Relations Between Seepage Velocities in Immiscible, Incompressible TwoPhase Flow in Porous Media
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Abstract
Based on thermodynamic considerations, we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible twophase flow in porous media. They necessitate the introduction of a new velocity function, the comoving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.
Keywords
Immiscible twophase flow Thermodynamics Seepage velocity Capillary tube model Network models1 Introduction
The simultaneous flow of immiscible fluids through porous media has been studied for a long time (Bear 1988). It is a problem that lies at the heart of many important geophysical and industrial processes. Often, the length scales in the problem span numerous decades, from the pores measured in micrometers to reservoir scales measured in kilometers. At the largest scales, the porous medium is treated as a continuum governed by effective equations that encode the physics at the pore scale.
The problem of tying the pore scale physics together with the effective description at large scale is the upscaling problem. In 1936, Wycoff and Botset proposed a generalization of the Darcy equation to immiscible twophase flow (Wyckoff and Botset 1936). It is instructive to reread Wycoff and Botset’s article. This is where the concept of relative permeability is introduced. The paper is eighty years old, and yet it is still remarkably modern. Capillary pressure was first considered by Richards as early as 1931 (Richards 1931). In 1940, Leverett combined capillary pressure with the concept of relative permeability, and the framework which dominates all later practical analyses of immiscible multiphase flow in porous media was in place (Leverett 1940).
However, other theories exist Larson et al. (1981), Gray and Hassanizadeh (1989), Hassanizadeh and Gray (1990), Hassanizadeh and Gray (1993a), Hassanizadeh and Gray (1993b), Hilfer (1998), Hilfer and Besserer (2000), Gray and Miller (2005), Hilfer (2006a, b, c), Hilfer and Döster (2010), Niessner et al. (2011), Döster et al. (2012), Bentsen and Trivedi (2013), Hilfer et al. (2015), Hassanizadeh (2015) and Ghanbarian et al. (2016). These theories are typically based on a number of detailed assumptions concerning the porous medium and concerning the physics involved.
It is the aim of this paper to present a new theory for flow in porous media that is based on thermodynamic considerations. In the same way as Buckley and Leverett’s analysis based on the conservation of the mass of the fluids in the porous medium led to their Buckley–Leverett equation (Buckley and Leverett 1942), the thermodynamical considerations presented here lead to a set of equations that are general in that they transcend detailed assumptions about the flow. This is in contrast to the relative permeability equations that rely on a number of specific physical assumptions.
The theory we present focuses on one aspect of thermodynamic theory. We see it as a starting point for a more general analysis based on nonequilibrium thermodynamics (Kondepudi and Prigogine 1998; Kjelstrup and Bedeaux 2008; Kjelstrup et al. 2017) which combines conservation laws with the laws of thermodynamics. The structure of the theory, already as it is presented here, is reminiscent of the structure of thermodynamics itself: We have a number of variables that are related through general thermodynamic principles leaving an equation of state to account for the detailed physics of the problem. To our knowledge, the analysis we present here has no predecessor. However, the framework in which it resides is that originally laid out by Gray and Hassanizadeh (1989), Hassanizadeh and Gray (1990), Hassanizadeh and Gray (1993a), Hassanizadeh and Gray (1993b) and Gray and Miller (2005). This approach, named Thermodynamically Constrained Averaging Theory (TCAT), has generated a large body of work since its inception; see Gray and Miller (2014).
The theory we present concerns relations between the flow rates, or equivalently, the seepage velocities of the immiscible fluids. We do not discuss relations between the seepage velocities and the driving forces that create them, such as a pressure gradient. In this sense, we are presenting a “kinetic” theory of immiscible twophase flow in porous media, leaving out the “dynamic” aspects which will be addressed elsewhere. Our main result is a set of equations between the seepage velocities that together with a constitutive relation between the average seepage velocity and the driving forces lead to a closed set of equations.
We consider only onedimensional flow in this paper, deferring to later the generalization to three dimensions. The fluids are assumed to be incompressible.
In Sect. 2, we describe the porous medium system we consider. We review the key concepts that will be used in the subsequent discussion. In particular, we discuss the relation between average seepage velocity and the seepage velocities of each of the two fluids.
In Sect. 3, we introduce our thermodynamic considerations. We focus on the observation that the total volumetric flow rate is an Euler homogeneous function of the first order. This allows us to define two thermodynamic velocities as derivatives of the total volumetric flow. We then go on in Sect. 4 to deriving several equations between the thermodynamic velocities, one of which is closely related to the Gibbs–Duhem equation in thermodynamics. In Sect. 4.1, we point out that the seepage velocity of each fluid is generally not equal to the corresponding thermodynamic velocity. This is due to the constraints that the geometry of the porous medium puts on how the immiscible fluids arrange themselves. In ordinary thermodynamics, such constraints are not present and questions of this type do not arise. We relate the thermodynamic and seepage velocities through the introduction of a comoving velocity function. This comoving velocity is a characteristic of the porous medium and depends on the driving forces only through the velocities and the saturation.
We discuss in Sect. 5 what happens when the thermodynamic and seepage velocities coincide, when the wetting and nonwetting seepage velocities coincide and when the thermodynamic wetting and nonwetting velocities coincide. Normally, the coincidences appear at different saturations. However, under certain conditions, the three coincide. When this happens, the fluids behave as if they were miscible.
In Sect. 6, we write down the full set of equations to describe immiscible twophase flow in porous media. These equations are valid on a large scale where the porous medium may be seen as a continuum.
In Sect. 7, we analyze four versions of the capillary tube model (Scheidegger 1953, 1974) within the concept of the representative elementary volume (REV) developed in the previous sections. This allows us to demonstrate these concepts in detail and to demonstrate the internal consistency of the theory.
In Sect. 8, we analyze a network model (Aker et al. 1998) for immiscible twophase flow within the framework of our theory. We calculate the comoving velocity for the model using a square and a hexagonal lattice. The comoving velocity is to within the level of the statistical fluctuations equal for the two lattice topologies. We compare successfully the measured seepage velocities for each fluid component with those calculated from our theory. Lastly, we use our theory to predict the coincidence of the three saturations defined in Sect. 5 and verify this prediction numerically.
Section 9 summarizes the results from the previous sections together with some further remarks on the difference between our approach and that of relative permeability. We may nevertheless anticipate here what will be our conclusions. Through our theory, we have accomplished two things. The first one is to construct a closed set of equations for the seepage velocities as a function of saturation based solely on thermodynamic principles. We do not propose relations between seepage velocities and driving forces such as pressure gradient here. Our discussion is solely based on relations between the seepage velocities. The second accomplishment is to pave the way for further thermodynamic analysis by identifying the proper thermodynamic variables that relate to the flow rates.
2 Defining the System
The aim of this paper is to derive a set of equations on the continuum level where differentiation makes sense. We define a representative elementary volume—REV—as a block of porous material with no internal structure filled with two immiscible and incompressible fluids: It is described by a small set of parameters which we will now proceed to define.
The extensive variables describing the REV, among them the wetting and nonwetting pore areas \(A_w\) and \(A_n\) and the wetting and nonwetting volumetric flow rates \(Q_w\) and \(Q_n\), are averages over the REV, and their corresponding intensive variables \(S_w\), \(S_n\), \(v_w\) and \(v_n\) are therefore by definition a property of the REV.
This definition of a REV is the same as that used by Gray and Miller (2005) in their thermodynamically constrained averaging theory and in the earlier literature on thermodynamics of multiphase flow, see, for example, Hassanizadeh and Gray (1990). The REV can be regarded as homogeneous only on the REV scale. With one driving force, the variables on this scale can be obtained, knowing the microscale ensemble distribution (Savani et al. 2017a, b). This averaging procedure must keep invariant the entropy production, a necessary condition listed already in the 1990s (Hassanizadeh and Gray 1990; Gray and Miller 2005).
3 The Volumetric Flow Rate Q is an Euler Homogeneous Function of Order One
4 Consequences of the Euler Theorem: New Equations
We note that Eqs. (20), (24) and (26) are interrelated in that any pair selected from the three equations will contain the third.
4.1 New Equations in Terms of the Seepage Velocities
5 Cross Points

A When \(S_w=S_w^A\) we have \(v_w={\hat{v}}_w\) and \(v_n={\hat{v}}_n\). By Eqs. (33) and (34), we have that \(v_m=0\) for this wetting saturation. We furthermore have that \(v_m\le 0\) for \(S_w\le S_w^A\) and \(v_m\ge 0\) for \(S_w\ge S_w^A\).

B When \(S_w=S_w^B\), we have that \({\hat{v}}_w={\hat{v}}_n\). From Eq. (32), we find \({\hat{v}}_w={\hat{v}}_n=v\). Equations (27) and (29) make \(\mathrm{d}v/\mathrm{d}S_w=0\) equivalent to the equality between the three velocities at this wetting saturation. Furthermore, at \(\mathrm{d}v/\mathrm{d}S_w=0\), v has its minimum value. We have that \(\mathrm{d}v/\mathrm{d}S_w\le 0\) for \(S_w\le S_w^B\) and \(\mathrm{d}v/\mathrm{d}S_w\ge 0\) for \(S_w\ge S_w^B\).

C When \(S_w=S_w^C\), we have that \(v_w=v_n\). This implies that \(v_w=v_n=v\) from Eq. (13) for this wetting saturation. By Eqs. (42) and (43), we have that this is equivalent to \(v_m=\mathrm{d}v/\mathrm{d}S_w\).
6 A Closed Set of Equations
We now consider the porous medium on scales larger than the REV scale. The properties of the porous medium may at these scales vary in space. We consider here only flow in the x direction, deferring the generalization to higher dimensions to later. Let x be a point somewhere in this porous medium. Hence, all the variables in the following will be functions of x.
Note that it is not enough to specify \(S_w\) and v to determine \(v_m\). It is also a function of \(\mathrm{d}v/\mathrm{d}S_w\), since \(\mathrm{d}v/\mathrm{d}S_w\) depends on how the external parameters are controlled as \(S_w\) is changed. For example, if the total volumetric flow rate \(Q=A_p v\) is held constant when \(S_w\) is changed, then \(\mathrm{d}v/\mathrm{d}S_w=0\) for all values of \(S_w\), and the system traces the curve \((S_w,v,v')\) space so that \(v'=0\). If, on the other hand, the driving forces are held constant when \(S_w\) is changed, then v and \(\mathrm{d}v/\mathrm{d}S_w\) will follow some nontrivial curve in \((S_w,v,v')\) space; see Sect. 8. A third example is to control the wetting and the nonwetting volumetric flow rates \(Q_w\) and \(Q_n\), making \(S_w\) and v and \(\mathrm{d}v/\mathrm{d}S_w\) dependent variables, following curves in \((S_w,v,v')\) space depending on how \(Q_w\) and \(Q_n\) are changed.
The equation set (13), (37), (44) and (46) together with the constitutive equation for \(v_m\) is independent of the driving forces. They are valid for any constitutive Eq. (48) that may be proposed. In particular, nonlinear constitutive equations such as those recently proposed which suggest that two immiscible fluids behave as if they were a single Bingham plastic (Tallakstad et al. 2009a, b; Rassi et al. 2011; Sinha and Hansen 2012; Sinha et al. 2017) fit well into this framework.
This is in stark contrast to, for example, the relative permeability framework that assumes a particular constitutive equation for \(v_w\) and \(v_n\) containing the two relative permeabilities \(k_{r,w}(S_w)\) and \(k_{r,n}(S_w)\) in addition to the capillary pressure function \(P_c(S_w)\). The equation set we present here is thus much more general than that of the relative permeability framework.
7 Analytically Tractable Models
We will in this section analyze four variants of the capillary tube model (Scheidegger 1953, 1974). In each case, we calculate the comoving velocity \(v_m\) and then use it to demonstrate the consistency of our theory. We also use this section to clarify the physical meaning of the area derivatives introduced in Sect. 3.
7.1 Parallel Capillaries Filled with Either Fluid
7.2 Parallel Capillaries with Bubbles
Combining Eq. (65) for \(v_m\) with Eqs. (33) and (34) gives Eq. (54) as it must.
7.3 Parallel Capillaries with a Subset of Smaller Ones
7.4 Large Capillaries with Bubbles and Small Capillaries with Wetting Fluid only
Hence, the discussion of these four models has not included the constitutive Eq. (48). In order to have the seepage velocities \(v_w\) and \(v_n\) as functions of the driving forces, we would need to include the constitutive equation and the incompressibility condition (46) in the analysis.
8 Network Model Studies
Our aim in this section is to map the comoving velocity \(v_m\) for a network model which we describe in the following. We compare the measured seepage velocities with those calculated using the formulas derived earlier. We use the theory to predict where the three cross points defined and discussed in Sect. 5 coincide and verify this prediction through direct numerical calculation.
The network model first proposed by Aker et al. has been refined over the years and is today a versatile model for immiscible twophase flow under steadystate conditions due to the implementation of biperiodic boundary conditions. The model tracks the interfaces between the immiscible fluids by solving the Kirchhoff equations with a capillary pressure in links created by the interfaces they contain due to a surface tension \(\sigma \). The links are hour glass shaped with average radii r drawn from a probability distribution. The minimum size of the bubbles of wetting or nonwetting fluid in a given link has a minimum length of r, the average radius of that link.
We show in Fig. 4 \(v_m=v'v_w+v_n\) (Eq. (36) versus \(v_m=S_w v_w'+S_n v_n'\) (Eq. 37). This verifies the main results of Sect. 4.1, besides being a measure of the quality of the numerical derivatives based on central difference for the internal points. Forward/backward difference was used at the endpoints of the curves where \(S_w=0\) or \(S_w=1\).
We compare in Fig. 6 the measured seepage velocities \(v_w\) and \(v_n\) to the seepage velocities calculated from Eqs. (42) and (43) where we have used the comoving velocity shown in Fig. 5. The spikes at low \(v_w\) and \(v_n\) are due to the transition from both fluids moving to only one fluid moving creating a jump in the derivative.
Figure 5 shows two closely spaced curves, one dotted and one dashed. The dashed curve passes through the point \((v_m,v')=(0,0)\) (marked by a nonfilled circle). We show in Fig. 7 v, \(v_w\), \(v_n\), \({\hat{v}}_w\) and \({\hat{v}}_n\) corresponding to these two curves. The thermodynamic velocities have been calculated using Eqs. (27) and (29). In Fig. 7a, the flow parameters have been chosen so that \(v'=0\) and \(v_m=0\) do not coincide as we see in Fig. 5. The ordering of the cross points then follows inequality (41): \(S_w^C\le S_w^B\le S_w^A\). In Fig. 7b, we have chosen the parameters so that \(v'=v_m=0\) for a given saturation, \(S_w=S_{w,0} \approx 0.6\); see Fig. 5. The theory then predicts \(S_w^C= S_w^B= S_w^A\), which is exactly what we find numerically. This constitutes a nontrivial test of the theory: From an experimental point of view, this is the point where \(v_w=v_n=v\) and v has its minimum. These are all measurable quantities.
9 Discussion and Conclusion
We have here introduced a new thermodynamic formalism to describe immiscible twophase flow in porous media. This work is to be seen as a first step toward a complete theory based on equilibrium and nonequilibrium thermodynamics. In fact, the only aspect of thermodynamic analysis that we have utilized here is the recognition that the central variables are Euler homogeneous functions. This has allowed us to derive a closed set of equations in Sect. 6. This was achieved by introducing the comoving velocity (47). This is a velocity function that relates the seepage velocities \(v_w\) and \(v_n\) to the thermodynamic velocities \({\hat{v}}_w\) and \({\hat{v}}_n\) defined through the Euler theorem in Sect. 3.
The theory we have developed here is “kinetic” in the sense that it only concerns relations between the velocities of the fluids in the porous medium and not relations between the driving forces that generate these velocities. We may use the term “dynamic” about these latter relations.
The theory rests on a description on two levels, or scales. The first one is the pore scale. Then, there is REV scale. This scale is much larger than the pore scale so that the porous medium is differentiable. At the same time, we assume the REV to be so small that we may assume that the flow through it is homogeneous and at steady state. At scales much larger than the REV scale, we do not assume steadystate flow. Rather, saturations and velocity fields may vary both in space and time. It is at this scale that the equations in Sect. 6 apply.
The necessity to introduce the comoving velocity \(v_m\) is surprising at first. However, it is a result of the homogenization procedure when going from the pore scale to the REV scale: The pore geometry imposes restrictions on the flow which must reflect itself at the REV scale.
One may at this point ask what has been gained in comparison with the relative permeability formalism? The relative permeability approach consists in making explicit assumptions about the functional form of \(v_w\) and \(v_n\) through the generalized Darcy equations Bear (1988). The relative permeability formalism thus reduces the immiscible twophase problem to the knowledge of three functions \(k_{r,w}=k_{r,w}(S_w)\), \(k_{r,n}=k_{r,w}(S_n)\) and \(P_c=P_c(S_w)\). However, strong assumptions about the flow have been made in order to reduce the problem to these three functions: Assumptions that are known to be at best approximative.
Our approach leading to the central equations in Sect. 6 does not make any assumptions about the flow field beyond the scaling assumption in Eq. (15). Hence, the constitutive Eq. (48) can have any form, for example the nonlinear one presented in Tallakstad et al. (2009a, b), Rassi et al. (2011), Sinha and Hansen (2012) and Sinha et al. (2017). Hence, our approach is in this sense much more general than the relative permeability one.
Footnotes
 1.
We do not scale the length of the REV L. By only scaling the area A orthogonal to the flow direction and, hence, to the driving forces generating the flow, the relations between the driving forces and the flow are not affected. For this reason, the driving forces never enter the discussion explicitly.
 2.
We clarify the physical meaning of these derivatives through a number of examples in Sect. 7.
Notes
Acknowledgements
The authors thank Carl Fredrik Berg, Eirik Grude Flekkøy, Knut Jørgen Måløy, Thomas Ramstad, Isha Savani, Per Arne Slotte, Marios Valavanides and Mathias Winkler for interesting discussions on this topic. We also thank the anonymous reviewers for helpful comments. AH and SS thank the Beijing Computational Science Research Center (CSRC) for financial support and hospitality. SS was supported by National Natural Science Foundation of China under Grant Number 11750110430. This work was partly supported by the Research Council of Norway through its Centres of Excellence funding scheme, Project Number 262644.
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