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Review of Steady-State Two-Phase Flow in Porous Media: Independent Variables, Universal Energy Efficiency Map, Critical Flow Conditions, Effective Characterization of Flow and Pore Network

Abstract

In many applications of two-phase flow in porous media, a wetting phase is used to displace through a network of pore conduits as much as possible of a non-wetting phase, residing in situ. The energy efficiency of this physical process may be assessed by the ratio of the flow rate of the non-wetting phase over the total mechanical power externally provided and irreversibly dissipated within the process. Fractional flow analysis, extensive simulations implementing the DeProF mechanistic model, as well as a recent retrospective examination of laboratory studies have revealed universal systematic trends of the energy efficiency in terms of the actual independent variables of the process, namely the capillary number, Ca, and the flow rate ratio, r. These trends can be cast into an energy efficiency map over the (Ca, r) domain of independent variables. The map is universal for all types of non-wetting/wetting phase porous medium systems. It demarcates the efficiency of steady-state two-phase flow processes in terms of pertinent system parameters. The map can be used as a tool for designing more efficient processes, as well as for the normative characterization of two-phase flows, as to the predominance of capillary or viscous effects. This concept is based on the existence of a unique locus of critical flow conditions, for which the energy efficiency takes locally maximum values. The locus shape depends on the physicochemical characteristics of the non-wetting phase/wetting phase/porous medium system, and it shows a significant mutation as the externally imposed flow conditions change the type of flow, from capillary- to viscosity-dominated. The locus can be approached by an S-type functional form in terms of the capillary number and the system properties (viscosity ratio, wettability, pore network geometry, etc.), suggesting that formative criteria can be derived for flow characterization in any system. A new, extended definition of the capillary number is also proposed that effectively takes into account the critical properties of all the system constituents. When loci of critical flow conditions pertaining to processes with different viscosity ratio in the same pore network, are expressed in terms of this true-to-mechanism capillary number, they collapse into a unique locus. In this context, a new methodology for the effective characterization of pore networks is proposed.

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Abbreviations

~:

A tilde embelishment indicates a dimensional variable

No tilde embelishment denotes a dimensionless variable

*:

An asterisk (as superscript) indicates a value corresponding to critical flow conditions

n :

Subscript indicating a non-wetting phase variable

w :

Subscript indicating a wetting phase variable

Ca :

Capillary number

f EU :

Energy utilization factor, or, energy efficiency

f :

Fractional flow

\( \tilde{k} \) :

Absolute permeability of the porous medium

k r :

Relative permeability

\( \tilde{p} \) :

Macroscopic pressure

r :

Non-wetting/wetting flow rate ratio

S w :

Wetting phase saturation

\( \tilde{q} \) :

Flow rate

\( U \) :

Superficial velocity, or, flow intensity

\( x \) :

Reduced macroscopic pressure gradient

x pm :

Vector containing the geometrical and topological parameters of the pore network

\( \tilde{z} \) :

Coordinate length along the macroscopic flow direction

\( \tilde{\gamma }_{nw} \) :

Interfacial tension between the wetting and the non-wetting phase

\( \theta_{0}^{a} \) :

Static contact angle for advancing non-wetting/wetting phase meniscus

\( \theta_{0}^{r} \) :

Static contact angle for receding non-wetting/wetting phase meniscus

\( \kappa \) :

Non-wetting/wetting phase viscosity ratio

\( \lambda \) :

Non-wetting/wetting phase mobility ratio

\( \tilde{\mu } \) :

Dynamic viscosity

CFC(s):

Critical flow condition(s)

NWP:

Non-wetting phase

N/W/PM:

Non-wetting phase/wetting phase/porous medium

WP:

Wetting phase

References

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Acknowledgements

Parts of this research work were carried out within the ImproDeProF project, financially supported by The European Union (European Social Fund) and Greek national resources in the frameworks of the ‘Archimedes III: Funding of Research Groups in TEI of Athens’ (MlS 379389) a RTD project of the ‘Education and Lifelong Learning’ Operational Program. The present work was also partly supported by the Research Council of Norway through its Centres of Excellence funding scheme, Project Number 262644. The author thanks Prof. Gregory Kamvyssas (TEI Western Greece), Dr. Christos Tsakiroglou (ICE/HT-FORTH), Prof. Marios Ioannides (Univ. of Waterloo), Dr. Zubair Kalam (ADCO), Professors Alex Hansen, Dick Bedeaux, Signe Kjelstrup (NTNU), Prof. Knut Jørgen Maløy (University of Oslo), Dr. Olga Vizika (IFP EN), Prof. Apostolos Kantzas (University of Calgary), Prof. Majid Hassanizadeh (Utrecht Univ.) and Prof. Michael Celia (Princeton Univ.) for stimulating discussions on topics of this work (in chronological order).

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Correspondence to Marios S. Valavanides.

Appendices

Appendix I

The definition of NWP/WP flow rate ratio, r, Eq. (8) when combined with the Darcy fractional flow Eq. (1),

$$ \tilde{U}_{n} = \frac{{\tilde{q}_{n} }}{{\tilde{A}}} = \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}k_{rn} \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\quad \tilde{U}_{w} = \frac{{\tilde{q}_{w} }}{{\tilde{A}}} = \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}k_{rw} \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\; $$
(I-1)

and the experimentally verified condition that in steady-state two-phase fully developed flow conditions the pressure gradient is the same for both fluids [see Eqs. (10) and (11) and Fig. 1 in Avraam and Payatakes 1999],

$$ \left. {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right|_{n} = \left. {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right|_{w} = \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} $$
(I-2)

yields relation (11)

$$ r = \frac{{\tilde{q}_{n} }}{{\tilde{q}_{w} }} = \frac{{\tilde{U}_{n} }}{{\tilde{U}_{w} }} = \frac{{{{k_{rn} } \mathord{\left/ {\vphantom {{k_{rn} } {\tilde{\mu }_{n} }}} \right. \kern-0pt} {\tilde{\mu }_{n} }}}}{{{{k_{rw} } \mathord{\left/ {\vphantom {{k_{rw} } {\tilde{\mu }_{w} }}} \right. \kern-0pt} {\tilde{\mu }_{w} }}}} = \frac{1}{\kappa }\frac{{k_{rn} }}{{k_{rw} }}\; $$
(I-3)

Let \( \tilde{W}^{1\varPhi } \) be the specific mechanical power dissipation (specific as per unit porous medium volume—p.u.v.p.m.) for one-phase flow of wetting phase at an equivalent flow rate, \( \tilde{q}_{w} \), against a porous medium cross section of surface, \( \tilde{A} \), and along a distance, \( \Delta \tilde{z} \), given by

$$ \tilde{W}^{1\varPhi } = \frac{{\tilde{q}_{w} \Delta \tilde{p}}}{{\tilde{A}\Delta \tilde{z}}} = \frac{{\tilde{q}_{w} }}{{\tilde{A}}}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} = \tilde{U}_{w} \frac{{\tilde{\mu }_{w} }}{{\tilde{k}}}\tilde{U}_{w} = \frac{{\tilde{\mu }_{w} }}{{\tilde{k}}}\left( {\frac{{\tilde{\mu }_{w} \tilde{U}_{w} }}{{\tilde{\gamma }_{ow} }}\frac{{\tilde{\gamma }_{ow} }}{{\tilde{\mu }_{w} }}} \right)^{2} = \frac{{\left( {\tilde{\gamma }_{nw} Ca} \right)^{2} }}{{\tilde{\mu }_{w} \tilde{k}}} $$
(I-4)

The specific (p.u.v.p.m.) mechanical power dissipation for the steady-state concurrent two-phase flow of non-wetting and wetting phases, at flow rates \( \tilde{q}_{n} \) and \( \tilde{q}_{w} \), with superficial velocities, \( \tilde{U}_{n} \) and \( \tilde{U}_{w} \), across a porous medium surface, \( \tilde{A} \), and along a distance, \( \Delta \tilde{z} \), is given by

$$ \begin{aligned} \tilde{W} & = \frac{{\tilde{q}_{n} \Delta \tilde{p}_{n} + \tilde{q}_{w} \Delta \tilde{p}_{w} }}{{\tilde{A}\Delta \tilde{z}}} = \frac{{\tilde{q}_{n} }}{{\tilde{A}}}\left. {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right|_{n} + \frac{{\tilde{q}_{w} }}{{\tilde{A}}}\left. {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right|_{w} = \tilde{U}_{n} \frac{{\tilde{\mu }_{n} }}{{\tilde{k}k_{rn} }}\tilde{U}_{n} + \tilde{U}_{w} \frac{{\tilde{\mu }_{w} }}{{\tilde{k}k_{rw} }}\tilde{U}_{w} \\ & = \tilde{U}_{w}^{2} \frac{{\tilde{\mu }_{w} }}{{\tilde{k}}}\left( {r^{2} \frac{\kappa }{{k_{rn} }} + \frac{1}{{k_{rw} }}} \right) = \frac{{\left( {\tilde{\gamma }_{nw} Ca} \right)^{2} }}{{\tilde{\mu }_{w} \tilde{k}}}\frac{1}{{k_{ro} }}\left( {r^{2} \kappa + \frac{{k_{ro} }}{{k_{rw} }}} \right) \\ & = \frac{{\left( {\tilde{\gamma }_{nw} Ca} \right)^{2} }}{{\tilde{\mu }_{w} \tilde{k}}}\kappa r\frac{1}{{k_{ro} }}\left( {r + 1} \right) = \frac{{\left( {\tilde{\gamma }_{nw} Ca} \right)^{2} }}{{\tilde{\mu }_{w} \tilde{k}}}\frac{1}{{k_{rw} }}\left( {r + 1} \right) \\ \end{aligned} $$
(I-5)

Therefore, the reduced mechanical power dissipation for the steady-state two-phase flow, W, may be expressed in any of the three equivalent forms

$$ W = \frac{{\tilde{W}}}{{\tilde{W}^{1\varPhi } }} = \frac{1}{{k_{rn} }}\left( {r^{2} \kappa + \frac{{k_{rn} }}{{k_{rw} }}} \right) = \kappa r\frac{1}{{k_{rn} }}\left( {r + 1} \right) = \frac{1}{{k_{rw} }}\left( {r + 1} \right) $$
(I-6)

and may be expressed in any of the three equivalent forms, expression (12),

$$ f_{EU} = \frac{r}{W} = \frac{{k_{rn} }}{{\kappa \left( {r + 1} \right)}} = k_{rw} \frac{r}{r + 1} = k_{rn} \left( {\frac{{k_{rn} }}{{k_{rw} }} + \kappa } \right)^{ - 1} $$
(I-7)

The above expressions are valid for any value of the capillary number.

Appendix II

Determination of the Asymptotic Values of r * and f EU (as \( Ca \to + \infty \))

The flow is regulated by the values of two independent variables, Ca and r. In the far end of the Ca spectrum (\( Ca \to + \infty \)), capillary effects (pertaining to the motion of interfaces) are negligible when compared to viscosity effects (pertaining to the strain rates of the bulk phases). This is so, because, by definition of the capillary number, at conditions of extremely large-Ca values, the capillary forces are relatively negligible when compared with the viscous forces. Consequently, the rate of energy dissipation caused by capillary effects is negligible when compared to that caused by viscous stresses in each phase. In addition, the total mechanical power dissipation per unit porous medium volume—whereby the viscous dissipation is the predominant term—is directly proportional to the saturation of each phase, S o∞ and S w∞ . This is so, irrespective of whether the NWP is connected or not, or to the degree of its disconnectedness (the case for non-wetting phase ganglia and droplets) and the magnitude of the N/W interface surface area. In flow conditions of large-Ca values, the latter do not have any discernible effect on the flow.

In the following, we will consider the case of very high-Ca values, and we will try to evaluate the particular flow rate ratio, \( r_{\infty }^{*} \), for which the process efficiency takes a maximum value.

In general, the simultaneous flow of NWP and WP at flow rates, \( \tilde{q}_{n} \) and \( \tilde{q}_{w} \), through a frontal area of porous medium, \( \tilde{A} \), can be formally expressed by the fractional Darcy (phenomenological) law as

$$ \tilde{q}_{n} = \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}k_{rn\infty } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A}\quad {\text{and}}\quad \tilde{q}_{w} = \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}k_{rw\infty } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A} $$
(II-1)

where \( k_{rn\infty } \) and \( k_{rw\infty } \) denote the value of relative permeability to NWP and WP as \( Ca \to + \infty \), under the common pressure gradient \( \left( {{{\Delta \tilde{p}} \mathord{\left/ {\vphantom {{\Delta \tilde{p}} {\Delta \tilde{z}}}} \right. \kern-0pt} {\Delta \tilde{z}}}} \right) \).

At extreme flow conditions (\( Ca \to + \infty \)), the flow of oil and water is internally balanced (regulated) and the N/W interfaces have a uniform influence—the capillary pressure is uniform along the flow direction. We may therefore express the flow rate of non-wetting and wetting phases as co-existing Darcian, saturated flows sharing the same pressure gradient,

$$ \tilde{q}_{n}^{1\varPhi } = \tilde{U}_{n}^{1\varPhi } \tilde{A}_{n\infty } = \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A}_{n\infty } \quad {\text{and}}\quad \tilde{q}_{w}^{1\varPhi } = \tilde{U}_{w}^{1\varPhi } \tilde{A}_{\infty w} = \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A}_{\infty w} $$
(II-2)

where\( \tilde{q}_{n}^{1\varPhi } = \tilde{q}_{n} \) and \( \tilde{q}_{w}^{1\varPhi } = \tilde{q}_{w} \) denote, respectively, the saturated flow of NWP and WP,

\( \tilde{A}_{n\infty } \) and \( \tilde{A}_{w\infty } \) denote the porous medium frontal area used by the NWP and WP flows. Considering fully developed flow, the frontal areas may not decompose into fixed shapes along the macroscopic flow direction; nevertheless, quantity-wise, they would be keeping fixed, Darcian-scale average values.

Considering the equivalent description of the flow through eqs (II—1) and (II—2) we get

$$ \tilde{q}_{n} = \tilde{q}_{n}^{1\varPhi } \quad \Rightarrow \quad \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}k_{rn\infty } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A} = \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A}_{n\infty } \quad \Rightarrow \quad k_{rn\infty } \tilde{A} = \tilde{A}_{n\infty } $$
(II-3)
$$ \tilde{q}_{w} = \tilde{q}_{w}^{1\varPhi } \quad \Rightarrow \quad \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}k_{rw\infty } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A} = \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}\tilde{A}_{w\infty } \quad \Rightarrow \quad k_{rw\infty } \tilde{A} = \tilde{A}_{w\infty } $$
(II-4)

and considering that the volume saturations are equivalent to the flow-front areal saturations,

$$ \tilde{A}_{n\infty } = S_{n\infty } \tilde{A}\quad {\text{and}}\quad \tilde{A}_{w\infty } = S_{w\infty } \tilde{A} $$
(II-5)

it follows that

$$ {\text{As}}\;Ca \to + \infty ,\quad k_{rn\infty } = S_{n\infty } \quad {\text{and}}\quad k_{rw\infty } = S_{w\infty } $$
(II-6)

This latter equivalence holds for any imposed value of N/W flow rate ratio, r, as long as the capillary number attains very large values (\( Ca \to + \infty \)).

We will now detect the value, \( r_{\infty }^{*} \), that turns the efficiency of the process maximum. We may proceed by considering two cases, A or B.

  1. A.

    Maximization of the energy efficiency of the process, \( \tilde{F}_{{\rm EU}\infty} \), expressed as \( F_{{{\text{EU}}\infty }} = {\text{\{ NWP}}\;{\text{flowrate}}\;{\text{per}}\;{\text{kW}}\;{\text{of}}\;{\text{total}}\;{\text{mechanical}}\;{\text{power}}\;{\text{dissipation\} }} = {{\tilde{q}_{n} } \mathord{\left/ {\vphantom {{\tilde{q}_{n} } {\left( {\tilde{W}_{n} + \tilde{W}_{w} } \right)}}} \right. \kern-0pt} {\left( {\tilde{W}_{n} + \tilde{W}_{w} } \right)}} \),

  2. B.

    Maximization of the reduced energy efficiency of the process, \( f_{EU\infty } \), expressed as \( f_{{{\text{EU}}\infty }} = {\text{reduced}}\;{\text{\{ NWP}}\;{\text{flowrate}}\;{\text{per}}\;{\text{kW}}\;{\text{of}}\;{\text{total}}\;{\text{mechanical}}\;{\text{power}}\;{\text{dissipation\} }} = {r \mathord{\left/ {\vphantom {r W}} \right. \kern-0pt} W} \)

In both cases, we will use expressions (II-3) to (II-6).

Case A

We will first write the expressions for the specific (as per unit porous medium volume, \( \Delta \tilde{V} \)) mechanical power required to maintain the flow of NWP and WP, denoted, respectively, by \( \tilde{W}_{n} \) and \( \tilde{W}_{w} \), as

$$ \tilde{W}_{n} = \tilde{W}_{n}^{1\varPhi } = \frac{{\tilde{q}_{n}^{1\varPhi } \Delta \tilde{p}}}{{\Delta \tilde{V}}} = \frac{{\tilde{q}_{n}^{1\varPhi } \Delta \tilde{p}}}{{\tilde{A}_{n\infty } \Delta \tilde{z}}} = \frac{{\tilde{q}_{n}^{1\varPhi } }}{{\tilde{A}_{n\infty } }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} = \tilde{U}_{n}^{1\varPhi } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} = \frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}\left( {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right)^{2} $$
(II-7)
$$ \tilde{W}_{w} = \tilde{W}_{w}^{1\varPhi } = \frac{{\tilde{q}_{w}^{1\varPhi } \Delta \tilde{p}}}{{\Delta \tilde{V}}} = \frac{{\tilde{q}_{w}^{1\varPhi } \Delta \tilde{p}}}{{\tilde{A}_{w\infty } \Delta \tilde{z}}} = \frac{{\tilde{q}_{w}^{1\varPhi } }}{{\tilde{A}_{w\infty } }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} = \tilde{U}_{w}^{1\varPhi } \frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}} = \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}\left( {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right)^{2} $$
(II-8)

Then, the energy efficiency of the process, \( \tilde{F}_{{\rm EU}\infty} \), expressed as NWP flow rate per kW of total mechanical power dissipated, at very high capillary number values, \( Ca \to + \infty \), i.e., for purely viscous dominated flow conditions, may be estimated as follows.

$$ \begin{aligned} \tilde{F}_{{{\text{EU}}\infty }} & = \frac{{{\text{Oil}}\;{\text{flowrate}}}}{{{\text{Mechanical}}\;{\text{power}}}} = \frac{{\tilde{q}_{n} }}{{\left( {\tilde{W}_{n} + \tilde{W}_{w} } \right)}} = \frac{{\tilde{U}_{n}^{1\varPhi } \tilde{A}_{n\infty } }}{{\left( {\tilde{W}_{n} + \tilde{W}_{w} } \right)}} \\ & = \frac{{\frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}S_{n\infty } \tilde{A}}}{{\left( {\frac{{\tilde{k}}}{{\tilde{\mu }_{n} }} + \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}} \right)\left( {\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}} \right)^{2} }} = \frac{{S_{n\infty } }}{{\left( {1 + \kappa } \right)}}\frac{{k_{rw\infty } \tilde{k}}}{{\tilde{\mu }_{w} \tilde{U}_{w} }}\tilde{A} = \frac{{\left( {1 - S_{w\infty } } \right)}}{{\left( {1 + \kappa } \right)}}S_{w\infty } \frac{{\tilde{k}}}{{\tilde{\gamma }_{nw} Ca}}\tilde{A} \\ \end{aligned} $$
(II-9)

The flow arrangement—in terms of water saturation—that maximizes energy efficiency, \( S_{w\infty }^{*} \), may be determined by setting the derivative of \( \tilde{F}_{EU\infty } \) with respect to S w∞ equal to zero:

$$ \left. {\frac{{{\text{d}}\left( {\tilde{F}_{{{\text{EU}}\infty }} } \right)}}{{{\text{d}}S_{w\infty } }}} \right|_{{S_{w\infty }^{*} }} = 0\quad \Rightarrow \quad \left. {\frac{\text{d}}{{{\text{d}}S_{w\infty } }}\left[ {\left( {1 - S_{w\infty } } \right)S_{w\infty } } \right]} \right|_{{S_{w\infty }^{*} }} = 0 $$
(II-10)
$$ 1 - 2S_{w\infty }^{*} = 0\quad \Rightarrow \quad S_{w\infty }^{*} = 0.5 $$
(II-11)

Therefore, we may infer that the flow rate ratio value, \( r_{\infty }^{*} \), that maximizes energy efficiency corresponds to a flow setup with \( S_{w\infty }^{*} \). In that respect,

$$ r_{\infty }^{*} = \frac{1}{\kappa }\frac{{k_{ro\infty }^{*} }}{{k_{rw\infty }^{*} }} = \frac{1}{\kappa }\frac{{S_{o\infty }^{*} }}{{S_{w\infty }^{*} }} = \frac{1}{\kappa }\frac{{\left( {1 - 0.5} \right)}}{0.5} {\text{i}} . {\text{e}} .\quad {\text{ as}}\quad Ca \to \infty ,\quad r_{\infty }^{*} = r_{x} = \frac{1}{\kappa } $$
(II-12)

Therefore, we may infer that the flow rate ratio value, \( r_{\infty }^{*} \), that maximizes energy efficiency at very large-Ca value flow conditions, \( f_{{{\text{EU}}\infty }} \), corresponds to a flow setup with \( S_{w\infty }^{*} \). Therefore,

$$ \tilde{F}_{{{\text{EU}}\infty }} = \frac{{\left( {1 - S_{w\infty } } \right)}}{{\left( {1 + \kappa } \right)}}S_{w\infty } \frac{{\tilde{k}}}{{\tilde{\gamma }_{nw} Ca}}\tilde{A} = \frac{{\left( {1 - 0.5} \right)}}{{\left( {1 + \kappa } \right)}}0.5\frac{{\tilde{k}}}{{\tilde{\gamma }_{nw} Ca}}\tilde{A}\quad \Rightarrow \quad \tilde{F}_{{{\text{EU}}\infty }} = \frac{1}{{4\left( {1 + \kappa } \right)}}\frac{{\tilde{k}}}{{\tilde{\gamma }_{nw} Ca}}\tilde{A} $$
(II-13)

Case B

The reduced energy efficiency of the process, \( f_{EU\infty } \), expressed as the reduced NWP flow rate (i.e., flow rate ratio) over the reduced total mechanical power dissipation (reduced with respect to power dissipation of the equivalent saturated flow of the wetting phase), at very high capillary number values, \( Ca \to + \infty \), i.e., for purely viscous dominated flow conditions, may be estimated as follows. We start by using Eq. (II-7)

$$ f_{\text{EU}} \frac{r}{W} = \frac{{k_{rn} }}{{\kappa \left( {r + 1} \right)}} = k_{rw} \frac{r}{r + 1} = k_{ro} \left( {\frac{{k_{rn} }}{{k_{rw} }} + \kappa } \right)^{ - 1} $$
(II-14)
$$ \begin{aligned} f_{\text{EU}} & = \left[ {\frac{r}{W}} \right]_{Ca \to \infty } = \frac{{k_{rn\infty } }}{{\kappa \left( {r_{\infty } + 1} \right)}} = \frac{{S_{n\infty } }}{{\kappa \left( {\frac{{U_{n}^{1\varPhi } S_{n\infty } }}{{U_{w}^{1\varPhi } S_{w\infty } }} + 1} \right)}} \\ & = \frac{{S_{n\infty } U_{w}^{1\varPhi } S_{w\infty } }}{{\kappa \left( {U_{n}^{1\varPhi } S_{n\infty } + U_{w}^{1\varPhi } S_{w\infty } } \right)}} = \frac{{S_{n\infty } \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}S_{w\infty } }}{{\kappa \left( {\frac{{\tilde{k}}}{{\tilde{\mu }_{n} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}S_{n\infty } + \frac{{\tilde{k}}}{{\tilde{\mu }_{w} }}\frac{{\Delta \tilde{p}}}{{\Delta \tilde{z}}}S_{w\infty } } \right)}} \\ & = \frac{{S_{n\infty } \frac{1}{{\tilde{\mu }_{w} }}S_{w\infty } }}{{\kappa \left( {\frac{1}{{\tilde{\mu }_{n} }}S_{n\infty } + \frac{1}{{\tilde{\mu }_{w} }}S_{w\infty } } \right)}} = \frac{{S_{n\infty } \kappa S_{w\infty } }}{{\kappa \left( {S_{n\infty } + \kappa S_{w\infty } } \right)}} = \frac{{\left( {1 - S_{w\infty } } \right)S_{w\infty } }}{{1 - S_{w\infty } \left( {1 - \kappa } \right)}} \\ \end{aligned} $$
(II-15)

The flow arrangement, in terms of water saturation, \( S_{w\infty }^{*} \), that maximizes the reduced energy efficiency, \( f_{EU\infty }^{{}} \), may be determined by setting its derivative with respect to S w∞ equal to zero:

$$ \begin{aligned} \left. {\frac{{{\text{d}}\left( {f_{{{\text{EU}}\infty }} } \right)}}{{{\text{d}}S_{w\infty } }}} \right|_{{S_{w\infty }^{*} }} & = \left. {\frac{\text{d}}{{{\text{d}}S_{w\infty } }}\left[ {\frac{{\left( {1 - S_{w\infty } } \right)S_{w\infty } }}{{1 - S_{w\infty } \left( {1 - \kappa } \right)}}} \right]} \right|_{{S_{w\infty }^{*} }} \\ & = \left. {\frac{{\left( {1 - 2S_{w\infty }^{*} } \right)\left[ {1 - S_{w\infty }^{*} \left( {1 - \kappa } \right)} \right] + \left( {1 - S_{w\infty }^{*} } \right)S_{w\infty }^{*} \left( {1 - \kappa } \right)}}{{\left[ {1 - S_{w\infty }^{*} \left( {1 - \kappa } \right)} \right]^{2} }}} \right|_{{S_{w\infty }^{*} }} = 0 \\ \end{aligned} $$
(II-16)
$$ \left( {S_{w\infty }^{*} } \right)^{2} \left( {1 - \kappa } \right) - 2S_{w\infty }^{*} + 1 = 0\quad \Rightarrow \quad S_{w\infty }^{*} = \frac{1}{{\left( {1 \pm \sqrt \kappa } \right)}}\quad \Rightarrow \quad 0 < S_{w\infty }^{*} = \frac{1}{{\left( {1 + \sqrt \kappa } \right)}} < 1 $$
(II-17)

Therefore, we may infer that the flow rate ratio value, \( r_{\infty }^{*} \), that maximizes energy efficiency at very large-Ca flow conditions, \( f_{EU\infty }^{*} \), corresponds to a flow setup with \( S_{w\infty }^{*} \). Therefore, as \( Ca \to \infty \),

$$ r_{\infty }^{*} = \frac{1}{\kappa }\frac{{k_{rn\infty }^{*} }}{{k_{rw\infty }^{*} }} = \frac{1}{\kappa }\frac{{S_{n\infty }^{*} }}{{S_{w\infty }^{*} }} = \frac{1}{\kappa }\frac{{\left( {1 - \frac{1}{{\left( {1 + \sqrt \kappa } \right)}}} \right)}}{{\frac{1}{{\left( {1 + \sqrt \kappa } \right)}}}}\quad \Rightarrow \quad r_{\infty }^{*} = \frac{1}{\sqrt \kappa } $$
(II-18)

and the corresponding energy efficiency takes the maximum value

$$ f_{{{\text{EU}}\infty }}^{*} = \left( {\frac{r}{W}} \right)_{w\infty }^{*} = \frac{{k_{rn\infty }^{*} }}{{\kappa \left( {r_{w\infty }^{*} + 1} \right)}} = \frac{{S_{n\infty }^{*} }}{{\kappa \left( {r_{w\infty }^{*} + 1} \right)}} = \frac{{1 - \frac{1}{1 + \sqrt \kappa }}}{{\kappa \left( {\frac{1}{\sqrt \kappa } + 1} \right)}}\quad \Rightarrow \quad f_{{{\text{EU}}\infty }}^{*} = \frac{1}{{\left( {1 + \sqrt \kappa } \right)^{2} }} $$
(II-19)

Appendix III

Determination of Checkpoint Values for Flow Characterization

The four parameters logistics function, used in Eq. (46), can be rewritten in a simpler form, by switching \( x \leftrightarrow \log Ca \) and \( y \leftrightarrow \log r \), as

$$ y = y_{\infty }^{*} + \frac{{y_{0}^{*} - y_{\infty }^{*} }}{{1 + \exp \left[ {\left( {x - x_{0} } \right)/z} \right]}} $$
(III-1)

where real number parameters \( y_{0}^{*} \), \( y_{\infty }^{*} \), z and x 0 stand for: \( y_{0}^{*} \) the maximum (asymptotic) value of function y, \( y_{\infty }^{*} \) the minimum (asymptotic) value of function y, x 0 the x value for the curve point that is midway between the maximum and minimum asymptotic values, \( y_{0}^{*} \) and \( y_{\infty }^{*} \), z the inclination of the curve at its midpoint; large values result in a steep curve, whereas small values a ‘shallow’ curve.

The first and second derivatives result in the following expressions

$$ y^{\prime} = \frac{{y_{0}^{*} - y_{\infty }^{*} }}{{\left( {1 + \exp \left[ {\left( {x - x_{0} } \right)/z} \right]} \right)^{2} }}\frac{1}{z}\exp \left[ {\left( {x - x_{0} } \right)/z} \right] $$
(III-2)

The first and second derivatives result in the following expressions

$$ y^{\prime\prime} = \frac{{y_{0}^{*} - y_{\infty }^{*} }}{{z^{2} }}\frac{{\exp \frac{{x - x_{0} }}{z}}}{{\left( {1 + \exp \frac{{x - x_{0} }}{z}} \right)^{3} }}\left( {1 - \exp \frac{{x - x_{0} }}{z}} \right) $$
(III-3)

In general, the curvature, K, of any function \( y = f\left( x \right) \) is given by

$$ K = \frac{{y^{\prime\prime}}}{{\left( {1 + y^{{{\prime }2}} } \right)^{3/2} }} $$
(III-4)

Substituting the expressions, we get the following expression for the curvature of the function at hand,

$$ K = \frac{{y_{0}^{*} - y_{\infty }^{*} }}{{z^{2} }}\frac{{\exp \frac{{x - x_{0} }}{z}}}{{\left( {1 + \exp \frac{{x - x_{0} }}{z}} \right)^{3} }}\frac{{\left( {1 - \exp \frac{{x - x_{0} }}{z}} \right)}}{{\left( {1 + \frac{1}{z}\exp \frac{{x - x_{0} }}{z}\frac{{y_{0}^{*} - y_{\infty }^{*} }}{{\left( {1 + \exp \frac{{x - x_{0} }}{z}} \right)^{2} }}} \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }} $$
(III-5)

The first derivative of the curvature is,

$$ K^{\prime} = \frac{{\left( {y_{0}^{*} - y_{\infty }^{*} } \right)\left( { - y_{0}^{*} + y_{\infty }^{*} - 8z + 4z\cosh \frac{{x - x_{0} }}{z}} \right)}}{{z^{2} \left( {y_{0}^{*} - y_{\infty }^{*} + 2z + 2z\cosh \frac{{x - x_{0} }}{z}} \right)^{2} \sqrt {4 + \frac{{2y_{0}^{*} - 2y_{\infty }^{*} }}{{z + z\cosh \frac{{x - x_{0} }}{z}}}} }} $$
(III-6)

Setting the first derivative equal to zero and solving the equation,

$$ K^{\prime} = 0\quad \Rightarrow \quad - y_{0}^{*} + y_{\infty }^{*} - 8z + 4z\cosh \frac{{x - x_{0} }}{z} = 0 $$
(III-7)

delivers the roots x 1 and x 2 , \( x_{1} < x_{0} < x_{2} \),

$$ x_{1,2} = x_{0} \mp z{\text{arc}}\cosh\left( {\frac{{y_{0}^{*} - y_{\infty }^{*} + 8z}}{4z}} \right) $$
(III-8)

for which the curvature takes locally extreme values.

The values of the second derivative corresponding to roots x 1,2 are

$$ K^{\prime\prime}\left( {x_{1} } \right) = - \frac{2}{9\sqrt 3 }\frac{{\left( {y_{0}^{*} - y_{\infty }^{*} } \right)\left( {y_{0}^{*} - y_{\infty }^{*} + 12z} \right)}}{{z^{3} \left( {y_{0}^{*} - y_{\infty }^{*} + 4z} \right)^{2} }} < 0 $$
(III-9)
$$ K^{\prime\prime}\left( {x_{2} } \right) = \frac{2}{9\sqrt 3 }\frac{{\left( {y_{0}^{*} - y_{\infty }^{*} } \right)\left( {y_{0}^{*} - y_{\infty }^{*} + 12z} \right)}}{{z^{3} \left( {y_{0}^{*} - y_{\infty }^{*} + 4z} \right)^{2} }} > 0 $$
(III-10)

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Valavanides, M.S. Review of Steady-State Two-Phase Flow in Porous Media: Independent Variables, Universal Energy Efficiency Map, Critical Flow Conditions, Effective Characterization of Flow and Pore Network. Transp Porous Med 123, 45–99 (2018). https://doi.org/10.1007/s11242-018-1026-1

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Keywords

  • Two-phase flow
  • Porous media
  • Steady-state
  • Energy efficiency
  • Critical flow conditions