Study On Numerical Modeling Of Effective Conductivity For Complex Multi-phase Medium

Effective conductivity of multi-phase rocks is an important topic in the field of rock physics and petroleum engineering. The theoretical solution of effective conductivity of multi-phase rock is only valid for fewer simple models of rocks. Some derivative formulas form the theory analysis and empirical formulas from experiments (eg, the Archie's law) have many limits. So, there are still a lot of problems needed to be solved for calculating effective conductivity of complex multi-phase rocks.Finite element method is valid for calculating effective conductivity of any complex multi-phase medium. On this basis, we introduce the generalized mixture rule (GMR) into the effective conductivity model. For stochastic model of two-phase medium we calculate the effective conductivity using finite element method and fit the data with GMR. For the first time, we obtain the formulas of the parameter of GMR——J with the ratio of conductivity of two phases——σ2 /σ1. So we can quickly predict (or calculate) the effective conductivity of any two-phase medium. The result is much more accurate than the existing effective conductivity models: random model and effective medium theory model.We study the application of the Modified Archie's Law (MAL) on stochastic model of two-phase medium. We calculate the effective conductivity using the finite element method, and get the value of the parameter (m) of MAL by fitting the data using genetic algorithm. Compare to the random medium model and the effective medium theory model, the formulas of MAL can predict the effective conductivity of any stochastic model of two-phase medium more accurately, too. We study the application of the GMR on the stochastic model of three-phase medium and obtain preliminary results. These results show that the parameter J is very complex on the stochastic model of three-phase medium and need further study.