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Porous Medium

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Abstract

The processes of flow, mass and heat refer to extensive quantities (such as mass, momentum, energy and entropy), cf. Sect. 2.2.2, which are transported through a spatial domain of interest. This spatial domain is said to behave as a continuum which is occupied by matter for which a continuous distribution can be postulated. The matter may take a number of M aggregate forms or phases α, particularly: solid s, liquid l and gaseous g. It retains their continuity regardless how small volume elements the matter is subdivided in and interior material interfaces or surfaces exist. Any mathematical point we select can be assigned to matter as a physical point of given finite size.

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Notes

  1. 1.

    It denotes a balance statement in its basic conservation formulation.

  2. 2.

    It denotes a balance statement in which mass conservation is substituted.

  3. 3.

    The equivalence of the area- and volume-averaged fluxes is shown for the interface term A α(ρ ψ) of (3.71), cf. [229]. The volume-averaged flux describes the REV average in the form:

    $$\displaystyle{[\langle \rho \rangle _{\alpha }{\overline{\psi }}^{\alpha }(\boldsymbol{{v}}^{\alpha } -\boldsymbol{ W}) -\boldsymbol{ {j}}^{\alpha }] \cdot \boldsymbol{ {n}}^{\mathrm{TB}} = \frac{1} {\mathit{dV}}\Bigl (\int {_{{\mathit{dV}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}]dv\Bigr ) \cdot \boldsymbol{ {n}}^{\mathrm{TB}}}$$

    Let us assume that the interface has a thickness D, the volume integral may be written

    $$\displaystyle{ \frac{1} {\mathit{dV}}\int _{-D/2}^{D/2}\Bigl (\int {_{{ \mathit{dS}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}\Bigr )dl \approx \frac{D} {\mathit{dV}}\int {_{{\mathit{dS}}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}}$$

    where mean values are used to replace the line integral. With dS = dV∕D we find finally

    $$\displaystyle{[\langle \rho \rangle _{\alpha }{\overline{\psi }}^{\alpha }(\boldsymbol{{v}}^{\alpha } -\boldsymbol{ W}) -\boldsymbol{ {j}}^{\alpha }] \cdot \boldsymbol{ {n}}^{\mathrm{TB}} = \frac{1} {\mathit{dS}}\int {_{d{S}^{\mathrm{TB}}}\gamma }^{\alpha }[\rho \psi (\boldsymbol{v} -\boldsymbol{ w}) -\boldsymbol{ j}] \cdot \boldsymbol{ {n}}^{\mathrm{TB}}\mathit{da}}$$

    which corresponds to a α(ρ ψ) in (3.71).

  4. 4.

    They represent constitutive relations originally found from the observation that fluxes of extensive quantities (e.g., mass, heat, momentum) are produced by the nonuniform distribution of their state variables (e.g., concentration gradient, temperature gradient, velocity difference). Frequently, a simple proportionality between fluxes and gradients of state variables is postulated using a parameter taken to be a property of the material (e.g., diffusivity, conductivity, friction).

  5. 5.

    In index notation we derive (dropping phase indices for the sake of simplicity)

    $$\displaystyle{\begin{array}{rcl} \frac{\partial } {\partial x_{j}}\bigl [\varepsilon \tfrac{1} {2}( \frac{\partial v_{i}} {\partial x_{j}} + \frac{\partial v_{j}} {\partial x_{i}})\bigr ] & = & \tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl [\frac{\partial (\varepsilon v_{i})} {\partial x_{j}} + \frac{\partial (\varepsilon v_{j})} {\partial x_{i}} - v_{i} \frac{\partial \varepsilon } {\partial x_{j}} - v_{j} \frac{\partial \varepsilon } {\partial x_{i}}\bigr ] \\ & = & \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{i})} {\partial x_{j}\partial x_{j}} + \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{j})} {\partial x_{i}\partial x_{j}} -\tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl (v_{i} \frac{\partial \varepsilon } {\partial x_{j}}\bigr ) -\tfrac{1} {2} \frac{\partial } {\partial x_{j}}\bigl (v_{j} \frac{\partial \varepsilon } {\partial x_{i}}\bigr ) \\ & = & \tfrac{1} {2} \frac{{\partial }^{2}(\varepsilon v_{i})} {\partial x_{j}\partial x_{j}} + \tfrac{1} {2}\varepsilon \frac{{\partial }^{2}v_{j}} {\partial x_{i}\partial x_{j}} -\tfrac{1} {2}v_{i} \frac{{\partial }^{2}\varepsilon } {\partial x_{j}\partial x_{j}} -\tfrac{1} {2}\bigl ( \frac{\partial v_{i}} {\partial v_{j}} -\frac{\partial v_{j}} {\partial v_{i}}\bigr ) \frac{\partial \varepsilon } {\partial x_{j}} \end{array} }$$
  6. 6.

    In 3D Cartesian coordinates, with v 1, v 2 and v 3 denoting the velocity components in the x 1, x 2 and x 3 directions, respectively, and \(v =\Vert \boldsymbol{ {v}}^{fs}\Vert\), we obtain from (3.182), dropping phase indices for convenience

    $$\displaystyle{\begin{array}{rcl} D_{\mathrm{mech},11} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{1}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{L}v_{1}^{2} +\beta _{T}v_{2}^{2} +\beta _{T}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},22} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{2}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{T}v_{1}^{2} +\beta _{L}v_{2}^{2} +\beta _{T}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},33} & = & \beta _{T}v + (\beta _{L} -\beta _{T})\frac{v_{3}^{2}} {v} = \frac{1} {v}{\bigl (\beta _{T}v_{1}^{2} +\beta _{T}v_{2}^{2} +\beta _{L}v_{3}^{2}\bigr )} \\ D_{\mathrm{mech},12} & = & (\beta _{L} -\beta _{T})\frac{v_{1}v_{2}} {v} = D_{\mathrm{mech},21} \\ D_{\mathrm{mech},13} & = & (\beta _{L} -\beta _{T})\frac{v_{1}v_{3}} {v} = D_{\mathrm{mech},31} \\ D_{\mathrm{mech},23} & = & (\beta _{L} -\beta _{T})\frac{v_{2}v_{3}} {v} = D_{\mathrm{mech},32} \end{array} }$$
  7. 7.

    Using calculus manipulations the material derivative of E f with respect to the density ρ f can be alternatively developed for the \(\tfrac{{D{}^{f}\rho }^{f}} {\mathit{Dt}}\) term:

    $$\displaystyle{{\frac{\varepsilon _{f}} {\rho }^{f}}{\Bigl ({p}^{f} - {T}^{f} \frac{\partial {p}^{f}} {\partial {T}^{f}}\Bigr )}\frac{{D{}^{f}\rho }^{f}} {\mathit{Dt}} ={ \frac{\varepsilon _{f}{p}^{f}} {\rho }^{f}} \frac{{D{}^{f}\rho }^{f}} {\mathit{Dt}} +\varepsilon _{f}{T{}^{f}\beta }^{f}\frac{{D}^{f}{p}^{f}} {\mathit{Dt}} }$$

    where the thermal expansion coefficient (3.197), \({\beta }^{f} = -(1{/\rho }^{f})({\partial \rho }^{f}/\partial {T}^{f})\), is inserted.

  8. 8.

    It can be alternatively expressed by introducing the relationships (3.95) and (3.100) of the solid displacement \(\boldsymbol{{u}}^{s}\):

    $$\displaystyle{ \frac{\partial } {\partial t}(\varepsilon {_{s}\rho }^{s}) +\varepsilon { _{s}\rho }^{s}\biggl (\boldsymbol{{m}}^{T} \cdot {\Bigl (\boldsymbol{ L} \cdot \frac{\partial \boldsymbol{{u}}^{s}} {\partial t} \Bigr )}\biggr )} + \nabla (\varepsilon {_{s}\rho }^{s}) \cdot \frac{\partial \boldsymbol{{u}}^{s}} {\partial t} ={\rho }^{s}Q_{s}$$
  9. 9.

    Sometimes, the volumetric flux density is simply represented by the so-called Dupuit-Forchheimer relationship [389], which is a bulk flux in the form \(\boldsymbol{v}_{f} =\varepsilon _{f}\boldsymbol{{v}}^{f}\). This quantity has been given various names by different authors (e.g., seepage or filtration velocity). We shall prefer the term Darcy velocity \(\boldsymbol{q}_{f}\) emphasizing the correct relationship (3.240) for the flux.

  10. 10.

    While a parallel behavior occurs in most of the natural porous media, there could be a porous-medium structure and orientation, where the heat conduction takes place in series. In this case, the heat flux can pass serially though the solid and the fluid, such that the overall thermal conductivity is a harmonic mean \(\boldsymbol{\varLambda }_{0}^{-1} =\varepsilon {s}^{f}{{(\varLambda }^{f}\boldsymbol{\delta })}^{-1} + (1-\varepsilon ){{(\varLambda }^{s}\boldsymbol{\delta })}^{-1}\). The arithmetic mean and harmonic mean represent upper and lower bounds, respectively, for the overall thermal conductivity \(\boldsymbol{\varLambda }_{0}\). Other, more empirical arrangements for \(\boldsymbol{\varLambda }_{0}\) can be made up for certain porous media as discussed in [305].

  11. 11.

    From (3.260) it is \({p}^{l} =\rho _{ 0}^{l}g({h}^{l} - x_{j})\) and with \(\boldsymbol{e} = \nabla x_{j}\) we find \(\nabla {p}^{l} =\rho _{ 0}^{l}g(\nabla {h}^{l} -\boldsymbol{ e})\). Now expanding

    $$\displaystyle\begin{array}{rcl}{ \frac{\boldsymbol{k}} {\mu }^{l}} =\underbrace{\mathop{ \frac{\boldsymbol{k}\rho _{0}^{l}g} {\mu _{0}^{l}} }}\limits _{\boldsymbol{{K}}^{l}}\,\underbrace{\mathop{{ \frac{\mu _{0}^{l}} {\mu }^{l}} }}\limits _{f_{\mu }^{l}}\, \frac{1} {\rho _{0}^{l}g} =\boldsymbol{ {K}}^{l}f_{\mu }^{l} \frac{1} {\rho _{0}^{l}g}& & {}\\ \end{array}$$

    and inserting into (3.258) with (3.261), we obtain

    $$\displaystyle{\boldsymbol{q}_{l} = -k_{r}^{l}\boldsymbol{{K}}^{l}f_{\mu }^{l} \cdot \bigl (\nabla {h}^{l} + \tfrac{{\rho }^{l}-\rho _{ 0}^{l}} {\rho _{0}^{l}} \boldsymbol{e}\bigr )}$$

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  1. Groundwater Modelling Centre DHI-WASY GmbH, Berlin, Germany

    Hans-Jörg G. Diersch

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Diersch, HJ.G. (2014). Porous Medium. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_3

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