Computing and Visualization in Science

, Volume 15, Issue 4, pp 169–190 | Cite as

Forchheimer’s correction in modelling flow and transport in fractured porous media

  • Alfio Grillo
  • Dmitriy Logashenko
  • Sabine Stichel
  • Gabriel Wittum
Article

Abstract

The scope of this manuscript is to investigate the role of the Forchheimer correction in the description of variable-density flow in fractured porous media. A fractured porous medium, which shall be also referred to as “the embedding medium”, represents a flow region that is made macroscopically heterogeneous by the presence of fractures. Fractures are assumed to be filled with a porous medium characterized by flow properties that differ appreciably from those of the embedding medium. The fluid, which is free to move in the pore space of the entire flow region, is a mixture of water and brine. Flow is assumed to be a consequence of the variability of the fluid mass density in response to the generally nonuniform distribution of brine, which is subject to diffusion and convection. The fractures are assumed to be thin in comparison with the characteristic sizes of the embedding medium. Within this framework, some benchmark problems are solved by adopting two approaches: (i) the fractures are treated as thin but \(\mathrm{d}\)-dimensional flow subregions, with \(d\) being the geometric dimension of the embedding medium; (ii) the fractures are regarded as \((\mathrm{d}-1)\)-dimensional manifolds. In the first approach, the equations of variable-density flow are written in the same, \(d\)-dimensional form both in the fractures and in the embedding medium. In the second approach, instead, new equations are obtained by averaging the \(\mathrm{d}\)-dimensional ones over the fracture width. The reliability of the second approach is discussed by comparing the results of the \((\mathrm{d}-1)\)-dimensional numerical simulations of the selected benchmark problems with those obtained by using the \(d\)-dimensional approach. Moreover, the deviations of the results determined by accounting for the Forchheimer correction to flow velocity are compared with those predicted by Darcy’s law.

Keywords

Porous media Density-driven flow Dissipation  Fractures Forchheimer’s correction 

List of symbols

Latin letters

\(a\)

Phenomenological coefficient [cf. (10) and (11)] \((\mathrm{kg}\,\mathrm{s}^{-1}\,\mathrm{m}^{-3})\)

\(A\)

Phenomenological coefficient [cf. (15)] \((\mathrm{m}^{-1}\,\mathrm{s})\)

\(A_{\alpha }\)

Phenomenological coefficient associated with the \(\alpha \hbox {th}\) subdomain \((\alpha = f,m)\)\((\mathrm{m}^{-1}\,\mathrm{s})\)

\({\varvec{A}}\)

Arbitrary differentiable vector field

\({\varvec{A}}^{(k)}\)

Restriction of \({\varvec{A}}\) to \({\fancyscript{S}}^{(k)}\)

\(A_{n}^{(k)}\)

Normal component of \({\varvec{A}}\) restricted to \({\fancyscript{S}}^{(k)}~(k=1,2)\)

\({\varvec{A}}^{\sigma }\)

Projection of \({\varvec{A}}\) onto the fracture mean plane

\(b\)

Phenomenological coefficient [cf. (10) and (11)] \((\mathrm{kg}\,\mathrm{m}^{-4})\)

\(\mathbf{B}\)

Generic tensor field

\(\tilde{\mathbf{B}}\)

Fluctuation of \(\mathbf{B}\)

\({\fancyscript{B}}\)

Band-shaped lateral boundary of a fracture

\(\hat{{\fancyscript{B}}}\)

Degenerated band-shaped lateral boundary of a fracture

\(B({\varvec{x}})\)

Balls surrounding the grid points, cf. Sect. 5.1

\(B_i\)

Parts of \(B({\varvec{x}})\), cf. Sect. 5.1

\(c_{f}\)

Concentration in the fractures \((\mathrm{kg}\,\mathrm{m}^{-3})\)

\(\bar{c}_{f}\)

Average-along-the-vertical of \(c_{f}\)\((\mathrm{kg}\,\mathrm{m}^{-3})\)

\(\tilde{c}_{f}\)

Fluctuation of \(c_{f}\)\((\mathrm{kg}\,\mathrm{m}^{-3})\)

\(c_{m}\)

Concentration in the embedding porous medium \((\mathrm{kg}\,\mathrm{m}^{-3})\)

\(\hat{c}_{m}\)

Concentration in the embedding porous medium in the \((d-1)\)-dimensional model \((\mathrm{kg}\,\mathrm{m}^{-3})\)

\(d\)

Dimensionality \((d=2,3)\)

\(\mathbf{D}\)

Effective diffusion tensor \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)

\(\mathbf{D}_{d}\)

Dispersion tensor \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)

\(D_{\!f}\)

Scalar diffusion coefficient of the brine in the fractures \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)

\(D_{m}\)

Scalar diffusion coefficient of the brine in the embedding porous medium \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)

\(\mathbf{D}_{T}\)

Diffusion–dispersion tensor \((\mathrm{m}^{2}\,\mathrm{s}^{-1})\)

\(e\)

Element of the triangulation of \(\varOmega ,\,{\varvec{T}}_{\varOmega }\)

\(f\)

Index associated with fractures

\(F\)

Forchheimer number \((-)\)

\({\mathfrak {F}}\)

Fracture or fracture network

\({\varvec{g}}\)

Gravity acceleration vector \((\mathrm{m}\,\mathrm{s}^{-2})\)

\({\varvec{g}}^{\sigma }\)

Tangential component of \({\varvec{g}}\)\((\mathrm{m}\,\mathrm{s}^{-2})\)

\({\varvec{J}}\)

Diffusive–dispersive mass flux of the brine \((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\({\varvec{J}}_{\!f}\)

Diffusive–dispersive mass flux of the brine in the fractures \((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\(J_{\!fn}^{(k)}\)

Normal component of \({\varvec{J}}_{\!f}\) computed on \({\fancyscript{S}}^{(k)}~(k =1,2)\)\((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\(\hat{J}_{\!fn}^{(k)}\)

Representation of \(J_{\!fn}^{(k)}\) in the \((d-1)\)-dimensional model \((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\({\varvec{J}}_{\!f}^{\sigma }\)

Tangential component of \({\varvec{J}}_{\!f}\)\((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\({\varvec{J}}_{m}\)

Diffusive–dispersive mass flux of the brine in the embedding porous medium \((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\({\varvec{J}}_{m} ^{\sigma }\)

Tangential component of \({\varvec{J}}_{m}\)\((\mathrm{kg}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1})\)

\(\hat{{\varvec{J}}}_{m}\)

Diffusive–dispersive mass flux of the brine in the embedding porous medium in the \((d-1)\)-dimensional model \((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\(K\)

Permeability \((\mathrm{m}^{2})\)

\(K_{f}\)

Permeability of the porous medium inside the fractures \((\mathrm{m}^{2})\)

\(K_{m}\)

Permeability of the embedding porous medium \((\mathrm{m^{2}})\)

\(L_{\ell },\,L_{t}\)

Longitudinal and transversal dispersion lengths \((\mathrm{m})\)

\(m\)

Measure of a set (e.g., in the sense of Lebesgue)

\({\mathfrak {M}}\)

Region of space occupied by the embedding porous medium

\(\hat{{\mathfrak {M}}}\)

Representation of \({\mathfrak {M}}\) in the \((d-1)\)-dimensional model

\(N\)

Number of degrees of freedom of grid functions defined in the enclosing medium

\(N_f\)

Number of the degrees of freedom of grid functions defined in the fractures

\({\fancyscript{N}}^e,\,{\fancyscript{N}}^s\)

Sets of indices of corner grid points for \(e \in {\varvec{T}}_\varOmega \) or \(s \in {\varvec{T}}_{\fancyscript{S}}\), resp

\({\fancyscript{N}}^e_i,\,{\fancyscript{N}}^s_i\)

\({\fancyscript{N}}^\cdot _i := {\fancyscript{N}}^\cdot \setminus \{ i \}\)

\(\hat{{\fancyscript{N}}}_i\)

Set of indices of degrees of freedom of the grid functions defined in the enclosing medium, associated with the fracture degree of freedom with index \(i\), and vice versa

\({\varvec{n}}^e_{ij}\)

Unit normal to \(\gamma ^e_{ij}\), pointing from \(V_i\) to \(V_j\)

\({\varvec{n}}^s_{ij}\)

Unit normal to \(\sigma ^s_{ij}\), pointing from \(S_i\) to \(S_j\)

\({\varvec{n}}^{(k)}\)

Unit vector normal to \({\fancyscript{S}}^{(k)}\) (in the \(d\)-dimensional model) and to \(\hat{{\fancyscript{S}}}^{(k)}\) (in the \((d-1)\)-dimensional model), \(k=1,2\)

\(p\)

Pressure \((\mathrm{N\,m^{-2}})\)

\(p_{\!f}\)

Pressure in the fractures \((\mathrm{N\,m^{-2}})\)

\(\bar{p}_{\!f}\)

Average-along-the-vertical of \(p_{\!f}\)\((\mathrm{N\,m^{-2}})\)

\(p_{m}\)

Pressure in the embedding porous medium \((\mathrm{N\,m^{-2}})\)

\(\hat{p}_{m}\)

Pressure in the embedding porous medium in the \((d-1)\)-dimensional model \((\mathrm{N\,m^{-2}})\)

\(\hat{p}^n_{mi}\)

Degrees of freedom of grid functions approximating \(\hat{p}_m\)\((\mathrm{N\,m^{-2}})\)

\(\hat{p}^n_{mh}\)

Grid functions approximating \(\hat{p}_m\)\((\mathrm{N\,m^{-2}})\)

\(\bar{p}^n_{fi}\)

Degrees of freedom of grid functions approximating \(\bar{p}_f\)\((\mathrm{N\,m^{-2}})\)

\(\bar{p}^n_{fh}\)

Grid functions approximating \(\bar{p}_f\)\((\mathrm{N\,m^{-2}})\)

\(\hat{p}_{m}^{(k)}\)

Restriction of \(\hat{p}_{m}\) on \(\hat{{\fancyscript{S}}}^{(k)}~(k=1,2)\)\((\mathrm{N\,m^{-2}})\)

\(P_{\alpha n}^{(k)}\)

Normal component of the total brine mass flux computed on the \(\alpha \)th side \((\alpha = f,m)\) of \({\fancyscript{S}}^{(k)}~(k=1,2)\)\((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\(\hat{P}_{\alpha n}^{(k)}\)

Representation of \(P_{\alpha n}^{(k)}\) in the \((d-1)\)-dimensional model \((\alpha = f,m,\,k = 1,2)\)\((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\({\varvec{q}}\)

Specific discharge \((\mathrm{m\,s^{-1}})\)

\(q\)

Euclidean norm of \({\varvec{q}}\)\((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{D}\)

Specific discharge computed according to Darcy’s law \((\mathrm{m\,s^{-1}})\)

\(q_{D}\)

Euclidean norm of \({\varvec{q}}_{D}\)\((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{\!f}\)

Specific discharge in the fractures \((\mathrm{m\,s^{-1}})\)

\(q_{\!f}\)

Euclidean norm of \({\varvec{q}}_{\!f}\)\((\mathrm{m\,s^{-1}})\)

\(q_{\!fn}^{(k)}\)

Normal component of \({\varvec{q}}_{\!f}\) computed on \({\fancyscript{S}}^{(k)}~(k\!=\!1,2)\)\((\mathrm{m\,s^{-1}})\)

\(\hat{q}_{\!fn}^{(k)}\)

Representation of \(q_{\!fn}^{(k)}\) in the \((d-1)\)-dimensional model \((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{\!fD}\)

Specific discharge computed according to Darcy’s law \((\mathrm{m\,s^{-1}})\)

\(q_{\!fD}\)

Euclidean norm of the specific discharge computed according to Darcy’s law \((\mathrm{m\,s^{-1}})\)

\(\hat{{\varvec{q}}}_{\!fD}\)

Representation of \({\varvec{q}}_{\!fD}\) in the \((d-1)\)-dimensional model \((\mathrm{m\,s^{-1}})\)

\(\hat{q}_{\!fD}\)

Representation of \(q_{\!fD}\) in the \((d-1)\)-dimensional model \((\mathrm{m\,s^{-1}})\)

\(\hat{q}_{\!fD}^{(k)}\)

Restriction of \(\hat{q}_{\!fD}\) to \(\hat{{\fancyscript{S}}}^{(k)}~(k=1,2)\) [cf. (74)] \((\mathrm{m\,s^{-1}})\)

\(\hat{q}_{\!fDn}^{(k)}\)

In the \((d-1)\)-dimensional model, approximation of the specific discharge normal to \(\hat{{\fancyscript{S}}}^{(k)}~(k=1,2)\), computed according to Darcy’s law \((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{\!f}^{\sigma }\)

Tangential component of \({\varvec{q}}_{\!f}\)\((\mathrm{m\,s^{-1}})\)

\(\tilde{{\varvec{q}}}_{\!f}^{\sigma }\)

Fluctuation of \({\varvec{q}}_{\!f}^{\sigma }\)\((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{\!fD}^{\sigma }\)

Tangential component of \({\varvec{q}}_{\!fD}\)\((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{\ell }\)

\((\equiv \phi _{\ell }{\varvec{v}}_{\ell s})\) specific discharge of the fluid phase \((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{m}\)

Specific discharge in the embedding porous medium \((\mathrm{m\,s^{-1}})\)

\({\varvec{q}}_{m}^{\sigma }\)

Tangential component of \({\varvec{q}}_{m}\)\((\mathrm{m\,s^{-1}})\)

\(\hat{{\varvec{q}}}_{m}\)

Specific discharge in the embedding porous medium in the \((d-1)\)-dimensional model \((\mathrm{m\,s^{-1}})\)

\(\hat{{\varvec{q}}}_{mD}\)

Specific discharge in the embedding porous medium as computed in the \((d-1)\)-dimensional model according to Darcy’s law \((\mathrm{m\,s^{-1}})\)

\(\hat{q}_{mD}\)

Euclidean norm of \(\hat{{\varvec{q}}}_{mD}\)\((\mathrm{m\,s^{-1}})\)

\(Q_{\alpha n}^{(k)}\)

Normal component of the mass flux of the fluid phase computed on the \(\alpha \)th side \((\alpha = f,m)\) of \({\fancyscript{S}}^{(k)}~(k=1,2)\)\((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\(\hat{Q}_{\alpha n}^{(k)}\)

Representation of \(Q_{\alpha n}^{(k)}\) in the \((d-1)\)-dimensional model \((\alpha = f,m,\,k = 1,2)\)\((\mathrm{kg\,m^{-2}\,s^{-1}})\)

\(s\)

An element of the triangulation \({\varvec{T}}_{{\fancyscript{S}}}\)

\({\fancyscript{S}}\)

Fracture mean plane

\({\fancyscript{S}}_h\)

Set of all grid points in the fractures

\(S_i\)

Control volume in the grid, dual to \({\varvec{T}}_{\fancyscript{S}}\), cf. Sect. 5.2

\({\fancyscript{S}}^{(k)}\)

Fracture-medium interface \((k=1,2)\)

\(\hat{{\fancyscript{S}}}^{(k)}\)

Representation of \({\fancyscript{S}}^{(k)}\) in the \((d-1)\)-dimensional model \((k=1,2)\)

\(t\)

Time \((\mathrm{s})\)

\(t^n\)

A point in the grid covering the time interval (s)

\(\mathbf{T}\)

Anisotropic tortuosity tensor \((-)\)

\(\mathbf{T}_{\alpha }\)

Anisotropic tortuosity tensor in the \(\alpha \hbox {th}\) subdomain \((\alpha = f,m)\)\((-)\)

\({\varvec{T}}_\varOmega \)

Triangulation of the domain \(\varOmega \)

\({\varvec{T}}_{\fancyscript{S}}\)

Triangulation of the fractures

\({\varvec{u}}\)

Velocity of brine relative to the velocity of the center of mass of the fluid phase \((\mathrm{m\,s^{-1}})\)

\({\varvec{v}}\)

Velocity of the center of mass of the fluid phase \((\mathrm{m\,s^{-1}})\)

\(V_i\)

Control volume in the grid, dual to \({\varvec{T}}_\varOmega \), cf. Sect. 5.2

\(V_i^e\)

The part of \(V_i\) lying in \(e \in {\varvec{T}}_\varOmega \)

\({\varvec{w}}\)

Velocity of the center of mass of the fluid phase relative to the solid phase \((\mathrm{m\,s^{-1}})\)

\({\varvec{x}}_i\)

Grid points from \(\varOmega _h\), indexed as introduced in Sect. 5.1\((\mathrm{m})\)

\(y\)

Relative chemical potential of the brine \((\mathrm{J\,kg^{-1}})\)

Greek letters

\(\alpha \)

Index \((\alpha = f,m )\)

\(\beta \)

Phenomenological parameter to be determined experimentally \((\mathrm{m}^{-1})\)

\(\gamma ^e_{ij}\)

A part of \(\partial V_i\), cf. Sect. 5.2

\(\epsilon \)

Fracture width \((\mathrm{m})\)

\(\varvec{\zeta }_{\alpha }\)

Vorticity in the \(\alpha \)th subdomain \((\alpha = f,m)\)\((\mathrm{s}^{-1})\)

\(\varvec{\zeta }_{\alpha D}\)

Vorticity in the \(\alpha \)th subdomain \((\alpha = f,m)\) computed according to Darcy’s law \((\mathrm{s}^{-1})\)

\({\varvec{\lambda }},\,{\varvec{\lambda }}_{r}\)

Dissipative forces (per unit mass) representing an exchange of momentum \((\mathrm{m\,s^{-2}})\)

\(\mu \)

Viscosity of the fluid phase

\(\phi \)

Volumetric fraction of the fluid phase \((-)\)

\(\phi _{f}\)

Volumetric fraction of the fluid phase in the fractures \((-)\)

\(\phi _{m}\)

Volumetric fraction of the fluid phase in the embedding medium \((-)\)

\(\varphi _{b}\)

Volumetric fraction of brine in the fluid phase \((-)\)

\(\varphi _{w}\)

Volumetric fraction of water in the fluid phase \((-)\)

\(\varrho \)

Mass density of the fluid phase \((\mathrm{kg\,m^{-3}})\)

\(\varrho _{b}\)

Brine mass density \((\mathrm{kg\,m^{-3}})\)

\(\varrho _{f}\)

Brine mass density in the fractures \((\mathrm{kg\,m^{-3}})\)

\(\varrho _{m}\)

Brine mass density in the embedding medium \((\mathrm{kg\,m^{-3}})\)

\(\varrho _{pb},\,\varrho _{pw}\)

True mass density of brine and water, respectively \((\mathrm{kg\,m^{-3}})\)

\(\varrho _{w}\)

Water mass density \((\mathrm{kg\,m^{-3}})\)

\(\sigma ^s_{ij}\)

A part of \(\partial S_i\) (analogous to \(\gamma ^e_{ij})\)

\(\tau ^n\)

Time step, \(\tau ^n = t^n - t^{n-1}\)\((\mathrm{s})\)

\(\omega \)

Brine mass fraction \((-)\)

\(\omega _{f}\)

Brine mass fraction in the fractures \((-)\)

\(\hat{\omega }^n_{fh}\)

Grid functions approximating \(\hat{\omega }_f\)\((-)\)

\(\omega _{m}\)

Brine mass fraction in the embedding medium \((-)\)

\(\hat{\omega }^n_{mh}\)

Grid functions approximating \(\hat{\omega }_m\)\((-)\)

\(\hat{\omega }^n_{mi}\)

Degrees of freedom of grid functions approximating \(\hat{\omega }_m\)\((-)\)

\(\varOmega \)

Region occupied by the porous media, including the fractures

\(\varOmega _h\)

Set of all grid points from \({\varvec{T}}_\varOmega \)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alfio Grillo
    • 1
  • Dmitriy Logashenko
    • 2
  • Sabine Stichel
    • 2
  • Gabriel Wittum
    • 2
  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly
  2. 2.G-CSC Goethe-Universität Frankfurt am MainFrankfurt am MainGermany

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