the problem domain into tworegions, an upstream region consisting of the fluid anda downstream region consisting of the porous medium.Across this interface, the porosity undergoes a jumpfrom its initial value into unity. Thus, the fluid–porousmedium interface propagation problem can be consid-ered as a Stefan moving boundary problem [21], so thatthe dimensionless governing equations of the planar in-terface between the fluid and porous medium domainsin both the upstream region and the downstream regioncan be expressed as follows:

φf vfront∂C∂x

+ ∇ •(∇C + C∇ p

)= 0, ∇2 p = 0,

φ = φ f(in the upstream region

)(47)

C = 0, ∇2 p = 0,

φ = φ f(in the upstream region

). (48)

If the planar interface between the fluid and porousmedium domains is denoted by S

(x, t

) = 0, then thedimensionless fluid pressure, acid concentration, andmass fluxes of both the acid and the fluid should be

158 Comput Geosci (2013) 17:151–166

continuous on S(x, t

) = 0. This leads to the followinginterface conditions for this moving interface problem.

limS→0−

C = limS→0+

C, limS→0−

p = limS→0+

p (49)

limS→0−

∂C∂n

= −vfront

ε

(φ f − φ0

),

limS→0−

∂ p∂n

= ψ∗ (φ0) limS→0+

∂ p∂n

− vfront(φ f − φ0

)(50)

where n is the dimensionless unit normal vector of themoving fluid–porous medium interface.

Note that since the propagation speed of thefluid–porous medium interface is usually much slowerthan the Darcy velocity of the fluid flow, the term,vfront

(φ f − φ0

), can be neglected from Eq. 50. When the

planar fluid–porous medium interface is under stableconditions, the base solutions for this problem can bederived from Eqs. 47 and 48 with the related boundaryconditions (i.e., Eqs. 35 and 36) and interface conditions(i.e., Eqs. 49 and 50). As a result, the following basesolutions are obtained:

C (ξ) = 1 − exp[−

(p ′

f x + φf vfront

)ξ],

p (ξ)= p ′f xξ + pC2, φ = φf

(in the upstream region

)

(51)C (ξ) = 0, p (ξ) = p ′

0xξ + pC1,

φ = φ0(in the upstream region

)(52)

where pC1 and pC2 are two constants to be determined.For example, pC1 can be determined by setting thedimensionless fluid pressure p (ξ) to be a constant ata prescribed location of the downstream region, while

pC2 can be determined using the pressure continuitycondition at the interface between the upstream anddownstream regions.

Equations 51 and 52 clearly indicates that althoughporosity may vary sharply (as a step function) at a fluid–porous medium interface, both the dimensionless acidconcentration and the dimensionless fluid pressure varycontinuously (as continuous functions) at the fluid–porous medium interface. This means that for the nu-merical simulation of an acid dissolution problem, thefocus should be on the appropriate simulation of thestep porosity front propagation.

3.2 Verification of the proposed numerical procedure

Since the conventional finite element method is onlysuitable for simulating the problem domain of finitesize, the derived analytical solutions for the fluid–porous medium interface propagation in the full spacecannot be directly used as benchmark solutions for ver-ifying the proposed numerical procedure. Therefore, itis necessary to establish a link between the analyticalsolutions for the fluid–porous medium interface prop-agation in the full space and those in a rectangulardomain, which can be easily simulated using both thefinite element and finite difference methods.

For the rectangular domain shown in Fig. 2, thecorresponding boundary conditions can be expressed asfollows:

C(t) = 1 − exp

[(p ′

f x + φf vfront

)vfrontt

],

φ = 1,∂ p∂x

= p ′f x, (x = 0) (53)

Fig. 2 Geometry andboundary conditions of thebenchmark problem

0,0y

p

y

C0

y

xL

fx

f

px

p

tCC )(

0,0y

p

y

C

yL00

C

Inflow

x

Comput Geosci (2013) 17:151–166 159

∂C∂x

= 0, p = p0

(x = Lx

)(54)

∂C∂y

= 0,∂ p∂y

= 0(at the top and bottom boundaries

)

(55)

where p0 is the dimensionless pressure of the fluid onthe downstream boundary.

Similarly, the initial conditions of the problem can bewritten as follows:

C = 0, φ = φ0(0 < x ≤ Lx

). (56)

From the derived analytical solutions for the fluid–porous medium interface propagation problem in thefull space, the corresponding analytical solutions for thefluid–porous medium interface propagation problem inthe rectangular domain can be expressed in the follow-ing form:

C(x, t

) = 0, p(x, t

) = −p ′0x

(Lx − x

) + p0,

φ(x, t

) = φ0(Lx ≥ x > vfrontt

)(57)

C(x, t

) = 1 − exp[−(

p ′f x + φ f v front

)(x − vfrontt

)],

p(x, t

) = p ′f x

(x − vfrontt

) − p ′0x

(Lx − vfrontt

) + p0,

φ(x, t

) = 1(0 ≤ x ≤ vfrontt

)(58)

Equation 58 indicates that for a rectangular domainof finite size, the dimensionless acid concentration atthe upstream boundary (x = 0) is no longer a constantof unity. To use this analytical solution as a bench-mark solution, a time-dependent boundary conditionshould be applied at the upstream boundary (x = 0). Asmentioned previously, since both vfront and p ′

0x can beexpressed explicitly, it is possible to apply such a time-dependent boundary condition at the left boundary ofa rectangular domain that is simulated by the numericalmodel.

The fluid–porous medium interface propagationproblem in the rectangular domain is then simulatedusing the proposed numerical procedure. For this pur-pose, the following parameters [11] are used in the com-putational simulation. The initial porosity (i.e., φ0) inthe porous medium domain and final porosity (i.e., φ f )

in the fluid domain are 0.38 and 0.98, respectively,which is very close to the idealized value of 1.0. Themineral dissolution ratio (i.e., ε) is equal to 0.076,while the permeability ratio (i.e., λ) is equal to 1,000,which results in a critical porosity (i.e., φC) of 0.925.

In addition, other two values of λ, namely λ = 100and λ = 5,000 are used to investigate its effect onthe numerical simulation results. Except for the leftboundary, the initial porosity of the porous mediumdomain is 0.38, while the initial dimensionless acidconcentration is zero within the rectangular domain.The final porosity is applied at the left boundary asa boundary condition of the computational domain.As indicated by Eq. 53, a time-dependent dimension-less acid concentration boundary condition (i.e., C(t) =1 − exp[(p ′

f x + φf vfront)vfrontt]) needs to be applied atthe left boundary of the rectangular domain. The di-mensionless fluid pressure (i.e., p0) applied at the rightboundary of the computational model is equal to 100.The dimensionless length and width of the rectangulardomain are 20 and 10, respectively. To simulate thepropagation of fluid–porous medium interfaces appro-priately, the finite element size has been varied toensure that the numerical dispersion does not affect thenumerical simulation results in a rectangular domain,for which the present analytical solutions can be usedfor comparison with the numerical solution. Throughthe mesh size sensitivity analysis, it is confirmed that aslong as the finite element size satisfies the mesh Pecletcriterion [8], the numerical dispersion can be minimizedin the computational simulation. As a result, the wholecomputational domain is simulated by 79,399 four-nodesquare finite elements with 80,000 nodal points in total.The dimensionless fluid pressure gradient (i.e., p ′

f x) ap-plied at the left boundary of the rectangular domain isequal to −0.02, implying that the acidic fluid is horizon-tally flowing from the left to the right boundaries of therectangular domain and that the Zhao number of thefluid–porous medium interface propagation problemunder consideration is equal to 0.02. Note that the Zhaonumber is a comprehensive dimensionless numberto represent the three major controlling mechanisms(i.e., advection, diffusion/dispersion, and chemical dis-solution) simultaneously taking place in the acid disso-lution system [24]. Since the critical Zhao number ofthis problem is equal to 0.085, which is greater than thecorresponding Zhao number, the fluid–porous mediuminterface is stable when it propagates in the rectangu-lar domain. Due to the slowness of the fluid–porousmedium interface propagation, the dimensionless time-step is set to be 45 in the corresponding computation.

Figures 3, 4, and 5 show the comparison of nu-merical solutions with analytical ones for the porosity,dimensionless acid concentration, and dimensionlessfluid pressure distributions within the whole computa-tional domain at four different time instants, namelyt = 1, 800, 3,600, 5,400, and 7,200, respectively, whenthe permeability ratio (i.e., λ) is equal to 1000. In these

160 Comput Geosci (2013) 17:151–166

Fig. 3 Comparison ofnumerical solutions withanalytical ones at fourdifferent time instants(porosity)

1.0

0.38 200

Poro

sity

1.0

0.38200

Poro

sity

1.0

0.38 200

Poro

sity

1.0

0.38200

Poro

sity

x)1800(t

x)3600(t

x)5400(t

x)7200(t

figures, the thick lines denote the numerical results,while the thin lines denote the corresponding analyticalsolutions, which are obtained from Eqs. 57 and 58

with the boundary condition of p0 = 100 at the rightboundary of the computational model. From these re-sults, it can be observed that the overall numerical

Fig. 4 Comparison ofnumerical solutions withanalytical ones at fourdifferent time instants(dimensionless acidconcentration)

)1800(t )3600(t

)5400(t )7200(t

20 0 x

Dim

ensi

onle

ss c

once

ntra

tion

0

0.07

20 0 x

Dim

ensi

onle

ss c

once

ntra

tion

0

0.13

20 0 x

Dim

ensi

onle

ss c

once

ntra

tion

0

0.20

20 0 x

Dim

ensi

onle

ss c

once

ntra

tion

0

0.25

Comput Geosci (2013) 17:151–166 161

Fig. 5 Comparison ofnumerical solutions withanalytical ones at fourdifferent time instants(dimensionless fluid pressure)

Dim

ensi

onle

ss p

ress

ure

400

100 20 0

x)1800(t

x0

Dim

ensi

onle

ss p

ress

ure

400

10020

330

0 x

Dim

ensi

onle

ss p

ress

ure

400

100 20 0

x

250

Dim

ensi

onle

ss p

ress

ure

400

10020

180

)3600(t

)5400(t )7200(t

solutions agree very well with the analytical solutions,except for a smooth effect on the numerically sim-ulated fluid–porous medium interface that is repre-sented by the porosity front. Nevertheless, such aneffect can only affect the sharpness of the fluid–porousmedium interface at a very narrow local level, so thatit has negligible influence on both the dimensionlessacid concentration and the dimensionless fluid pressuredistributions within the whole computational domain.This indicates that except for losing the sharpness ofthe fluid–porous medium interface, the proposed nu-merical procedure is capable of simulating the pla-nar fluid–porous medium interface propagation withinthe coupled fluid flow–chemical dissolution system.Clearly, the numerically simulated dimensionless acidconcentrations are overlapped with the correspond-ing analytical solutions (see Fig. 4), indicating thatthe proposed numerical procedure can produce highlyaccurate numerical results for the dimensionless acidconcentration in the computational model. Althoughthere are some smooth effects on the numerically sim-ulated fluid–porous medium interface due to numericaldispersion, the propagation speed of the numericallysimulated fluid–porous medium interface is in goodcoincidence with that of the analytically predicted one.For this benchmark problem, the overall accuracy ofthe numerical results is indicated by the dimensionlessfluid pressure. The maximum analytical value of thedimensionless fluid pressure is 397, 322, 246, and 171at the four different time instants, while the maxi-mum numerically simulated value of the dimensionless

fluid pressure is 403, 327, 252, and 176 at the samefour different time instances, respectively, Thus, themaximum relative error of the numerically simulateddimensionless fluid pressure is 1.5, 1.6, 2.4, and 2.9 %for the four dimensionless time instants being 1,800,3,600, 5,400, and 7,200, respectively. This quantitativelydemonstrates that the proposed numerical procedurecan produce accurate numerical solutions for the pla-nar fluid–porous medium interface propagation prob-lem within a coupled fluid flow–chemical dissolutionsystem.

4 Application of the proposed numerical procedure

The proposed numerical procedure is firstly used tosimulate the morphological evolution of a fluid–porousmedium interface in a supercritical system. Toward thisend, a dimensionless pressure gradient (i.e., p ′

f x = −2)

is applied at the left boundary of the computationaldomain that is shown in Fig. 2. This means that theZhao number of the system is equal to 2, which isgreater than the corresponding critical Zhao numberof 0.085, so that the system is in a supercritical state.The dimensionless length and width of the rectangulardomain are 200 and 100, respectively. Due to thesechanges, the dimensionless time-step is changed into 4.5in the computation. The values of other parameters areexactly the same as those used in the previous bench-mark problem. In order to simulate heterogeneity ofthe porous rock, a perturbation of 20 % initial porosity

162 Comput Geosci (2013) 17:151–166

is randomly added to the initial porosity field in thecomputational domain.

Figure 6 shows the morphological evolution of thefluid–porous medium interface in the computationaldomain. It is observed that although the initial fluid–porous medium interface that is located at the leftboundary of the computational domain is of a planarshape, it has gradually changed into an irregular one.With the increase of the dimensionless time, the am-plitude of the resulting irregular fluid–porous mediuminterface increases significantly, indicating that thefluid–porous medium interface is morphologically un-

stable during its propagation within the computationalmodel. This demonstrates that the proposed numericalprocedure is capable of simulating the morphologicalinstability of the fluid–porous medium interface whenthe coupled fluid flow–chemical reaction system con-sisting of the fluid domain and porous medium domainis in a supercritical state.

It is interesting to investigate how the fluid flowevolves with time during the propagation of thefluid–porous medium interface in the computationalmodel. Figure 7 shows the streamline evolution dur-ing the morphological evolution of the unstable fluid–

Fig. 6 Morphologicalevolution of the fluid–porousmedium interface in asupercritical system(Zh = 2.0)

(t- = 45) (t

- = 90)

(t- = 135) (t

- = 180)

(t- = 225) (t

- = 270)

(t- = 315) (t

- = 360)

Comput Geosci (2013) 17:151–166 163

(t- = 135)

(t- = 225)

(t- = 315)

Fig. 7 Streamline distributions due to morphological evolutionof the fluid–porous medium interface in a supercritical system(Zh = 2.0)

porous medium interface between the fluid domainand porous medium domain within the coupled fluidflow–chemical dissolution system. Due to the growthof the amplitude of the irregular fluid–porous mediuminterface, the fluid flow focusing takes place in thepeak region of the porosity, which can be identifiedby the streamline density (in Fig. 7). Generally, thegreater the streamline density, the stronger the fluidflow focusing. It is obvious that the width of the flow-focusing zone is clearly dependent on the peak valuesof the irregular fluid–porous medium interface in thecomputational model. Since the proposed numericalprocedure can be used to simulate both the porositygeneration and the fluid flow focusing, it provides a use-

ful tool for the better understanding of the related phys-ical and chemical mechanisms associated with karstformation and mineralization within the upper crust ofthe Earth.

As the second application example, the proposednumerical procedure is used to examine the effects ofthe Zhao numbers on the morphological evolution ofthe fluid–porous medium interface in a coupled fluidflow–chemical dissolution system. For this purpose,four different Zhao numbers, namely Zh = 0.02, 0.2,2, and 20, have been used in the corresponding compu-tational simulations, respectively. Figure 8 shows howthe four different Zhao numbers affect the morpho-logical evolution of the fluid–porous medium interfacein the coupled fluid flow–chemical dissolution system.It is noted that the computational simulation resultshave displayed the following four possible evolvedmorphologies for the fluid–porous medium interface:the face or compact morphology (Zh = 0.02), conicalwormhole morphology (Zh = 0.2), dominant worm-hole morphology (Zh = 2), and ramified wormholemorphology (Zh = 20). As expected theoretically, theface morphology is formed when the coupled fluidflow–chemical dissolution system is in a stable state,while other three morphologies are formed when thecoupled fluid flow–chemical reaction system is in unsta-ble states. This demonstrated that through comparingthe Zhao number with the critical Zhao number of acoupled fluid flow–chemical reaction system, differentevolved morphologies of the fluid–porous medium in-terface can be predicted beforehand in the system.

The third application example of using the proposednumerical procedure is to investigate the effects ofpermeability ratios on the morphological evolution ofthe fluid–porous medium interface in a coupled fluidflow–chemical dissolution system consisting of a fluiddomain and a porous medium domain. Toward this end,three different values of the permeability ratio, namelyλ = 100, λ = 1,000, and λ = 5,000, respectively, areused in the corresponding computational simulations.The corresponding critical porosities associated withthese three permeability ratios are 0.808, 0.925, and0.963. Figure 9 shows the related computational simu-lation results. It is observed that the permeability ratiohas a considerable effect on the evolved morphologiesof the fluid–porous medium interface in the coupledfluid flow–chemical dissolution system. Generally, thegreater the permeability ratio, the smaller the thicknessof the resulting flow channel. If the time taken bythe flow channel to horizontally penetrate the wholecomputational domain is defined as the breakthroughtime of the acidic fluid, then the breakthrough timesare about 495, 360, and 225, respectively. This indi-

164 Comput Geosci (2013) 17:151–166

Fig. 8 Effects of the Zhaonumbers on morphologicalevolution of the fluid–porousmedium interface

(Zh = 0.02, t- = 45000) (Zh = 0.02, t- = 85500)

(Zh = 0.2, t- = 4500) (Zh = 0.2, t- = 8550)

(Zh = 2, t- = 45) (Zh = 2, t- = 315)

(Zh = 20, t- = 4.5) (Zh = 20, t- = 45)

cates that if the breakthrough time of acidic fluid isof major interest, just like what is concerned about inthe case of developing a new technology to increasethe secondary oil recovery in petroleum engineering[7, 12, 13, 17], then great care should be taken when thepermeability ratio is determined in the computationalsimulation of a coupled fluid flow–chemical dissolutionsystem.

It should be pointed out that according to the linearinstability theory, when the acid dissolution system isin a supercritical state, the morphology of the fluid–porous medium interface is strongly dependent on the

small perturbation applied to the system. If the truevalue of the permeability ratio (of the fluid domain tothe porous medium domain) is λtrue, then the differencebetween λtrue and an assigned λ (in the numerical sim-ulation) is equivalent to a small perturbation. This isthe theoretical reason why the different values of λ canhave a considerable effect on both the breakthroughtime and the morphology of the fluid–porous mediuminterface. Nevertheless, since the breakthrough timedecreases with an increase in the value of λ, it is over-estimated if the value of λ is smaller than that of λtrue.In such a case, the breakthrough time determined from

Comput Geosci (2013) 17:151–166 165

Fig. 9 Effects of permeabilityratios on morphologicalevolution of the fluid-porousmedium interface

(λ = 100, t- = 90) (λ = 100, t- = 495)

(λ = 1000, t- = 45) (λ = 1000, t- = 360)

(λ = 5000, t- = 45) (λ = 5000, t- = 225)

the present numerical simulation is always greater thanthe breakthrough time of the real acid dissolution sys-tem. This means that the present numerical simula-tion can provide a conservative solution for the break-through time of the fluid–porous medium interface, sothat it may have some useful value for determiningthe breakthrough time of the acid dissolution frontused in petroleum engineering. However, based on thedefinition of the permeability ratio (λ = ∇ p porous

∇ p fluid, where

∇ pporous and ∇ p fluid are the fluid pressure gradients in theporous medium and fluid domains, respectively), it is pos-sible to measure the pressure gradients in both the fluidand porous medium domains through laboratory or fieldexperiments, so that the true value of λ can be determinedfor the acid dissolution system. This indicates that oncethe true value of λ is used in the proposed numericalsimulation in this study, the true breakthrough time ofthe fluid–porous medium interface can be determined.

5 Conclusions

A numerical procedure, which is used to simulate mov-ing interfaces between fluid and porous medium do-mains is presented in this study. To avoid the directdescription of the interface boundary conditions at theinterface between the fluid and porous medium do-mains, Darcy’s law is used to simulate fluid flows inboth the fluid and porous medium domains. For thepurpose of effectively simulating the fluid flow usingDarcy’s law, the artificial permeability of the fluid do-main is used to establish a permeability ratio betweenthe artificial permeability in the fluid domain and thereal permeability in the porous medium domain. Us-ing the proposed permeability ratio, the ratio of fluidpressure gradient in the porous medium domain to thatin the fluid domain can be appropriately simulated.To verify the proposed numerical procedure, analytical

166 Comput Geosci (2013) 17:151–166

solutions have been derived for a benchmark problem,the geometry of which can be easily simulated using theproposed numerical procedure. Comparison of the nu-merical solutions with the derived analytical solutionshas demonstrated the correctness and accuracy of theproposed numerical procedure.

Through applying the proposed numerical proce-dure to several examples associated with the fluid–porous medium interface propagation problems, therelated numerical solutions have demonstrated that:(1) the proposed numerical procedure is capable ofsimulating the morphological instability of the fluid–porous medium interface when the coupled fluid flow–chemical dissolution system consisting of a fluid domainand a porous medium domain is in a supercritical state;(2) through comparing the Zhao number with the crit-ical Zhao number of a coupled fluid flow–chemicaldissolution system, different evolved morphologies ofthe fluid–porous medium interface can be predictedbeforehand in the system; (3) the permeability ratio canhave a considerable effect on the evolved morphologiesof the fluid–porous medium interface in the coupledfluid flow–chemical reaction. Generally, the greaterthe permeability ratio, the smaller the thickness of theresulting flow channel.

Acknowledgements This work is financially supported by theNatural Science Foundation of China (Grant No: 11272359). Theauthors express their thanks to the anonymous referees for theirvaluable comments, which led to a significant improvement overan early version of the paper.

References

1. Beavers, G.S., Joseph, D.D.: Boundary conditions at a natu-rally permeable wall. J. Fluid Mech. 30, 197–207 (1967)

2. Brinkman, H.C.: A calculation of the viscous force exerted bya flowing fluid on dense swarm of particles. Appl. Sci. Res.A1, 27–34 (1947)

3. Chadam, J., Hoff, D., Merino, E., Ortoleva, P., Sen, A.: Reac-tive infiltration instabilities. IMA J. Appl. Math. 36, 207–221(1986)

4. Chadam, J., Ortoleva, P., Sen, A.: A weekly nonlinear stabil-ity analysis of the reactive infiltration interface. IMA J. Appl.Math. 48, 1362–1378 (1988)

5. Chen, J.S., Liu, C.W.: Numerical simulation of the evolutionof aquifer porosity and species concentrations during reactivetransport. Comput. Geosci. 28, 485–499 (2002)

6. Chen, J.S., Liu, C.W., Lai, G.X., Ni, C.F.: Effects of mechan-ical dispersion on the morphological evolution of a chemicaldissolution front in a fluid-saturated porous medium.J. Hydrol. 373, 96–102 (2009)

7. Cohen, C.E., Ding, D., Quintard, M., Bazin, B.: From porescale to wellbore scale: impact of geometry on wormholegrowth in carbonate acidization. Chem. Eng. Sci. 63, 3088–3099 (2008)

8. Daus, A.D., Frid, E.O., Sudicky, E.A.: Comparative erroranalysis in finite element formulations of the advection-dispersion equation. Adv. Water Resour. 8, 86–95 (1985)

9. Detournay, E., Cheng, A.H.D.: Fundamentals of poroelas-ticity. In: Hudson, J.A., Fairhurst, C. (eds.) ComprehensiveRock Engineering, vol. 2. Analysis and Design Methods,Pergamon, New York (1993)

10. Fredd, C.N., Fogler, H.S.: Influence of transport and reactionon wormhole formation in porous media. AIChE J. 44, 1933–1949 (1998)

11. Golfier, F., Zarcone, C., Bazin, B., Lenormand, R., Lasseux,D., Quintard, M.: On the ability of a Darcy-scale modelto capture wormhole formation during the dissolution of aporous medium. J. Fluid Mech. 457, 213–254 (2002)

12. Kalia, N., Balakotaiah, V.: Modeling and analysis of worm-hole formation in reactive dissolution in carbonate rocks.Chem. Eng. Sci. 62, 919–928 (2007)

13. Kalia, N., Balakotaiah, V.: Effect of medium heterogeneitieson reactive dissolution of carbonates. Chem. Eng. Sci. 64,376–390 (2009)

14. Nield, D.A., Bejan, A.: Convection in Porous Media.Springer, New York (1992)

15. Ormond, A., Ortoleva, P.: Numerical modeling of reaction-induced cavities in a porous rock. J. Geophys. Res. 105,16737–16747 (2000)

16. Ortoleva, P., Chadam, J., Merino, E., Sen, A.: Geochemicalself-organization II: the reactive-infiltration instability. Am.J. Sci. 287, 1008–1040 (1987)

17. Panga, M.K.R., Ziauddin, M., Balakotaiah, V.: Two-scalecontinuum model for simulation of wormholes in carbonateacidization. AIChE J. 51, 3231–3248 (2005)

18. Rubinstein, J.: Effective equations for flow in random porousmedia with a large number of scales. J. Fluid Mech. 170, 379–383 (1986)

19. Scheidegger, A.E.: The Physics of Flow through PorousMedia. University of Toronto Press, Toronto (1974)

20. Tam, C.T.: The drag on a cloud of spherical particles inlow Raynolds number flows. J. Fluid Mech. 38, 537–546(1969)

21. Turcotte, D.L., Schubert, G.: Geodynamics: Applicationsof Continuum Physics to Geological Problems. Wiley,New York (1982)

22. Vafai, K., Thiyagaraja, R.: Analysis of flow and heat transferat the interface region of a porous medium. Int. J. Heat MassTransfer 25, 1183–1190 (1987)

23. Zhao, C., Hobbs, B.E., Hornby, P., Ord, A., Peng, S., Liu, L.:Theoretical and numerical analyses of chemical-dissolutionfront instability in fluid-saturated porous rocks. Int. J. Numer.Anal. Methods Geomech. 32, 1107–1130 (2008)

24. Zhao, C., Hobbs, B.E., Ord, A.: Fundamentals of Compu-tational Geoscience: Numerical Methods and Algorithms.Springer, Berlin (2009)

25. Zhao, C., Hobbs, B.E., Regenauer-Lieb, K., Ord, A.:Computational simulation for the morphological evolu-tion of nonaqueous-phase-liquid dissolution fronts in two-dimensional fluid-saturated porous media. Comput. Geosci.15, 167–183 (2011)