Research Papers

Modeling Flow in Porous Media With Double Porosity/Permeability: Mathematical Model, Properties, and Analytical Solutions

[+] Author and Article Information
Kalyana B. Nakshatrala

Department of Civil and Environmental
University of Houston,
Houston, TX 77204
e-mail: knakshatrala@uh.edu

Seyedeh Hanie S. Joodat

Department of Civil and Environmental
University of Houston,
Houston, TX 77204
e-mail: sseyedjoodat@uh.edu

Roberto Ballarini

Thomas and Laura Hsu Professor and Chair,
Department of Civil and Environmental
University of Houston,
Houston, TX 77204
e-mail: rballarini@uh.edu

1Corresponding author.

Manuscript received February 14, 2018; final manuscript received April 24, 2018; published online June 4, 2018. Assoc. Editor: N.R. Aluru.

J. Appl. Mech 85(8), 081009 (Jun 04, 2018) (17 pages) Paper No: JAM-18-1094; doi: 10.1115/1.4040116 History: Received February 14, 2018; Revised April 24, 2018

Geomaterials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores with different hydromechanical properties. Moreover, these pore-networks are connected through fissures and conduits. Although some models are available in the literature to describe flows in such media, they lack a strong theoretical basis. This paper aims to fill this gap in knowledge by providing the theoretical foundation for the flow of incompressible single-phase fluids in rigid porous media that exhibit double porosity/permeability. We first obtain a mathematical model by combining the theory of interacting continua and the maximization of rate of dissipation (MRD) hypothesis, and thereby provide a firm thermodynamic underpinning. The governing equations of the model are a system of elliptic partial differential equations (PDEs) under a steady-state response and a system of parabolic PDEs under a transient response. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the model satisfy. We also present several canonical problems with analytical solutions which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding ones under Darcy equations.

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Grahic Jump Location
Fig. 1

Conceptualization of a synthetic medium with two pore-scales

Grahic Jump Location
Fig. 2

Porous media and their idealizations: top part of the figure displays the idealization of a fractured porous medium using the dual-porosity model and the bottom part shows the idealization of a porous medium with two distinct pore-networks using the double porosity/permeability model. The arrows represent the fluid pathways and the mass transfer within the domain.

Grahic Jump Location
Fig. 3

One-dimensional problem #1: variation of micropressure and macropressure in the one-dimensional (1D) domain. For comparison, the analytical solution under Darcy equations is also plotted. The maximum and minimum pressures in the pore-networks need not occur on the boundary in the case of double porosity/permeability model. The parameter η is defined in equation (75): (a) p2R<p2L<1and (b) 1<p2R<p2L.

Grahic Jump Location
Fig. 4

One-dimensional problem #1: This figure numerically verifies the maximum principle given by Theorem 4. According to the maximum principle, p1(x) – p2(x) in the entire domain lies between the non-negative maximum and nonpositive minimum values on the boundary. Note that the medium properties are isotropic and homogeneous: (a) 0=p1R<0.3=p2R<p2L=0.9<p1L=1 and (b) 0=p1R<0.9=p2R<p1L=1<p2L=1.5.

Grahic Jump Location
Fig. 5

One-dimensional problem #2: Variation of the microvelocity and mass transfer for various η values for the cases k1 < k2 and k1 > k2. Although there is no supply of fluid on the boundaries of the micropore network, there is still a discharge (i.e., nonzero velocity) in the micropore network, and there is a mass transfer across the pore-networks: (a) microvelocity for k1 < k2, (b) microvelocity for k1 > k2, (c) mass transfer for k1 < k2, and (d) mass transfer for k1 > k2.

Grahic Jump Location
Fig. 6

One-dimensional problem #2: This figure compares the velocities under double porosity/permeability model and Darcy model for the cases k1 > k2 and k1 < k2. Macro- and microvelocities and their summation under the double porosity/permeability model as well as the velocity under the Darcy model with k = keff are displayed. As can be seen, keff obtained by the classical Darcy experiment cannot capture the complex internal pore-structure: (a) case 1: k1 = 1.0 and k2 = 0.1 and (b) case 2: k1 = 0.1 and k2 = 1.0.

Grahic Jump Location
Fig. 7

The top figure provides a pictorial description of the boundary value problem. There is no discharge on the inner and outer surfaces of the micropore network. For the macropore network, the inner surface is subjected to a pressure of unity, and the outer surface is subjected to a pressure of zero. The bottom figure illustrates that the macropressure under the double porosity/permeability model is qualitatively different from the pressure under Darcy equations.



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