Abstract
This chapter deals with the finite element solutions for variably saturated porous media (unsaturated-saturated flow). The different formulations of Richards equations with the favorite solution strategies, including the computation of hysterestic effects and time-varying porosity, are discussed. Typical examples and benchmark tests are described to illustrate the usefulness, efficiency and accuracy of the proposed numerical techniques.
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Notes
- 1.
Optionally, FEFLOW suppresses the time derivative terms ∂ s∕∂ t and ∂ h∕∂ t for solving steady-state solutions.
- 2.
The saturation relation s(ψ) depends on the porous-medium properties, such as parameters α, n and m appearing in the van Genuchten relationship (D.4). In heterogeneous media the parameters can vary in space, i.e., \(\alpha =\alpha (\boldsymbol{x})\), \(n = n(\boldsymbol{x})\) and \(m = m(\boldsymbol{x})\). Then, the chain rule applied to ∇s yields for a van Genuchten relationship
$$\displaystyle{\nabla s = \frac{\partial s} {\partial \psi } \nabla \psi + \frac{\partial s} {\partial \alpha } \nabla \alpha + \frac{\partial s} {\partial n}\nabla n + \frac{\partial s} {\partial m}\nabla m}$$and contrary to (10.14), the correct s−form of the Richards’ equation reads for heterogeneous porous media:
$$\displaystyle{{\bigl (s\,S_{o}\,{C}^{-1} +\varepsilon \bigr )} \frac{\partial s} {\partial t} -\nabla \cdot {\bigl [\boldsymbol{D}\cdot (\nabla s-\tfrac{\partial s} {\partial \alpha } \nabla \alpha -\tfrac{\partial s} {\partial n}\nabla n- \tfrac{\partial s} {\partial m}\nabla m)+k_{r}\boldsymbol{K}f_{\mu }(1+\chi )\boldsymbol{e}\bigr ]} = Q_{h}+Q_{\mathit{hw}} +Q_{\mathrm{EOB}}}$$exemplified for a van Genuchten relationship. Similar expressions result for other empirical s(ψ)−relations, see Appendix D. The terms \(\tfrac{\partial s} {\partial \alpha } \nabla \alpha\), \(\tfrac{\partial s} {\partial n}\nabla n\) and \(\tfrac{\partial s} {\partial m}\nabla m\) additionally appearing in the s−based form of the Richards’ equation need a specific treatment in the numerical solution, e.g., [311]. More discussions are given by LaBolle and Clausnitzer [327].
- 3.
BC’s for the transformed ADE (10.22) can be equivalently found for (10.6) when written by the new F variable:
$$\displaystyle{\begin{array}{rcll} F & = & \tfrac{1} {\alpha } {e}^{\alpha (h_{D}-z)} & \;\;\mbox{ on}\quad \varGamma _{D} \times t[t_{0},\infty ) \\ - (\boldsymbol{K}f_{\mu } \cdot \nabla F -\boldsymbol{ v}F) \cdot \boldsymbol{ n}& = & q_{F} &\;\;\mbox{ on}\quad \varGamma _{N} \times t[t_{0},\infty ) \\ - [\boldsymbol{K}f_{\mu } \cdot (1+\chi )\boldsymbol{e})] \cdot \boldsymbol{ n}& = & \tfrac{1} {\alpha } \boldsymbol{v} \cdot \boldsymbol{ n} = q_{F}^{\mbox{ $\nabla $}} &\;\;\mbox{ on}\quad \varGamma _{N}^{\mbox{ $\nabla $}}\times t[t_{0},\infty ) \\ - (\boldsymbol{K}f_{\mu } \cdot \nabla F -\boldsymbol{ v}F) \cdot \boldsymbol{ n}& = & -\varPhi _{h}[h_{C} - z -\tfrac{1} {\alpha } \ln (\alpha F)] & \;\;\mbox{ on}\quad \varGamma _{C} \times t[t_{0},\infty ) \\ Q_{\mathit{hw}} & = & -\sum _{w}Q_{w}(t)\delta (\boldsymbol{x} -\boldsymbol{ x}_{w}) & \;\;\mbox{ on}\quad \boldsymbol{x}_{w} \in \varOmega \times t[t_{0},\infty ) \end{array} }$$additionally, the seepage face BC for (10.7) as
$$\displaystyle{F = \tfrac{1} {\alpha } \quad \mbox{ at}\quad Q_{n_{h}} > 0\quad \mbox{ on}\quad \varGamma _{S} \times t[t_{0},\infty )}$$and the IC (10.8) in the form
$$\displaystyle{F(\boldsymbol{x},t_{0}) = \tfrac{1} {\alpha } {e}^{\alpha [h_{0}(\boldsymbol{x})-z]}\quad \mbox{ in}\quad \bar{\varOmega }}$$We note that the Cauchy-type BC on Γ C introduces a nonlinear expression in F.
- 4.
It can be shown that the h−based formulation of the Picard method in form of (10.55) deduces from the more general \(h - s-\) based formulation of the Picard method in form of (10.39) if the saturation terms on the RHS of (10.39) are expressed by their derivatives with respect to the hydraulic head, viz.,
$$\displaystyle{\boldsymbol{s}_{n+1}^{\tau +1} -\boldsymbol{ s}_{n} - (1-\theta )\varDelta t_{n}\dot{\boldsymbol{s}}_{n} =\boldsymbol{ C}_{ n+1}^{\tau } \cdot {\bigl [\boldsymbol{ h}_{ n+1}^{\tau } -\boldsymbol{ h}_{n} - (1-\theta )\varDelta t_{n}\dot{\boldsymbol{h}}_{n}\bigr ]}}$$so that the storage matrix \(\boldsymbol{{O}}^{\dag }\) of the h−form results in
$$\displaystyle{\boldsymbol{{O}}^{\dag }(\boldsymbol{s}_{ n+1}^{\tau }) =\boldsymbol{ O}(\boldsymbol{s}_{ n+1}^{\tau }) +\boldsymbol{ B} \cdot \boldsymbol{ C}_{ n+1}^{\tau }}$$where the matrices \(\boldsymbol{O}\), \(\boldsymbol{B}\) and \(\boldsymbol{C}\) are given from the \(h - s-\) form by (10.32) and (10.38), respectively.
- 5.
Empirical target-based time step control: If Newton iterations have converged a new provisional step size Δ t n+1 can be computed in the following way [141]:
$$\displaystyle{\varDelta t_{n+1} =\varXi \;\varDelta t_{n}}$$where Ξ is a time step multiplier, which is determined by the minimum ratio of prescribed target change parameters DXWISH (DSWISH for the saturations \(\boldsymbol{s}_{n+1}\) and DPWISH for the pressure head \(\boldsymbol{\psi }_{n+1}\)) to the Newton correction, viz.,
$$\displaystyle{\varXi =\min _{i} \frac{\mathrm{DXWISH}} {\vert X_{i,n+1}^{\tau +1} - X_{i,n}\vert }}$$Typically used values are DSWISH = 0. 4 and DPWISH = 400 m. Additionally, it can be useful to constrain Ξ by a maximum multiplier Ξ ≤Ξ max, where Ξ max = 1. 1, …, 5. If the Newton scheme does not converge within a maximum number of iterations τ ≤ ITMAX, where ITMAX is typically 12, the current time step has to be rejected. A reduced time step size is then computed by \(\varDelta t_{n}^{\mathrm{red}} =\varDelta t_{n}/\mathrm{TDIV}\) and the solution process is restarted for the current time plane n + 1, but with Δ t n =Δ t n red. The time step divider TDIV is usually 2.
- 6.
Of primary interests are the schemes no.1, no.3 and no.4, providing a full residual control and best mass-conservative properties. Scheme no.1 is very effective for dry porous media, however, it is not well applicable to hysteretic porous-media problems. The Picard method of scheme no.4 is potentially more robust compared to the Newton scheme no.3, however, to the disadvantage of only a linear convergence rate. In solving the mixed \(\psi -s-\) form (or the equivalent \(h - s-\) form) of Richards’ equation, the moisture capacity C is usually evaluated analytically. On the other hand, for the standard h−based forms of the Richards’ equation the chord slope evaluation (see Sect. J.3 of Appendix J) of the moisture capacity is often preferred due to a potentially better discrete mass conservation property. The h−form of schemes no.9 and no.7 are suited for classic seepage simulations (at moderate capillary pressure conditions) involving free surface(s). Scheme no.9 with θ = 1 can be used to approach to steady-state solutions (whenever exist).
- 7.
Two interesting results can be detected from (10.129):
-
1.
We can ask which flux is concerned to force the pressure head zero everywhere? It can be easily shown from (10.129) that such a situation occurs if the infiltration has the amount of the saturated conductivity, i.e., \(v = -K\)
-
2.
We also can ask which flux is concerned to make the pressure head ψ infinity at the soil surface z = 0, i.e., ψ(0) = ∞? This should occur for a certain rate v which represents the theoretically maximum evaporative flux v max. The pressure head ψ becomes infinity at z = 0 if the argument of the logarithm of (10.129) goes to zero. It implies that
$$\displaystyle{\frac{v_{\mathrm{max}}} {K} = {e}^{-\alpha L}{\bigl (\frac{v_{\mathrm{max}}} {K} + 1\bigr )}}$$and leads to a solution of the theoretically maximum evaporative flux as
$$\displaystyle{v_{\mathrm{max}} = \frac{K} {{e}^{\alpha L} - 1}.}$$
-
1.
- 8.
Using the exponential relationship (D.39) in the form of k r = e α ψ we can integrate (10.139) analytically. The BC’s at the top and bottom of the fine layer are the following: At the top, the relative permeability is simply the infiltration rate v divided by the saturated hydraulic conductivity K of the fine layer, i.e., \(k_{r} = v/K =\exp (\alpha \psi _{2})\) so that \(\psi _{2} = \tfrac{1} {\alpha } \ln ( \tfrac{v} {K})\). At the bottom of the fine layer we find the pressure head equal to the value at the top of the coarse layer in a similar relation: \(k_{r}^{\star } = v/{K}^{\star } =\exp {(\alpha }^{\star }\psi _{1})\) so that \(\psi _{1} ={ \tfrac{1} {\alpha }^{\star }}\ln ( \tfrac{v} {{K}^{\star }})\), where K ⋆ and α ⋆ are the saturated hydraulic conductivity and sorptive number, respectively, of the coarse layer. Applying these BC’s for ψ 2 and ψ 1 we find the analytical solution of (10.139) for the diversion length as [448]
$$\displaystyle{L = K\, \frac{\tan \varphi } {v\alpha }{\Bigl [{\Bigl ({ \frac{v} {{K}^{\star }}\Bigr )}}^{\alpha {/\alpha }^{\star }} -{\Bigl ( \frac{v} {K}\Bigr )}\Bigr ]}.}$$ - 9.
In 2D and under steady-state conditions equipotential lines are given by the interval of hydraulic head Δ h. The interval of streamlines (actually, interval of the streamfunction, cf. Sect. 2.1.11) Δ Ψ is determined from
$$\displaystyle{ \varDelta \varPsi = K \frac{\varDelta h} {\varDelta l} \varDelta q }$$where Δ l is the distance between two neighboring equipotential lines and Δ q is the width of the stream tube. A flow net can be constructed if setting Δ l =Δ q so that streamlines and equipotential lines form ‘curvilinear squares’. For such a flow net configuration it is
$$\displaystyle{\varDelta \varPsi = K\varDelta h.}$$ - 10.
The present example is easily solvable for classic free-surface flow modeling with moving mesh (cf. Sect. 9.5.3), even with only a small number of elements. Contrarily, the classic free-surface modeling strategy with fixed mesh and pseudo-unsaturated conditions (cf. Sect. 9.5.4.4) will not give reasonable results for such type of a vertically dominant drainage because the free-surface BC assigned unmovably to the upper element slice becomes ineffective when all underlying elements fall dry in time.
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Diersch, HJ.G. (2014). Flow in Variably Saturated Porous Media. In: FEFLOW. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38739-5_10
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