2.1 Forward Modeling
Use the constitutive equation to explain the hydraulic problems of heterogeneous and anisotropic porous materials:
\(\left\{\par \begin{matrix}a\frac{\partial h}{\partial t}+\nabla\bullet\mathbf{u=}Q_{\mathbf{s}}\\ \mathbf{u=-}K\nabla h\\ \end{matrix}\right.\ \) (2.1)
Where \(h\) denotes the hydraulic head (in m), \(a\)(m-1) denotes the storage ratio, Qs denotes the external sources,\(\ \mathbf{u}\) denotes the Darcy velocity (m\(\bullet\)s−1), and K(m\(\bullet\)s−1) means the penetration rate. The pressure of pore fluid \(p\) (in pa) can be expressed as the relationship between the mass density of pore water (in kg/m3), the acceleration of gravity g (in m/s2), and the hydraulic head \(h\) (in m)\(\text{\ p}\mathbf{=}\rho_{\mathbf{f}}g(h-z)\). \(\mathbf{z}\) (in m) is the constant elevation above a given datum. Eq 2.1 should follow the first type boundary (Dirichlet) condition and the second type boundary (Neumann) condition while following the initial term.
Boundary \(\left\{\par \begin{matrix}h=h_{D}\text{\ at\ }\Gamma_{D}\\ -nK\nabla h=q_{0}\text{\ at\ }\Gamma_{N}\\ \end{matrix}\right.\ \) (2.2)
Initial \(\ h=h_{0}\ at\ t=0\) (2.3)
After solving hydraulic problems, it is essential to analyze electrical problems. The sum of conductive current density (Ohm’s Law)\(-\sigma\nabla\psi\ \)and the current source density\(\mathbf{j}_{s}\)is the total current density \(\mathbf{j}\)(A/m2) (Ahmed &Jardani, 2013).
\(\mathbf{j}=-\sigma\nabla\psi+\mathbf{j}_{s}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\)(2.4)
\(\mathbf{j}_{s}=Q_{V}\mathbf{u}\) (2.5)
Where \(\psi\) (in V) denotes the electrical potential, σ is the electrical conductivity tensor.\(\ Q_{V}\) is the effective excess charge density tensor of the pore water per unit pore volume.\(Q_{V}\ \)can be predicted from the permeability tensor according to (Jardani et al., 2007):
\(\log_{10}Q_{V}=-9.2-0.82\log_{10}K\) (2.6)
According to the continuity equation of charge\(\nabla\mathbf{j}=0\)
\(\nabla\bullet(\sigma\nabla\psi)={\nabla\bullet\mathbf{j}}_{s}\)(2.7)
Where a Neumann boundary condition \(\Gamma_{\text{NV}}\) is applied to the upper interface (air-ground) and a Dirichlet boundary condition\(\ \Gamma_{\text{DV}}\) is used to other boundaries. It can ensure that the response of the underground potential is received on the ground surface. The boundary problem indicates that the anomaly contains some information related to the hydraulic flow path (Revil et al., 2010).