# Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Scope of Work

## 2. Hierarchical Mechanistic Modeling for Steady-State Two-Phase Flow in Pore Networks

_{o}and q

_{w}, or, in reduced form, the set of capillary number, Ca = μ

_{w}U

_{w}/γ

_{ow}, and oil/water flowrate ratio, r = q

_{0}/q

_{w}. Here, μ

_{w}is the viscosity of water, U

_{w}is the superficial velocity of water, and γ

_{ow}is the oil/water interfacial tension.

_{0}/μ

_{w}, the advancing and receding contact angle of the oil/water interface on the pore walls, θ

_{A}and θ

_{R}, and a parameter vector,

**x**

_{pm}, composed of all the dimensionless geometrical and topological parameters of the porous medium affecting the flow (e.g., porosity, genus, coordination number, normalized chamber and throat size distributions, chamber-to-throat size correlation factors, etc.).

#### 2.1. The Concept of Decomposition into Prototype Flows

_{o}and q

_{w}, so that the operational (dimensionless) parameters, i.e., the capillary number, Ca, and the oil/water flowrate ratio, r, have constant values.

_{w}. Here, a remark must be made with reference to the conventional use of saturation as the independent variable of the process. It is based on the perception that disconnected oil blobs and other fluidic elements (ganglia and/or droplets) do not move with the average flow but remain stranded in the pore medium matrix. This situation arises when flow conditions of relatively “small values” of the capillary number are maintained. Nevertheless, there is ample experimental evidence that disconnected flow is a substantial and sometimes prevailing flow pattern. A particular value in saturation does not necessarily imply that a unique disconnected structure (arrangement) of the non-wetting phase will settle in. In this context, saturation represents macroscopic static information and cannot adequately (or uniquely) describe the flow conditions (it brings no definite input to the momentum balance). The issue is discussed by Valavanides et al. (Section 2 in [16]).

- Connected-oil Pathway Flow (CPF)
- Disconnected Oil Flow (DOF).

- Ganglion Dynamics (GD) flow
- Drop Traffic Flow (DTF).

**Figure 1.**(

**a**) Schematic representation of the actual flow (left sketch) and its theoretical decomposition into prototype flows: Connected-oil Pathway Flow (CPF) and Disconnected-Oil Flow (DOF) (right sketch). (

**b**) A microscopic scale representation (snapshot) of a DOF region. An oil ganglion of size class 5 is shown. For simplicity, all cells are shown identical and the lattice constant is shown expanded (chambers and throats have prescribed size distributions). The dashed, light grid lines define the network unit-cells. The thick dashed line separates the Ganglion Dynamics (GD) cells domain and the Drop-Traffic Flow (DTF) cells domain.

_{w}, β and ω are called flow arrangement variables (FAV) because they provide a coarse indication of the structure (or “immiscible mixing”) of the flow pattern. One of the objectives of the DeProF model algorithm is to determine the values of S

_{w}, β and ω that conform to the externally imposed flow conditions (Ca, r).

^{12}pores, or more) and a microscopic scale, and produces a system of equations that includes macroscopic water and oil mass balances, flow arrangement relations at the macroscopic scale, equations expressing the consistency between the microscopic and macroscopic scale representations of these balances in the DOF region and an equation that is obtained by applying effective medium theory [24] to the “equivalent one-phase flow” in the DOF (GD and DTF) region—implicitly representing the transfer function for this region [18,19]. The system is closed by imposing an appropriate type of (sharply decreasing) distribution function for the ganglion volumes, which is dictated by the physics of ganglion dynamics, in compliance with experimental observations [20,21] and numerical/pore network simulations [25,26]. In reality, large ganglia cannot survive because, as they migrate downstream the tortuous paths of the pore network, they break-up into smaller ones.

#### 2.2. Physically Admissible Flow Configurations

^{4}–10

^{9}pores), the actual flow at any given region of the porous medium “wanders” over the domain of physically admissible flow configurations “visiting” any one with equal probability or frequency (ergodicity).

_{w}, β, ω} of all possible flow arrangement parameter values is partitioned using sufficiently fine steps to obtain a 3D grid (Figure 2). Then, for each grid set of the flow arrangement parameters values (S

_{w}, β, ω), the DeProF algorithm is asked to detect (if) a solution of the DeProF equations (exists). In such case the selected set of (S

_{w}, β, ω) values, is allowed as a physically admissible solution (PAS) and is denoted as (S

_{w}', β’, ω’). Otherwise, the set is rejected. In the end, the subdomain of the {S

_{w}, β, ω} space, denoted Ω

_{PAS}, is detected as the set of physically admissible solutions.

_{PAS}is given by:

_{PAS}(Ca,r) stands for integration carried over the physically admissible ranges in (S

_{w}, β, ω) for the imposed Ca, r values. The volume of Ω

_{PAS}is a measure of the degrees of freedom of the process at the mesoscopic scale; it is also related to the rate of entropy production at the mesoscopic scale (configurational entropy).

_{PAS}, a unique solution for the macroscopic flow is obtained. For any quantity, Φ’, the corresponding expected mean macroscopic flow quantity, Φ, is defined as:

_{w}, β, ω) can be obtained by averaging over the domain of physically admissible solutions, using Equation (3), with Φ replaced by S

_{w}, β and ω. These values define the flow configuration of the mean macroscopic flow (ergodicity).

**Figure 2.**Typical domains of physically admissible flow configurations, as predicted by DeProF model simulations. For any fixed, externally imposed, flow conditions, (Ca, r), the two-phase flow visits a canonical ensemble of physically admissible flow configurations (S

_{w}', β’, ω’) depicted by the cloud of small, red balls. Here, Ca = 1.19 × 10

^{−6}, κ = 1.45 and flowrate ratio values, r, span 2 orders of magnitude. A unique set of values for (S

_{w}, β, ω), the large, black ball, is obtained by averaging over the ensemble and defines the mass center of the red cloud. The shape and position of the red cloud changes with flow conditions.

_{ΩPAS}, as well as any expected mean macroscopic flow quantity, Φ, can be determined by any coarse graining (discretization) procedure used to detect the ensemble of physically admissible solutions and delineate its envelope. The phase space {S

_{w}× β × ω} pertaining to the FAVs, is partitioned in corresponding N

_{S}× N

_{β}× N

_{ω}phase space voxels. The finer the graining, the smaller is the size of these voxels. Nevertheless, the smallest discretization size of voxels has a limit; it cannot become smaller than a minimum, critical size in each prototype flow. Suppose that we detect two solutions (S

_{w}', β’, ω’)

_{i}and (S

_{w}', β’, ω’)

_{i+}

_{1}associated with two adjacent voxels, i and i+1. These solutions are virtual, in the sense that they have been detected by a numerical algorithm implementing a particular discretization. In essence they have been picked-up from the complete ensemble of the physically admissible solutions. So, there is a great chance that other flow configurations exist as well—other than those detected (as solutions) by the discretization scheme. This possibility could be verified by re-discretizing the phase space into a finer grid implementing smaller voxels. The new virtual ensemble of physically admissible solutions would eventually contain more solutions. Nevertheless, it would not occupy a volume in the FAV phase space larger than the previous ensemble (associated to the coarse grid). The only difference between the two virtual ensembles would be on the “texture” of their envelopes. The latter would have a finer surface than the former. We would be able to repeat the re-discretization with finer and finer grids, resulting in smaller and smaller voxels. But we would eventually reach a critical size whereby solutions would not have discernible characteristics. This limiting critical size (threshold) depends on the structure of the pore network (its fractal characteristics) and on the modeling of the prototype flows (the characteristic sizes of the disconnected fluidic elements). We will address this issue in Section 4.

#### 2.3. Discussion on Assumption of the Process Ergodicity

_{w}', β’, ω’), is replaced by the average macroscopic flow configuration (S

_{w}, β, ω), Equation (3). To test the specificity of the DeProF model predictions, that are based on the underlying assumption of ergodicity, the average mechanical power dissipation, W, estimated through Equation (3), was benchmarked against the available laboratory data. In these simulations, the DeProF algorithm was implemented to account for the particular geometry of the pore network model. The comparison is depicted in the diagrams of Figure 5 in [12] (originally of Figure 3 in [19]). It is obvious that there is an excellent agreement between the DeProF model predicted values and the values measured in the laboratory study.

#### 2.4. Statistical Thermodynamics Aspects of Two-Phase Flow in Pore Networks

_{0}). A justification of the existence of optimum operating conditions was recently proposed [17] along the lines of the postulate stating that the efficiency of a stationary process in dynamic equilibrium is proportional to its spontaneity [28]. Spontaneity, the notional inverse for irreversibility, may be quantitatively assessed by the amount of entropy produced globally. To evaluate the entropy, we need first to define the physical domains in which the process of steady-state two-phase flow in pore networks is maintained at dynamic equilibrium under fixed Ca and r.

_{0}. The infinite heat reservoir can absorb all the heat released by the System. The Universe comprises the System and the Surroundings.

_{UNIV}, is the sum of two terms: a term representing the entropy released from the System to the Surroundings, S

_{SUR}, and a term representing the entropy produced within the System, S

_{SYS}:

_{0}), S

_{SUR}, is due to the rate with which mechanical energy is dissipated within the System, W = (q

_{0}+ q

_{w}) Δp, irreversibly transformed into heat, Q = W, a source of microscopic chaos, and then released to the Surroundings:

_{SYS}, may be directly related to the multitude of the physically admissible flow configurations (the ensemble of physically admissible solutions) that are maintained for so long as the (steady-state) process is kept at conditions of dynamic equilibrium. It can be interpreted as the production of chaos over mesoscopic scales (configurational entropy). It can be expressed, similar to the Boltzmann entropy formulation in statistical mechanics, as:

_{ss2fpm}is a constant quantity, similar to Boltzmann’s constant in the statistical thermodynamics definition of entropy, that would have to be appropriately defined for the sought process, (index “ss2fpm” stands for “steady-state 2-phase flow in porous media”), and N

_{PAS}is the actual number of different mesoscopic flow arrangements (or microstates) consistent with the macroscopic flow at (Ca, r) conditions.

_{SUR}, is considered to be known (or given); it can be measured or it can be estimated if the DeProF, or any other model algorithm is implemented in the process analysis and the detection of the physically admissible flow configurations. The second term, S

_{SYS}, is not readily available. In the following, we will present the methodology for estimating, the number of microstates of the process for fixed macroscopic flow conditions, given the ensemble of physically admissible flow arrangements.

## 3. Pore Network Geometry and Topology—Discrete Fluidic Elements

#### 3.1. Pore Network Geometry and Topology

_{3D}= 6. In general, chambers and throats are classified according to their shape, size, etc. into an appropriate number of classes. The latter is a modeling parameter and depends on the form of the pore size distribution of the real network or the structure of the porous medium. We consider there is no correlation between classes of chambers and throats therefore the network is isotropic on the macroscopic scale. Let $\ell $ denote the lattice constant. Each unit-cell occupies an equilateral octahedron of pore network space with diagonal lengths equal to $\ell $, edge lengths equal to $\ell \sqrt{2}/2$ and volume ${\ell}^{3}/6$. The pore network volume density of unit-cells is $M=6/{\ell}^{3}$. The geometry of the pore space within every unit-cell is defined by the shape of the chambers and throats. For this particular type of cubic lattice networks, each unit-cell comprises one eighth of each one of the two adjacent chambers that are interconnected with the throat (see Appendix). There are many variations on the shapes and forms of chambers and throats; the only universal characteristic is that—on average—the cross-sections of throats are significantly smaller than those of chambers. The exact shape of the cross-sections is appropriately described by a set of geometric parameters. The geometry of the pore unit-cell and of the chambers and throats, as well as their size distributions, in a typical network—like the one used in the DeProF simulations—are provided in the Appendix. The analysis carried out in the present work can be implemented in all types of cubic lattice networks.

**Figure 4.**(

**a**) A typical 3D cubic pore network. The lattice constant, $\ell $, compared to the size of the throats and chambers, is shown expanded. The thick arrow indicates the direction of the macroscopic flow; (

**b**) The pore network cubic lattice with one main diagonal parallel to the macroscopic flow direction. The two parallel equilateral triangles are perpendicular to the macroscopic flow; (

**c**) Equilateral triangles align to form a checker-board pattern and exactly fill any frontal area perpendicular to the macroscopic flow.

_{z}× (M

_{x}× M

_{y}) unit-cells, with z indicating the direction of macroscopic flow and x, y the other two directions perpendicular to z. This reference volume includes M = M

_{x}× M

_{y}× M

_{z}pore unit-cells in total. If the reference volume is a cube, parameters K and M are related as $K/M=2\ell /\left(3\sqrt{3}\right)$.

^{−3}M. To secure a macroscopic scale description of the process, a sufficiently large reference volume would be 1 liter (1000 cm

^{3}). There are M = 3.296129 × 10

^{6}unit-cells per liter of network space and K = 1.540592631 × 10

^{4}pathways per square decimeter (dm

^{2}) of network frontal area. These figures are increased to 3.296129 × 10

^{9}unit-cells and 1.540592631 × 10

^{6}pathways if the reference volume and frontal area are extended to 1m

^{3}and 1m

^{2}, respectively. We provide these figures here because, in Section 4, we will consider two contributions in accounting the microstates, an extensive contribution (depending on the area of the reference surface and the size of the reference volume) and an intensive contribution (depending on the structure of physically admissible flow arrangements).

#### 3.2. Process and Discrete Fluidic Elements

_{i}denote the ratio of the total number of i-class ganglion cells over the total number of all ganglion cells in the DOF region. The distribution of the population of ganglia of size i, per reference volume, is given by:

_{1}, n

_{2}, I

_{max}and ζ are parameters that define the particular physically admissible flow configuration (as in [18]). Their values can be estimated either numerically, implementing the DeProF model algorithm [18,19] or in the laboratory, implementing any of the modern imaging/scanning technologies [21,22,23]. I

_{max}is the maximum attainable size of a ganglion (I

_{max}<< M

_{z}) and 0 < ζ < 1 provides a measure of the sharpness of the ganglion size distribution. Indicative ganglion size distributions are presented in the top row diagrams of Figure 7 and Figure 8. The number of different microstates in the DOF region can be evaluated by considering an equivalent “chains-in-barbs” problem in combinatorial analysis (see Section 4.3).

## 4. Definition and Counting of Microstates

#### 4.1. Countability of Process Microstates

_{w}’, β’, ω’}, namely, the water saturation, S

_{w}’, the volume fraction of connected pathway flow cells, β’, and the volume fraction of ganglion unit-cells within the disconnected oil flow unit-cells, ω’. Each one of these flow arrangements, described by any unique triple value {S

_{w}’, β’, ω’} has a countable set of microstates (or degrees of freedom) that may be evaluated by combinatorial analysis.

#### 4.2. Counting of the Connected Pathway Flow Microstates

_{x}M

_{y}Mz = M unit-cells (Section 3.1). Connected-Oil Pathways (COPs) occupy any β’ region of the pore network and remain aligned to the macroscopic flow. Of all the unit-cells in the reference volume, β’M

_{x}M

_{y}Mz = β’M unit-cells are occupied by connected oil. Now, since Connected-Oil Pathways are aligned to the macroscopic flow, and they have an infinite length, or at least, long enough to be laid across the reference volume, volume fractions and area fractions are equal. COPs occupy β’M

_{x}M

_{y}part of the frontal area (perpendicular to the macroscopic flow), see Figure 5a,b. At any instance, each one of the COPs occupies a connected pathway of unit-cells but not necessarily the same. As time evolves, COPs may rearrange their tracks between parallel pathways. Since the process is considered to be ergodic, during a large time frame, COPs may occupy any of the available connected pathways with equal probability. In that context the flow may take a number of different and equi-probable microstates. The problem of counting the CPF microstates is equivalent to the problem of counting the number of different ways in placing a number of indistinguishable balls (the COPs) in a larger number of distinguishable boxes (the pathways) with exclusion. The pathways are distinguishable because they provide the reference framework for comparing two instants (snapshots) of COP arrangements. Exclusion means that no identical snapshots of COP arrangements can be counted more than once.

**Figure 5.**(

**a**) Side view of the macroscopic flow comprising a mixture of connected and disconnected prototype flows; (

**b**) Frontal, microstate snapshot of the connected-oil pathway flow (CPF); (

**c**) Side, microstate snapshot of the Disconnected-Oil Flow (DOF); Focus is on the middle-row, colored ganglia. (

**c1**) & (

**c2**) Separation into two complementary problems in combinatorial analysis. Microstate snapshot of repositioning of ganglia (

**c1**) for a possible permutation (

**c2**) (a 1-2-1-3-2-1-2-3-2 snapshot).

_{x}M

_{y}unit-cells or pathways, then, for any physically admissible flow arrangement, N

_{COP}= β’K of the pathways are occupied by indistinguishable connected-oil pathways that may take any possible arrangement parallel to the macroscopic flow (recall that K is the number of the pore network virtual pathways), Figure 5b. Combinatorial analysis provides the solution to this particular “balls-in-boxes” problem. The number, P

_{CPF}, of different ways (excluding double counting of two identical arrangements) in placing N

_{COP}indistinguishable balls (the connected-oil pathways) into K distinguishable boxes (the pore network virtual pathways), is given by:

#### 4.3. Counting of the Disconnected Oil Flow Microstates

_{max}links (the ganglia), may engage onto an available number of barbs (the ganglia fitting into the available unit-cells).

_{max}}, i.e., a population density distribution of an integer multiple of i-linked ganglion unit-cells (oil saturated unit-cells). Ganglia of different size classes may arrange themselves anywhere within the (1−β’) volume fraction of the pore network unit-cells comprising the Disconnected-Oil Flow (DOF) domain.

_{max}. Given the ganglion size distribution, the distribution of the population of i-size ganglia within the reference volume, is directly derived from Equation (7) as:

_{1}, n

_{2}, I

_{max}and ζ given (better, determined as a physically admissible solution, e.g., implementing the DeProF model algorithm) and 0 < ζ < 1, e.g., ζ ∈ {0.3, 0.5, 0.7} and (I

_{max}<< M

_{z}).

_{C}, the total number of links in these chains (ganglion unit-cells), N

_{GUC}, and the total number of Drop Traffic Flow (DTF) unit-cells, N

_{DTF}, are given respectively by the following expressions:

_{GUC}unit-cells saturated with oil (ganglia cells) and N

_{DTF}unit-cells containing water and oil droplets. Obviously, N

_{GUC}+ N

_{DTF}= M(1−β’).

_{GUC}+ N

_{DTF}.

_{C}empty cells from a total of N

_{DTF}+N

_{C}cells. This is a problem of repositioning N

_{C}balls in (N

_{DTF}+ N

_{C}) cells. In the 2nd part, we find the number of permutations (ordering) of the N

_{C}ganglia. Then, by applying the so called “basic counting principle” [30], we derive the result by multiplying these two numbers.

_{DTF}empty unit-cells plus N

_{C}empty i-cells (of various i sizes) and we have to place, in these cells, N

_{C}chains (of various i-chain sizes) in such a way that each i-cell can contain at most one i-chain (Figure 5c1) . Since the N

_{C}i-cells are identical, the number of ways we can place N

_{C}i-cells in a total of N

_{DTF}+ N

_{C}cells (order does not matter) is equal to:

_{1}chains of size 1, N

_{2}chains of size 2 and so on, N

_{Imax}chains of size I

_{max}. In other words, there are N

_{C}objects, grouped in N

_{1}indistinguishable objects, N

_{2}indistinguishable objects and so on, up to N

_{Imax}indistinguishable objects. These needs to be placed in corresponding N

_{C}receptors, considering no receptor is to contain more than one object. We seek the number of different permutations in allocating the N

_{C}objects in the N

_{C}receptors, Figure 5(c2). This problem can be tackled as the classical problem of estimating the number of different permutations, P

_{D}

_{2}’ of N

_{C}objects (ganglia), of which N

_{1}are alike, N

_{2}are alike, …, N

_{Imax}are alike [30]. This is equal to:

_{C}chains in the N

_{DTF}+N

_{C}empty hooks (barbs) or, equivalently, the number of different ways that the given population of ganglia may be arranged to occupy the available empty network unit-cells, P

_{DOF}’, is equal to the multiplication of P

_{D}

_{1}’ by P

_{D}

_{2}’:

_{C}= 9 ganglia (chains) of different size (i-cells), with N

_{GUC}= 1 + 2 + 1 + 3 + 2 + 3 + 2 + 1 + 2 = 17 links. There are N

_{EUC}= N

_{DTF}+ N

_{GUC}= 40 empty unit-cells available for hosting these links (receptor cells), Figure 5c. Of the N

_{EUC}cells, N

_{GUC}will be occupied whereas N

_{DTF}= 23 unit-cells will be available for repositioning the chains. Therefore the number of different positioning of the particular array of nine chains (i-cells) within a total of 32 receptors (nine i-cells and 23 unit-cells), Figure 5c1, is equal to P

_{D}

_{1}’ = (23+9)!/(9! × 23!) = 32!/(9! × 23!) = 28,048,800.

_{C}= 9chains (chains of the same size are indistinguishable). This problem is equivalent to the problem of estimating the number of times we can reorder a group of N

_{C}= 9 balls comprising N

_{1}= 3 red balls, N

_{2}= 4 green balls and N

_{3}= 2 blue balls (Figure 5c2). This number is equal to P

_{D}

_{2}’ = 9!/(3! × 4! × 2!) = 1,260. Therefore, the number of different microstates of the array in Figure 5c is estimated at P

_{DOF}’ = P

_{D}

_{1}’P

_{D}

_{2}’ = 28,048,800 × 1260 = 35,341,488,000.

#### 4.4. Counting of Microstates per Physically Admissible Solution

_{w}', β’, ω’) is given by:

^{9}microstates that we computed in the disconnected oil flow taking place within just a small strip of 40 cells. Nevertheless, we do not have to actually compute such large figures. We only need to estimate the natural logarithm of expression Equation (16) and use it in the Boltzmann type expression, Equation (6). Therefore instead of direct computation of P’, it is possible to tackle its natural logarithm, lnP’.

_{COP}= β’K, and using expressions Equations (9)–(12) yields (by straightforward algebra) a form similar to the Boltzmann–Gibbs entropy [31] expression:

- (a)
- The extensive contribution of the size of the reference frame (volume and frontal area), expressed by the values of “user-selected” parameters M and K respectively. Their value depend on the aspect ratio of the reference volume and the type of the pore network. For the sought network, considering a cube as reference volume, parameters K and M are related as $K/M=2\ell /\left(3\sqrt{3}\right)$. For a lattice constant, $\ell =$1.221 mm, K = 4.699631 × 10
^{−3}. - (b)
- The non-extensive contribution of the actual flow configuration, expressed by reduced variables, namely the flow arrangement variables β’, ω’ and the reduced ganglion size distribution, n’
_{i}, i = 1, …, I_{max}.

_{i}’ lnn

_{i}’), mutually excluding each other from fully occupying the available ω’(1−β’) domain. In other words, if all ganglia were exclusively and equally partitioned into singlets, then the number of microstates would equal (−ω'lnω'); nevertheless this number is actually reduced by the accumulated number of microstates of ganglia of size i, (−n

_{i}’ lnn

_{i}’), for i=1 to I

_{max}. This complementarity is expressed by the B-G expression, $\left({\mathsf{\omega}}^{\prime}\mathrm{ln}{\mathsf{\omega}}^{\prime}-{\mathsf{\omega}}^{\prime}{\displaystyle {\sum}_{1}^{{\mathrm{I}}_{\text{max}}}{n}_{i}^{\prime}\mathrm{ln}{n}_{i}^{\prime}}\right)$, in the second term of Equations (20) and (25).

## 5. Configurational Entropy

_{SYS}(Ca, r), we need first to estimate the number of microstates of the canonical ensemble of physically admissible solutions, P(Ca,r).

_{j}’, for j∈{1, …, N

_{PAS}(Ca, r)}:

_{SYS,j}, for j∈{1,…,N

_{PAS}}. This is justified by considering that, essentially, configurational entropy provides an appropriately reduced measure of the process’ degrees of freedom. The term “appropriately reduced” refers to the Boltzmann type constant for the particular process, k

_{DeProF}. So long as any two flow configurations (members of the ensemble of physically admissible solutions) are discernibly different, their unified degrees of freedom add. In addition, since any two different flow configurations have the same physical structure, they share the same constant, k

_{DeProF}. Therefore we may write:

_{j}', as well as the contribution of other microstates stemming from the plurality of all physically admissible flow arrangements, N

_{PAS}(Ca,r), have all been taken into account.

_{DeProF}appearing in the expressions of Equations (28) and (29). Actually the k

_{DeProF}constant must have an effective character, in the sense that it must blend the contributions of two types of microstates, Equation (21), extending over different domains. In specific, the connected-oil pathway (CPF) microstates, Equation (22), extending over a reference surface (frontal area), and the Disconnected-Oil Flow (DOF) microstates, Equations (23)–(25), extending over a reference volume.

_{CPF}and k

_{DOF}, because of the different energy density associated with each prototype flow. Then, taking into account Equation (21), the Boltzmann-Gibbs expression for the configurational entropy should be expressed as:

## 6. Results and Discussion

^{−6}, and flowrate ratio values, r = 0.1, 1.0 and 10.0, within a 3D model pore network and for fluid systems with a typical oil/water viscosity ratio, κ = 1.45. The specific geometrical characteristics of the network as well as the physicochemical characteristics of fluids are presented in the Appendix. The ensembles of physically admissible solutions, corresponding to the above flow conditions are presented in the diagrams of Figure 6. The projections of the cloud of all detected solutions, over the (S

_{w}× β), (S

_{w}× ω), (β × ω) planes of the FAV phase space are depicted with small markers. The projections of the average flow configuration (S

_{w}, β, ω) are identified with large markers.

**Figure 6.**Projections of the domain of physically admissible flow configurations, (S

_{w}', β’, ω’), and of the average, macroscopic flow configuration, (S

_{w}, β, ω), for three different macroscopic flow conditions, pertaining to same capillary number value, Ca = 1.0 × 10

^{−6}and to flowrate values (

**a**) r = 0.1 (

**b**) r = 1.0 and (

**c**) r = 10.0. In every ordered pair of values, (x,y), x is the abcissa and y the ordinate.

_{w}', β’, ω’), and the associated ganglion size distributions parameters, (n

_{1}, n

_{2}, ζ), on the number of microstates as well as on the relative and absolute contributions of the CPF and Disconnected-Oil Flow (DOF) microstates, P

_{CPF}’ and P

_{DOF}’. For that purpose, we have elected to show the microstates for two types of physically admissible flow arrangements and different sets of macroscopic flow conditions, (Ca, r). “Moderate” flow arrangements are characterized by flow arrangement values (S

_{w}’, β’, ω’)close to the ensemble average, (S

_{w}, β, ω) and are presented in Figure 7. “Extreme” flow arrangements are characterized by flow arrangement values (S

_{w}’, β’, ω’) pertaining to the extreme borderline corners of the ensemble envelope, and are presented in Figure 8.

_{w}’, β’, ω’), and of the critical parameters, (n

_{1}, n

_{2}, ζ), that define ganglion size distributions, n

_{i}, Equation (7). Each row pertains to one physically admissible configuration. The data from each row are used as input to corresponding expressions estimating the values of the physical quantities plotted in the diagrams.

_{i}, corresponding to the examined physically admissible flow arrangements. The distributions have been determined directly from the associated tabulated data using Equation (7). The diagrams on the middle row present the non-extensive, reduced entropy contributions of the CPF and DOF prototype flows, as computed from Equations (22) and (23)–(25). The contributions of repositions (D

_{1}), Equation (24), and permutations (D

_{2}), Equation (25), of ganglia in DOF are also presented in the same diagrams. Similarly, the diagrams at the bottom row present the extensive, reduced entropy contributions of the CPF and DOF prototype flows (as well as the D

_{1}D

_{2}contributions).

**Figure 7.**Diagrams of the ganglion size distributions (2nd row), the non-extensive (3rd row) and the extensive (4th row) reduced entropy contributions of moderate physically admissible flows for different macroscopic flow conditions, columns (

**a**), (

**b**), (

**c**). Tabulated data indicate values of the flow arrangement variables, (S

_{w}’, β’, ω’) and the corresponding parameters, (n

_{1}’, n

_{2}’, ζ’) of ganglion size distributions. In the ordinate titles of the 3rd and 4th row diagrams, P’ denotes any of the P

_{CPF}’, P

_{D1}’, P

_{D2}’, P

_{DOF}’ contributions, whereas "A" in denominator denotes either K or M.

**Figure 8.**Diagrams of the ganglion size distributions (2nd row), the non-extensive (3rd row) and the extensive (4th row) reduced entropy contributions of extreme physically admissible flows for same macroscopic flow conditions. Tabulated data indicate values of the flow arrangement variables, (S

_{w}’, β’, ω’) and the corresponding parameters, (n

_{1}', n

_{2}', ζ') of ganglion size distributions. In the ordinate titles of the 3rd and 4th row diagrams, P’ denotes any of the P

_{CPF}', P

_{D}

_{1}', P

_{D2}', P

_{DOF}' contributions, whereas "A" in denominator denotes either K or M.

^{−6}. The diagrams in Figure 7 pertain to flowrate ratio values, r = 0.10, r = 1 and r = 10.0, in corresponding columns (a), (b) and (c). In each column, the flow arrangement values, (S

_{w}’, β’, ω’) are selected in proximity to the corresponding average, macroscopic flow arrangement, (S

_{w}’, β’, ω’). The diagrams in Figure 8 share a common value for the flowrate ratio, r = 1 ; diagrams in columns (a), (b) and (c) correspond to extreme physically admissible flow configurations, i.e., the flow arrangement variables (FAV) are as distant as possible from the FAV of the physically admissible ensemble average.

_{w}', β’, ω’) that are compatible with the externally imposed flow conditions, determined by the values of the independent variables, i.e., the superficial velocities of oil and water, U

_{o}, U

_{w}, or, equivalently expressed in reduced form, by the capillary number, Ca, and the flowrate ratio, r. The triplet values (S

_{w}’, β’, ω’) describe the partitioning of the total flow into volume fractions of prototype flows. Of these three variables, S

_{w}' and β', pertain to prototype flow configurations that have no particular interstitial structure in the sense that no additional/detailed description at the pore scale is required. In contrast, ω' pertains to the volume fraction of Ganglion Dynamics, a prototype flow configuration that has a characteristic interstitial structure, stemming from the different ganglion size distributions. The latter must satisfy the micro- to macro- scale consistency equations for mass and flowrate of oil. And this is because the flowrate in each flow domain is related to (depends on) the macroscopic pressure gradient, i.e., the solution. A simplistic approach would be to deprive ganglia velocity of their actual dependence on the locally induced macroscopic pressure gradient. But this would not only violate basic modeling principles in fluid mechanics (considering bulk viscosity and capillary hysteresis phenomena have a significant contribution), it would also not be consistent with the basic phenomenology of ganglion migration downstream the pore network—whereby migration velocity depends on the locally induced pressure gradient. Considering that disconnected oil remains stranded (immobile), provides an—optional—oversimplifying modeling approach, suggesting water saturation, S

_{w}, is to be considered as the independent variable of the process. Yet, this “conventional wisdom” fails to provide a rational explanation of the process phenomenology when disconnected-oil flow is substantial. And still today, the true/accurate description of the phenomenon relays to the classical Darcy fractional flow representation of the cause-effect relation (pressure gradient vs. flowrates) only if laboratory measured values of relative permeability to oil and water are provided, i.e., an entirely phenomenological approach. The delivery and use of relative permeability diagrams implementing saturation as the independent variable is only conventional. Another overly simplistic situation where there is no flow of oil based on Ganglion Dynamics is—according to numerous laboratory studies—physically impossible if conditions of simultaneous flow of oil and water are to be maintained.

## 7. Conclusions

- (a)
- molecular level: entropy is accounted as thermal entropy within the continuum mechanics scale;
- (b)
- configurational level: entropy is accounted within a domain of discrete and, presumably, countable flow microstates, inherent in every flow configuration that is physically admissible with the macroscopic, steady-state flow condition.

_{CPF}and k

_{DOF}appearing in the B-G expressions for the configurational entropy, Equation (30), and pertain to the connected- and disconnected-oil flows. To the authors’ confidence, delivery of an appropriate expression for the effective physical constant, k

_{DeProF}, would eventually provide a sound theoretical justification of the phenomenology of steady-state two-phase flow in porous media, based on statistical thermodynamics principles.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Geometrical Characteristics of the Pore Network Unit-Cell

^{2}.

_{Cj}and D

_{Ck}, interconnected with throat-i, a right circular cylinder of diameter D

_{Ti}. The thick dashed line rectangle represents one of the three diagonal planes of the octahedron, while the dashed-dotted lines are the two of the three octahedron diagonals.

_{o}= 1.36 × 10

^{−3}Pa s, and water, μ

_{w}= 0.94 × 10

^{−3}Pa s, providing an oil/water viscosity ratio κ = μ

_{o}/μ

_{w}= 1.45, oil/water interfacial tension γ

_{ow}= 25 × 10

^{−3}N/m, advancing (A) and receding (R) oil-water/porous medium contact angles, ${\mathsf{\theta}}_{\mathrm{A}}^{0}=$45 deg and ${\mathsf{\theta}}_{\mathrm{R}}^{0}=$39 deg respectively.

**Figure A1.**Typical jik-class unit cell of the examined model porous network. The thick inclined arrow represents the orientation of the lattice skeleton relative to the macroscopic pressure gradient.

**Table A1.**Occurrence probabilities, f

_{j}, and respective reduced size diameters for the 5 classes of chambers, D

_{Cj}, and throats, D

_{Tj}, of the 3D pore network used in the DeProF simulations. All dimensions are normalized to the lattice constant, $\ell =$ 1221 μm.

j | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

f_{j} (%) | 16 | 21 | 26 | 21 | 16 |

D_{Cj} | 0.2703 | 0.3849 | 0.4996 | 0.6143 | 0.7289 |

D_{Tj} | 0.0929 | 0.1036 | 0.1127 | 0.1192 | 0.1251 |

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Valavanides, M.S.; Daras, T.
Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks. *Entropy* **2016**, *18*, 54.
https://doi.org/10.3390/e18020054

**AMA Style**

Valavanides MS, Daras T.
Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks. *Entropy*. 2016; 18(2):54.
https://doi.org/10.3390/e18020054

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Valavanides, Marios S., and Tryfon Daras.
2016. "Definition and Counting of Configurational Microstates in Steady-State Two-Phase Flows in Pore Networks" *Entropy* 18, no. 2: 54.
https://doi.org/10.3390/e18020054