On oscillating flows in randomly heterogeneous porous media

Abstract

The emergence of structure in reactive geofluid systems is of current interest. In geofluid systems, the fluids are supported by a porous medium whose physical and chemical properties may vary in space and time, sometimes sharply, and which may also evolve in reaction with the local fluids. Geofluids may also experience pressure and temperature conditions within the porous medium that drive their momentum relations beyond the normal Darcy regime. Furthermore, natural geofluid systems may experience forcings that are periodic in nature, or at least episodic. The combination of transient forcing, nearcritical fluid dynamics and heterogeneous porous media yields a rich array of emergent geofluid phenomena that are only now beginning to be understood. One of the barriers to forward analysis in these geofluid systems is the problem of data scarcity. It is most often the case that fluid properties are reasonably well known, but that data on porous medium properties are measured with much less precision and spatial density. It is common to seek to perform an estimation of the porous medium properties by an inverse approach, that is, by expressing porous medium properties in terms of observed fluid characteristics. In this paper, we move toward such an inversion for the case of a generalized geofluid momentum equation in the context of time-periodic boundary conditions. We show that the generalized momentum equation results in frequency-domain responses that are governed by a second-order equation which is amenable to numerical solution. A stochastic perturbation approach demonstrates that frequency-domain responses of the fluids migrating in heterogeneous domains have spatial spectral densities that can be expressed in terms of the spectral densities of porous media properties.

Stochastic relationships for periodic responses in randomly heterogeneous aquifers

“…It is also possible to represent the storativity S as a spatial random process, but here, for simplicity, we assume that S is uniform and constant throughout the domain. In this derivation we follow the general approach of Trefry et al [2010] to deduce a spectral transfer function between the stochastic form of κ and the resulting head distribution. If κ is statistically stationary, then we can write , where the effective mean of the conductivity satisfies .…”

“…To progress further, we consider the spatial spectral characteristics of determined by application of the spatial Fourier transformation , defined as where k is the wave vector associated with coordinate vector x. Gradients of the mean solution appear in , which complicates further reduction. The second‐order term may be simplified by integrating by parts and employing a proximity assumption [ Trefry et al , 2010], yielding In , k 2 = | k | 2 , and is a constant vector chosen to approximate the slope of the mean solution near the forcing boundary (the proximity assumption; see for an estimation procedure for ). This simplifying assumption allows the stationary spectral approach of Bakr et al [1978] to be employed.…”

“…where k is the wave vector associated with coordinate vector x. Gradients of the mean solution <ĥ > appear in (16), which complicates further reduction. The second-order term may be simplified by integrating by parts and employing a proximity assumption [Trefry et al, 2010], yielding…”

“… Srivastava et al [2006] employed a stochastic Boussinesq equation to model transient stream‐aquifer interactions. More recently, Trefry et al [2010] used a perturbation approach to estimate the stochastic filtration relationship between the permeability and dependent variable spectral densities for a model viscoelastic fluid subject to oscillating boundaries. In that work, Darcian flows were represented as a special case of a linear viscoelastic filtration law.…”

A Single-Pole Approximation to Interfacial Mass Transfer in Porous Media Augmented with Bulk Reactions

“…Therefore, we consider a Darcy-Brinckman flow equation (rather than Darcy flow equation) Fig. 1 Sketch of the problem adopted from Sookhak Lari et al (2015), including two immiscible fluids (with the thickness ratio h 1 / h 2 ) flowing through an inclined homogeneous isotropic porous medium to better resolve the velocity profiles at the interface between the fluids (Vafai and Tien 1981;Wang 2009;Brinkman 1949;Trefry et al 2010). An integral Laplace transformation approach is adopted to solve the mass transport equation.…”