^{1}

^{*}

^{2}

^{1}

^{2}

^{1}

^{2}

Edited by: Daniel Bonamy, Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA), France

Reviewed by: Wenzheng Yue, China University of Petroleum, China; Eric Josef Ribeiro Parteli, University of Cologne, Germany

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We present a theoretical framework for immiscible incompressible two-phase flow in homogeneous porous media that connects the distribution of local fluid velocities to the average seepage velocities. By dividing the pore area along a cut transversal to the average flow direction up into differential areas associated with the local flow velocities, we construct a distribution function that allows us to not only re-establish existing relationships of between the seepage velocities of the immiscible fluids, but also to find new relations between their higher moments. We support and demonstrate the formalism through numerical simulations using a dynamic pore-network model for immiscible two-phase flow with two- and three-dimensional pore networks. Our numerical results are in agreement with the theoretical considerations.

When two immiscible fluids compete for the same pore space, we are dealing with immiscible two-phase flow in porous media [

Thermodynamically Constrained Averaging Theory (TCAT) [

A further development somewhat along the same lines, based on non-equilibrium thermodynamics uses Euler homogeneity, more about this later, to define the up-scaled pressure. From this, Kjelstrup et al. derive constitutive equations for the flow [

Another class of theories is based on detailed and specific assumptions concerning the physics involved. An example is Local Porosity Theory [

A recent work [

Thermodynamics is a theory that is valid on scales large enough so that the system it refers to may be regarded as a continuum. Statistical mechanics is then the theory that makes the connection between thermodynamics and the underlying atomistic picture.

It is the aim of this paper to formulate a description of immiscible two-phase flow in porous media that may form a link between the continuum-level approach of Hansen et al.[

After defining the system and the variables involved in section 2, we will in section 3 review the pseudo-thermodynamic approach [

In two-phase flow, the steady state [

Our REV is a block of homogeneous porous material of length

where _{p} is the pore volume and

There is also a solid matrix area fluctuating around

The homogeneity assumption consists in the fluctuations being so small that they can be ignored.

There is a time averaged volumetric flow rate _{w} and _{n}, which are the volumetric flow rates of the more wetting (

In the porous medium, there is a volume _{w} of the wetting fluid and a volume _{n} of the non-wetting fluid so that _{p} = _{w} + _{n}. We define the wetting and non-wetting saturations _{w} = _{w}/_{p} and _{n} = _{n}/_{p}, so that _{w} + _{n} = 1.

We define the wetting and non-wetting transversal pore areas _{w} and _{n} as the parts of the transversal pore area _{p} which occupied by the wetting or the non-wetting fluids, respectively. We have that

As the porous medium is homogeneous, we will find the same averages _{w} and _{n} in any cross section through the porous medium orthogonal to the flow direction. We have therefore _{w}/_{p} = (_{w}_{p}_{w}/_{p} = _{w}, so that

Likewise,

We define the seepage velocities, i.e., the average flow velocities in the pores, for the two immiscible fluids, _{w} and _{n} as

and

The seepage velocity associated with the total flow rate

We may express Equation (4) in terms of the seepage velocities,

Hansen et al.[_{w} and _{n}. We present here a short review of the main results in that paper for completeness. The meaning of the statement that the volumetric flow rate is an Euler homogeneous function of order one is that it obeys the scaling relation

where λ is a scale factor. By taking the derivative of this equation with respect to λ and then setting λ = 1, we find

By dividing this expression by the transversal pore area _{p} and using Equations (5)–(7), we may write this equation as

The two partial derivatives have the units of velocity, and Hansen et al.[

and

We use Equations (6) and (7) and the chain rule to derive

Likewise, we find

We now combine these two equations with the definitions (15) and (16), and use that _{p}_{p}, i.e., Equation (10), to find

and

Combining the definitions (15) and (16) with Equation (14) gives

which should be compared to Equation (11). We see that

The seepage and thermodynamic velocities are related through a transformation _{m},

and

We now calculate

Using Equations (6) and (7) together with Equations (19) and (20), we transform this equation into

where we have used that _{p} = _{p}, i.e., Equation (10). We now use Equation (21) to calculate

Compare this equation to Equation (26) and we get an analog to the Gibbs-Duhem equation,

Using Equations (23) and (24), we find that the seepage velocities obey

and

where we have combined Equations (23) and (24) with Equation (28).

By combining Equations (15), (16), (23), and (24), one finds

and

These two equations, (31) and (32), may be seen as a transformation (_{p}, _{m}) → (_{w}, _{n}). The inverse of this transformation, i.e., (_{w}, _{n}) → (_{p}, _{m}) are given by Equations (11) and (29), i.e.,

where

But, what is the co-moving velocity _{m} physically? We first need to understand the thermodynamic velocities _{w} leads to a change in the average seepage velocity _{p} which is the difference in seepage velocities of the two fluids. However, the two fluids are

From Equation (26) onwards to the end of this sections, none of the equations contain the size of the REV. If we now imagine a REV associated with each point in the porous medium, we have a continuum description. We may then add equations that transport the fluids between these points. Assuming that the fluids are incompressible, these equations are [

where

We add the two equations and get

The generalization to three dimensions is straight forward.

In order to connect the equations that now have been derived to a given porous medium, constitutive equations for _{p} and _{m} need to be supplied, linking the flow to the driving forces. These may in the simplest case be pressure gradient and saturation gradient.

In this section, we connect the pseudo-thermodynamic results of section 3 to the properties of an underlying ensemble distribution. This concept in the context of immiscible two-phase flow was first considered by Savani et al.[

We define a _{p} = _{p}(_{w}, _{p}_{p}—and the other differential transversal pore areas that we will proceed to construct—are

We must have that

where the integral runs over the entire range of negative and positive velocities since there may be local areas where the flow direction is opposite to the global flow. The total flow rate

and the see page velocity defined in Equation (10) is then given by

Likewise, we define a wetting differential pore area _{w} and a non-wetting differential pore area _{n}. They have the same properties except that they are restricted to the wetting or the non-wetting fluids only. That is, we have

and

They relate to the wetting and non-wetting seepage velocities defined in Equations (8) and (9) as

and

We have that

We now combine this equation with Equation (39) to find

which is Equation (11). We have here used Equations (6) and (7).

We may associate a differential area _{m} to the co-moving velocity _{m} defined in Equation (29). By using Equations (39), (42), and (43) in combination with Equation (29), we find

so that

where we have used Equation (44). Equation (47) may be rewritten as

Averaging this equation over

It follows that

where _{m} is the pore area associated with co-moving velocity _{m}. This is to expected as the areas _{w}, _{n}, _{p} and _{m} are ways to partition the transversal pore area _{p}; and we have that _{w} + _{n} = _{p} + 0. This implies that there is no volumetric flow rate associated with the co-moving velocity since

Lastly, we may associate differential transversal areas to the thermodynamic velocities defined in Equations (19) and (20). We use Equations (23) and (24) to find

and

where _{m} is given in Equation (48). The thermodynamic velocities are then given by

and

We find as expected that

and

Summing the two differential transversal areas for the thermodynamic areas gives

This leads us to an important remark. The differential transversal areas are statistical velocity distributions at the pore level. We see that the differential transversal areas that are associated with the thermodynamic velocities are different from those associated with the seepage velocities. However, Equation (57) shows that the _{w} and _{n} are preserved. We see the same from Equation (49) showing that _{m} is zero and combining this Equations (51) and (52).

We see from Equation (47) that _{m} is only zero if _{w} and _{n} are linear in _{w} and _{n}, respectively, i.e., _{w} = _{w}_{w} where _{w} is independent of _{w} and _{n} = _{n}_{n} where _{n} is independent of _{n}. Hence, this is the condition for the thermodynamic velocities to be equal to the seepage velocities.

We define the

By using Equation (44) we find immediately

where we have defined

and

We may work out the moments of the co-moving velocity are given by

where we have used (47).

The thermodynamic velocity moments may be defined as in a similar manner as the moments of the seepage velocities, (60) and (61),

and

and we find

where we have used Equations (52) and (55).

We may Fourier transform _{p}, _{w}, and _{n},

and

From Equation (44) we find

and

We write 〈exp(_{p} as a cumulant expansion,

where _{p} as a moment expansion

By expanding the cumulant expression in Equation (71) and equating each power in ^{2},

Noting that

We may follow this procedure for any of the cumulants.

The corresponding equation between the second cumulants of the thermodynamic velocities is

The relations presented in sections 4, 5 provide the bridge between the velocity distributions at the pore level and the pseudo-thermodynamic theory outlined in section 3. In order to test these relations, and to show how they may be used, we use a dynamic pore network simulator [

In pore network modeling, the porous medium is represented by a network of pores which transport two immiscible fluids. The pore-network model we consider here can be applied to regular networks such as a regular lattice with an artificial disorder as well as to irregular networks such as a reconstructed network from real samples. The flow of the two immiscible fluids is described in this model by keeping the track of all interface positions with time. This approach of pore network modeling was first introduced by Aker et al.[

The porous medium is represented by a network of links that are connected at nodes. All the pore space in this model is assigned to the links and, hence, the nodes do not contain any volume, they only represent the positions where the links meet. The flow rate _{j} inside any link

where Δ_{j} is the pressure drop across link, _{j} is the link length and _{j} is the link mobility which depends on the cross section of the link. The viscosity term μ_{j} is the saturation-weighted viscosity of the fluids inside the link given by μ_{j} = _{j,w}μ_{w} + _{j,n}μ_{n} where μ_{w} and μ_{n} are the wetting and non-wetting viscosities and _{j,w} and _{j,n} are the wetting and non-wetting fluid saturations inside the link, respectively. The term ∑_{c,j} corresponds to the sum of all the interfacial pressures inside the

where _{j} is the average radius of the link and _{j}] is the position of the interface inside the link. Here γ is the surface tension between the fluids and θ is the contact angle between the interface and the pore wall. These two Equations (77) and (76), together with the Kirchhoff relations, that is, the sum of the net volume flux at every node at each time step will be zero, provide a set of linear equations. We solve these equations with conjugate gradient solver [

We construct a diamond lattice with 64 × 64 links in two dimensions (2D) with link lengths _{j} = 1 mm for each link. Disorder is introduced by choosing the link radii _{j} randomly from a uniform distribution in the range 0.1 mm and 0.4 mm. We use 10 different realizations of such network for our simulations in 2D. In three dimensions (3D), we use a network reconstructed from a 1.8 × 1.8 × 1.8 mm^{3} sample of Berea sandstone that contains 2, 274 links and 1, 163 nodes [_{n}/μ_{w}) = 0.5, 1.0, and 2.0 are considered. These values are chosen in such a way that the capillary number, defined as

falls in a range of around 10^{−3} to 10^{−1}. Here μ_{e} is the saturation weighted effective viscosity of the system given by μ_{e} = _{w}μ_{w} + _{n}μ_{n}. Specifically, we find Ca in the range of 0.004–0.074 for 2D and 0.001–0.271 for 3D in the steady state. As the simulations are performed under constant pressure drop, the capillary number fluctuates.

The simulations are continued to the steady state which is defined by the global measurable quantities, such as the fractional flow or the total flow rate _{w}, and _{n}) and the pore areas (_{p}, _{w}, and _{n}) through any cross section orthogonal to the applied pressure drop and then use Equations (8)–(10) to calculate the seepage velocities. Next, we perform the measurements using the differential pore areas (_{p}, _{w}, and _{n}) and calculate the seepage velocities using description given in section 4. We then compare the results from the two measurements and calculate the co-moving velocities. We then verify the relation between the seepage velocities and their higher moments.

For the direct measurements, imagine a cross section at any place of the network orthogonal to the overall direction of flow. For the regular diamond lattice in 2D, all the links have the same length. Different moments of the seepage velocities can therefore be calculated by

and

where _{j} is the projection of the pore area of the

and

where _{x,j} = _{j} cos α_{j} is the projection of the link length (_{j}) to the direction of the overall flow.

If we consider every link having the same length _{j} = _{j} = α in these equations, we retrieve the Equations (79)–(81). For the first moment (_{p}, _{w}, and _{n}, respectively, in both 2D and 3D.

For the second approach, we construct the distribution of differential transversal pore areas _{p}, _{w}, and _{n} such that _{p}_{w}_{n}

where _{j} < _{j} being the local velocity of link _{x,j}s are same for any

For any saturation, the seepage velocities and their higher moments should follow the relations (11), (45), and (59). We plot our numerical measurements in

Verification of the relations (11), (45), and (59) between the steady-state seepage velocities _{p}, _{w}, and _{n}, and their higher moments for the 2D regular network. The top row represents the direct approach of measurements using Equations (79), (80), and (81). The bottom row corresponds to the velocities measured from the differential area distributions defined in Equations (58), (60), and (61). _{p} has a unit mm/s. Subsequently, for ^{2}/s and mm^{3}/s, respectively.

Verification of the relations (11), (45), and (59) between the steady-state seepage velocities _{p}, _{w}, and _{n}, and their higher moments for the 3D Berea network. The direct approach of measurements using Equations (82)–(84) are presented in the top row. The measurements using the differential area distributions defined in Equations (58), (60), and (61) are presented in the bottom row. _{p} has a unit mm/s. Subsequently, for ^{2}/s and mm^{3}/s, respectively.

Next we measure the fluctuations in the seepage velocities which obey Equation (74). Numerically,

In

Numerical verification of Equation (74) between the fluctuations in the seepage velocities. ^{2}/s.

Finally, we verify the relations between seepage velocities and their higher moments while varying the fluid saturation as given by the Equations (29), (30), and (62). For this, we first calculated the co-moving velocity (_{m}) and its higher moments from Equations (29) and (30) where we used the values of the seepage velocities measured with the direct approach. This is shown in the top rows of _{m} corresponding to the co-moving velocity from Equation (47) where we have used the variations of _{p}, _{w}, and _{n} with the saturation _{w}. For this purpose, we have considered 21 different values of saturations within 0 and 1 with an interval of 0.05. We then integrate _{m} from −∞ to ∞, weighted by the velocity and normalized by the total pore area to obtain the desired co-moving velocity with Equation (46). These results are plotted in the bottom row of

Measurement of the co-moving velocity (_{m}) and its higher moments for the 2D network. The top row corresponds to the calculations using Equations (29) and (30) with the direct measurements. The bottom row shows the measurements of _{m} has a unit mm/s. Subsequently, for ^{2}/s and mm^{3}/s, respectively.

Measurements of _{m} has a unit mm/s. Subsequently, for ^{2}/s and mm^{3}/s, respectively.

The aim of this paper is to provide the link between the pseudo-thermodynamic theory at the continuum level developed in Hansen et al. [_{p}, the wetting fluid differential velocity transversal pore area _{w}, the non-wetting fluid velocity differential transversal pore area _{n}, and the co-moving velocity differential transversal pore area _{m}. We also consider the thermodynamic velocity differential transversal pore areas _{w} and _{n}. The relations found by Hansen et al.[

The theoretical derivations are then in section 6 validated by numerical simulations. We used dynamic pore-network modeling where an interface-tracking model is used to simulate steady-state two-phase flow. We used both regular pore networks and an irregular pore network reconstructed from a Berea sandstone for our simulations. By measuring the seepage velocities from the differential area distributions and comparing them with the direct measurements, we validated the essential predictions from the earlier theoretical sections.

Both Hansen et al.[_{p} and _{m}, which provide the specifics of the porous medium.

The resulting set of equations may then be solved for structured porous media where the structure are associated with length scales larger than that set by the REV. This is e.g., seen in the explicit appearance of the porosity ϕ in Equations (34)–(36).

An open question, though, is what happens when there is non-trivial structure in the porous medium all the way from the pore scale to the continuum scale, see [

The datasets generated for this study are available on request to the corresponding author.

SR did the numerical simulations and analysis. SS developed the codes and performed the 3D simulations. AH developed the theory.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Carl Fredrik Berg, M. Aa. Gjennestad, Knut Jørgen Måløy, Per Arne Slotte, Ole Torsæter, Morten Vassvik, and Mathias Winkler for interesting discussions. AH thanks the Beijing Computational Science Research Center CSRC for the hospitality and financial support.