Convergence of the Finite Element Method for the Porous Media Equation with Variable Exponent

2013 ◽  
Vol 51 (6) ◽  
pp. 3483-3504 ◽  
Author(s):  
José C. M. Duque ◽  
Rui M. P. Almeida ◽  
Stanislav N. Antontsev
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Variable Exponent ◽  
Porous Media Equation ◽  
The Finite Element Method ◽  
Media Equation ◽  
Element Method

scholarly journals Finite element modeling of fluid flow in fractured porous media using unified approach

2020 ◽  
Vol 43 (1) ◽  
pp. 13-22
Author(s):  
Hai-Bang Ly ◽  
Hoang-Long Nguyen ◽  
Minh-Ngoc Do
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Fractured Porous Media ◽  
Flow In Porous Media ◽  
Soil Science ◽  
The Finite Element Method ◽  
Element Method

Understanding fluid flow in fractured porous media is of great importance in the fields of civil engineering in general or in soil science particular. This study is devoted to the development and validation of a numerical tool based on the use of the finite element method. To this aim, the problem of fluid flow in fractured porous media is considered as a problem of coupling free fluid and fluid flow in porous media or coupling of the Stokes and Darcy equations. The strong formulation of the problem is constructed, highlighting the condition at the free surface between the Stokes and Darcy regions, following by the variational formulation and numerical integration using the finite element method. Besides, the analytical solutions of the problem are constructed and compared with the numerical solutions given by the finite element approach. Both local properties and macroscopic responses of the two solutions are in excellent agreement, on condition that the porous media are sufficiently discretized by a certain level of finesse. The developed finite element tool of this study could pave the way to investigate many interesting flow problems in the field of soil science.

Application of the Finite Element Method to Transient Flow in Porous Media

1968 ◽  
Vol 8 (03) ◽  
pp. 241-252 ◽  
Author(s):  
Iraj Javandel ◽  
P.A. Witherspoon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Steady State ◽  
Transient Flow ◽  
Energy Functional ◽  
Flow In Porous Media ◽  
The Finite Element Method ◽  
Potential Energy Functional ◽  
Element Method

Abstract The finite element method was originally developed in the aircraft industry to handle problems of stress distribution in complex airframe configurations. This paper describes how the method can be extended to problems of transient flow in porous media. In this approach, the continuum is replaced by a system of finite elements. By employing the variational principle, one can obtain time dependent solutions for the potential at every point in the system by minimizing a potential energy functional. The theory of the method is reviewed. To demonstrate its validity, nonsteady-state results obtained by the finite element method are compared with those of typical boundary value problems for which rigorous analytical solutions are available. To demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. The generality of this approach with respect to arbitrary boundary conditions and changes in rock properties provides a new method of handling properties provides a new method of handling problems of fluid flow in complex systems. problems of fluid flow in complex systems Introduction Problems of transient flow in porous media often can be handled by the methods of analytical mathematics as long as the geometry or properties of the flow system do not become too complex. When the analytical approach becomes intractable, it is customary to resort to numerical methods, and a great variety of problems have been handled in this manner. One such method relies on the finite difference approach Wherein the system is divided into a network of elements, and a finite difference equation for the flow into and out of each element is developed. The solution of the resulting set of equations usually requires a high speed computer. When heterogeneous systems of arbitrary geometry must be considered, however, this approach is sometimes difficult to apply and may require large amounts of computer time. The finite element method is a new approach that avoids these difficulties. It was developed originally in the aircraft industry to provide a refined solution for stress distributions in extremely complex airframe configurations. Clough has recently reviewed the application of the finite element method in the field of structural mechanics The technique has been applied successfully in the stress analysis of many complex structures. Recognition that this procedure can be interpreted in terms of variational procedures involving minimizing a potential energy functional leads naturally to its extension to other boundary value problems. problems. In the field of heat flow, there recently have been introduced several approximate methods of solution that are based on variational principles. By employing the variational principle in conjunction with the finite element idealization, a powerful solution technique is now available for determining the potential distribution within complex bodies of arbitrary geometry. In the finite element approximation of solids, the continuum is replaced by a system of elements. An approximate solution for the potential field within each element is assumed, and flux equilibrium equations are developed at a discrete number of points within the network of finite elements. For the case of steady-state heat flow, the technique is completely described by Zienkiewicz and Cheung. Since the flow of fluids in porous media is analogous to the flow of heat, Zienkiewicz et al. have employed the finite element method in obtaining steady-state solutions to heterogeneous and anisotropic seepage problems. Taylor and Brown have used this method to investigate steady-state flow problems involving a free surface. The work of Gurtin has been instrumental in laying the groundwork for the application of finite element methods to linear initial-value problems. SPEJ P. 241

Finite-Element Analysis of Temperature and Thermal Stresses Induced by Hot Water Injection

1978 ◽  
Vol 18 (06) ◽  
pp. 457-469 ◽  
Author(s):  
T. Aktan ◽  
S.M. Farouq Ali
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Fluid Flow ◽  
Thermal Stresses ◽  
Water Injection ◽  
Hot Water ◽  
The Finite Element Method ◽  
Heat And Fluid Flow ◽  
Element Method

Abstract Results of an experimental study of the temperature and pressure dependence of rock elastic moduli were used in a numerical mathematical model that describes the unsteady-state, two-dimensional flow of a single-phase, single-component, slightly compressible hot fluid and that calculates the state of stress in porous media. The numerical formulation was conducted using the finite-element method of triangular elements to discretize the space domain and backward differencing for discretizing the time domain. Sections of the model dealing with the flow of mass and heat, and stress calculations were tested separately then, the complete model was used to investigate the effect of temperature-dependent Young's moduli and Poisson's ratios on the state of stress and the propagation of thermal and pore stresses with an advancing hot water front in porous media. porous media Introduction The finite-element method forms the basis for a versatile analysis procedure applicable to problems in several different fields. Essentially, the finite-element idealization replaces the continuum with a finite number of discrete elements. Geometry of the elements is defined by a set of spatial points (called nodal points) of the system. Heat and fluid-flow problems are analyzed by using interpolation functions for the unknown temperatures and pressures. Thus, the variables within elements are pressures. Thus, the variables within elements are defined completely in terms of the values at nodal points. The isoparametric element concept, where points. The isoparametric element concept, where the geometry and displacements of the element are described in terms of the same parameters and are of the same order, is particularly useful for stress analysis in continuous bodies. The investigation of thermal stresses may be found helpful when answering questions associated with heated reservoir rocks. Creation of microfactures in formations subjected to elevated temperatures often results in increased permeabilities. A correlation between the thermal stresses induced by these high temperatures and the degree of microfracturing affecting the rock absolute permeability could be useful in understanding fluid flow permeability could be useful in understanding fluid flow in heated reservoirs. It also would be interesting to see if thermal stresses have any effect on the casing failures in steam injection or geothermal wells. The analysis under consideration involves the determination of pressures, temperatures, and stresses in a porous medium undergoing hot-water injection. The study is complicated further by the transient nature of the variables. Background The finite-element method has been used widely for studying stresses in various kinds of structures. The basic concepts of the method, first laid down by Turner et al, essentially are composed of the idealization of continuous bodies by a set of interconnected finite elements with known behavioral characteristics. Recently, this versatile method was applied to diverse kinds of engineering problems, mainly in heat and fluid-flow areas. Desai problems, mainly in heat and fluid-flow areas. Desai and Abel list the fields where the method may be applied. Visser solved the unsteady-state heat conduction equations, using a variational approach. He solved for temperature distribution, also calculating the associated stresses by determining the displacements occurring in the body due to this temperature field. Later, Zienkiewicz and Parekh used two- and three-dimensional curved isoparametric elements to solve the same equation by using the Galerkin principle. SPEJ P. 457

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