On the Importance of the Stokes-Brinkman Equations for Computing Effective Permeability in Karst Reservoirs

Abstract

Cavities and fractures significantly affect the flow paths in carbonate reservoirs and should be accurately accounted for in numerical models. Herein, we consider the problem of computing the effective permeability of rock samples based on high-resolution 3D CT scans containing millions of voxels. We use the Stokes-Brinkman equations in the entire domain, covering regions of free flow governed by the Stokes equations, porous Darcy flow, and transitions between them. The presence of different length scales and large (ten orders of magnitude) contrasts in permeability leads to highly ill-conditioned linear systems of equations, which are difficult to solve. To obtain a problem that is computationally tractable, we first analyze the relative importance of the Stokes and Darcy terms for a set of idealized 2D models. We find that, in terms of effective permeability, the Stokes-Brinkman equations are only applicable for a special parameter set where the effective free-flow permeability is less than four orders of magnitude different from the matrix permeability. All other cases can be accurately modeled with either the Stokes or the Darcy end-member flows, depending on if there do or do not exist percolating free-flow regions. The insights obtained are used to perform a direct computation of the effective permeability of a rock sample model with more than 8 million cells.

Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

“…Their consistent formulation, using the TRT-LBM framework, was established in a series of works Ginzburg 2008Ginzburg , 2014. At the same time, the FEM simulation of Brinkman flows has also found vast application (Krotkiewski et al 2011;Hannukainen et al 2011). A detailed comparison on the specificities of both numerical approaches in this class of problems was performed in the recent works Silva and Ginzburg 2015).…”

“…To conclude with this introductory section, let us physically illustrate the above set of parameters by considering a porous medium, as described in Krotkiewski et al (2011), which is composed of an ensemble of fractured calcite rocks of dimensionā ∼ 0.01 [m] and permeabilityk 1 ∼ 10 −7 [m 2 ] embedded in a sandstone environment of permeabilitȳ k 2 ∼ 10 −10 [m 2 ]. Then, typical related Darcy numbers for this configuration are σ 2 1 10 3 and σ 2 2 10 6 (assuming for simplicity f 1 (φ 1 ) = f 2 (φ 2 ) = 1).…”

“…The aforementioned reasons explain why the Stokes-Brinkman-Darcy model is widely used, both as theoretical and numerical framework to describe the conservation equations of fluid flow in porous media, e.g. Krotkiewski et al (2011).…”

Flow simulation through porous ceramics used as a throttle in an implantable infusion pump

“…In the permeable regions, the equation is governed by Darcy flow, whereas in large voids or complete free flow regions, tends to infinity, reducing equation 2 (in vector notation) to Stokes flow [5].…”

Stokes–Brinkman formulation for prediction of void formation in dual-scale fibrous reinforcements: a BEM/DR-BEM simulation

“…According to Ochoa-Tapia and Whitaker [40], Brinkman approximation can be applied when the following three length scale constrains are simultaneously satisfied: r 0 2 / l ε l p 1 , r 0 2 / (l ε l v ) 1, l f r 0 , where l ε , l p , l v and l f are the characteristic lengths associated to the porosity, pressure, velocity gradients and fluid phase, respectively, and r 0 is the characteristic length scale of the Representative Elementary Volume (REV). Krotkiewski et al [28] reported a direct numerical simulation of the flow field in homogeneous two dimensional porous media with characteristic length L and permeability K , concluding that the Stokes solution is dominant for K /L 2 ≥ 10, Darcy law is representative of the flow field if K /L 2 ≤ 10 −4 , while for 10 −4 ≤ K /L 2 ≤ 10 the Brinkman approximation should be used to account for the transition between both flow regimes, Stokes and Darcy.…”