0
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
Engineering,
University of Houston,
Houston, TX 77204
e-mail: knakshatrala@uh.edu

Seyedeh Hanie S. Joodat

Department of Civil and Environmental
Engineering,
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
Engineering,
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Al-Mukhtar, M. , 1995, “Macroscopic Behavior and Microstructural Properties of a Kaolinite Clay Under Controlled Mechanical and Hydraulic State,” First International Conference on Unsaturated Soils/unsat'95, Paris, France, Sept. 6–8, pp. 3–9. https://trid.trb.org/view/468401
Delage, P. , Audiguier, M. , Cui, Y. J. , and Howat, M. D. , 1996, “Microstructure of a Compacted Silt,” Can. Geotech. J., 33(1), pp. 150–158. [CrossRef]
Didwania, A. K. , 2002, “Micromechanical Basis of Concept of Effective Stress,” J. Eng. Mech., 128(8), pp. 864–868. [CrossRef]
Koliji, A. , Laloui, L. , Cusinier, O. , and Vulliet, L. , 2006, “Suction Induced Effects on the Fabric of a Structured Soil,” Transp. Porous Media, 64(2), pp. 261–278. [CrossRef]
Borja, R. I. , and Koliji, A. , 2009, “On the Effective Stress in Unsaturated Porous Continua With Double Porosity,” J. Mech. Phys. Solids, 57(8), pp. 1182–1193. [CrossRef]
Straughan, B. , 2017, Mathematical Aspects of Multi-Porosity Continua, Springer, Cham, Switzerland. [CrossRef]
Pruess, K. , and Narasimhan, T. N. , 1985, “A Practical Method for Modeling Fluid and Heat Flow in Fractured Porous Media,” Soc. Pet. Eng. J., 25(1), pp. 14–26.
van Genuchten, M. T. , and Wierenga, P. J. , 1976, “Mass Transfer Studies in Sorbing Porous Media—I: Analytical Solutions,” Soil Sci. Soc. Am. J., 40(4), pp. 473–480. [CrossRef]
Šimunek, J. , Jarvis, N. J. , van Genuchten, M. T. , and Gärdenäs, A. , 2003, “Review and Comparison of Models for Describing Non-Equilibrium and Preferential Flow and Transport in the Vadose Zone,” J. Hydrol., 272(1–4), pp. 14–35. [CrossRef]
Geiger, S. , Dentz, M. , and Neuweiler, I. , 2013, “A Novel Multi-Rate Dual-Porosity Model for Improved Simulation of Fractured and Multiporosity Reservoirs,” SPE J., 18(04), pp. 670–684. [CrossRef]
Warren, J. E. , and Root, P. J. , 1963, “The Behavior of Naturally Fractured Reservoirs,” Soc. Pet. Eng. J., 3(03), pp. 245–255. [CrossRef]
Hayes, J. B. , 1979, “Sandstone Diagenesis—The Hole Truth,” Aspects of Diagenesis, Vol. 26, Society of Economic Paleontologists and Mineralogists, Tulsa, OK, pp. 127–139.
Schmidt, V. , and Mcdonald, D. A. , 1979, “Texture and Recognition of Secondary Porosity in Sandstones,” Aspects of Diagenesis, Vol. 26, Society of Economic Paleontologists and Mineralogists, Tulsa, OK, pp. 209–225.
Cuisinier, O. , and Laloui, L. , 2004, “Fabric Evolution During Hydromechanical Loading of a Compacted Silt,” Int. J. Numer. Anal. Methods Geomech., 28(6), pp. 483–499. [CrossRef]
Barenblatt, G. I. , Zheltov, I. P. , and Kochina, I. N. , 1960, “Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks (Strata),” J. Appl. Math. Mech., 24(5), pp. 1286–1303. [CrossRef]
Dykhuizen, R. C. , 1990, “A New Coupling Term for Dual-Porosity Models,” Water Resour. Res., 26(2), pp. 351–356.
Vogel, T. , Gerke, H. H. , Zhang, R. , and van Genuchten, M. T. , 2000, “Modeling Flow and Transport in a Two-Dimensional Dual-Permeability System With Spatially Variable Hydraulic Properties,” J. Hydrol., 238(1–2), pp. 78–89. [CrossRef]
Balogun, A. S. , Kazemi, H. , Ozkan, E. , Al-Kobaisi, M. , and Ramirez, B. A. , 2007, “Verification and Proper Use of Water-Oil Transfer Function for Dual-Porosity and Dual-Permeability Reservoirs,” SPE Middle East Oil and Gas Show and Conference, Manama, Bahrain, Mar. 11–14, SPE Paper No. SPE-104580-MS.
Lowell, S. , Shields, J. E. , Thomas, M. A. , and Thommes, M. , 2012, Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density, Springer Science and Business Media, New York.
Stock, S. R. , 2008, Microcomputed Tomography: Methodology and Applications, CRC Press, Boca Raton, FL. [CrossRef]
Arbogast, T. , Douglas, J. J. , and Hornung, U. , 1990, “Derivation of the Double Porosity Model of Single Phase Flow Via Homogenization Theory,” SIAM J. Math. Anal., 21(4), pp. 823–836. [CrossRef]
Amaziane, B. , and Pankratov, L. , 2015, “Homogenization of a Model for Water–Gas Flow Through Double-Porosity Media,” Math. Methods Appl. Sci., 39(3), pp. 425–451. [CrossRef]
Boutin, C. , and Royer, P. , 2015, “On Models of Double Porosity Poroelastic Media,” Geophys. J. Int., 203(3), pp. 1694–1725. [CrossRef]
Amaziane, B. , Antontsev, S. , Pankratov, L. A. , and Piatnitski, A. , 2010, “Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository,” Multiscale Model. Simul., 8(5), pp. 2023–2047. [CrossRef]
Hornung, U. , 1996, Homogenization and Porous Media, Springer-Verlag, New York.
Lubliner, J. , 2008, Plasticity Theory, Dover Publications Inc., Mineola, NY.
Borja, R. I. , 2013, Plasticity: Modeling and Computation, Springer Science and Business Media, New York. [PubMed] [PubMed]
Evans, L. C. , 1998, Partial Differential Equations, American Mathematical Society, Providence, RI.
Bowen, R. , 2014, Porous Elasticity: Lectures on the Elasticity of Porous Materials as an Application of the Theory of Mixtures, Texas A&M University, College Station, TX.
Chen, Z. X. , 1989, “Transient Flow of Slightly Compressible Fluids Through Double-Porosity, Double-Permeability Systems-a State-of-the-Art Review,” Transp. Porous Media, 4(2), pp. 147–184. [CrossRef]
Haggerty, R. , and Gorelick, S. M. , 1995, “ Multiple-Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media With Pore-Scale Heterogeneity,” Water Resour. Res., 31(10), pp. 2383–2400. [CrossRef]
Bowen, R. M. , 1976, “Theory of Mixtures,” Continuum Physics, A. C. Eringen , ed., Vol. III, Academic Press, New York. [CrossRef]
Pekař, M. , and Samohýl, I. , 2014, The Thermodynamics of Linear Fluids and Fluid Mixtures, Springer, Cham, Switzerland. [CrossRef]
de Boer, R. , 2012, Theory of Porous Media: Highlights in Historical Development and Current State, Springer Science & Business Media, New York.
Atkin, R. J. , and Craine, R. E. , 1976, “Continuum Theories of Mixtures: Basic Theory and Historical Development,” Q. J. Mech. Appl. Math., 29(2), pp. 209–244. [CrossRef]
Ziegler, H. , 1983, An Introduction to Thermomechanics, North Holland Publishing Company, Amsterdam, The Netherlands.
Ziegler, H. , and Wehrli, C. , 1987, “The Derivation of Constitutive Relations From the Free Energy and the Dissipation Function,” Adv. Appl. Mech., 25, pp. 183–238. [CrossRef]
Srinivasa, A. R. , and Srinivasan, S. M. , 2009, Inelasticity of Materials: An Engineering Approach and a Practical Guide, Vol. 80, World Scientific Publishing, Singapore. [CrossRef]
Rajagopal, K. R. , and Srinivasa, A. R. , 2001, “Modeling Anisotropic Fluids Within the Framework of Bodies With Multiple Natural Configurations,” J. Non-Newtonian Fluid Mech., 99(2–3), pp. 109–124. [CrossRef]
Xu, C. , Mudunuru, M. K. , and Nakshatrala, K. B. , 2016, “Material Degradation Due to Moisture and Temperature. part 1: Mathematical Model, Analysis, and Analytical Solutions,” Continuum Mech. Thermodyn., 28(6), pp. 1847–1885. [CrossRef]
Karra, S. , 2013, “Modeling the Diffusion of a Fluid Through Viscoelastic Polyimides,” Mech. Mater., 66, pp. 120–133. [CrossRef]
Truesdell, C. , 1991, A First Course in Rational Continuum Mechanics, Vol. I, Academic Press, New York.
Callen, H. B. , 1985, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York.
Batchelor, G. K. , 2000, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, UK. [CrossRef]
Brinkman, H. C. , 1947, “On the Permeability of the Media Consisting of Closely Packed Porous Particles,” Appl. Sci. Res., A1(1), pp. 81–86.
Rajagopal, K. R. , 2007, “On a Hierarchy of Approximate Models for Flows of Incompressible Fluids Through Porous Solids,” Math. Models Methods Appl. Sci., 17(02), pp. 215–252. [CrossRef]
Joodat, S. H. S. , Nakshatrala, K. B. , and Ballarini, R. , 2018, “Modeling Flow in Porous Media With Double Porosity/Permeability: A Stabilized Mixed Formulation, Error Analysis, and Numerical Solutions,” Comput. Methods Appl. Mech. Eng., 337(1), pp. 632–676. [CrossRef]
Shabouei, M. , and Nakshatrala, K. B. , 2016, “ Mechanics-Based Solution Verification for Porous Media Models,” Commun. Comput. Phys., 20(05), pp. 1127–1162. [CrossRef]
Love, A. E. H. , 1920, A Treatise on the Mathematical Theory of Elasticity, 3rd ed., Cambridge University Press, New York.
Sadd, M. H. , 2009, Elasticity: Theory, Applications, and Numerics, Academic Press, Burlington, MA.
Gilbarg, D. , and Trudinger, N. S. , 2001, Elliptic Partial Differential Equations of Second Order, Springer, New York.
Lighthill, M. J. , 1958, An Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press, Cambridge, UK. [CrossRef]
Tricomi, F. G. , 1957, Integral Equations, Interscience Publishers, New York.
Stackgold, I. , 1998, Green's Functions and Boundary Value Problems. Wiley, Interscience, New York.
Atkinson, K. E. , 1997, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK. [CrossRef]
Polyanin, A. D. , and Manzhirov, A. V. , 2008, Handbook of Integral Equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL. [CrossRef]
Dickenson, T. C. , 1997, Filters and Filtration Handbook, 4th ed., Elsevier, New York.
Bowman, F. , 2010, Introduction to Bessel Functions, Dover Publications, Mineola, NY.

Figures

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. 1

Conceptualization of a synthetic medium with two pore-scales

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.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In