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\section{Constitutive relationships}  \subsection{Water retention curve, van Genuchten (1980):}  \begin{align}  \theta(\psi) = \theta_{r} + (\theta_{s} - \theta_{r}) (1+(a_{VG} \psi)^{n_{VG}})^{-m_{VG}} \\  \end{align}  \subsection{Hydraulic conductivity function, van Genuchten et al. (1991):}  \begin{align}  k(\psi) = k_{sat} \frac {(1-((a_{VG} \psi)^{n_{VG}m_{VG}}) (1+(a_{VG} \psi)^{n_{VG}})^{-m_{VG}}))^2} { (1+(a_{VG} \psi)^{n_{VG}})^{m_{VG}l_{VG}}}  \end{align}  \section{Partial differential equation}  Mass conservation law.  \begin{align}  \frac{\partial \theta}{\partial t} = - \nabla \cdot \vec{q} \\  \end{align}  Darcy law.  \begin{align}  \vec{q} = k \left( h, P \right) \nabla H \\  \end{align}  Where h is the pore pressure, P is the position in 3D space and H is the total head.  \begin{align}  P = (x, y, z)\\  h= - \psi \\  \nabla H = \nabla h + \nabla z \\  \end{align}  The partial differential equation can be written as:  \begin{align}  \frac{\partial \theta}{\partial t} = -\nabla \cdot k \left( h, P \right) \nabla H + A, \\  \end{align}  where $A$ is a source/sink term.  \section{Weak formulation}  Integrate on both sides and multiply by the test function $v$.  \begin{align}  \int_{\Omega}\frac{\partial \theta}{\partial t} v\,d\Omega = -\int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega + \int_{\Omega}A v\,d\Omega \\  \end{align}  Integrate by parts.  \begin{align}  \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \\  \int_{\Omega}\nabla \cdot \left( k \left( h, P \right) \nabla H v\right) \,d\Omega \\  \end{align}  Apply Gauss divergence theorem.  \begin{align}  \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N} \cup \Gamma_{D}} k \left( h, P \right) \nabla H v n\,d\Gamma \\  \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N}} k \left( h, P \right) \nabla H v n\,d\Gamma + \int_{\Gamma_{D}} k \left( h, P \right) \nabla H v n\,d\Gamma  \end{align}  Because $v = 0$ on $\Gamma_{D}$.  \begin{align}  \int_{\Omega}\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = -\int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega + \int_{\Gamma_{N}} k \left( h, P \right) \nabla H v n\,d\Gamma \\  \end{align}  Put in the initial integral.  \begin{align}  \int_{\Omega} \frac{\partial \theta}{\partial t} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\  \end{align}  $\theta \left( h \right)$ is a constitutive relationship. The weak formulation can thus be written on a $h$ basis.  \begin{align}  \int_{\Omega} \frac{\partial \theta}{\partial t} \times \frac{\partial h}{\partial h} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\  \\  \frac{\partial \theta}{\partial h} \int_{\Omega} \frac{\partial h}{\partial t} v\,d\Omega = \int_{\Omega} k \left( h, P \right) \nabla H \nabla v\,d\Omega - \int_{\Gamma_{N}} \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + \int_{\Omega}A v\,d\Omega \\  \end{align}  \section{Notes}  Integration by parts.  \begin{align}  \int_{\Omega} \left(\nabla \cdot u \right) v\,d\Omega = -\int_{\Omega} u \nabla v\,d\Omega + \int_{\Omega} \nabla \cdot \left( u v \right) n\,d\Omega \\  \end{align}  Gauss divergence theorem.  \begin{align}  \int_{\Omega} \nabla \cdot F\, d\Omega = \int_{\partial \Omega = \Gamma_{N} \cup \Gamma_{D}} F \cdot n\, d\Gamma  \end{align}  The problem in steady-state with no Neumann boundary conditions [can be defined in Sfepy](http://sfepy.org/doc-devel/examples/diffusion/poisson_field_dependent_material.html).