Numerical solutions for flow in porous media

2003 ◽  
Vol 27 (7) ◽  
pp. 565-583 ◽  
Author(s):  
J.G. Wang ◽  
C.F. Leung ◽  
Y.K. Chow
Keyword(s):  
Porous Media ◽  
Numerical Solutions ◽  
Flow In Porous Media

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.

Evaluation of Machine Learning Methodologies Using Simple Physics Based Conceptual Models For Flow in Porous Media

2021 ◽  
Author(s):  
Daulet Magzymov ◽  
Ram R. Ratnakar ◽  
Birol Dindoruk ◽  
Russell T. Johns
Keyword(s):  
Machine Learning ◽  
Porous Media ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Building Blocks ◽  
Flow In Porous Media ◽  
Computational Techniques ◽  
Physical Parameters ◽  
Flow Physics ◽  
Suitable Form

Abstract Machine learning (ML) techniques have drawn much attention in the engineering community due to recent advances in computational techniques and an enabling environment. However, often they are treated as black-box tools, which should be examined for their robustness and range of validity/applicability. This research presents an evaluation of their application to flow/transport in porous media, where exact solutions (obtained from physics-based models) are used to train ML algorithms to establish when and how these ML algorithms fail to predict the first order flow-physics. Exact solutions are used so as not to introduce artifacts from the numerical solutions. To test, validate, and predict the physics of flow in porous media using ML algorithms, one needs a reliable set of data that may not be readily available and/or the data might not be in suitable form (i.e. incomplete/missing reporting, metadata, or other relevant peripheral information). To overcome this, we first generate structured datasets for flow in porous media using simple representative building blocks of flow physics such as Buckley-Leverett, convection-dispersion equations, and viscous fingering. Then, the outcomes from those equations are fed into ML algorithms to examine their robustness and predictive strength of the key features, such as breakthrough time, and saturation and component profiles. In this research, we show that a physics-informed ML algorithm can capture the physical behavior and effects of various physical parameters (even when shocks and sharp gradients are present). Further the ML approach can be utilized to solve inverse problems to estimate physical parameters.

Numerical Solutions of the Equations for One-Dimensional Multi-Phase Flow in Porous Media

1966 ◽  
Vol 6 (01) ◽  
pp. 62-72 ◽  
Author(s):  
Byron S. Gottfried ◽  
W.H. Guilinger ◽  
R.W. Snyder
Keyword(s):  
Porous Media ◽  
Numerical Methods ◽  
Numerical Solutions ◽  
Flow In Porous Media ◽  
Phase Flow ◽  
Two Phase ◽  
One Dimensional ◽  
Three Phase ◽  
Three Phase Flow ◽  
The One

Abstract Two numerical methods are presented for solving the equations for one-dimensional, multiphase flow in porous media. The case of variable physical properties is included in the formulation, although gravity and capillarity are ignored. Both methods are analyzed mathematically, resulting in upper and lower bounds for the ratio of time step to mesh spacing. The methods are applied to two- and three-phase waterflooding problems in laboratory-size cores, and resulting saturation and pressure distributions and production histories are presented graphically. Results of the two-phase flow problem are in agreement with the predictions of the Buckley-Leverett theory. Several three-phase flow problems are presented which consider variations in the water injection rate and changes in the initial oil- and water-saturation distributions. The results are different physically from the two-phase case; however, it is shown that the Buckley-Leverett theory can accurately predict fluid interface velocities and displacing-fluid frontal saturations for three-phase flow, providing the correct assumptions are made. The above solutions are used as a basis for evaluating the numerical methods with respect to machine time requirements and allowable time step for a fixed mesh spacing. Introduction Considerable progress has been made in recent years in obtaining numerical solutions of the equations for two-phase flow in porous media. Douglas, Blair and Wagner2 and McEwen11 present different methods for solving the one-dimensional case for incompressible fluids with capillarity (the former using finite differences, the latter with an approach based upon characteristics). Fayers and Sheldon4 and Hovanesian and Fayers8 have extended these studies to include the effects of gravity. West, Garvin and Sheldon,14 in a pioneer paper, treat linear and radial systems with both capillarity and gravity and they also include the effects of compressibility. Douglas, Peaceman and Rachford3 consider two-dimensional, two-phase, incompressible flow with gravity and capillarity and Blair and Peaceman1 have extended this method to allow for compressible fluids. No one, however, has examined the case of three-phase flow, even for the relatively simple case of one-dimensional flow of incompressible fluids in the absence of gravity and capillarity. In obtaining a numerical technique for simulating forward in situ combustion laboratory experiments, Gottfried5 has developed a method for solving the one-dimensional, compressible flow equations with any number of flowing phases. Gravity and capillarity are not included in the formulation. The method has been used successfully, however, for two- and three-phase problems in a variable-temperature field with sources and sinks. This paper examines the algorithm of Gottfried more critically. Two numerical methods are presented for solving the one-dimensional, multi-phase flow equations with variable physical properties. Both methods are analyzed mathematically, and are used to simulate two- and three-phase waterflooding problems. The numerical solutions are then taken as a basis for comparing the utility of the methods. Problem Statement Consider a one-dimensional system in which capillarity, gravity and molecular diffusion are negligible. If n immiscible phases are present, n 2, the equation describing the flow of the ith phase is:12Equation 1 where all terms can vary with x and t.

scholarly journals Macroscopic model for unsteady flow in porous media

2019 ◽  
Vol 862 ◽  
pp. 283-311 ◽  
Author(s):  
Didier Lasseux ◽  
Francisco J. Valdés-Parada ◽  
Fabien Bellet
Keyword(s):  
Porous Media ◽  
Unsteady Flow ◽  
Numerical Solutions ◽  
Creeping Flow ◽  
Flow In Porous Media ◽  
Macroscopic Model ◽  
Initial Flow ◽  
Effective Coefficients ◽  
Wide Range ◽  
Non Local

The present article reports on a formal derivation of a macroscopic model for unsteady one-phase incompressible flow in rigid and periodic porous media using an upscaling technique. The derivation is carried out in the time domain in the general situation where inertia may have a significant impact. The resulting model is non-local in time and involves two effective coefficients in the macroscopic filtration law, namely a dynamic apparent permeability tensor,$\unicode[STIX]{x1D643}_{t}$, and a vector,$\unicode[STIX]{x1D736}$, accounting for the time-decaying influence of the flow initial condition. This model generalizes previous non-local macroscale models restricted to creeping flow conditions. Ancillary closure problems are provided, which allow the effective coefficients to be computed. Symmetry and positiveness analyses of$\unicode[STIX]{x1D643}_{t}$are carried out, showing that this tensor is symmetric only in the creeping regime. The effective coefficients are functions of time, geometry, macroscopic forcings and the initial flow condition. This is illustrated through numerical solutions of the closure problems. Predictions are made on a simple periodic structure for a wide range of Reynolds numbers smaller than the critical value characterizing the first Hopf bifurcation. Finally, the performance of the macroscopic model for a variety of macroscopic forcings and initial conditions is examined in several case studies. Validation through comparisons with direct numerical simulations is performed. It is shown that the purely heuristic classical model, widely used for unsteady flow, consisting of a Darcy-like model complemented with an accumulation term on the filtration velocity, is inappropriate.

scholarly journals Numerical Simulation Using Finite-Difference Schemes with Continuous Symmetries for Processes of Gas Flow in Porous Media

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 45 ◽  
Author(s):  
Markov ◽  
Rodionov
Keyword(s):  
Porous Media ◽  
Gas Flow ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Flow In Porous Media ◽  
Symmetry Groups ◽  
Continuous Symmetry ◽  
Parabolic Differential Equations ◽  
Solution Generation ◽  
Continuous Symmetries

This article presents the applications of continuous symmetry groups to the computational fluid dynamics simulation of gas flow in porous media. The family of equations for one-phase flow in porous media, such as equations of gas flow with the Klinkenberg effect, is considered. This consideration has been made in terms of difference scheme constructions with the preservation of continuous symmetries, which are presented in original parabolic differential equations. A new method of numerical solution generation using continuous symmetry groups has been developed for the equation of gas flow in porous media. Four classes of invariant difference schemes have been found by using known group classifications of parabolic differential equations with partial derivatives. Invariance of necessary conditions for stability has been shown for the difference schemes from the presented classes. Comparison with the classical approach for seeking numerical solutions for a particular case from the presented classes has shown that the calculation speed is greater by several orders than for the classical approach. Analysis of the accuracy for the presented method of numerical solution generation on the basis of continuous symmetries shows that the accuracy of generated numerical solutions depends on the accuracy of initial solutions for generations.

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