Network-inspired versus Kozeny-Carman based permeability-porosity relations applied to Biot's poroelasticity model

Water injection in the aquifer induces deformations in the soil. These mechanical deformations give rise to a change in porosity and permeability, which results in non-linearity of the mathematical problem. Assuming that the deformations are very small, the model provided by Biot's theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. In this continuum scale model, the Kozeny-Carman equation is commonly used to determine the permeability of the porous medium from the porosity. The Kozeny-Carman relation states that flow through the pores is possible at a certain location as long as the porosity is larger than zero at this location in the aquifer. However, from network models it is known that percolation thresholds exist, indicating that the permeability will be equal to zero if the porosity becomes smaller than these thresholds. In this paper, the relationship between permeability and porosity is investigated. A new permeability-porosity relation, based on the percolation theory, is derived and compared with the Kozeny-Carman relation. The strongest feature of the new approach is related to its capability to give a good description of the permeability in case of low porosities. However, with this network-inspired approach small values of the permeability are more likely to occur. Since we show that the solution of Biot's model converges to the solution of a saddle point problem for small time steps and low permeability, we need stabilisation in the finite element approximation.