Mathematical modeling and numerical simulation of porous media single-phase fluid flow problem: a scientific review

The complexity of porous media makes the classical methods used to study hydrocarbon reservoirs inaccurate and insufficient to predict the performance and behavior of the reservoir. Recently, fluid flow simulation and modeling used to decrease the risks in the decision of the evaluation of the reservoir and achieve the best possible economic feasibility. This study deals with a brief review of the fundamental equations required to simulate fluid flow through porous media. In this study, we review the derivative of partial differential equations governing the fluid flow through pores media. The physical interpretation of partial differential equations (especially the pressures diffusive nature) and discretization with finite differences are studied. We restricted theoretic research to slightly compressible fluids, single-phase flow through porous media, and these are sufficient to show various typical aspects of subsurface flow numerical simulation. Moreover, only spatial and time discretization with finite differences will be considered. In this study, a mathematical model is formulated to express single-phase fluid flow in a one-dimensional porous medium. The formulated mathematical model is a partial differential equation of pressure change concerning distance and time. Then this mathematical model converted into a numerical model using the finite differences method.

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Estimating the Coefficient of Inertial Resistance in Fluid Flow Through Porous Media

Society of Petroleum Engineers Journal ◽

10.2118/4706-pa ◽

1974 ◽

Vol 14 (05) ◽

pp. 445-450 ◽

Cited By ~ 157

Author(s):

J. Geertsma

Keyword(s):

Fluid Flow ◽

Flow Resistance ◽

Single Phase ◽

Darcy’S Law ◽

Darcy's Law ◽

High Flow ◽

Flow Through Porous Media ◽

Flow Rates ◽

Flow Through ◽

Phase Fluid

Abstract The object of this paper is to introduce an empirical, time-honored relationship between inertia coefficient - frequently misnamed "turbulence factor" - permeability, and porosity, based on a combination of experimental data, dimensional analysis, and other physical considerations. The formula can be used effectively for, among other things, the preliminary evaluation of the number of wells in a new gas field and the spacing between them. Introduction It has long been recognized that Darcy's law for single-phase fluid flow through porous media,Equation 1 in which ?=superficial velocity µ=fluid viscosity k=formation permeability p=pressure head, is approximately correct only in a specific flow regime where the velocity ? is low. Single-phase fluid flow in reservoir rocks is often characterized by conditions in favor of this linearized flow law, but important exceptions do occur. They are in particular related to the surroundings of wells producing at high flow rates such as gas wells. For the prediction or analysis of the production behavior of such wells it is necessary to apply a more general nonlinear flow law. The appropriate formula was given in 1901 by Forchheimer1; it readsEquation 2 in which ?=density a=coefficient of viscous flow resistance 1/k ß=coefficient of inertial flow resistance. This equation indicates that in single-phase fluid flow through a porous medium two forces counteract the external force simultaneously - namely, viscous and inertial forces - the latter continuously gaining importance as the velocity ? increases. For low flow rates the viscous term dominates, whereas for high flow rates the inertia term does. The upper limit of practical applicability of Darcy's law can best be specified by some "critical value" orf the dimensionless ratio.Equation 3 which has a close resemblance to the Reynolds number. Observe that ß/a has the dimension of a length. Inertia and Turbulence As the Reynolds number is commonly used as an indicator for either laminar or turbulent flow conditions, the coefficient ß is often referred to as the turbulence coefficient. However, the phenomenon we are interested in has nothing to do with turbulence. The flow regime of concern is usually fully laminar. The observed departure from Darcy's law is the result of convective accelerations and decelerations of the fluid particles on their way through the pore space. Within the flow range normally experienced in oil and gas reservoirs, including the well's surroundings, energy losses caused by actual turbulence can be safely ignored.

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