Numerical Simulation of Constitutive Relations for Unsaturated Flow in Fractured Porous Media

In this study we determine constitutive relations (relations between saturation, hydraulic conductivity and pressure head) for unsaturated fractured rocks using a computational procedure that mimics the laboratory technique of measurement. Isotropic three-dimensional rock samples with random vertical and horizontal fractures are computationally constructed. Each fracture is conceptualized as a thick plate of porous medium whose hydraulic properties are described by the well-known van Genuchten model. The procedure used to obtain simulated values of saturation and relative hydraulic conductivity is based on the numerical solution of the steady-state unsaturated flow equation in a three-dimensional domain. A constant value of pressure head (effective pressure head) is prescribed at the top and bottom boundaries and non flow conditions are imposed at side boundaries of the synthetic rock sample. The unsaturated flow equation is solved on a regular mesh of one million elements using a hybridized mixed finite element method. For the prescribed value of pressure head the corresponding values of water saturation and hydraulic conductivity are numerically computed. Then, by selecting different values of pressure head as boundary conditions a complete set of constitutive relations can be defined from the simulation results. The simulated relations are fitted using two recently proposed closed-form analytical expressions for unsaturated fractured rocks. A numerical test designed for fractured basalt shows that both analytical models can match reasonably well the simulated relations.