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Variable-Density Flow, Mass and Heat Transport in Porous Media

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Abstract

In Chap. 3 the continuum approach of the porous medium has been described. A fluid (or better a phase) appears there as an effectively continuous medium with a mass density ρ (fluid mass per unit volume of fluid) as a fundamental bulk property. The density of a fluid is often not uniform. In general, the fluid is composed of N miscible chemical species with a partial density ρ k (mass of the constituent k per unit volume of fluid), so that for the mixture \(\rho =\sum _{ k}^{N}\rho _{k}\) (density increases when dissolved mass of constituents increases). Moreover, the density of a fluid can be influenced by the temperature T (density decreases when temperature increases) and by the pressure p (density increases when pressure increases due to compressibility). In a formal manner, the density is to be regarded as a dependent thermodynamic variable for which an equation of state (EOS) ρ =ρ(p, ρ k , T) holds, cf. Sect. 3.8.6.1.

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Notes

  1. 1.

    We note that the impact of p, ρ k and T on ρ does not lead to the same flow effects. Compression effects caused by pressure changes will not feature a new physical characteristic, quite contrary to variable concentration or/and temperature fields which are governed by distinct balance statements subjected to advection and dispersion/conduction. Only the presence of at least one of these quantities is capable of forming complex convective flow phenomena such as flow recirculations, stratified and physically oscillating flow patterns. Flow processes affected exclusively by compression due to pressure changes will not belong to the distinct category of variable-density flow.

  2. 2.

    Alternatively, by using the divergence forms of the governing transport equations for mass and heat, the coupled PDE system reads

    $$\displaystyle{\begin{array}{rcl} s\,S_{o}\frac{\partial h} {\partial t} +\varepsilon \frac{\partial s} {\partial t} + \nabla \cdot \boldsymbol{ q}& = & Q + Q_{\mathrm{EOB}} \\ \boldsymbol{q}& = & - k_{r}\boldsymbol{K}f_{\mu } \cdot \bigl (\nabla h +\chi \boldsymbol{ e}\bigr ) \\ \frac{\partial } {\partial t}(\varepsilon s\mathfrak{R}_{k}C_{k}) + \nabla \cdot (\boldsymbol{q}C_{k}) -\nabla \cdot (\boldsymbol{D}_{k} \cdot \nabla C_{k}) +\varepsilon s\vartheta _{k}\mathfrak{R}_{k}C_{k}& = & \tilde{R}_{k}\quad (k = 1,\ldots,N) \\ \frac{\partial } {\partial t}\bigl [\bigl (\varepsilon s\rho c + {(1-\varepsilon )\rho }^{s}{c}^{s}\bigl )(T - T_{0})\bigr ] + \nabla \cdot (\rho c\boldsymbol{q}(T - T_{0})) -\nabla \cdot (\boldsymbol{\varLambda }\cdot \nabla T) & = & H_{e} \end{array} }$$

    Note that the divergence form of heat transport in terms of the temperature T assumes that the specific heat capacities c and c s are independent of T (cf. discussions in Sect. 3.9.1).

  3. 3.

    Optionally, FEFLOW suppresses all time derivative terms ∂ h∕∂ t, ∂ s∕∂ t, ∂ C k ∕∂ t and ∂ T∕∂ t for solving steady-state solutions. A specific option exists, named steady flow – transient transport, in which only the derivative terms of the flow equation ∂ h∕∂ t and ∂ s∕∂ t are dropped to exclude flow storage effects in the variable-density flow simulation. Note, however, due to the nonlinearity in the flow equation the solution of the flow must be updated at each time t once the concentration C k and/or the temperature T change. As the result, h, s and \(\boldsymbol{q}\) remain time-dependent.

  4. 4.

    The time integration of (11.38) by using the simple θ−method (Sect. 8.13.4) gives

    $$\displaystyle{\begin{array}{l} \Bigl (\frac{\boldsymbol{G}(\boldsymbol{U}_{n+1})} {\varDelta t_{n}} +\boldsymbol{ K}(\boldsymbol{U}_{n+1})\theta \Bigr ) \cdot \boldsymbol{ U}_{n+1} = \\ \Bigl (\frac{\boldsymbol{G}(\boldsymbol{U}_{n+1})} {\varDelta t_{n}} -\boldsymbol{ K}(\boldsymbol{U}_{n+1})(1-\theta )\Bigr ) \cdot \boldsymbol{ U}_{n} +\bigl (\boldsymbol{ Q}(\boldsymbol{U}_{n+1})\theta +\boldsymbol{ Q}(\boldsymbol{U}_{n})(1-\theta )\bigr ) \end{array} }$$

    where \(\theta \in (\tfrac{1} {2}, 1)\) for the Crank-Nicolson and the fully implicit scheme,respectively. A nonlinear matrix system \(\boldsymbol{R}_{n+1} =\boldsymbol{ A}(\boldsymbol{U}_{n+1}) \cdot \boldsymbol{ U}_{n+1} -\boldsymbol{ Z}(\boldsymbol{U}_{n+1},\boldsymbol{U}_{n}) = \mathbf{0}\) results, which must be iteratively solved either via the Picard method (Sect. 8.18.1)

    $$\displaystyle{\boldsymbol{A}(\boldsymbol{U}_{n+1}^{\tau }) \cdot \boldsymbol{ U}_{ n+1}^{\tau +1} =\boldsymbol{ Z}(\boldsymbol{U}_{ n+1}^{\tau },\boldsymbol{U}_{n})\quad \tau = 0, 1, 2,\ldots }$$

    or via the Newton method (Sect. 8.18.2)

    $$\displaystyle{\begin{array}{rcl} \boldsymbol{J}(\boldsymbol{U}_{n+1}^{\tau }) \cdot \varDelta \boldsymbol{ U}_{n+1}^{\tau } & = & -\boldsymbol{ R}_{n+1}(\boldsymbol{U}_{n+1}^{\tau },\boldsymbol{U}_{n})\quad \tau = 0, 1, 2,\ldots \\ \varDelta \boldsymbol{U}_{n+1}^{\tau } & = & \boldsymbol{U}_{n+1}^{\tau +1} -\boldsymbol{ U}_{n+1}^{\tau } \\ \boldsymbol{J}(\boldsymbol{U}_{n+1}^{\tau }) & = & \frac{\partial \boldsymbol{R}_{n+1}(\boldsymbol{U}_{n+1}^{\tau },\boldsymbol{U}_{n})} {\partial \boldsymbol{U}_{n+1}^{\tau }} \end{array} }$$

    until satisfactory convergence is achieved for the iterations τ at each given time stage n + 1. Note that this iterative solution strategy is also applicable to steady-state variable-density problems if setting θ = 1 and Δ t n →∞.

  5. 5.

    For the divergence form of the governing species mass and heat ADE’s it results

    $$\displaystyle{\begin{array}{rcl} \hat{J}_{k,\mathit{ij}} & = & \sum _{e}\int _{{\varOmega }^{e}}\bigl (k_{r}^{e}\boldsymbol{{K}}^{e}f_{\mu }^{e} \cdot \boldsymbol{ e}\bigr ) \cdot \nabla N_{ i}N_{j}\sum _{k}\beta _{c_{k}}^{e}\sum _{ l}(N_{l}C_{k,l}^{p}){d\varOmega }^{e} \\ \hat{J}_{T,\mathit{ij}} & = & -\sum _{e}\int _{{\varOmega }^{e}}\bigl (k_{r}^{e}\boldsymbol{{K}}^{e}f_{\mu }^{e} \cdot \boldsymbol{ e}\bigr ) \cdot \nabla N_{ i}N{_{j}\beta }^{e}({T}^{e})\sum _{ l}(N_{l}T_{l}^{p}){d\varOmega }^{e} \end{array} }$$

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Diersch, HJ.G. (2014). Variable-Density Flow, Mass and Heat Transport in Porous Media. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_11

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