Abstract
In this chapter the computation of multispecies (including single-species) mass transport in porous media with chemical reaction in particular is examined. The complexity of those reactive transport processes arising in natural and engineered porous media requires some specific treatment due to their nonlinearity and the occurrence of multiple unknowns. In the preceding Chap. 5 the constitutive relations in form of reversible reaction and irreversible chemical kinetics have been developed. It ends up with a set of mass transport equations for each chemical species k = 1, …, N of an arbitrary number, nonlinearly coupled by the rate expressions of chemical reaction in form of degradation type, Arrhenius type, Monod type or freely editable kinetics. A given species k can be either mobile associated with a liquid (aqueous) phase l or immobile associated with a solid phase s, so that N = N l + N s. Chemicals in the liquid phase are subject to advection and dispersion, while in a solid phase there is no advection and dispersion. We solve the reactive multispecies mass transport processes in multi-dimensional porous media under variably saturated, variable-density and nonisothermal conditions. The focus of this chapter is on the treatment of the species mass transport PDE system, while for the flow computations we refer to Chap. 9 for saturated porous media, to Chap. 10 for variably saturated porous media and to Chap. 11 for density-coupled problems. Nonisothermal aspects are subject of Chaps. 11 and 13.
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Notes
- 1.
In 3D Cartesian coordinates the components of the mechanical dispersion tensor \(\boldsymbol{D}_{\mathrm{mech}}\) for the classic Scheidegger-Bear dispersion model, cf. (3.182), are
$$\displaystyle{\begin{array}{rcl} D_{\mathrm{mech},11} & = & \frac{1} {q}{\bigl (\beta _{L}q_{1}^{2} +\beta _{T}q_{2}^{2} +\beta _{T}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},22} & = & \frac{1} {q}{\bigl (\beta _{T}q_{1}^{2} +\beta _{L}q_{2}^{2} +\beta _{T}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},33} & = & \frac{1} {q}{\bigl (\beta _{T}q_{1}^{2} +\beta _{T}q_{2}^{2} +\beta _{L}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},12} & = & D_{\mathrm{mech},21} = (\beta _{L} -\beta _{T})\frac{q_{1}q_{2}} {q} \\ D_{\mathrm{mech},13} & = & D_{\mathrm{mech},31} = (\beta _{L} -\beta _{T})\frac{q_{1}q_{3}} {q} \\ D_{\mathrm{mech},23} & = & D_{\mathrm{mech},32} = (\beta _{L} -\beta _{T})\frac{q_{2}q_{3}} {q} \end{array} }$$where \(\boldsymbol{{q}}^{T} = \left (\begin{array}{ccc} q_{1} & q_{2} & q_{3} \end{array} \right )\) and \(q =\Vert \boldsymbol{ q}\Vert\). In strictly stratified aquifer system, where the transverse dispersion in the vertical x 3−direction can be much smaller than in the horizontal, Burnett and Frind [65] proposed the 3D mechanical dispersion tensor in an alternative form
$$\displaystyle{\begin{array}{rcl} D_{\mathrm{mech},11} & = & \frac{1} {q}{\bigl (\beta _{L}q_{1}^{2} +\beta _{TH}q_{2}^{2} +\beta _{\mathit{TV}}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},22} & = & \frac{1} {q}{\bigl (\beta _{TH}q_{1}^{2} +\beta _{L}q_{2}^{2} +\beta _{\mathit{TV}}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},33} & = & \frac{1} {q}{\bigl (\beta _{\mathit{TV}}q_{1}^{2} +\beta _{\mathit{TV}}q_{2}^{2} +\beta _{L}q_{3}^{2}\bigr )} \\ D_{\mathrm{mech},12} & = & D_{\mathrm{mech},21} = (\beta _{L} -\beta _{TH})\frac{q_{1}q_{2}} {q} \\ D_{\mathrm{mech},13} & = & D_{\mathrm{mech},31} = (\beta _{L} -\beta _{\mathit{TV}})\frac{q_{1}q_{3}} {q} \\ D_{\mathrm{mech},23} & = & D_{\mathrm{mech},32} = (\beta _{L} -\beta _{\mathit{TV}})\frac{q_{2}q_{3}} {q} \end{array} }$$splitting the transverse dispersivity into a horizontal transverse dispersivity β TH and a vertical transverse dispersivity β TV , where it is assumed that β TH ≫β TV . However, as noted by Bear and Cheng [38], Burnett and Find’s mechanical dispersion tensor is not consistent with the basic constitutive relations, such as derived in Sects. 3.8.5.4 and 3.8.5.5, and not conform with tensor transformation rules shown by Lichtner et al. [349].
- 2.
The separation of the linear decay term allows its numerically implicit treatment in the LHS of the resulting discrete equation system, while nonlinearities appearing in \(\hat{R}_{k}\) require an appropriate iterative approach. Indeed, the reaction rate \(\hat{R}_{k}\) can also incorporate a linear degradation term (in this case \(\vartheta _{k}\) should be zero), however, its numerical computation can be less effective than in the direct separation.
- 3.
In the special case of a single-species solute, where only one dissolved component exists, we can drop the species indicator k and write the governing mass transport equation (12.1) and (12.2), respectively, simply as
$$\displaystyle{\begin{array}{rcll} \frac{\partial } {\partial t}(\varepsilon s\mathfrak{R}C) + \nabla \cdot (\boldsymbol{q}C) -\nabla \cdot (\boldsymbol{D} \cdot \nabla C) +\varepsilon s\vartheta \mathfrak{R}C & = & \hat{R} + Q_{Cw} + Q_{C}\quad \mbox{ divergence form} \\ \varepsilon s\acute{\mathfrak{R}}\frac{\partial C} {\partial t} +\boldsymbol{ q} \cdot \nabla C -\nabla \cdot (\boldsymbol{D} \cdot \nabla C) + (\varepsilon s\vartheta \mathfrak{R} + Q_{h})C & = & \hat{R} + Q_{Cw} + Q_{C}\quad \mbox{ convective form} \end{array} }$$with
$$\displaystyle{\mbox{ $\begin{array}{rcl} \boldsymbol{D}& = & \varepsilon sD\boldsymbol{\delta } +\boldsymbol{ D}_{\mathrm{mech}} \\ \mathfrak{R}& = & 1 +{\bigl ( \frac{1-\varepsilon } {\varepsilon } \bigr )}\varphi \\ \acute{\mathfrak{R}}& = & 1 +{\bigl ( \frac{1-\varepsilon } {\varepsilon } \bigr )}\,\frac{\partial (\varphi C)} {\partial C} \\ \varphi & = & \left \{\begin{array}{ll} \kappa & \mbox{ Henry} \\ {b}^{\dag }\,{C}^{{b}^{\ddag }-1 }& \mbox{ Freundlich} \\ \frac{{k}^{\dag }} {1+{k}^{\ddag }\,C} & \mbox{ Langmuir} \end{array} \right. \mbox{ (Table 3.8)} \\ \hat{R}& = & \hat{R}(\varepsilon,s,C,T) \end{array} $}}$$for solving the solute concentration C associated with the liquid phase l, where Q Cw and Q C denote the well-type SPC term and the zero-order mass sink/source term, respectively, for the single-species solute.
- 4.
Optionally, FEFLOW suppresses the time derivative term ∂ C k ∕∂ t for solving steady-state solutions. A specific option exists, named steady flow – transient transport, in which the advective flow vector \(\boldsymbol{q}\) is invariant with time.
- 5.
A boundary with OBC on \(\varGamma _{N_{kO}}\) can be separated from the Neumann boundary \(\varGamma _{N_{k}}\) so that for the divergence form
$$\displaystyle{\int _{\varGamma _{N_{ k}}}wq_{\mathit{kC}}^{\dag }d\varGamma =\int _{\varGamma _{ N_{k}}\setminus \varGamma _{N_{kO}}}wq_{\mathit{kC}}^{\dag }d\varGamma +\int _{\varGamma _{ N_{kO}}}w(C_{k}\boldsymbol{q} -\boldsymbol{ D}_{k} \cdot \nabla C_{k}) \cdot \boldsymbol{ n}d\varGamma }$$and for the convective form
$$\displaystyle{\int _{\varGamma _{N_{ k}}}wq_{\mathit{kC}}d\varGamma =\int _{\varGamma _{N_{ k}}\setminus \varGamma _{N_{kO}}}wq_{\mathit{kC}}d\varGamma -\int _{\varGamma _{N_{kO}}}w(\boldsymbol{D}_{k} \cdot \nabla C_{k}) \cdot \boldsymbol{ n}d\varGamma }$$The implicit treatment of OBC requires the incorporation of the \(\varGamma _{N_{kO}}-\) integrals into the LHS of the resulting matrix system (see below). In contrast, a natural Neumann-type BC with \(-(\boldsymbol{D}_{k} \cdot \nabla C_{k}) \cdot \boldsymbol{ n} \approx 0\) on \(\varGamma _{N_{kO}}\) is often the preferred alternative formulation for an OBC. Note, however, that for both cases in the divergence form the boundary flux \(\boldsymbol{q} \cdot \boldsymbol{ n}\) must be known a priori. The boundary flux \(\boldsymbol{q} \cdot \boldsymbol{ n}\) can be either explicitly given from a Neumann-type BC \(q_{h} =\boldsymbol{ q} \cdot \boldsymbol{ n}\) for flow or must be computed by a postprocessing budget evaluation of the flow equation on the corresponding outflowing boundary section imposed by Dirichlet-type or Cauchy-type BC of flow.
- 6.
- 7.
Alternatively to the GLS predictor-corrector method, the time integration of (12.22) for each species k by using the simple θ−method (Sect. 8.13.4) gives
$$\displaystyle{\begin{array}{l} {\Bigl (\frac{\boldsymbol{H}_{k}(\boldsymbol{C}_{n+1})} {\varDelta t_{n}} +\boldsymbol{ E}_{k}(\boldsymbol{C}_{n+1})\theta \Bigr )} \cdot \boldsymbol{ C}_{k,n+1} = \\ {\Bigl (\frac{\boldsymbol{H}_{k}(\boldsymbol{C}_{n+1})} {\varDelta t_{n}} -\boldsymbol{ E}_{k}(\boldsymbol{C}_{n+1})(1-\theta )\Bigr )} \cdot \boldsymbol{ C}_{k,n} +{\bigl (\boldsymbol{ R}_{k}(\boldsymbol{C}_{n+1})\theta +\boldsymbol{ R}_{k}(\boldsymbol{C}_{n})(1-\theta )\bigr )} \end{array} }$$where \(\theta \in (\tfrac{1} {2}, 1)\) for the Crank-Nicolson and the fully implicit scheme, respectively. For chemically reactive processes a nonlinear matrix system \(\boldsymbol{R}_{k,n+1}^{\star } =\boldsymbol{ A}_{k}(\boldsymbol{C}_{n+1}) \cdot \boldsymbol{ C}_{k,n+1} -\boldsymbol{ Z}_{k}(\boldsymbol{C}_{n+1},\boldsymbol{C}_{n}) = \mathbf{0}\) results, which must be iteratively solved either via the Picard method (Sect. 8.18.1)
$$\displaystyle{ \boldsymbol{A}_{k}(\boldsymbol{C}_{n+1}^{\tau }) \cdot \boldsymbol{ C}_{ k,n+1}^{\tau +1} =\boldsymbol{ Z}_{ k}(\boldsymbol{C}_{n+1}^{\tau },\boldsymbol{C}_{n})\quad \tau = 0, 1, 2,\ldots }$$or via the Newton method (Sect. 8.18.2)
$$\displaystyle{\begin{array}{rcl} \boldsymbol{J}_{k}(\boldsymbol{C}_{n+1}^{\tau }) \cdot \varDelta \boldsymbol{ C}_{k,n+1}^{\tau } & = & -\boldsymbol{ R}_{k,n+1}^{\star }(\boldsymbol{C}_{n+1}^{\tau },\boldsymbol{C}_{n})\quad \tau = 0, 1, 2,\ldots \\ \varDelta \boldsymbol{C}_{k,n+1}^{\tau } & = & \boldsymbol{C}_{k,n+1}^{\tau +1} -\boldsymbol{ C}_{k,n+1}^{\tau } \\ \boldsymbol{J}_{k}(\boldsymbol{C}_{n+1}^{\tau }) & = & \frac{\partial \boldsymbol{R}_{k,n+1}^{\star }(\boldsymbol{C}_{n+1}^{\tau },\boldsymbol{C}_{n})} {\partial \boldsymbol{C}_{k,n+1}^{\tau }} \end{array} }$$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 mass transport problems if setting θ = 1 and Δ t n →∞.
- 8.
Since the complementary error function erfc() is often in combination with exp(), it is numerically useful to introduce the function exf(a, b) defined as
$$\displaystyle{\mathrm{exf}(a,b) =\exp (a)\mathrm{erfc}(b)}$$which is suitably approximated as follows [540]:
$$\displaystyle{\mathrm{exf}(a,b) \approx \left \{\begin{array}{ll} \exp (a - {b}^{2})(a_{1}\tau + a{_{2}\tau }^{2} + a{_{3}\tau }^{3} + a{_{4}\tau }^{4} + a{_{5}\tau }^{5}) & \;\mbox{ if}\;\;0 \leq b \leq 3 \\ \frac{1} {\sqrt{\pi }}\exp (a - {b}^{2})/(b + 0.5/(b + 1/(b + 1.5/(b + 2/(b + 2.5/(b + 1)))))) & \;\mbox{ if}\;\;b > 3 \\ 2\exp (a) -\mathrm{ exf}(a,-b) & \;\mbox{ if}\;\;b < 0 \\ 0\qquad \qquad \mbox{ if}\;\;\vert a\vert > 170\;\;\mbox{ and}\;\;b \leq 0\quad \mbox{ or}\quad \vert a - {b}^{2}\vert > 170\;\;\mbox{ and}\;\;b > 0 & \end{array} \right.}$$where τ = 1∕(1 + 0. 3275911b) and a 1 = 0. 2548296, a 2 = −0. 2844967, a 3 = 1. 421414, a 4 = −1. 453152 and a 5 = 1. 061405.
- 9.
FEFLOW results described in this section were obtained by D. Etcheverry † and Y. Rossier (France).
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Diersch, HJ.G. (2014). Mass Transport in Porous Media with and Without Chemical Reactions. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_12
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DOI: https://doi.org/10.1007/978-3-642-38739-5_12
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