Digital Rock Physics - Effective Permeability of Rocks
Further info
Have a look at the digital rock portal to get a better idea of what Digital Rock Physics is all about!
Theoretical Background
Flow in porous media occurs at various scales, and different theoretical frameworks are used to describe flow depending on the scale. Direct simulations are based on solving the Navier-Stokes equations using a discretized representation (mesh) of the pore space. These simulations provide a detailed description of the flow field within the pore network. Naturally, such simulations are computationally demanding, as the nano- to micrometer-sized pores must be fully resolved. The simplified Navier-Stokes equations for incompressible, steady-state flow are:
Here, equation (12) is the continuity equation (expressing mass conservation), and equation (13) is the momentum balance equation. \(\nu\) is the kinematic viscosity, and \(p\) is the kinematic pressure.
Tip
There are many resources on the derivation of the Navier–Stokes equations, including several tailored to OpenFOAM. We particularly recommend Tobias Holzmann’s book. Also see the new book by CFD Direct. There is also a great pdf on the relationship between Navier-Stokes and Darcy flow on the webpages of Cyprien Soulaine.
An alternative approach is continuum simulation. In continuum models, volume-averaged properties are used to describe flow behavior. The two key parameters in such models are porosity and permeability. Porosity represents the void fraction within a control volume, while permeability quantifies how easily fluids can flow through it under a pressure gradient. Although these properties are correlated, the exact relationship often needs to be determined empirically or estimated using Digital Rock Physics methods.
Flow at the continuum scale is described by Darcy’s law:
Here, \(\vec{u}\) is the Darcy velocity, which is the volume-averaged flow velocity through the porous medium. We will discuss this further, but note that it relates to the pore velocity via \(\vec{u} = \epsilon \vec{v}\), where \(\epsilon\) is the porosity and \(\vec{v}\) the actual fluid velocity in the pores. The parameters \(\mu_f\) and \(k\) are the fluid’s dynamic viscosity and the medium’s permeability, respectively.
Extensive theoretical work has clarified the relationship between Navier-Stokes flow and Darcy flow, the latter originally proposed as a phenomenological equation.
Note that direct and continuum simulations are not per se related to absolute scales (like mm vs meters) but simply refer to differing relative length scales / processes that are resolved.
Important note
It is important to note that the absolute values of porosity and permeability in porous media are not universal constants but depend on the choice of the Representative Elementary Volume (REV). Since the REV defines the scale over which microscale heterogeneities are averaged, different choices can lead to significantly different estimates of these effective properties. This scale dependence means that porosity and permeability are not always independently observable — observational data often reflect a combined, scale-dependent response of the medium, making it difficult to infer one property without assumptions about the other. In media with strong heterogeneities or multi-scale structures, identifying a meaningful REV is particularly challenging. These limitations should be kept in mind when interpreting or applying continuum-scale flow models like Darcy’s law.