CONSTITUTIVE RELATIONSHIPS Water retention curve, van Genuchten (1980): \theta(\psi) = + ( - ) (1+(a_{VG} \psi)^{n_{VG}})^{-m_{VG}} \\ Hydraulic conductivity function, van Genuchten et al. (1991): 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}}} PARTIAL DIFFERENTIAL EQUATION Mass conservation law. {\partial t} = - \nabla \cdot \\ Darcy law. = k \left( h, P \right) \nabla H \\ Where h is the pore pressure, P is the position in 3D space and H is the total head. P = (x, y, z)\\ h= - \psi \\ \nabla H = \nabla h + \nabla z \\ The partial differential equation can be written as: {\partial t} = -\nabla \cdot k \left( h, P \right) \nabla H + A, \\ where A is a source/sink term. WEAK FORMULATION Integrate on both sides and multiply by the test function v. {\partial t} v\,d\Omega = -\nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega + A v\,d\Omega \\ Integrate by parts. \nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = - k \left( h, P \right) \nabla H \nabla v\,d\Omega + \\ \nabla \cdot \left( k \left( h, P \right) \nabla H v\right) \,d\Omega \\ Apply Gauss divergence theorem. \nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = - k \left( h, P \right) \nabla H \nabla v\,d\Omega + \cup } k \left( h, P \right) \nabla H v n\,d\Gamma \\ \nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = - k \left( h, P \right) \nabla H \nabla v\,d\Omega + } k \left( h, P \right) \nabla H v n\,d\Gamma + } k \left( h, P \right) \nabla H v n\,d\Gamma Because v = 0 on ΓD. \nabla \cdot k \left( h, P \right) \nabla H v\,d\Omega = - k \left( h, P \right) \nabla H \nabla v\,d\Omega + } k \left( h, P \right) \nabla H v n\,d\Gamma \\ Put in the initial integral. {\partial t} v\,d\Omega = k \left( h, P \right) \nabla H \nabla v\,d\Omega - } \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + A v\,d\Omega \\ θ(h) is a constitutive relationship. The weak formulation can thus be written on a h basis. {\partial t} \times {\partial h} v\,d\Omega = k \left( h, P \right) \nabla H \nabla v\,d\Omega - } \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + A v\,d\Omega \\ \\ {\partial h} {\partial t} v\,d\Omega = k \left( h, P \right) \nabla H \nabla v\,d\Omega - } \left( k \left( h, P \right) \nabla H n \right) v\,d\Gamma + A v\,d\Omega \\ NOTES Integration by parts. \left(\nabla \cdot u \right) v\,d\Omega = - u \nabla v\,d\Omega + \nabla \cdot \left( u v \right) n\,d\Omega \\ Gauss divergence theorem. \nabla \cdot F\, d\Omega = \cup } F \cdot n\, d\Gamma The problem in steady-state with no Neumann boundary conditions can be defined in Sfepy.