Thermodynamically consistent modeling of two-phaseincompressible flows in heterogeneous and fractured mediaHuicai Gao1,2, Jisheng Kou1,2,3,*, Shuyu Sun4,*, and Xiuhua Wang3

1 School of Civil Engineering, Shaoxing University, Shaoxing, 312000 Zhejiang, China2Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province, Shaoxing, 312000 Zhejiang, China3 School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000 Hubei, China4Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering, King Abdullah University ofScience and Technology, 23955-6900 Thuwal, Kingdom of Saudi Arabia

Received: 29 January 2020 / Accepted: 2 April 2020

Abstract. Numerical modeling of two-phase flows in heterogeneous and fractured media is of great interest inpetroleum reservoir engineering. The classical model for two-phase flows in porous media is not completelythermodynamically consistent since the energy reconstructed from the capillary pressure does not involvethe ideal fluid energy of both phases and attraction effect between two phases. On the other hand, the satura-tion may be discontinuous in heterogeneous and fractured media, and thus the saturation gradient may be notwell defined. Consequently, the classical phase-field models can not be applied due to the use of diffuse inter-faces. In this paper, we propose a new thermodynamically consistent energy-based model for two-phase flows inheterogeneous and fractured media, which is free of the gradient energy. Meanwhile, the model inherits the keyfeatures of the traditional models of two-phase flows in porous media, including relative permeability,volumetric phase velocity and capillarity effect. To characterize the capillarity effect, a logarithmic energypotential is proposed as the free energy function, which is more realistic than the commonly used double wellpotential. The model combines with the discrete fracture model to describe two-phase flows in fractured media.The popularly used implicit pressure explicit saturation method is used to simulate the model. Finally, theexperimental verification of the model and numerical simulation results are provided.

1 Introduction

Modeling of two-phase flows in porous media plays acrucial role in petroleum reservoir engineering; for instance,in the secondary oil recovery, water is injected to displaceoil from reservoirs. In fractured media, there exist two dis-tinct pore spaces: the fractures and the matrix. Comparedto the matrix, the fractures usually have the higherporosity and permeability, but occupy very small volumes.Modeling of two-phase flows in fractured media is a challen-ging task in petroleum reservoir engineering and subsurfaceflow [1–3].

The classical two-phase flow models [2, 4–7] have beendeveloped based on the single-phase flow model throughintroducing several new quantities, such as saturation, rela-tive permeability, and capillary pressure. Relative perme-ability considerably effects on oil and water two-phase

flows in porous media.Measurements and estimation formu-lations on relative permeabilities have been widely studiedand successfully applied in petroleum reservoir engineering.Capillary effect [2, 3, 6, 8–12] is frequently viewed as the lead-ing mechanism of oil recovery in fractured reservoirs, whichresults from the surface tension between immiscible fluids atthe pore scales [13]. In fact, at the pore scales, the interfacesalways exist between any binary fluids. Interactions betweenfluid molecules on the interfaces lead to surface tension,which can be calculated by the thermodynamically consis-tent diffuse interface models [13–15]. Recently, the energyconcept has been taken into account in the conventionalcapillary formulations of two-phase flows in porous media(see [16] and the references therein). A phase-field modelfor fracture propagation in porous media filled with two-phase fluids has been proposed in [17]. However, the energyreconstructed from the classical model involves the idealcontribution of the wetting-phase saturation only, but theideal fluid energy of both phases and attraction effectbetween two phases have not been taken into account,so the classical two-phase flow model is not completely

Advanced modeling and simulation of flow in subsurface reservoirs with fractures and wells for a sustainable industryS. Sun, M.G. Edwards, F. Frank, J.F. Li, A. Salama, B. Yu (Guest editors)

* Corresponding authors: jishengkou@163.com;shuyu.sun@kaust.edu.sa

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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REGULAR ARTICLEREGULAR ARTICLE

thermodynamically consistent. Phase-field models for two-phase flows have been extensively studied in the literature,for instance [18–22]. Thermodynamically consistent phase-field-based diffuse interface models and simulation of two-phase flows in porous media have been reported in the lit-erature, for instance [23–30], which introduce free energypotentials to characterize capillarity effect caused by thesurface tension. The key ingredient of phase-field modelsof binary fluids is that a free energy potential is introducedto characterize phase behaviors, especially capillarity effectcaused by the surface tension. In phase-field models, thegeneral free energy is usually composed of two terms: thebulk free energy and gradient energy. The latter is intro-duced to account for the gradual phase transmissionbetween binary bulk fluids, which results in clear, but verythin diffusive interfaces rather than sharp interfaces. For arealistic two-phase system, the interface thickness is usuallyless than ten nanometers [13–15, 31, 32]. However, there areno clear diffuse interfaces for binary immiscible fluids inporous media; actually, clear interfaces only exist at themicroscopic pore scales. The phase saturations, as well asDarcy’s velocities, are the volumetric variables at themacroscopic Darcy’s scales. This means that the gradientenergy has no physical basis for two-phase flows in porousmedia. Moreover, the saturation may be discontinuous inheterogeneous media and in fractured media [3, 6], and thusthe gradient of saturation may be not well defined from amathematical point of view. In addition, the effect of rela-tive permeabilities is never taken into account in somephase-field models of two-phase flows in porous media. Tofix the above defects, we will propose a new energy-basedmodel of two-phase flows in porous media, which will beproved to be thermodynamically consistent and have greatpotentials in practical engineering applications.

For multi-phase flows in fractured media, severalmathematical models have been developed in the literature[2, 3, 6, 33–40], such as the single-porosity model, the dual-porosity/dual-permeability model, and the discrete-fracturemodel. The single-porosity model treats the fractures as thematrix, and thus the excessive gridcells are needed due tothe geometrical scale contrast between the matrix and thefractures. In the dual-porosity/dual-permeability model[34, 37, 39, 40], empirical functions to describe thematrix-fracture mass transfer incorporate ad hoc shape fac-tors, and moreover, no theory is available for the calculationof shape factors in two-phase flows with capillary [3]. A two-phase flow model with local nonequilibrium in doubleporosity media has been derived in [41] using the homogeni-zation approach, and this model includes the long-memoryeffects of the microscopic disequilibrium and the mass trans-fer between fractures and blocks. The discrete fracturemodel [35, 36] is based on the conception that the fractureaperture is very small compared to the matrix blocks, andas a consequence, we can simplify the fracture as the lowerdimensional domain to reduce the contrast of geometricscales occurring in the single-porosity model. It is demon-strated in [3, 6] that the discrete fracture model is morepreferable in computational efficiency. In this paper, we willpresent a discrete fracture energy-based model for two-phase flows in fractured media.

The IMplicit Pressure Explicit Saturation (IMPES)method [42–51] is an extensively used time-steppingapproach for multi-phase flow simulation. Based on multi-physics processes of two-phase flows, the IMPES methodtreats the pressure implicitly and the saturation explicitlyin the pressure equation, and thus leads to a linear pressureequation. The treatments result in the decoupling relation-ship between the pressure equation and the saturationequation, and as a result, the saturation can be calculatedexplicitly. In this paper, an IMPES scheme will be devel-oped for the proposed model of two-phase flows in heteroge-neous media and fractured media.

The rest of this paper is structured as follows. InSection 2, we present the model formulations for two-phaseflows in heterogeneous and fractured media. In Section 3,the IMPES scheme is formulated for the proposed model.In Section 4, the experimental verification of the model isgiven. In Section 5, numerical examples are provided anddiscussed. Finally, the concluding remarks are provided inSection 6.

2 Mathematical model

In this section, we present a thermodynamically consistentenergy-based model of incompressible, immiscible two-phase flows in heterogeneous porous media and fracturedmedia.

2.1 Basic model equations

Here, the wetting phase is denoted by the subscript w, whilethe non-wetting phase is denoted by the subscript n. Forthe oil/water two-phase system, water and oil are the wet-ting phase and non-wetting phase, respectively. We let / bethe porosity of a porous medium, which may vary in space,but keeps constant with time. The saturation of phase a,denoted by Sa, where a = w, n, is the volume fraction ofthe void of a porous medium filled by phase a. The voidvolume of porous medium is jointly filled by binary fluids,so the saturations satisfy the constraint,

Sw þ Sn ¼ 1: ð1ÞThe mass conservation equation of each fluid is formulatedas,

/oSa

otþr � ua ¼ qa; a ¼ w; n; ð2Þ

where t is the time, ua denotes Darcy’s velocity of phase a,and qa stands for the injection/production rate of phase a.

The phase-field models introduce the free energy poten-tials to characterize phase behaviors, and capillarity effectcaused by the surface tension is also modeled therein.In phase-field models, the general free energy usually con-tains the bulk free energy and gradient energy. The latteraccounts for the gradient energy contributions on thediffuse interfaces. For a realistic two-phase system, the inter-face thickness is usually no more than a few nanometers[13–15, 32]. So the diffuse interfaces between two phases

H. Gao et al.: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 32 (2020)2

only exist at the microscopic pore scales, and the gradientenergy has no physical meaning for two-phase flows inporous media. What’s more, the saturation may be discon-tinuous in heterogeneous and fractured media [3, 6], andthus the gradient of saturation may be not well definedfrom a mathematical point of view. Consequently, thegradient free energy in the classical phase-field modelsshould not be taken into account for two-phase flows inporous media.

There have been several bulk free energy functions usedfor phase-field models in the literature, for instance, thedouble well potential and logarithmic Flory–Hugginsenergy potential. From a physical view of point, the loga-rithmic energy potential is more realistic than the doublewell potential [52–54]; in fact, it can be viewed as a simpli-fied approximation of the Helmholtz free energy [55] deter-mined by realistic equations of state, e.g. van der Waalsequation of state and Peng–Robinson equation of state[56], which are widely employed in physics and oil–gas engi-neering respectively. Let Sra be the residual saturation ofphase a and let 1 = 1�Srw�Srn. We propose the followinglogarithmic free energy function F as,

F Sw; Snð Þ ¼Xa¼w;n

1caSea ln Se

a

� �� 1� �þ 1cwnS

ewS

en; ð3Þ

where ca and cwn are the energy parameters that measurethe capillary intensity, and Se

a is the normalized satura-tion defined by,

Sea ¼

Sa � Sra

1:

The free energy in equation (3) is a combination of Flory–Huggins energy potential and Helmholtz free energy deter-mined by Peng–Robinson equation of state. The first partin the energy function equation (3) accounts for the idealfluid energy, while the attraction effect is taken intoaccount in the second part of F [55]. The gravitationalpotential energy is expressed as,

GðSw; SnÞ ¼ gzXa¼w;n

Saqa; ð4Þ

where g is the gravity acceleration, z is the depth, and qa isthe mass density of phase a. The total energy U is the sumof the free energy and gravitational potential energy,

UðSw; SnÞ ¼ FðSw; SnÞ þGðSw; SnÞ: ð5Þ

We remark that ca and cwn may depend on the mediumproperties, such as porosity and permeability, and thus itmay vary in different spaces of heterogeneous and fracturedmedia. Indeed, if we take c = cw = cn = cwn, c can be viewedas the macroscopic mean value of microscopic capillarypressure caused by the surface tension between binaryfluids. There have been a few formulations to describe themicroscopic capillary pressure, for instance, the famousYoung–Laplace equation:

c ¼ 2re; ð6Þ

where r is the surface tension and e measures the poreradius at pore scales.

The chemical potentials are defined as the partial deri-vatives of F with respect to Sw and Sn,

lw Sw; Snð Þ ¼ oF Sw; Snð ÞoSw

¼ cw ln Sew

� �þ cwnSen; ð7Þ

ln Sw; Snð Þ ¼ oF Sw; Snð ÞoSn

¼ cn ln Sen

� �þ cwnSew: ð8Þ

In heterogeneous media and fractured media, la may becontinuous in space, whereas ca and cwn may take differentvalues for different types of media, and as a consequence,the saturation may be discontinuous in space. The deriva-tive of the gravitational potential energy G with respectto Sa is expressed as,

ma ¼ oGoSa

¼ qagz: ð9Þ

The modified Darcy’s law for phase a is formulated as,

ua ¼ � kraga

Krðpþ la þ maÞ; a ¼ w; n; ð10Þ

where p is the pressure, K represents the absolute perme-ability tensor of the porous medium, kra is the relative per-meability of phase a and ga is the viscosity of phase a. Wenote that the physical meaning of the pressure p is relevantto the free energy, which will be discussed in detail inSection 2.3. The absolute permeabilityK is the symmetricpositive definite tensor. The relative permeability kra ofphase a is a nonnegative function of Sa, for instance [3],

krw ¼ ðSewÞlw ; krn ¼ ðSe

nÞln ; ð11Þwhere lw and ln are positive integers. It is distinguishedfrom the traditional Darcy model of two-phase flows inporous media that the chemical potential is involved inequation (12) as one of primary driving forces in additionto pressure. To simplify the notations, we define the phasemobility ka ¼ kra

gaand rewrite equation (10) as,

ua ¼ �kaKrðpþ la þ maÞ; a ¼ w; n: ð12ÞIn summary, the phase-field model of two-phase flows inporous media is a system of equations (2) and (12) asso-ciated with the constraint equation (1) and the formula-tions of the free energy and chemical potential. Theproper boundary conditions are also required to completethe model, which rely on specific problems.

2.2 Thermodynamical consistency

The free energy of the proposed model has been constructedbased on thermodynamical principles, and thus it is ther-modynamically consistent. We now prove that the modelobeys the second law of thermodynamics; that is, the totalenergy U of isothermal fluids in a closed domain would be

H. Gao et al.: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 32 (2020) 3

decreasing with time until an equilibrium state is reached[55, 57]. More precisely, in a spatial domain X, we considerthe simplified model,

/oSa

otþr � ua ¼ 0; a ¼ w; n; ð13Þ

ua ¼ �kaKrðpþ la þ maÞ; a ¼ w; n; ð14Þassociated with the boundary conditions,

ua � n ¼ 0; on oX; a ¼ w; n; ð15Þwhere n is the outward unit normal vector to oX.

Using the constraint Sw + Sn = 1, we derive the diver-gence-free condition from equation (13) as,

r � ðuw þ unÞ ¼ 0: ð16ÞWe define the auxiliary phase velocity,

va ¼ �Krðpþ la þ maÞ; a ¼ w; n: ð17ÞThen equation (14) is rewritten as,

ua ¼ kava; a ¼ w; n: ð18ÞIt is deduced from equations (17) and (18) usingequation (16) that,P

a¼w;n

RX kava �K�1vadx ¼ P

a¼w;n

RX ua �K�1vadx

¼ �Xa¼w;n

ZXua � rðpþ la þ maÞdx

¼ Pa¼w;n

RX pþ la þ mað Þr � uadx

¼Xa¼w;n

ZX

la þ mað Þr � uadx ð19Þ

Since ka� 0 andK is the symmetric positive definite tensor,we deduce using equation (13) that,

RX /

oUot dx ¼ R

X /oðFþGÞ

ot dx ¼ Pa¼w;n

RX /

oðFþGÞoSa

oSaot dx

¼ Pa¼w;n

RX ðla þ maÞ/ oSa

ot dx

¼ �Xa¼w;n

ZX

la þ mað Þr � uadx

¼ �Xa¼w;n

ZXkava �K�1vadx � 0: ð20Þ

Thus, the proposed model is thermodynamically consistent.

2.3 Special cases

We now present several special cases of the proposed model,one of which is the classical two-phase flow model.

The a-phase pressure is denoted by pa, where a = w, n.The capillary pressure pc is defined as the differencebetween two phase pressures as,

pc ¼ pn � pw: ð21Þ

As usual, the capillary pressure can be described as a func-tion of saturations. For the proposed model, we take,

pc ¼ ln � lw: ð22Þ

In the first special case, we take cn = cwn = 0, and as aresult, ln = 0. From equation (22), the capillary pressurechanges into pc ¼ �lw. Darcy’s velocity of the wettingphase is rewritten as,

uw ¼ �kwKrðpþ lw þ mwÞ¼ �kwKrðp� pc þ mwÞ¼ �kwKrðpþ pw � pn þ mwÞ;

ð23Þ

which leads to p ¼ pn as a result of,

uw ¼ �kwKrðpw þ mwÞ: ð24Þ

Thus, we obtain the Darcy velocity of the non-wettingphase as,

un ¼ �knKrðpþ mnÞ ¼ �knKrðpn þ mnÞ: ð25Þ

From equation (7), the capillary pressure has the formpc ¼ �cw lnðSe

wÞ. So this case is the classical two-phase flowmodel [3, 6, 42, 46–48]. In this case, the model usuallydescribes an imbibition displacement process; for instance,water is displacing oil.

In the second special case, we take cw ¼ cwn ¼ 0, andthen the capillary pressure becomes pc ¼ ln. Darcy’svelocity of the non-wetting phase is rewritten as,

un ¼ �knKrðpþ ln þ mnÞ¼ �knKrðpþ pc þ mnÞ

¼ �knKrðpþ pn � pw þ mnÞ; ð26Þ

which results in p ¼ pw. Consequently, we obtain theformulation of Darcy’s velocities as,

ua ¼ �kaKrðpa þ maÞ; a ¼ w; n: ð27Þ

In this case, the model can describe a drainage displacementprocess, for instance, oil displacing water [2].

In the third special case, we take cw ¼ cn ¼ c andcwn ¼ 0, where c is a constant. Moreover, Srw ¼ Srn ¼ 0.In this case, we have la ¼ c lnðSaÞ and then we deduce,

Swrlw þ Snrln ¼ cSwr lnðSwÞ þ cSnr lnðSnÞ¼ crSw þ crSn ¼ 0: ð28Þ

Taking into account equations (21), (22) and (28), wededuce the formulations of Darcy’s velocities as,

H. Gao et al.: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 32 (2020)4

uw ¼ �kwKrðpþ lw þ mwÞ¼ �kwKðrpþrlw � Snrln � Swrlw þrmwÞ¼ �kwKðrpþ Snrðlw � lnÞ þ rmwÞ¼ �kwKðrp� Snrpc þrmwÞ¼ �kwKðrp� Snrðpn � pwÞ þ rmwÞ¼ �kwKðrpw þrp� Snrpn � Swrpw þrmwÞ; ð29Þ

un ¼ �knKrðpþ ln þ mnÞ¼ �knKðrpþrln � Snrln � Swrlw þrmnÞ¼ �knKðrpþ Swrðln � lwÞ þ rmnÞ¼ �knKðrpþ Swrpc þrmnÞ¼ �knKðrpþ Swrðpn � pwÞ þ rmnÞ¼ �knKðrpn þrp� Snrpn � Swrpw þrmnÞ; ð30Þ

which lead to the weighted pressure as,

rp ¼ Snrpn þ Swrpw; ð31Þas a result of,

ua ¼ �kaKrðpa þ maÞ; a ¼ w; n: ð32ÞIn this case, the model describes the imbibition anddrainage mixed displacement process.

The above derivations of special cases show that speci-fied pressure and capillary pressure can be derived fromspecified free energy. This means that for the energy-basedmodel proposed in this paper, the major effort will bedevoted to designing the physically reasonable free energyformulations rather than caring the specific pressures andcapillary pressure. Generally speaking, the free energy canbe constructed using relevant physical laws and relationsrather than simple empirical formulations. The resultantmodel will be naturally consistent with thermodynamics.This is a pronounced feature of the proposed energy-basedmodel.

2.4 Alternative formulation

There is not a pressure equation in the system of two-phaseflow model, so we need to reconstruct an equation forpressure. Following the two-phase flow formulation forthe classical two-phase flow model [3, 6], we provide analternative formulation for the proposed model. Thealternative formulation is a system composed of a pressureequation and a saturation equation, which can be solvedmore conveniently than the original system.

We define the general phase potential,

Ua ¼ pþ la þ ma; a ¼ w; n; ð33Þand the general capillary potential,

Uc ¼ Un � Uw ¼ ln � lw þ mn � mw: ð34ÞThen we rewrite the velocity of phase a as,

ua ¼ �kaKrUa; a ¼ w; n: ð35ÞThe total velocity ut is defined as the sum of velocities oftwo phases: ut ¼ un þ uw, which has the equivalent form,

ut ¼ �kwKrUw � knKrUn

¼ �ktKrUw � knKrðUn � UwÞ¼ �ktKrUw � knKrUc; ð36Þ

where kt ¼ kw þ kn is the total mobility. Furthermore, wedefine,

ua ¼ �ktKrUw; uc ¼ �knKrUc: ð36ÞAs a result, ut is reformulated as,

ut ¼ ua þ uc: ð38ÞWe define the fractional flow function nw ¼ kw=kt . Thewetting-phase velocity is rewritten as,

uw ¼ nwua: ð39ÞThe model is reformulated as the following governingequations,

r � ut ¼ �r � ktKrUw �r � knKrUc ¼ qt; ð40Þ

/oSw

otþr � ðnwuaÞ ¼ qw; ð41Þ

where qt ¼ qw þ qn.The boundary oX of the spatial domain X is decom-

posed into the Dirichlet part CD and Neumann part CN ,where oX ¼ CD

SCN . The boundary conditions associated

to equations (40) and (41) are given by,

Uw ¼ UD on CD; ð42Þ

ðua þ ucÞ � n ¼ qN on CN ; ð43Þwhere n is the outward unit normal vector to oX. Thewetting-phase saturation on the boundary oX is subjectto,

Sw ¼ SNw on CN : ð44Þ

The initial saturation is provided by,

Sw ¼ S0w in X: ð45Þ

2.5 Discrete-fracture model

Here, the discrete-fracture model [3, 6] is applied to describetwo-phase flows in fractured media based on the proposedmodel. The key idea of the discrete-fracture model [3, 6] isthat for a n-dimensional matrix-fracture domain, thematrix is n-dimensional, and the fractures are approxi-mated as the (n�1)-dimension.

H. Gao et al.: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 32 (2020) 5

We use the subscript m to denote the variables definedin the matrix; for example, Uw;m denotes the form of Uw inthe matrix. The pressure equation (40) in the matrixchanges into,

�r � kt;mKmrUw;m �r � kn;mKmrUc;m ¼ qt;m: ð46ÞIn the fracture system, the potentials and saturations areconstant along the fracture width, and the pressure equa-tion in the fractures is expressed as,

�r � kt;fKfrUw;f �r � kn;fKfrUc;f ¼ qt;f þQt;f ; ð47Þ

where the subscript f denotes the fracture and Qt;f standsfor the total mass transfer across the matrix-fractureboundaries. The velocities ua in the matrix and fracturesare expressed as,

ua;m ¼ �kt;mKmrUw;m; ua;f ¼ �kt;fKfrUw;f : ð48ÞSimilarly, we can formulate the saturation equation in thematrix as,

/moSw;m

otþr � ðnw;mua;mÞ ¼ qw;m; ð49Þ

and the saturation equation in the fractures as,

/foSw;f

otþr � ðnw;fua;f Þ ¼ qw;f þQw;f ; ð50Þ

where Qw;f represents the mass transfer of wetting phaseacross the matrix-fracture boundaries.

3 Numerical method

The IMplicit Pressure Explicit Saturation (IMPES) method[2, 42–44, 50] is a splitting approach based on physics, inwhich only a pressure equation is solved implicitly, andthe saturation is updated explicitly as long as the pressureis obtained. Compared to the fully implicit scheme, theIMPES method requires smaller computational cost andmemory at each time step. This advantage is more pro-nounced for large-scale numerical simulations.

The total time interval ½0;T � is divided into M timesteps as 0 ¼ t0 < t1 < . . . < tM ¼ T . We denote the timestep length by �ti ¼ tiþ1 � ti. The value of a function vat the time ti is denoted by vi.

3.1 General IMPES method

Following the IMPES method for the classical two-phaseflow model [3, 6], we present the IMPES scheme for the pro-posed model. The relative permeabilities kra are calculatedfrom the saturations at the previous time step, and thusthe variables kw, kn, and kt in the pressure equation aretreated explicitly. The chemical potentials are computedexplicitly as well. The pressure equation is semi-implicitlydiscretized as,

�r � kitKrUiþ1w �r � kinKrUi

c ¼ qiþ1t ; ð51Þ

which is a linear, symmetric positive definite equationof pressure and can be easy to be solved. Once thepressure is calculated, the velocity uiþ1

a can be evaluatedexplicitly as,

uiþ1a ¼ �kitKrUiþ1

w : ð52ÞThe saturation equation of the wetting phase is discretizedby the forward Euler time scheme as,

/Siþ1w � Si

w

�tiþr � ðniwuiþ1

a Þ ¼ qiþ1w : ð53Þ

As a result, the saturations are updated explicitly.

3.2 IMPES method for the discrete-fracture model

The semi-implicit time schemes for the pressureequations (46) and (47) in the matrix domain and fracturesare formulated as,

�r � kit;mKmrUiþ1w;m �r � kin;mKmrUi

c;m ¼ qiþ1t;m ; ð54Þ

�r � kit;fKfrpiþ1f �r � kin;fKfrUi

c;f ¼ qiþ1t;f þQiþ1

t;f : ð55Þ

Once the pressure equations are solved, the velocities arecalculated as,

uiþ1a;m ¼ �kit;mKmrUiþ1

w;m; uiþ1a;f ¼ �kit;fKfrUiþ1

w;f : ð56Þ

The saturations in the matrix domain and fractures areupdated as,

/m

Siþ1w;m � Si

w;m

�tiþr � niw;mu

iþ1a;m

� �¼ qiþ1

w;m; ð57Þ

/f

Siþ1w;f � Si

w;f

�tiþr � ðniw;fuiþ1

a;f Þ ¼ qiþ1w;f þQiþ1

w;f : ð58Þ

4 Model verification

In this section, we verify the proposed model using theexperimental data. We recall that a physically reasonablefree energy has already introduced in this paper, and thenthe corresponding chemical potentials are included in themodified Darcy law. As shown in equation (22), the capil-lary pressure can be obtained from the difference betweenchemical potentials of two phases. To validate the model,we employ the capillary pressure formulation equation(22) to fit the experimental data of water–oil systems pro-vided in [58]. Using the constraint Sw þ Sn ¼ 1, we canexpress the capillary pressure function given in equation(22) as a function of Sw, i.e., pcðSwÞ. The energy parameterscan be calculated by the least squares method: solve theminimizer of,

J cw; cn; cnwð Þ ¼Xmi¼1

pc Sw;ið Þ � pc;ið Þ2; ð59Þ

H. Gao et al.: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 32 (2020)6

where m is the sample size and pc;i is the experimentalvalue of capillary pressure measured at Sw;i.

Figure 1 shows the capability and performance of thecapillary pressure function equation (22) in approximatingthe experimental data of three samples in [58]. Here, thewetting-phase residual saturations are taken as Srw ¼0:098; 0:039; 0:03 for three samples respectively, whileSrn ¼ 0 is taken for all samples. It can be observed thatthe capillary pressure functions are essentially in agreementwith the experimental data. The capillary pressure curvesclearly demonstrate that the capillary pressure functionsare non-convex functions with respect to the saturationSw, and thus the conventional convex functions [2, 4–7, 16]for the capillary pressure would not agree with the experi-mental data.

5 Numerical results

The Cell-Centered Finite Difference (CCFD) method isused as the spatial discretization method in numerical simu-lation. The detailed formulations of CCFD for the discrete-fracture model can be found in [59].

In numerical examples, we consider the horizontal layersin fractured media with the spatial domain dimensions10 m � 10 m � 1 m, but containing different fracture net-works. The fracture aperture is 0.01 m. The porosities of thematrix and fractures are taken as /m ¼ 0:2 and /f ¼ 1.The absolute permeabilities in the matrix blocks and thefractures are 1 md and 105 md, respectively. The quadraticrelative permeabilities are applied in both fractures and ma-trix; that is, lw ¼ ln ¼ 2 in equation (11). The viscosities ofwater and oil are gw ¼ 1 cP and gn ¼ 0:5 cP, respectively.The residual saturations are taken as Srw ¼ 0:03 andSrn ¼ 0. The energy parameters are taken as cmw ¼0:1166 bar, cmn ¼ 0:0108 bar, cmnw ¼ 0:0742 bar in the matrixand cfw ¼ 1:166e� 3 bar, cfn ¼ 1:08e� 4 bar, cfnw ¼7:42e� 4 bar in the fractures. The effect of gravity isneglected in the horizontal media.

5.1 Example 1

In this example, we verify thermodynamical consistency ofthe proposed model; that is, the total free energy of isother-mal fluids in a closed domain is decreasing with time. Themass fluxes towards outsides of all boundaries vanish. Thereis no injection or extraction to the interior of the domain.The distribution of three parallel fractures in the fracturedmedium is illustrated in Figure 2.

The diffusion between oil and water takes place due tothe capillarity effect. The dynamical process is simulateduntil 1 Pore Volume Injection (PVI). The water distribu-tions at different times are shown in Figure 3. The variationof total energy with time is depicted in Figure 4. It is appar-ent that total energy is decreasing with time, and as aconsequence, thermodynamical consistency of the proposedmodel is verified.

5.2 Example 2

The fractured medium is the same to that in Example 1 asillustrated in Figure 2. The void of the media is initially

Fig. 1. Experimental verification of the capillary pressure formulation

Fig. 2. Example 1: distribution of fractures.

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Fig. 3. Example 1: water saturation contours at different times.

Fig. 4. Example 1: verification of thermodynamical consistency.

Fig. 5. Example 2: water saturation contours with free energy at different times.

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saturated with oil and residual water (Srw ¼ 0:03). Water isinjected at the left end of the medium, while oil is producedat the right-hand side. The injection rate is 0.2 PV/year.The mass fluxes towards outsides of the other boundariesvanish. There is no other injection or extraction to theinterior of the domain.

The displacement process of oil by water is simulateduntil 0.39 PVI. The water distributions at different timesare shown in Figure 5. For comparison, the case without freeenergy, i.e., with zero energy parameters, is simulated as welland the corresponding results are depicted in Figure 6.

Figure 6 shows that water flows quickly through thefractures and the breakthrough takes place early. In con-trast, Figure 5 demonstrates that the capillarity effectcaused by surface tension significantly delays breakthrough,and thus it leads to more efficiency of the displacement ofoil by water. Figure 7 illustrates that the free energy inthe regions occupied by oil and fractures is clearly higherthan the other regions, which leads to the energy transmis-sion and dissipation, and as a result, water tends to flowfrom fractures to matrix regions. This reveals that capillar-ity effect is a result of the free energy transmission anddissipation.

5.3 Example 3

In this example, we consider a fractured medium with afracture network that is illustrated in Figure 8. The displa-cement process of oil by water is simulated until 0.33 PVI.

Fig. 6. Example 2: water saturation contours without free energy at different times.

Fig. 7. Example 2: free energy contours at different times.

Fig. 8. Example 3: distribution of fractures.

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Fig. 9. Example 3: water saturation contours with free energy at different times.

Fig. 10. Example 3: water saturation contours without free energy at different times.

Fig. 11. Example 3: energy contours at different times.

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The initial and boundary conditions are the same toExample 2. The water distributions at different times aredepicted in Figure 9. For comparison, the results withoutfree energy are also shown in Figure 10. The energy distri-butions at different times are illustrated in Figure 11.

From Figure 9, we can see that in the displacementprocess, water is driven to cross from fractures to matrixregions due to the capillarity effect caused by surfacetension. The contrast phenomena is observed in Figure 10that water flows quickly through the fractures and it isaccumulating at the ends of fractures. Consequently, capil-larity can improve the oil recovery largely. Figure 11 illus-trates that the fluids in the oil regions and fractures havethe higher free energy, which leads to more water flowingfrom fractures to the matrix.

6 Conclusion

A thermodynamically consistent energy-based model hasbeen developed for two-phase flows in heterogeneous andfractured media. The model inherits the framework of thetraditional model of two-phase flows in porous media, suchas relative permeability, volumetric phase velocity, andcapillarity effect. A logarithmic energy potential, which ismore realistic than the commonly used double well poten-tial, is proposed to characterize the capillarity effect. Wedevelop the energy-based discrete fracture model todescribe two-phase flows in fractured media. The popularlyused implicit pressure explicit saturation method is appliedto simulate the model equations. The model is verified bythe experimental data. Numerical simulation results areprovided to demonstrate that the proposed model can pro-vide physically reasonable results.

Acknowledgments. This work was supported by the NationalNatural Science Foundation of China (No. 41907167), theNatural Science Foundation of Zhejiang Province (No.LY18E090004), the Scientific and Technical Research Projectof Hubei Provincial Department of Education (No. D20192703)and the Technology Creative Project of Excellent Middle &Young Team of Hubei Province (No. T201920).

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