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

Abstract

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.

The influence of grain shape and size on the relationship between porosity and permeability in sandstone: a digital approach

“…Whilst consideration of grain characteristics is shown to be detrimental in the range of study samples analysed here, further investigation of a wider range of materials would allow it to be determined whether this is representative of all sedimentary materials. The poorer quality of fit determined using grain characteristics may be a result of using a Kozeny–Carman fit equation which makes the assumption that grains are spherical producing a simple pore structure 49 . Bear 8 describes how this assumption arises from the transformation of the specific surface area term 2 to a characteristic grain size term.…”

A decoupling and linearizing discretization for weakly coupled poroelasticity with nonlinear permeability

“…Therein, the permeability depends quadratically on the dilatation, which turns out to be suitable to describe the flow of a Newtonian fluid through cylindrical pores. Finally, we also mention the networkinspired permeability as described in [RLVM20] and the exponential dependence on the displacement as investigated in [LM80,HM90], again in the context of biological tissues.…”

“…[Bal87,Won88]. Considering the possibility of channels being randomly closed or open, a permeability-porosity relation is presented in [RLVM20] for an arbitrary network topology that reads…”

“…As computational domain we consider the unit square D = (0, 1) 2 . Inspired by the examples in [RLVM20], we further choose the remaining parameters as λ = 2.019 • 10 7 , µ = 1.346 • 10 7 , α = 0.8, M = 10, ν = 10 −6 , T = 1.…”

A moving finite element framework for fast infiltration in nonlinear poroelastic media

“…When using the finite element method to solve the poroelastic equations, the main challenge is to ensure convergence of the method and to prevent numerical instabilities that often manifest themselves in the form of spurious oscillations in the pressure field. It has been proved that this problem is caused by the saddle point structure in the coupled equations resulting in a violation of the famous Ladyzhenskaya-Babuska-Brezzi (LBB) condition [34], thus highlighting the need for a stable combination of mixed finite elements [22] such as the popular Taylor-Hood element.…”