Skip to main content
SpringerLink
Log in

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

  • Published:
Computing and Visualization in Science

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

\(a\) :

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 \((\mathrm{J\,kg^{-1}})\)

\(\alpha \) :

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

References

  1. Angot, P., Boyer, F., Hubert, F.: Asymptotic and numerical modelling of flow in fractured porous media. ESAIM: M2AN. Math. Model. Numer. Anal. 43, 239–275 (2009)

    MATH  MathSciNet  Google Scholar 

  2. Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA (1993)

    MATH  Google Scholar 

  3. Bastian, P., Birken, K., Johannsen, K., Lang, S., Reichenberger, V., Wieners, C., Wittum, G., Wrobel, C.: Parallel solution of partial differential equations with adaptive multigrid methods on unstructured grids. In: Jäger, W., Krause, E. (eds.) High Performance Computing in Science and Engineering, pp. 506–519. Springer, Berlin (2000)

    Google Scholar 

  4. Bear, J.: Dynamics of Fluid in Porous Media. Dover, New York (1972)

    Google Scholar 

  5. Bear, J.: On the aquifer’s integrated balance equations. Adv. Water Resour. 1(1), 15–23 (1977)

    Article  MathSciNet  Google Scholar 

  6. Bear, J.: Hydraulics of Groundwater. Dover, Mineola (1979)

    Google Scholar 

  7. Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic, Dordrecht, Boston, London (1990)

    Book  Google Scholar 

  8. Bear, J., Tsang, C.F., deMarsily, G.: Flow and Contaminant Transport in Fractured Rocks. Academic, New York (1993)

    Google Scholar 

  9. Bennethum, L.S., Giorgi, T.: Generlized Forchheimer equation for two-phase flow based on hybrid mixture theory. Transp. Porous Media 26, 261–275 (1997)

    Article  Google Scholar 

  10. Bennethum, L.S., Murad, M.A., Cushman, J.H.: Macroscale thermodynamics and the chemical potential for swelling porous media. Transp. Porous Media 39, 187–225 (2000)

    Article  MathSciNet  Google Scholar 

  11. Brenner, H.: The diffusion model of longitudianl mixing in beds of finite length: numerical values. Chem. Eng. Sci. 17, 229–243 (1962)

    Article  Google Scholar 

  12. Cai, Z.: On the finite volume element method. Numer. Math. 58(1), 713–735 (1990)

    Article  Google Scholar 

  13. Cermelli, P., Fried, E., Sellers, S.: Configurational stress, yield and flow in rate-independent plasticity. Proc. R. Soc. Lond. A 457, 1447–1467 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. D’Angelo, C., Scotti, A.: A mixed finite element method for darcy flow in fractured porous media with non-matching grids. ESAIM: Math. Model. Numer. Anal. 46(02), 465–489 (2012). doi:10.1051/m2an/2011148

    MATH  Google Scholar 

  15. Diersch, H.J.G., Kolditz, O.: Variable-density flow and transport in porous media: approaches and challenges. In: Wasy Software FEFLOW—Finite Element Subsurface flow and Transport Simulation System, vol. 2. Wasy GmbH (2005)

  16. Eringen, A.C.: Mechanics of Continua. Wiley, New York (1980)

    Google Scholar 

  17. Fein, E.: Ein Programmpaket zur Modellierung von Dichteströmungen. GRS, Braunschweig 139 (1999). In German

  18. Frih, N., Roberts, J., Saada, A.: Modeling fractures as interfaces: a model for Forchheimer fractures. Comput. Geosci. 12(1), 91–104 (2008). doi:10.1007/s10596-007-9062-x

    Article  MATH  MathSciNet  Google Scholar 

  19. Frolkovic, P., De Schepper, H.: Numerical modelling of convection dominated transport coupled with density driven flow in porous media. Adv. Water Resour. 24(1), 63–72 (2000)

    Article  Google Scholar 

  20. Frolkovič, P.: Consistent velocity approximation for density driven flow and transport. In: Van Keer, R., et al. (eds.) Advanced Computational Methods in Engineering, Part 2: Contributed papers, pp. 603–611. Shaker Publishing, Maastricht (1998)

  21. Frolkovič, P.: Maximum principle and local mass balance for numerical solutions of transport equation coupled with variable density flow. Acta Math. Univ. Comenianae 1(68), 137–157 (1998)

    Google Scholar 

  22. Frolkovič, P., Knabner, P.: Consistent velocity approximations in finite element or volume discretizations of density driven flow. In: Aldama, A.A., et al. (eds.) Computational Methods in Water Resources XI, pp. 93–100. Computational Mechanics Publication, Southhampten (1996)

    Google Scholar 

  23. Graf, T., Degener, L.: Grid convergence of variable-density flow simulations in discretely fractured porous media. Adv. Water Resour. 34, 760–769 (2011)

    Article  Google Scholar 

  24. Graf, T., Therrier, R.: Variable-density groundwater flow and solute transport in porous media containing nonuniform discrete fractures. Adv. Water Resour. 28, 1351–1367 (2005)

    Article  Google Scholar 

  25. Graf, T., Therrier, R.: Variable-density groundwater flow and solute transport in irregular 2D fracture networks. Adv. Water Resour. 30, 455–468 (2007)

    Article  Google Scholar 

  26. Graf, T., Therrier, R.: Stable-unstable flow of geothermal fluids in fractured rock. Geofluids 9, 138–152 (2009)

    Article  Google Scholar 

  27. Grillo, A., Lampe, M., Logashenko, D., Stichel, S., Wittum, G.: Simulation of salinity- and thermohaline-driven flow in fractured porous media. J. Porous Media 15(5), 439–458 (2012). doi:10.1615/JPorMedia.v15.i5.40

    Article  Google Scholar 

  28. Grillo, A., Lampe, M., Wittum, G.: Three-dimensional simulation of the thermohaline-driven buoyancy of a brine parcel. Comput. Vis. Sci. 13(6), 287–297 (2010)

    Article  MATH  Google Scholar 

  29. Grillo, A., Lampe, M., Wittum, G.: Modelling and simulation of temperature-density-driven flow and thermodiffusion in porous media. J. Porous Media 14(8), 671–690 (2011)

    Article  Google Scholar 

  30. Grillo, A., Logashenko, D., Stichel, S., Wittum, G.: Simulation of density-driven flow in fractured porous media. Adv. Water Resour. 33(12), 1494–1507 (2010). doi:10.1016/j.advwatres.2010.08.004

    Article  Google Scholar 

  31. Grisak, G., Pickens, J.F.: Solute transport through fractured media. Water Resour. Res. 16(4), 719–730 (1980)

    Article  Google Scholar 

  32. Grisak, G., Pickens, J.F.: An analytic solution for solute transport through fractured media with matrix diffusion. J. Hydrol. 52, 47–57 (1981)

    Article  Google Scholar 

  33. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New-York (1994)

    Book  MATH  Google Scholar 

  34. Hassanizadeh, S.M.: Derivation of basic equations of mass transport in porous media, part 1. Generalized Darcy’s and Fick’s laws. Adv. Water Resour. 9, 196–206 (1986)

    Article  Google Scholar 

  35. Hassanizadeh, S.M.: Derivation of basic equations of mass transport in porous media, part 2. Generalized Darcy’s and Fick’s laws. Adv. Water Resour. 9, 207–222 (1986)

    Article  Google Scholar 

  36. Hassanizadeh, S.M., Gray, W.G.: Reply to comments by Barak on “High velocity flow in porous media”. Transp. Porous Media 25(3), 319–321 (1988)

    Google Scholar 

  37. Hassanizadeh, S.M., Gray, W.G.: Derivation of conditions describing transport across zones of reduced dynamics within multiphase systems. Water Resour. Res. 25, 529–539 (1989)

    Article  Google Scholar 

  38. Henry, H.R.: Effects of dispersion on salt encroachment in coastal aquifers. In: Sea water in coastal aquifers, pp. 70–84. USGS Water Supply Paper 1613-c (1964)

  39. Hoteit, H., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31(6), 891–905 (2008). doi:10.1016/j.advwatres.2008.02.004

    Article  Google Scholar 

  40. Karimian, S.M., Schneider, G.E.: Pressure-based control-volume finite element method for flow at all speeds. AIAA J. 33(9), 1611–1618 (1995)

    Article  MATH  Google Scholar 

  41. Kempers, L.J.T.M., Haas, H.: The dispersion zone between fluids with different density and viscosity in a heterogeneous porous medium. J. Fluid Mech. 267, 299–324 (1994)

    Article  Google Scholar 

  42. Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005). doi:10.1137/S1064827503429363

  43. Martinez-Landa, L., Carrera, J.: A methodology to interpret cross-hole tests in a granite block. J. Hydrogeol. 325(1–4), 222–240 (2006)

    Google Scholar 

  44. Murphy, H.D.: Convective instabilities in vertical fractures and faults. J. Geophys. Res. 84, 6121–6130 (1979)

    Article  Google Scholar 

  45. Musuuza, J.L., Radu, F.A., Attinger, S.: An extended stability criterion for density-driven flows in homogeneous porous media. Adv. Water Resour. 32, 796–808 (2009)

    Article  Google Scholar 

  46. Musuuza, J.L., Radu, F.A., Attinger, S.: The effect of dispersion on the stability of density-driven flows in saturated homogeneous porous media. Adv. Water Resour. 34, 417–432 (2011)

    Article  Google Scholar 

  47. Musuuza, J.L., Radu, F.A., Attinger, S.: The stability of density-driven flows in saturated homogeneous porous media. Adv. Water Resour. 34, 1464–1482 (2011)

    Article  Google Scholar 

  48. Ogden, R.W.: Non-linear Elastic Deformations. Wiley, New York (1984)

    Google Scholar 

  49. Oldenburg, C.M., Pruess, K.: Layered thermohaline convection in hypersaline geothermal systems. Transp. Porous Media 33, 29–63 (1998)

    Article  Google Scholar 

  50. Reichenberger, V., Jakobs, H., Bastian, P., Helmig, R.: A mixed-dimensional finite volume method for two-phase flow in fractured porous media. Adv. Water Resour. 29(7), 1020–1036 (2006). doi:10.1016/j.advwatres.2005.09.001

    Article  Google Scholar 

  51. Reiter, S., Logashenko, D., Stichel, S., Wittum, G., Grillo, A.: Models and simulations of variable-density flow in fractured porous media. Int. J. Comput. Sci. Eng. (2012) (in press)

  52. Sharp, J.M.J., Shi, M.: Heterogeneity effects on possible salinity-driven free convection in low-permeability strata. Geofluids 34, 263–274 (2009)

    Article  Google Scholar 

  53. Shikaze, S.G., Sudicky, E.A., Schwartz, F.W.: Density-dependent solute transport in discretely-fractured geologic media: is prediction possible? J. Contam. Hydrogeol. 34, 273–291 (1998)

    Google Scholar 

  54. Simpson, M., Clement, T.: Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models. Water Resour. Res. 40, W01504 (2004). doi:10.1029/2003WR002199

  55. Sorek, S., Borisov, V., Yakirevich, A.: A two-dimensional areal model for density dependent flow regime. Transp. Porous Media 43, 87–105 (2001)

    Article  Google Scholar 

  56. Stichel, S., Logashenko, D., Grillo, A., Reiter, S., Lampe, M., Wittum, G.: Numerical methods for flow in fractured porous media. In: Delgado, J. (ed.) Heat and Mass Transfer in Porous Media, Advanced Structured Materials, vol. 13, pp. 83–113. Springer, Berlin, Heidelberg (2012). doi:10.1007/978-3-642-21966-5_4

  57. Tang, D., Frind, E., Sudicky, E.: Contaminant transport in fractured porous media: analytical solution for a single fracture. Water Resour. Res. 17(3), 555–564 (1981)

    Article  Google Scholar 

  58. Thauvin, F., Mohanty, K.K.: Network modeling of non-darcy flow through porous media. Transp. Porous Media 31, 19–37 (1998)

    Article  Google Scholar 

  59. Whitaker, S.: The Method of Volume Averaging—Theory and Applications of Transport in Porous Media, 1 edn. Kluwer Academic, Dordrecht, The Netherlands (1999)

  60. Zeng, Z., Grigg, R.: A criterion for non-darcy flow in porous media. Transp. Porous Media 63, 57–69 (2006). doi:10.1007/s11242-005-2720-3

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 , Turin, Italy

    Alfio Grillo

  2. G-CSC Goethe-Universität Frankfurt am Main, Kettenhofweg 139, 60325 , Frankfurt am Main, Germany

    Dmitriy Logashenko, Sabine Stichel & Gabriel Wittum

Authors
  1. Alfio Grillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitriy Logashenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sabine Stichel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gabriel Wittum
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfio Grillo.

Additional information

Communicated by Peter Frolkovič.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grillo, A., Logashenko, D., Stichel, S. et al. Forchheimer’s correction in modelling flow and transport in fractured porous media. Comput. Visual Sci. 15, 169–190 (2012). https://doi.org/10.1007/s00791-013-0208-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-013-0208-1

Keywords

Search

Navigation