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Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

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Abstract

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

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Abbreviations

\(C,C_{1} ,C_{2} ,D\) :

Integration constants

\(f\left( {\varvec{x}} \right),g\left( {\varvec{x}} \right)\) :

Unknown functions of space

\(h\) :

Fracture aperture (m)

\(k\) :

Matrix permeability (m2)

\(L\) :

Fracture length (m)

\(N\left( {\varvec{x}} \right)\) :

Standard shape function

\({\mathbf{n}}_{{{\Gamma }_{{\text{f}}} }}\) :

Unit vector normal to fracture (–)

\(p\) :

Fluid pressure (Pa)

\(\overline{p}_{I}\) :

Standard degree of freedom for pressure (Pa)

\(\tilde{p}_{I}\) :

Enriched degree of freedom for pressure (Pa)

\(R^{ + }\) :

Reynolds number for the wall with maximum normal velocity (–)

\({\text{Re}}\) :

Reynolds number for the fluid flow inside fracture (–)

\({\mathbf{t}}_{{{\Gamma }_{{\text{f}}} }}\) :

Unit vector tangential to fracture (–)

\({\mathbf{v}}\) :

Fluid velocity vector (m/s)

\(\overline{v}\) :

Mean velocity inside the fracture (m/s)

\(v_{{\text{C}}}\) :

Mean velocity based on cubic law (m/s)

\(\alpha\) :

Ratio of fracture aperture to square root of matrix permeability (–)

\(\alpha^{ + }\) :

Non-dimensional number indicating ratio of normal velocity on one fracture wall to the other (–)

\({\Gamma }\) :

Symbol indicating the boundary or interface

\(\gamma\) :

Non-dimensional coefficient for Beavers–Joseph condition (–)

\(\zeta\) :

Normalized length (–)

\(\lambda\) :

Normalized width (–)

\(\mu\) :

Fluid viscosity (Pa.s)

\(\rho\) :

Fluid density (kg/m3)

\({\Phi }\) :

Correction factor for cubic law (–)

\(\varphi \left( {\varvec{x}} \right)\) :

Level-set function

\(\phi\) :

Porosity (–)

\(\psi \left( {\varvec{x}} \right)\) :

Enrichment ridge function

\({\Omega }\) :

Symbol indicating domain of fracture or matrix

\({\mathcal{N}}\) :

Set of all nodal points

\({\tilde{\mathcal{N}}}\) :

Set enriched nodal points

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  1. Department of Civil Engineering, Center of Excellence in Structures and Earthquake Engineering, Sharif University of Technology, P.O. Box, Tehran, 11365-9313, Iran

    N. Hosseini & A. R. Khoei

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  1. N. Hosseini
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  2. A. R. Khoei
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Hosseini, N., Khoei, A.R. Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture. Transp Porous Med 139, 109–129 (2021). https://doi.org/10.1007/s11242-021-01648-5

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