Modelling Flow through Porous Media under Large Pressure Gradients

The most interesting and technologically important problems in the study of flow through porous media involve very high pressures and pressure gradients in the flow do- main such as enhanced oil recovery and carbon dioxide sequestration. The popular Darcy or Brinkman models do not take into account the changes in the fluid properties (like viscosity) due to high pressures and temperatures, or the deformation of the solid itself as the fluid flows through it. We focus on the pressure dependence of viscosity and show that its significance in these problems cannot be neglected. Mixture theory is employed as the tool to develop models for this task. The popular models due to Darcy and Brinkman (and their generalizations) are derived using a general thermodynamic framework which appeals to the criterion of maximal rate of entropy production. Such a thermodynamic approach has been used with great success to describe various classes of material response and here we demonstrate its use within the context of mixture theory. One such generalization of the Brinkman model takes into account the variation of the viscosity and the drag coefficient with the pressure and is used in the problems studied subsequently. We then consider the steady flow of a fluid through a porous slab, driven by a large pressure gradient, and show that the traditional approach that ignores the variation of the viscosity and drag with the pressure greatly over-predicts the mass flux taking place through the porous structure. While incorporating the pressure dependence of viscosity and drag leads to a ceiling flux, the traditional approaches lead to a continued increase in the flux with the pressure difference. The effect of inhomogeneities and anisotropy of the porous medium is investigated by modifying the previous problem to prescribe the drag coefficient as a piecewise constant, positive definite second order tensor. Finally, we allow for the possibility that the flow is unsteady, the viscosity and drag are dependent on the pressure and consider the flow of a fluid due to a pulsatile forcing pressure at one end of a rigid, homogenoues, isotropic solid while the other end is open to the atmosphere. In contrast to certain non-Newtonian fluids where the volumetric flux is enhanced by pulsating the pressure gradient about a non-zero mean value, we find that pulsations in the pressure diminish the volumetric flux in case of the flow through a porous medium when the fluid viscosity is considered to be pressure dependent

Real-Time Maps of Fluid Flow Fields in Porous Biomaterials

Mechanical forces such as fluid shear have been shown to enhance cell growth and differentiation, but knowledge of their mechanistic effect on cells is limited because the local flow patterns and associated metrics are not precisely known. Here we present real-time, noninvasive measures of local hydrodynamics in 3D biomaterials based on nuclear magnetic resonance. Microflow maps were further used to derive pressure, shear and fluid permeability fields. Finally, remodeling of collagen gels in response to precise fluid flow parameters was correlated with structural changes. It is anticipated that accurate flow maps within 3D matrices will be a critical step towards understanding cell behavior in response to controlled flow dynamics.Comment: 23 pages, 4 figure

Pore-scale dynamics and the multiphase Darcy law

Synchrotron x-ray microtomography combined with sensitive pressure differential measurements were used to study flow during steady-state injection of equal volume fractions of two immiscible fluids of similar viscosity through a 57-mm-long porous sandstone sample for a wide range of flow rates. We found three flow regimes. (1) At low capillary numbers, Ca, representing the balance of viscous to capillary forces, the pressure gradient, ∇ P , across the sample was stable and proportional to the flow rate (total Darcy flux) q t (and hence capillary number), confirming the traditional conceptual picture of fixed multiphase flow pathways in porous media. (2) Beyond Ca ∗ ≈ 10 − 6 , pressure fluctuations were observed, while retaining a linear dependence between flow rate and pressure gradient for the same fractional flow. (3) Above a critical value Ca > Ca i ≈ 10 − 5 we observed a power-law dependence with ∇ P ∼ q a t with a ≈ 0.6 associated with rapid fluctuations of the pressure differential of a magnitude equal to the capillary pressure. At the pore scale a transient or intermittent occupancy of portions of the pore space was captured, where locally flow paths were opened to increase the conductivity of the phases. We quantify the amount of this intermittent flow and identify the onset of rapid pore-space rearrangements as the point when the Darcy law becomes nonlinear. We suggest an empirical form of the multiphase Darcy law applicable for all flow rates, consistent with the experimental results

A numerical study of fluids with pressure dependent viscosity flowing through a rigid porous medium

In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution

Phreatic seepage flow through an earth dam with an impeding strip

New mathematical models are developed and corresponding boundary value problems are analytically and numerically solved for Darcian flows in earth (rock)–filled dams, which have a vertical impermeable barrier on the downstream slope. For saturated flow, a 2-D potential model considers a free boundary problem to Laplace’s equation with a traveling-wave phreatic line generated by a linear drawup of a water level in the dam reservoir. The barrier re-directs seepage from purely horizontal (a seepage face outlet) to purely vertical (a no-flow boundary). An alternative model is also used for a hydraulic approximation of a 3-D steady flow when the barrier is only a partial obstruction to seepage. The Poisson equation is solved with respect to Strack’s potential, which predicts the position of the phreatic surface and hydraulic gradient in the dam body. Simulations with HYDRUS, a FEM-code for solving Richards’ PDE, i.e., saturated-unsaturated flows without free boundaries, are carried out for both 2-D and 3-D regimes in rectangular and hexagonal domains. The Barenblatt and Kalashnikov closed-form analytical solutions in non-capillarity soils are compared with the HYDRUS results. Analytical and numerical solutions match well when soil capillarity is minor. The found distributions of the Darcian velocity, the pore pressure, and total hydraulic heads in the vicinity of the barrier corroborate serious concerns about a high risk to the structural stability of the dam due to seepage. The modeling results are related to a “forensic” review of the recent collapse of the spillway of the Oroville Dam, CA, USA

Two-fluid model for the simultaneous flow of colloids and fluids in porous media

To describe the velocities of particles such as ions, protein molecules and colloids dispersed or dissolved in a fluid, it is important to also describe the forces acting on the fluid, including pressure gradients and friction of the fluid with the particles and with the porous media through which the fluid flows. To account for this problem, the use of a two-fluid model is described, familiar in the field of fluid mechanics, extended to include osmotic effects. We show how familiar relationships follow in various situations and give examples of combined fluid/particle transport in neutral and charged membranes driven by a combination of electrostatic, diffusional and pressure forces. The analysis shows how the same modeling framework can be generally used both for multidimensional electrokinetic flow through macroscopic channels and around macroscopic objects, as well as for mean-field modeling of transport through porous media such as gels and membranes