Saturation overshoot and hysteresisfor twophase flow in porous media

R. Hilfer and R. Steinle

Institute for Computational PhysicsUniversitat Stuttgart, 70569 Stuttgart, Germany

Abstract

Saturation overshoot and hysteresis for two phase flow in porous media

are briefly reviewed. Old and new challenges are discussed. It is widely

accepted that the traditional Richards model for twophase flow in porous

media does not support non-monotone travelling wave solutions for the

saturation profile. As a concequence various extensions and generaliza-

tions have been recently discussed. The review highlights different limits

within the traditional theory. It emphasizes the relevance of hysteresis

in the Buckley-Leverett limit with jump-type hysteresis in the relative

permeabilities. Reviewing the situation it emerges that the traditional

theory may have been abandoned prematurely because of its inability to

predict saturation overshoot in the Richards limit.

Contents

1. Introduction 22. Open questions 63. Saturation Overshoot 73.1. Experimental Observations 73.2. Mathematical Formulation in d = 1 73.3. Hysteresis 94. Approximate Analytical Solution 104.1. Step Function Approximation 104.2. Travelling wave solutions 104.3. General overshoot solutions with two wave speeds 115. Approximate Numerical Solution 135.1. Initial conditions 135.2. Numerical Methods 135.3. Results 136. Critical perspective and challenges 15Acknowledgement 16References 16

1

2

[page 2323, §1]

1. Introduction

[2323.1.1] A macroscopic theory of two phase flow inside a rigid porous medium poses notonly challenges to nonequilibrium statistical physics and geometry [1], but is also crucialfor many applications [1–10]. [2323.1.2] Despite its popularity the accepted macroscopictheory of two phase flow seems unable to reproduce the experimentally observed phenom-enon of saturation overshoot [11].

[2323.2.1] Models for twophase flow in porous media can be divided into macroscopic (lab-oratory or field scale) models popular in engineering, and microscopic (pore scale) modelssuch as network models [12–17] that are popular in physics. [2323.2.2] As of today no rig-orous connection exists between microscopic and macroscopic models [1,18]. [2323.2.3] Inview of the predominantly non-specialist readership with a physics background it is appro-priate to remind the reader of the traditional theory, introduced between 1907 and 1941by Buckingham, Richards, Muskat, Meres, Wyckoff, Botset, Leverett and others [19–23]a.[2323.2.4] One formulation of the traditional macroscopic theory starts from the funda-mental balance laws of continuum mechanics for two fluids (called water W and oil O)inside the pore space (called P) of a porous sample S = P ∪ M with a rigid [page 2324,

§0] solid matrix (called M). [2324.0.1] Recall the law of mass balance in differential form

∂(φi%i)

∂t+∇ · (φi%ivi) = Mi (1)

where %i(x, t), φi(x, t),vi(x, t) denote mass density, volume fraction and velocity of phasei = W,O as functions of position x ∈ S ⊂ R3 and time t ∈ R+. [2324.0.2] Exchange ofmass between the two phases is described by mass transfer rates Mi giving the amount ofmass by which phase i changes per unit time and volume. [2324.0.3] Momentum balancefor the two fluids requires in addition

φi%iDi

Dtvi − φi∇ · Σi − φiFi = mi − viMi (2)

where Σi is the stress tensor in the ith phase, Fi is the body force per unit volume actingon the ith phase, mi is the momentum transfer into phase i from all the other phases,and Di/Dt = ∂/∂t+ vi · ∇ denotes the material derivative for phase i = W,O.

[2324.1.1] Defining the saturations Si(x, t) as the volume fraction of pore space P filled withphase i one has the relation φi = φSi where φ is the porosity of the sample. [2324.1.2] Ex-pressing volume conservation φW + φO = φ in terms of saturations yields

SW + SO = 1. (3)

[2324.1.3] In order to obtain the traditional theory these balance laws for mass, momentumand volume have to be combined with specific constitutive assumptions for Mi,mi,Fi andΣi.

[2324.2.1] Great simplification is afforded by assuming that the porous medium is rigidand macroscopically homogeneous

φ(x, t) = φ = const (4)

aThe following introductory paragraphs are quoted from Ref. [24] for convenience of the interdisci-

plinary readership and following an explicit request from the editor.

3

although this is often violated in applications [25]. [2324.2.2] Let us focus first on themomentum balance (2). [2324.2.3] One assumes that the stress tensor of the fluids isdiagonal

ΣW = −PW1 (5a)

ΣO = −PO1 (5b)

where PW, PO are the fluid pressures. [2324.2.4] Realistic subsurface flows have lowReynolds numbers so that the inertial term

Di

Dtvi = 0 (6)

can be neglected in the momentum balance equation (2). [2324.2.5] It is further assumedthat the body forces

FW = −%Wgez (7a)

FO = −%Ogez (7b)

are given by gravity. [2324.2.6] As long as there are no chemical reactions between thefluids the mass transfer rates vanish, so that MW = −MO = 0 holds. [2324.2.7] Momentumtransfer between the fluids and the rigid matrix is dominated by viscous drag in the form

mW = −k−1 µW φ2WkrW(SW)

vW (8a)

mO = −k−1µO φ

2O

krO(SW)vO (8b)

[page 2325, §0] where µW, µO are the constant fluid viscosities, k is the absolute permeabilitytensor, and krW(SW), krO(SW) are the nonlinear relative permeabilitiy functions for water

and oil b.

[2325.1.1] Inserting the constitutive assumptions (4)–(8) into the mass balance eq. (1)yields

∂(%WSW)

∂t+∇ · (%WSWvW) = 0 (9a)

∂(%OSO)

∂t+∇ · (%OSOvO) = 0 (9b)

while the momentum balance eq. (2)

φWvW = − k

µWkrW(SW)(∇PW − %Wgez) (10a)

φOvO = − k

µOkrO(SW)(∇PO − %Ogez) (10b)

give the generalized Darcy laws for the Darcy velocities φivi [3, p. 155]. [2325.1.2] Equa-tions (9) and (10) together with eq. (3) provide 9 equations for 12 primary unknownsSW, SO, %W, %O, PWPO,vW,vO. Additional equations are needed.

bkrW(SW), krO(SW) account for the fact, that the experimentally observed permability of two immiscible

fluids deviates from their partial (or mean field) permeabilities kSW, kSO obtained from volume averaging

of the absolute permeability.

4

[2325.2.1] Observations of capillary rise in regular packings [26] suggest that the pressuredifference between oil and water should in general depend only on saturation [23]

PO − PW = σWOκ(SW) = Pc(SW) (11)

where σWO is the oil-water interfacial tension and κ(SW) is the mean curvature of theoil-water interface. [2325.2.2] The system of equations is closed with two equations ofstate relating the phase pressures and densities. [2325.2.3] In petroleum engineering thetwo fluids are usually assumed to be incompressible

%W(x, t) = %W = const (12a)

%O(x, t) = %O = const (12b)

while in hydrology one thinks of water W and air O setting

%W(x, t) = %W = const (12c)

%O(x, t) =PO(x, t)

RsT(12d)

where Rs ≈ 287J kg−1K−1 is the specific gas constant and the temperature T is assumedto be constant throughout S.

[2325.3.1] When the fluids (water and oil) are incompressible (as in petroleum engineering)eqs. (12a) and (12b) hold. In this case, adding equations (9a) and (9b), using eq. (3) andintegrating the result shows

qW(x, t) + qO(x, t) = Q(t) (13)

where the total volume flux Q is independent of x and qi = φivi = φSivi with i = W,Oare the volume flux of water and oil. [2325.3.2] Inserting eqs. (10) into eq. (13) and usingeq. (11) to eliminate PW gives

Q = −kλ [∇PO − fW∇Pc − (fW%W + fO%O) gez] (14)

[page 2326, §0] where (with i = W,O)

λi =kriµi

; λ = λW + λO; fi =λiλ

(15)

are the mobilities λi, total mobility λ and fractional flow functions fi, respectively.[2326.0.1] Multiplying eq. (10a) with fO, eq. (10b) with fW and subtracting eq. (10b)from eq. (10a), using eq. (13) and fW + fO = 1 to eliminate qO gives the result

qW = fW [Q + kλO(%W − %O)gez] + kλWλOλ∇Pc (16)

which can be inserted into eq. (9a) to give

φ∂SW

∂t+∇·{fW(SW) [Q + kλO(SW)(%W − %O)gez + kλO(SW)∇Pc(SW)]} = 0 (17)

a nonlinear partial differential equation for the saturation field SW(x, t). [2326.0.2] Forsmall k or when gravity and capillarity effects can be neglected the last two terms vanishand eq. (17) reduces to the Buckley-Leverett equation [27]

φ∂SW

∂t+ Q · ∇fW(SW) = 0 (18)

a quasilinear hyperbolic partial differential equation. [2326.0.3] Equation (17) supple-mented with a (quasilinear elliptic) equation obtained from ∇·Q = 0 by defining a globalpressure in such a way that the total flux Q obeys a Darcy law with respect to the global

5

pressure provides, for incompressible fluids, an equivalent formulation of eqs. (9)-(12b).[2326.0.4] For compressible fluids the situation is different.

[2326.1.1] When W corresponds to water and O to air (as for applications in hydrology)eqs. (12c) and (12d) hold. [2326.1.2] The large density difference %O � %W suggests toconsider the case %O ≈ 0 of a very rarified gas or vacuum as a first approximation.[2326.1.3] For %O = 0 eq. (9b) is identically fulfilled, eq. (12d) implies PO = 0 and theneq. (10b) implies vO = 0. [2326.1.4] In this way the O-phase vanishes from the problemand one is left only with the W-phase. [2326.1.5] Inserting eq. (10a) into eq. (9a) andusing eq. (11) gives the Richards equation [20]

φ∂SW

∂t+∇ · {k λW(SW) [∇Pc(SW) + %Wgez]} = 0 (19a)

for saturation or

φ∂θ(PW)

∂t= ∇ · [k λW(θ(PW)) {∇PW − %Wgez}] (19b)

for pressure after writing SW(x, t) = Pc−1(−PW(x, t)) = θ(PW(x, t)) with the help of

eq. (11). [2326.1.6] To define the nonlinear function θ(PW) the typical sigmoidal shapehas been assumed for Pc(x).

[2326.2.1] The quasilinear elliptic-parabolic Richards equation (19) is the basic equation inhydrology, while the quasilinear hyperbolic Buckley-Leverett equation (18) is fundamentalfor applications in petroleum engineering. [2326.2.2] Both equations, (18) and (19), differfrom the general fractional flow formulation (17) in terms of saturation SW and global pres-sure P . [2326.2.3] They differ also from the formulation in terms of SW, SO,vW,vO, PW, POgiven by (3),(9),(10) and (11). [2326.2.4] These latter equations appropriately supple-mented with initial and boundary conditions and spaces of functions resp. generalizedfunctions for the unknowns constitute the traditional theory of macroscopic capillarity inporous media.

[page 2327, §1] [2327.1.1] The question of domains is important for wellposedness and numer-ical solution. [2327.1.2] For eq. (18) it is well known that classical solutions, i.e. locallyLipschitz continuous functions, will in general exist only for a finite length of time [28–30].[2327.1.3] Hence it is necessary to consider also weak solutions [31]. [2327.1.4] Weak solu-tions are locally bounded, measurable functions satisfying eq. (18) in the sense of distri-butions. [2327.1.5] Weak solutions are frequently constructed by the method of vanishingviscosity or the theory of contraction semigroups. [2327.1.6] For the Richards equation(19) a domain of definition in the space of Bochner-square-integrable Sobolev-space-valuedfunctions has been discussed in [32]. [2327.1.7] In many engineering applications formu-lations such as eqs. (17), (18) or (19) with (11) augmented with appropriate initial andboundary conditions are solved by computer programs [33–35]. [2327.1.8] This concludesour brief introduction into the traditional theory.

6

2. Open questions

[2327.2.1] Numerous open questions surround the traditional model above. [2327.2.2] Im-portant examples are the macroscopic effects of capillarity and surface tensions, the spa-tiotemporal variability of residual saturations, hysteresis and saturation overshoot (seee.g. [1, 18, 24, 36–50] and references therein). [2327.2.3] In this paper we focus on theinterplay and relation between hysteresis and saturation overshoot.

[2327.3.1] Saturation overshoot refers to the experimental observation of non-monotonesaturation profiles during certain classes of infiltration problems, where fingers of infiltrat-ing water develop due to gravitational instabilities [51–55]. [2327.3.2] Within the fingers,the profile of water saturation is non-monotone [53]. [2327.3.3] Experiments in relativelythin tubes, with diameter less than the characteristic finger width, show the existence ofsimilar non-monotone profiles [53–55].

[2327.4.1] Many theoretical and numerical investigations have in recent years addressedsaturation overshoot and gravity driven fingering and their relation to each other [11,56–66]. [2327.4.2] Ref. [56] reported saturation overshoot in numerical solutions of Richardsequation when the hydraulic conductivities between collocation points weighted appro-priately in the numerical scheme. [2327.4.3] In [11] it was argued that these overshootsolutions are numerical artifacts generated by truncation errors. [2327.4.4] This findingagreed with earlier mathematical proofs of existence, uniqueness and stability of solutionswithin an L1-theory for a class of quasilinear elliptic-parabolic equations [32, 57, 58, 67].[2327.4.5] These were applied to the Richards equation [32, 61, 68] leading to the con-clusion that the conventional theory is inadequate to represent fingering and overshoot[61,63,68–70].

[2327.5.1] Accordingly, new approaches were proposed [48, 60, 63–65, 71, 72]. Ref. [60]discussed additional terms in Richards equation, meant to represent a so called hold-back-pile-up effect. [2327.5.2] Other suggestions are based on dynamic extensions of capillarypressure [63, 71, 72]. [2327.5.3] Some authors proposed an additional term based on theyet to be observed “effective macroscopic surface tension” [64, 65]. [2327.5.4] Saturationovershoot profiles at rest when the velocities of all fluids vanish were recently predictedwithin an approach based on distinguishing percolating and nonpercolating (trapped,immobile) fluid parts in [48].

[2327.6.1] The overshoot phenomenon continues to challenge researchers in the field.[2327.6.2] There is an ongoing and lively scientific debate on the subject as witnessedby numerous publications (see also [70] for more references). [2327.6.3] Very recently, Di-Carlo and coworkers [69] investigated the physics behind the displacement front using thetraditional model of two phase flow. [2327.6.4] They argue that gas is drawn in behind theovershoot tip and that the viscosity of this gas plays an observable role when the infiltrat-ing flux is large. [2327.6.5] They emphasize, however, that in their paper they are “notconcerned with why the overshoot occurs, or in other words why the saturation jumps”and that “adding in the viscosity of the [page 2328, §0] displaced phase...does not change thearguments” of van Duijn et al. [63]. [2328.0.1] According to these arguments the tradi-tional standard model leads to parabolic equations and hence overshoot behaviour is notallowed [69, p.965].

7

[2328.1.1] The objective for the rest of this paper is to review analytical and numerical evi-dence for the possible existence of non-monotone solutions (saturation overshoot profiles)within the traditional standard theory outlined above.

3. Saturation Overshoot

3.1. Experimental Observations

[2328.2.1] Infiltration experiments [59] on constant flux imbibition into a very dry porousmedium report existence of non-monotone travelling wave profiles for the saturation [55,62]

S(z, t) = s(y) (20)

as a function of time t ≥ 0 and position 0 ≤ z ≤ 1 along the column. [2328.2.2] Here−∞ < y <∞ denotes the similarity variable

y = z − c∗t (21)

and the parameter −∞ < c∗ <∞ is the constant wave velocity. [2328.2.3] From here onall quantitites z, t, y, S, c∗ are dimensionless. [2328.2.4] The relation

z = zL (22a)

defines the dimensional depth coordinate 0 ≤ z ≤ L increasing along the orientation ofgravity. [2328.2.5] The length L is the system size (length of column). [2328.2.6] Thedimensional time is

t =tL

Q(22b)

where Q denotes the total (i.e. wetting plus nonwetting) spatially constant flux throughthe column in m/s (see eq. (13)). [2328.2.7] The dimensional similarity variable reads

y = yL = z − c∗Qt. (22c)

[2328.2.8] Experimental observations show fluctuating profiles with an overshoot region[54,55, 59,73–75]. [2328.2.9] It can be viewed as a travelling wave profile consisting of animbibition front followed by a drainage front.

3.2. Mathematical Formulation in d = 1

[2328.3.1] The problem is to determine the height S∗ of the overshoot region (=tip) and itsvelocity c∗ given an initial profile S(z, 0), the outlet saturation Sout, and the saturationSin at the inlet as data of the problem. [2328.3.2] Constant Q and constant Sin areassumed for the boundary conditions at the left boundary. [2328.3.3] Note, that thisdiffers from the experiment, where the flux of the wetting phase and the pressure of thenonwetting phase are controlled.

[2328.4.1] The leading (imbibition) front is a solution of the nondimensionalized fractionalflow equation (obtained from eq. (17) for d = 1)

φ∂S

∂t+

∂

∂z

[fim(S)−Dim(S)

∂S

∂z

]= 0 (23a)

8

[page 2329, §0] while

φ∂S

∂t+

∂

∂z

[fdr(S)−Ddr(S)

∂S

∂z

]= 0 (23b)

must be fulfilled at the trailing (drainage) front. [2329.0.1] Here φ is the porosity andthe variables z and t have been nondimensionalized using the system size L and the totalflux Q. The latter is assumed to be constant. [2329.0.2] The functions fim, fdr are thefractional flow functions for primary imbibition and secondary drainage. [2329.0.3] Theyare given as

fi(S) =

µO

µW

krWi(S)

krOi(S)+krWi(S)

GrW

(1− %O

%W

)1 +

µO

µW

krWi(S)

krOi(S)

(24a)

=

1 +krOi(S)

GrW

µW

µO

(1− %O

%W

)1 +

µW

µO

krOi(S)

krWi(S)

(24b)

with i ∈ {im,dr} and the dimensionless gravity number [37,76]

GrW =µWQ

%Wgk(25)

defined in terms of total flux Q, wetting viscosity µW, density %W, acceleration of gravityg and absolute permeability k of the medium. [2329.0.4] The functions krWi(S), krOi(S)

with i ∈ {im,dr} are the relative permeabilities. [2329.0.5] The capillary flux functionsfor drainage and imbibition are defined as

Di(S) = −

krWi(S)

CaW

dPci(S)

dS

1 +µO

µW

krWi(S)

krOi(S)

(26)

with i ∈ {im,dr} and a minus sign was introduced to make them positive. [2329.0.6] Thefunctions Pci are capillary pressure saturation relations for drainage and imbibition.[2329.0.7] The dimensionless number

CaW =µWQL

Pbk(27)

is the macroscopic capillary number [37,76] with Pb representing a typical (mean) capillarypressure at S = 0.5 (see eq. (34)). [2329.0.8] For Ca = ∞ one has Di = 0 and eqs. (23)reduce to two nondimensionalized Buckley-Leverett equations

φ∂S

∂t+

∂

∂zfim(S) = 0 (28a)

for the leading (imbibition) front and

φ∂S

∂t+

∂

∂zfdr(S) = 0 (28b)

for the trailing (drainage) front.

9

[page 2330, §1]

3.3. Hysteresis

[2330.1.1] Conventional hysteresis models for the traditional theory require to store theprocess history for each location inside the sample [77–79]. [2330.1.2] Usually this meansto store the pressure and saturation history (i.e. the reversal points, where the processswitches between drainage and imbibition). [2330.1.3] A simple jump-type hysteresismodel can be formulated locally in time based on eq. (23) as

φ∂S

∂t+ Ξ(S)

∂

∂z

[fim(S)−Dim(S)

∂S

∂z

]+ [1− Ξ(S)]

∂

∂z

[fdr(S)−Ddr(S)

∂S

∂z

]= 0. (29)

[2330.1.4] Here Ξ(S) denotes the left sided limit

Ξ(S) = limε→0

Θ

[∂S

∂t(z, t− ε)

], (30)

Θ(x) is the Heaviside step function (see eq. (35)), and the parameter functions fi(S), Di(S)with i ∈ {dr, im} require a pair of capillary pressure and two pairs of relative permeabilityfunctions as input. [2330.1.5] The relative permeability functions employed for computa-tions are of van Genuchten form [80,81]

krWi(Sei) = KeWiSe

1/2i

[1− (1− Se1/αi

i )αi

]2(31a)

krOi(Sei) = KeOi(1− Sei)

1/2(1− Se1/βi

i )2βi (31b)

with i ∈ {im,dr} and

Sei(S) =S − SWi

i

1− SOri − SWi

i(32)

the effective saturation. [2330.1.6] The resulting fractional flow functions with parametersfrom Table 1 are shown in Figure 1b. [2330.1.7] The capillary pressure functions used inthe computations are

Pci(Sei) = Pbi

(Se−1/αi

i − 1)1−αi

(33)

with i = dr, im and the typical pressure Pb in (27) is defined as

Pb =Pcim(0.5) + Pcdr(0.5)

2. (34)

[2330.1.8] Equation (29) with initial and boundary conditions for S and an initial con-dition for ∂S/∂t is conjectured to be a well defined semigroup of bounded operators onL1([0, L]) on a finite interval [0, T ] of time. [2330.1.9] The conjecture is supported by thefact that each of the equations (23) individually defines such a semigroup, and becausemultiplication by Ξ or (1− Ξ) are projection operators.

10

4. Approximate Analytical Solution

4.1. Step Function Approximation

[2330.2.1] The basic idea of the analysis below is to approximate the travelling wave profilefor long times t → ∞ with piecewise constant functions (step functions). [2330.2.2] Forlarge [page 2331, §0] Ca → ∞ (i.e. in the Buckley-Leverett limit) one may view this ap-proximate profile as a superposition of two Buckley-Leverett shock fronts. [2331.0.1] Thisis possible by virtue of the fact, that the Heaviside step function

Θ(y) =

1 , y > 0 or z/t > c∗

0 , y ≤ 0 or z/t ≤ c∗(35)

can also be regarded as a function of the similarity variable z/t of the Buckley-Leverettproblem.

[2331.1.1] In the crudest approximation one can split the total profile for sufficiently larget into the sum

s(y) = sim(y) + sdr(y) (36a)

of an imbibition front

sim(y) = Sout + (Sin − Sout) [1−Θ(y − z∗)] (36b)

located at zim = c∗t+ z∗ and a drainage front profile located at zdr = c∗t

sdr(y) = (S∗ − Sin) Θ(y) [1−Θ(y − z∗)] (36c)

both moving with the same speed c∗, where Sin = s(−∞) resp. Sout = s(∞) are the upper(inlet) resp. lower (outlet) saturations and S∗ is the maximum (overshoot) saturation.[2331.1.2] The quantity z∗ = zim − zdr is the distance by which the imbibition frontprecedes the drainage front, i.e. the width of the tip (=overshoot) region.

4.2. Travelling wave solutions

[2331.2.1] The two equations (28) become coupled, if eq. (20) holds true, because thenthere is only a single wave speed c∗ for both fronts. [2331.2.2] At the imbibition disconti-nuity the Rankine-Hugoniot condition demands

c∗ =fim(S∗)− fim(Sout)

S∗ − Sout:= cim(S∗) (37a)

and the second equality (with colon) defines the function cim(S). [2331.2.3] Similarly

c∗ =fdr(S

∗)− fdr(Sin)

S∗ − Sin:= cdr(S

∗) (37b)

defines the drainage front velocity as a function cdr(S) of the overshoot S∗. Examples ofthe velocities ci used in the compuations are shown in Figure 1a. [2331.2.4] For a travellingwave both fronts move with the same velocity so that the mathematical problem is tofind a solution S∗ of the equation

cim(S∗) = cdr(S∗) (38)

obtained from equating eqs. (37b) and (37a) (See Fig. 1a). [2331.2.5] The wave velocityc∗ is then obtained as cim(S∗) or equivalently as cdr(S

∗).

11

0 0.2 0.4 0.6 0.8 1

1

3

5

7

9

c im,c

dr

S

a)

0.5

1

1.5

2

2.5f im

,f drb)

0.1 0.3 0.5 0.7 0.9

0.2

0.4

0.6

0.8

1

S

z

c)

Figure 1. a) Graphical solution of eq. (38). The dashed curve shows cdr(S),the solid line shows cim(S). b) Fractional flow functions for drainage (dashed) andimbibition (solid). The two secants (solid/dashed) show the graphical constructionof the travelling wave solution with cim = cdr = c∗. For better visibility, their linestyles have been reversed. Their slopes are equal to each other. c) Numerical solu-tion of eq. (29) using Open∇FOAM for the travelling wave constructed graphicallyin subfigure b). The solution is displayed at time t = 0.053222. The monotoneinitial profile at t = 0 is shown as a dash-dotted step.

[page 2332, §1]

4.3. General overshoot solutions with two wave speeds

[2332.1.1] Equation (38) provides a necessary condition for the existence of a travellingwave solution of the form of eq. (36) with velocity c∗ and overshoot S∗. [2332.1.2] Moregenerally, if the saturation plateau SP is larger or smaller than S∗, one expects to findnon-monotone profiles that are, however, not travelling waves. [2332.1.3] Instead thedrainage and imbibition fronts are expected to have different velocities. [2332.1.4] Thefractional flow functions with relative permeabilities from eqs. (31) and the parametersfrom Table 1 give for SP < S∗ the result

cim(SP) < cdr(SP) (39)

while for SP > S∗ one has

cim(SP) > cdr(SP). (40)

[2332.1.5] In this case, for plateau saturations SP < S∗, the leading (imbibition) fronthas a smaller velocity than the trailing (drainage) front. [2332.1.6] Thus the trailingfront catches up and the profile approaches a single front at long times. [2332.1.7] Forplateau saturations SP > S∗ on the other hand the trailing drainage front moves slower

12

than the leading imbibition front. [2332.1.8] In this case a non-monotone profile persistsindefinitely, albeit with a plateau (tip) width that increases linearly with time.

Parameter Symbol Value Units

system size L 1.0 mporosity φ 0.38 –permeability k 2 · 10−10 m2

density W %W 1000 kg/m3

density O %O 800 kg/m3

viscosity W µW 0.001 Pa·sviscosity O µO 0.0003 Pa·simbibition exp. αim = βim 0.85 –drainage exp. αdr = βdr 0.98 –end pnt. rel.p. Ke

Wim 0.35 –end pnt. rel.p. Ke

Oim1 –

end pnt. rel.p. KeWdr 0.35 –

end pnt. rel.p. KeOdr

0.75 –

imb. cap. press. Pbim 55.55 Padr. cap. press. Pbdr 100 Pa

end pnt. sat. SWiim 0 –

end pnt. sat. SWidr 0.07 –

end pnt. sat. SOrim 0.045 –

end pnt. sat. SOrdr 0.045 –

boundary sat. Sout 0.01 –boundary sat. Sin 0.60 –total flux Q 1.196 10−5 m/s

Table 1. Parameter values, their symbols and units

13

[page 2333, §1]

5. Approximate Numerical Solution

5.1. Initial conditions

[2333.1.1] The preceding theoretical considerations are confirmed by numerical solutionsof eq. (23) with a simple jump-type hysteresis, as formulated in eq. (29). [2333.1.2] Theinitial conditions are monotone and of the form

s(z, 0) = Sout + (SP − Sout)(1−Θ(z − z∗)) (41)

where Sout is given in Table 1. [2333.1.3] Note that all initial conditions are monotone anddo not have an overshoot. [2333.1.4] Two cases are, labelled A and B, will be illustratedin the figures, namely

A : SP = 0.6554 = S∗ with width z∗ = 0.25,B : SP = 0.9540 6= S∗ with width z∗ = 0.015.

[2333.1.5] Case A is chosen to illustrate travelling wave solutions with cim = cdr. [2333.1.6]

The second initial condition illustrates a general overshoot solution with two wave speedscim 6= cdr. [2333.1.7] Moreover it illustrates one possibility for the limit z∗ → 0 andSP → 1 − SOr

im in which a very thin saturated layer (that may arise from sprinklingwater on top of the column) initiates an overshoot profile. [2333.1.8] This could providea partial answer to the question of how saturation overshoot profiles can be initiated.

5.2. Numerical Methods

[2333.2.1] The numerical solution of eq. (29) was performed using the open source toolkitfor computational fluid mechanics Open∇FOAM [82]. [2333.2.2] This requires to developan appropriate solver routine. The solver routine employed here was derived from thesolver [page 2334, §0] scalarTransportFoam of the toolkit. [2334.0.1] Two different dis-cretizations of eq. (29) have been implemented. One is based on a direct discretization of∂fi(S)/∂z using the fvc::div-operators, the other on f ′i(S)× ∂S/∂z using fvc::grad-operators. [2334.0.2] For the discretization of the time derivatives an implicit Euler schemewas chosen, for the discretization of ∂S/∂z an explicit least square scheme, and for thediscretization of the second order term an implicit Gauss linear corrected scheme has beenselected. [2334.0.3] The second order term had to be regularized by replacing the functionDi(S) with maxS∈[0,1]{Di(S), 10−3}. [2334.0.4] This avoids oscillations at the imbibitionfront. [2334.0.5] The discretized system was solved with an incomplete Cholesky conjugategradient solver in Open∇FOAM. [2334.0.6] The numerical solutions were found to differonly in the numerical diffusion of the algorithms. [2334.0.7] The divergence formulationseems to be numerically more stable and accurate.

5.3. Results

[2334.1.1] Equation (38) can have one solution, several solutions or no solution. [2334.1.2]

This can be seen numerically, but also analytically. [2334.1.3] Depending on the values ofthe parameters the velocity of the imbibition front may be larger or smaller than that ofthe drainage front. [2334.1.4] With the parameters from Table 1 the overshoot saturation

14

for a travelling wave as computed from (38) is found to be S∗ = 0.6554 and its velocityis cdr(S

∗) = cim(S∗) = 4.2.

[2334.2.1] The travelling wave solution expected from the graphical construction in Fig-ure 1a) and b) for the initial condition A with SP = S∗ = 0.6554 and width z∗ = 0.25 isdisplayed in Figure 1c) at the initial time t = 0 and after dimensionless time t = 0.053222corresponding to 4450s. [2334.2.2] It confirms a travelling wave of the form (36) whosevelocity c∗ = 4.2 agrees perfectly with that predicted from eq. (38). [2334.2.3] We havealso checked that the solution does not change with grid refinement.

[2334.3.1] For different initial conditions the numerical solutions also agree with the the-oretical considerations. [2334.3.2] Saturation overshoot is found also when initially athin saturated layer with steplike or linear profile is present. [2334.3.3] These solutions,however, do not form travelling waves. [2334.3.4] Instead they have cim 6= cdr for their im-bibition and drainage front as predicted analytically. [2334.3.5] For SP > S∗ the overshootregion grows linearly with time at the rate cim−cdr. [2334.3.6] Such an overshoot solutionis generated by the second initial condition B and compared with the travelling wavesolution in Figure 2. [2334.3.7] The non-travelling overshoot for initial condition B withz∗ = 0.015 and SP = 0.954 is shown as the solid profiles at times t = 0, 0.02392, 0.053222corresponding to 0, 2000, 4450s. [2334.3.8] The profile quickly approaches an overshootsolution with plateau value S ≈ 0.682 close to the saturation of the Welge tangent con-struction. [2334.3.9] This is higher than the plateau value S∗ = 0.6554. [2334.3.10] Thedifference will be difficult to observe experimentally because of the large uncertaintiesin the measurements [54, 55]. [2334.3.11] The velocities of the numerical imbibition anddrainage fronts in the case of initial conditions B with z∗ = 0.015 and SP = 0.954 (solidcurves in Fig. 2) again agree perfectly with cim(0.682) = 4.27 and cdr(0.682) = 2.813from the theoretical analysis.

[2334.3.12] Equation (29) with initial and boundary conditions for S and an initial con-dition for ∂S/∂t is conjectured to be a well defined semigroup of bounded operators onL1([0, L]) on a finite interval [0, T ] of time. [2334.3.13] The conjecture is supported by thefact that each of the equations (23) individually defines such a semigroup, and becausemultiplication by Ξ or (1− Ξ) are projection operators.

[2334.3.14] Equation (29) with initial and boundary conditions for S and an initial con-dition for ∂S/∂t is conjectured to be a well defined semigroup of bounded operators onL1([0, L]) on a finite interval [0, T ] of time. [2334.3.15] The conjecture is supported by thefact that each of the equations (23) individually defines such a semigroup, and becausemultiplication by Ξ or (1− Ξ) are projection operators.

[2334.4.1] Other values of the initial saturation SP have also been investigated. [2334.4.2] Arich phenomenology of possibilities indicates that the existence or nonexistence of over-shoot solutions depends very sensitively on the parameters of the problem and the initialconditions.

[2334.5.1] Note also the asymmetric shape of the overshoot solutions (travelling or not).[2334.5.2] This asymmetry resembles the asymmetry seen in experiment [54,55]. [2334.5.3]

While the leading imbibition front is steep, the trailing drainage front is more smeared

15

0.1 0.3 0.5 0.7 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S

dept

h z

Figure 2. Numerical solutions S(z, 0), S(z, 0.02392), S(z, 0.053222) of eq. (29)at dimensionless times t = 0, 0.02392, 0.053222 with parameters from Table 1 forinitial condition A (dashed) and B (solid). For case A (shown as a short dashedline) the profile for t = 0.053222 is the same as in Fig.1c). For the solid profiles (caseB) the imbibition front moves with speed cim = 4.27 while the trailing drainagefront moves with cdr = 2.813.

out. [2334.5.4] This results from the capillary flux term Di(S) whose values at Sout aresmaller than at Sin.

[page 2335, §1]

6. Critical perspective and challenges

[2335.1.1] Theoretical analysis and the numerical results suggest, that propagation oftravelling and non-travelling saturation overshoots must be expected in the traditionalsetting for hysteretic relative permeability functions of jump-type. [2335.1.2] Althoughhysteresis in the relative permeabilities may be small the hysteresis in the flux functionscan be significant and the saturation overshoot can be pronounced depending on theparameters. [2335.1.3] Note, that hysteresis in Pc is not needed to obtain propagation ofthe overshoot.

[2335.2.1] At first glance this situation seems to be at variance with mathematical proofs ofthe unconditional stability of the traditional theory [11,57,58,61,68] where it is argued thatsaturation overshoot is impossible with or without capillary hysteresis. [2335.2.2] Notehowever, that the mathematical proofs are based on the elliptic-parabolic Richards equa-tion (19). [2335.2.3] Analytical arguments and numerical solutions seem to show thatovershoot saturations (travelling or non-travelling) are possible in the more general caseof eq. (17) with hysteresis of jump type in the relative permeabilities. [2335.2.4] Thus,failure to reproduce saturation overshoot cannot be considered a valid criticism of the

16

established traditional theory. [2335.2.5] More theoretical and analytical work is neededto clarify the situation and the crossover between these regimes.

[2335.3.1] Extensive numerical solutions of eq. (29) reveal a strong dependence on theinitial conditions. [2335.3.2] This could provide partial answers to the question how satu-ration overshoot can be initiated. [2335.3.3] It suggests also that the experimental obser-vations of saturation overshoot could be largely explained as a consequence of hysteresisin the relative permeability and fractional flow functions. [2335.3.4] There are, however,numerous other experimental phenomena (related to the residual, irreducible and trapped[page 2336, §0] phase fractions) that cannot be adequately predicted by the traditional modelwith hysteresis [24,41,46,47,49,50].

[2336.1.1] The current situation with respect to saturation overshoot is still unsatisfactoryin several respects and poses a number of challenges for future research. [2336.1.2] Exper-imentally, the assumption of travelling waves [55, 62], i.e. the assumption cim = cdr, hasto the best of our knowledge, not been clearly demonstrated [54]. [2336.1.3] The uncer-tainties in saturation measurement and the relatively short columns make it difficult toestablish that the two fronts move at the same speed. [2336.1.4] Theoretically, the exis-tence of overshoot solutions needs more rigorous analysis, especially in higher dimensions.[2336.1.5] Also the relation with front stability and fingering poses challenges for theoryand experiment.

Acknowledgement

The authors are grateful to M. Celia and R. Helmig for discussions. They acknowledgethe Deutsche Forschungsgemeinschaft for financial support.

References

[1] R. Hilfer, “Transport and relaxation phenomena in porous media,” Advances in Chemical PhysicsXCII, 299 (1996).

[2] M. Muskat, The Flow of Homogeneous Fluids through Porous Media (McGraw Hill, New York,

1937).[3] A.E. Scheidegger, The Physics of Flow Through Porous Media (University of Toronto Press, Canada,

1957).[4] R.E. Collins, Flow of Fluids through Porous Materials (Reinhold Publishing Co., New York, 1961).

[5] J. Bear, Dynamics of Fluids in Porous Media (Elsevier Publ. Co., New York, 1972).[6] G.d. Marsily, Quantitative Hydrogeology – Groundwater Hydrology for Engineers (Academic Press,

San Diego, 1986).

[7] G.P. Willhite, Waterflooding, SPE Textbook Series, Vol. 3 (Society of Petroleum Engineers, USA,

1986).[8] L.W. Lake, Enhanced Oil Recovery (Prentice Hall, Englewood Cliffs, 1989).

[9] F.A.L. Dullien, Porous Media - Fluid Transport and Pore Structure (Academic Press, San Diego,1992).

[10] M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH Verlagsgesellschaft mbH,

Weinheim, 1995).

[11] M. Eliassi and R.J. Glass, “On the continuum-scale modeling of gravity-driven fingers in unsatu-rated porous media: The inadequacy of the Richards equation with standard monotonic constitutive

relations and hysteretic equations of state,” Water Resour. Res. 37, 2019 (2001).[12] M.I.J.van Dijke and K. Sorbie, “Pore-scale network model for three-phase flow in mixed-wet porous

media,” Phys.Rev.E 66, 046302 (2002).

17

[13] I. Fatt, “The network model of porous media I. capillary pressure characteristics,” AIME PetroleumTransactions 207, 144 (1956).

[14] M.M. Dias and A.C. Payatakes, “Network models for two-phase flow in porous media I: Immiscible

microdisplacement of non-wetting fluids,” J. Fluid Mech. 164, 305 (1986).[15] M. Blunt and P. King, “Macroscopic parameters from simulations of pore scale flow,” Phys. Rev. A

42, 4780 (1990).[16] M. Blunt, M.J. King, and H. Scher, “Simulation and theory of two-phase flow in porous media,”

Phys. Rev. A 46, 7680 (1992).

[17] M. Ferer, G.S. Bromhal, and D.H. Smith, “Pore-level modeling of drainage: Crossover from invasionpercolation fingering to compact flow,” Phys.Rev.E 67, 051601 (2003).

[18] R. Hilfer and H. Besserer, “Old problems and new solutions for multiphase flow in porous media,”

in Porous Media: Physics, Models, Simulation, edited by A. Dmitrievsky and M. Panfilov (WorldScientific Publ. Co., Singapore, 2000) p. 133.

[19] E. Buckingham, “Studies on the movement of soil moisture,” Tech. Rep. (U.S.Department of Agri-

culture, Bureau of Soils, 1907).[20] L.A. Richards, “Capillary conduction of liquids through porous mediums,” Physics 1, 318 (1931).

[21] R.D. Wyckoff and H. Botset, “Flow of gas-liquid mixtures through unconsolidated sands,” Physics

7, 325 (1936).[22] M. Muskat and M. Meres, “Flow of heterogeneous fluids through porous media,” Physics 7, 346

(1936).

[23] M. Leverett, “Capillary behaviour in porous solids,” Trans. AIME 142, 152 (1941).[24] R. Hilfer, “Macroscopic capillarity and hysteresis for flow in porous media,” Phys.Rev.E 73, 016307

(2006).[25] R. Hilfer and R. Helmig, “Dimensional analysis and upscaling of two-phase flow in porous media

with piecewise constant heterogeneities,” Adv. Water Res. 27, 1033 (2004).

[26] W.O. Smith, P.D. Foote, and P.F. Busang, “Capillary rise in sands of uniform spherical grains,”Physics 1, 18 (1931).

[27] S. Buckley and M. Leverett, “Mechanism of fluid displacement in sands,” Trans. AIME 146, 107

(1942).[28] J. Challis, “On the velocity of sound, in reply to the remarks of the astronomer royal,” Phil.Mag.Ser.3

32, 494 (1848).

[29] G. Stokes, “On a difficulty in the theory of sound,” Phil.Mag.Ser.3 33, 349 (1848).[30] P. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation,” Comm.

Pure Appl.Math. 7, 159 (1954).

[31] P. LeFloch, Hyperbolic Systems of Conservation Laws (Birkhauser, 2002).[32] H. Alt, S. Luckhaus, and A. Visintin, “On nonstationary flow through porous media,” Annali di

Matematica Pura ed Applicata 136, 303–316 (1984).[33] K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science Publishers Ltd., London,

1979).

[34] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997).[35] Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flow in Porous Media (SIAM,

2006).

[36] B. Biswal, P.E. Øren, R.J. Held, S. Bakke, and R. Hilfer, “Modeling of multiscale porous media,”Image Analysis and Stereology 28, 23–34 (2009).

[37] L. Anton and R. Hilfer, “Trapping and mobilization of residual fluid during capillary desaturationin porous media,” Physical Review E 59, 6819 (1999).

[38] R. Hilfer and H. Besserer, “Macroscopic two phase flow in porous media,” Physica B 279, 125 (2000).

[39] R. Hilfer, “Capillary pressure, hysteresis and residual saturation in porous media,” Physica A 359,

119 (2006).[40] E. Gerolymatou, I. Vardoulakis, and R. Hilfer, “Modeling infiltration by means of a nonlinear frac-

tional diffusion model,” J.Phys.D 39, 4104 (2006).[41] R. Hilfer, “Macroscopic capillarity without a constitutive capillary pressure function,” Physica A

371, 209 (2006).

[42] S. Manthey, M. Hassanizadeh, R. Helmig, and R. Hilfer, “Dimensional analysis of two-phase flowincluding a rate-dependent capillary pressure-saturation relationship,” Adv. Water Res. 31, 1137

(2008).

[43] R. Hilfer, “Modeling and simulation of macrocapillarity,” in CP1091, Modeling and Simulation ofMaterials, edited by P. Garrido, P. Hurtado, and J. Marro (American Institute of Physics, New York,

2009) p. 141.

[44] R. Hilfer and F. Doster, “Percolation as a basic concept for macroscopic capillarity,” Transport inPorous Media 82, 507 (2010).

18

[45] F. Doster, P. Zegeling, and R. Hilfer, “Numerical solutions of a generalized theory for macroscopiccapillarity,” Physical Review E 81, 036307 (2010).

[46] F. Doster and R. Hilfer, “Generalized Buckley-Leverett theory for two phase flow in porous media,”

New Journal of Physics 13, 123030 (2011).[47] F. Doster, O. Honig, and R. Hilfer, “Horizontal flows and capillarity driven redistribution,”

Phys.Rev.E 86, 016317 (2012).[48] R. Hilfer, F. Doster, and P. Zegeling, “Nonmonotone saturation profiles for hydrostatic equilibrium

in homogeneous media,” Vadose Zone Journal 11, vzj2012.0021 (2012).

[49] O. Honig, F. Doster, and R. Hilfer, “Travelling wave solutions in a generalized theory for macroscopiccapillarity,” Transport in Porous Media DOI, 196 (2013).

[50] F. Doster and R. Hilfer, “A comparison between simulation and expriment for hysteretic phenomena

during two phase immiscible displacement,” Water Resources Research 50, 1 (2014).[51] E. G. Youngs, “Redistribution of moisture in porous materials after infiltration: 2,” Soil Sci. 86,

202–207 (1958).

[52] J. Briggs and D. Katz, “Drainage of water from sand in developing aquifer storage,” (1966), paperSPE1501 presented 1966 at the 41st Annual Fall Meeting of the SPE, Dallas, USA.

[53] R. Glass and T. Steenhuis ad J. Parlange, “Mechanism for finger persistence in homogeneous unsat-

urated, porous media: theory and verification,” Soil Science 148, 60 (1989).[54] S. Shiozawa and H. Fujimaki, “Unexpected water content profiles und flux-limited one-dimensional

downward infiltration in initially dry granular media,” Water Resources Research 40, W07404 (2004).

[55] D. DiCarlo, “Experimental measurements of saturation overshoot on infiltration,” Water ResourcesResearch 40, W04215 (2004).

[56] J.L. Nieber, “Modeling of finger development and persistence in initially dry porous media,” Geo-derma 70, 207 (1996).

[57] F. Otto, “L1-contraction and uniqueness for quasilinear elliptic-parabolic equations,” Journal of

Differential Equations 131, 20 (1996).[58] F. Otto, “L1-contraction and uniqueness for unstationary saturated-unsaturated porous media flow,”

Adv.Math.Sci.Appl. 7, 537 (1997).

[59] S. Geiger and D. Durnford, “Infiltration in homogeneous sands and a mechanistic model of unstableflow,” Soil Sci.Soc.Am.J. 64, 460 (2000).

[60] M. Eliassi and R.J. Glass, “On the porous-continuum modeling of gravity-driven fingers in unsatu-

rated materials: Extension of standard theory with a hold-back-pile-up effect,” Water Resour. Res.38, 1234 (2002).

[61] A. Egorov, R. Dautov, J. Nieber, and A. Sheshukov, “Stability analysis of gravity-driven infiltrating

flow,” Water Resources Research 39, 1266 (2003).[62] D. DiCarlo, “Modeling observed saturation overshoot with continuum additions to standard unsat-

ured theory,” Adv. Water Res. 28, 1021 (2005).[63] C.van Duijn, L. Peletier, and I. Pop, “A new class of entropy solutions of the Buckley-Leverett

equation,” SIAM J.Math.Anal. 39, 507–536 (2007).

[64] L. Cueto-Felgueroso and R. Juanes, “Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media,” Phys.Rev.Lett. 101, 244504 (2008).

[65] L. Cueto-Felgueroso and R. Juanes, “Stability analysis of a phase-field model of gravity-driven un-

saturated flow through porous media,” Phys.Rev.E 79, 036301 (2009).[66] D. diCarlo, “Stability of gravity driven multiphase flow in porous media: 40 years of advancements,”

Water Resources Research 49, 4531 (2013).[67] H. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Math. Z. 183, 311–

341 (1983).

[68] T. Furst, R. Vodak, M. Sir, and M. Bil, “On the incompatibility of Richards’ equation and finger-

like infiltration in unsaturated homogeneous porous media,” Water Resources Research 45, W03408(2009).

[69] D. DiCarlo, M. Mirzaei, B. Aminzadeh, and H. Dehghanpur, “Fractional flow approach to saturationovershoot,” Transport in Porous Media 91, 955 (2012).

[70] Y. Xiong, “Flow of water in porous medi with saturation overshoot: A review,” J. Hydrology 510,

353–362 (2014).[71] J. Nieber, R. Dautov, E. Egorov, and A. Sheshukov, “Dynamic capillary pressure mechanism for

instability in gravity-driven flows; review and extension of very dry conditions,” Transp.Porous

Med. 58, 147–172 (2005).[72] C.van Duijn, Y. Fan, L. Peletier, and I. Pop, “Travelling wave solutions for degenerate pseudo-

parabolic equations modelling two-phase flow in porous media,” Nonlinear Analysis: Real World

Applications 14, 1361–1383 (2013).

19

[73] T. Bauters, D. DiCarlo, T. Steenhuis, and Y. Parlange, “Soil water content dependent wetting frontcharacteristics in sands,” J. Hydrology 231-232, 244–254 (2000).

[74] M. Deinert, J. Parlange, K. Cady, T. Steenhuis, and J. Selker, “Comment on ”on the continuum-

scale modeling of gravity-driven fingers in unsaturated porous media: The inadequacy of the Richardsequation with standard monotonic constitutive relations and hysteretic equations of state” by Mehdi

Eliassi and Robert J. Glass,” Water Resources Research 39, 1263 (2003).[75] S. Bottero, M. Hassanizadeh, P. Kleingeld, and T. Heimorvaara, “Nonequilibrium capillarity effects

in two-phase flow through porous media at different scales,” Water Resources Research 47, W10505

(2011).[76] R. Hilfer and P.E. Øren, “Dimensional analysis of pore scale and field scale immiscible displacement,”

Transport in Porous Media 22, 53 (1996).

[77] Y. Mualem, “A conceptual model for hysteresis,” Water Resour.Res. 10, 514 (1974).[78] J.C. Parker and R.J. Lenhard, “A model for hysteresic constitutive relations governing multiphase

flow. 1. saturation-pressure relations,” Water Resour.Res. 23, 2187 (1987).

[79] R. Lenhard and J. Parker, “A model for hysteretic constitutive relations governing multiphase flow2. permeability-saturation relations,” Water Resources Research 23, 2197 (1987).

[80] M.T.v. Genuchten, “A closed form equation for predicting the hydraulic conductivity of unsaturated

soils,” Soil Sci.Soc.Am.J. 44, 892–898 (1980).[81] L. Luckner, M.T.v. Genuchten, and D. Nielsen, “A consistent set of parametric models for the two-

phase flow of immiscible fluids in the subsurface,” Water Resources Research 25, 2187–2193 (1989).

[82] http://www.openfoam.com/.