Upscaling of Forchheimer flows

Transport in Porous Media, 2015

In this work, we revisit the upscaling process of diffusive mass transfer of a solute undergoing a homogeneous reaction in porous media using the method of volume averaging. For linear reaction rate kinetics, the upscaled model exhibits a vis-à-vis correspondence with the mass transfer governing equation at the microscale. When nonlinear reactions are present, other methods must be adopted to upscale the nonlinear term. In this work, we explore a linearization approach for the purpose of solving the associated closure problem. For large rates of nonlinear reaction relative to diffusion, the effective diffusion tensor is shown to be a function of the reaction rate, and this dependence is illustrated by both numerical and analytical means. This approach leads to a macroscale model that also has a similar structure as the microscale counterpart. The necessary conditions for the vis-à-vis correspondence are clearly identified. The validation of the macroscale model is carried out by comparison with pore-scale simulations of the microscale transport process. The predictions of both concentration profiles and effectiveness factors were found to be in acceptable agreement. In an appendix, we also briefly discuss an integral formulation of the nonlinear problem that may be useful in developing more accurate results for the upscaled transport and reaction equations; this approach requires computing the Green function corresponding to the linear transport problem.

2010

Transport in Porous Media, 2007

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.

2012

Abstract We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure.

2011

The upscaling process of mass transport with chemical reaction in porous media is carried out using the method of volume averaging under diffusive and dispersive conditions. We study cases in which the (first-order) reaction takes place in the fluid phase that saturates the porous medium or when the reaction occurs at the solid-fluid interface. The upsca1ing effort leads to average transport equations, which are expressed in terms of effective medium coefficients for ( diffusive or dispersive) mass transport and reaction that are computed by solving the associated closure problems in representative unit cells. Our derivations show that mass transport effective coefficients depend, in general, of the nature and magnitude of the microscopic reaction rate coefficient as well as of the essential geometrical structure of the sol id matrix and the flow rate. Furthermore, if the chemical reaction is homogeneous, the effective reaction rate coefficient is found to be simply the multiplication of its microscopic counterpart with the porosity; whereas, if the reaction is heterogeneous, the effective reaction coefficient is determined from a closure problem solution.

2005

This book provides concise, up-to-date and easy-to-follow information on certain aspects of an ever important research area: multiphase flow in porous media. This flow type is of great significance in many petroleum and environmental engineering problems, such as in secondary and tertiary oil recovery, subsurface remediation and CO2 sequestration. This book contains a collection of selected papers (all refereed) from a number of well-known experts on multiphase flow. The papers describe both recent and state-of-the-art modeling ...

Mathematical Models and Methods in Applied Sciences, 2009

Motivated by the reservoir engineering concept of the well Productivity Index, we introduced and analyzed a functional, denoted as \di®usive capacity", for the solution of the initi-alÀboundary value problem (IBVP) for a linear parabolic equation. 21 This IBVP described laminar (linear) Darcy°ow in porous media; the considered boundary conditions corresponded to di®erent regimes of the well production. The di®usive capacities were then computed as steady-state invariants of the solutions to the corresponding time dependent boundary value problem.

A brief reference to various non-linear forms of relation between hydraulic gradient and velocity of flow through porous media is presented, followed by the justification of the use of Forchheimer equation. In order to study the nature of coefficients of this equation, an experimental programme was carried out under steady state conditions, using a specially designed permeameter. Eight sizes of coarse material and three sizes of glass spheres are used as media with water as the fluid medium. Equations for linear and non-linear parameters of Forchheimer equation are proposed in terms of easily measurable media properties. These equations are presented in the form of graphs as quick reckoners.

Computational Geosciences

Geological models are becoming increasingly large and detailed to account for heterogeneous structures on different spatial scales. To obtain simulation models that are computationally tractable, it is common to remove spatial detail from the geological description by upscaling. Pressure and transport equations are different in nature and generally require different strategies for optimal upgridding. To optimize the accuracy of a transport calculation, the coarsened grid should generally be constructed based on a posteriori error estimates and adapt to the flow patterns predicted by the pressure equation. However, sharp and rigorous estimates are generally hard to obtain, and herein we therefore consider various ad hoc methods for generating flow-adapted grids. Common for all is that they start by solving a single-phase flow problem once and then continue to form a coarsened grid by amalgamating cells from an underlying fine-scale grid. We present several variations of the original method. First, we discuss how to include a priori information in the coarsening process, e.g. to adapt to special geological features or to obtain less irregular grids in regions where flow-adaption is not crucial. Second, we discuss the use of bi-directional versus net fluxes over the coarse blocks and show how the latter gives systems that better represent the causality in the flow equations, which can be exploited to develop very efficient nonlinear solvers. Finally, we demonstrate how to improve simulation accuracy by dynamically adding local resolution near strong saturation fronts.