A Graph-theoretic Pipe Network Method for water flow simulation in a porous medium: GPNM

A new numerical simulation method for water flow in a porous medium is proposed. A porous medium is discretized graph-theoretically into a discrete pipe network. Each pipe in the oriented network is defined as a weighted element with a starting node and an ending node. Equivalent hydraulic parameters are derived based on the Darcy's Law. A node law of flow rate and a pipe law of pressure are derived based on the conservation of mass and energy, as well as the graph-theoretic network theory. A unified governing equation for both the inner pipes and the boundary pipes are deduced. A conversion law of flow rate/velocity is proposed and discussed. A few case studies are analyzed and compared with those from analytical solutions and finite element analysis. It shows that the proposed Graph-theoretic Pipe Network Method (GPNM) is effective in analyzing water flow in a porous medium. The advantage of the proposed GPNM is that a continuous porous medium is discretized into a discrete pipe network, which is analyzed same as for a discrete fracture network. Solutions of water pressures and flow rates in the discrete pipe network are obtained by solving a system of nonhomogeneous linear equations. It is demonstrated with high efficiency and accuracy. The developed method can be extended to analyzing water flow in fractured and porous media in 3-D conditions.