A universal expression of the equivalent permeability of heterogeneous porous media

Flow and transport inside porous media is a common physical process, and it is related to subsurface engineering, geothermal energy extraction, environment engineering and so on. Due to the complexity of geological conditions, strata suffer diagenesis, compaction, weathering, biological effect, etc. The feature of permeability is rather complex, and numerical simulation is utilized to supply data and key information for engineering design and construction, with a larger amount of time and computation. However, the accuracy of simulation directly relies on the key parameters such as permeability. This paper aims to study the relationship of the equivalent permeability and permeability distribution function in the space of heterogeneous porous media. Based on the continuum assumption, Darcy’s law and permeability distribution function, the mathematical model of the Darcy seepage problem from 1D to 3D is established, and the theoretical solution of the equivalent permeability is obtained by solving the partial differential equation. Besides, the theoretical solution and the numerical solution are obtained by using finite element simulation. The results show that they are closed to each other, and the error is quite small, indicating that the expression of the equivalent permeability is correct and valid. The permeability distribution function can be established through multi-points permeability measurements of media, then the seepage properties of the whole porous media can be simulated and evaluated rapidly. The complex geological models are simplified to reduce the computation with the expressions. It is of great significance for the rapid simulation of engineering and the evaluation of simulation results.