Abstract
The purpose of this paper is to develop the numerical equations for the Finite Difference Method (FDM), the Integral Finite Difference Method (IFDM) and the Finite Element Method (FEM). The numerical equations formulated will incorporate appropriate provisions for handling such special features as boundary conditions, initial conditions, heterogeneities, anisotropy, sources, time- or potential-dependent changes in parameters and so on. The next step is to describe the methods by which various differencing schemes can be employed to march in the time domain. The end-product of these efforts is an assemblage of conservation equations for each discrete subspace of the flow region of interest. If desired, these equations can be organized as a matrix of equations which can be solved by direct or iterative methods of solution.
Numerical equations are sufficiently general in their structure to include porous or/and fractured materials. Therefore, a discussion about the manner in which either type of material can be handled in a generalized numerical formulation is of considerable interest.
The objective of this study is to provide an overall conceptual-mathematical framework in which the different numerical methods can be compared and contrasted. The treatment will be restricted to isothermal fluid-flow in saturated-unsaturated deformable media. This class of problems is usually written in the form of a non-linear parabolic differential equation. It is hoped that this section will serve as a useful introduction to the next two papers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
List of Pertinent References
Barenblatt, G.E., Zheltov, I.P. and Kochina, I.N. (1960) Basic Concepts in the Theory of Homogeneous Liquids. J. Applied Math, USSR, 24: 5, 1286–1303.
Duguid, J.O. and Lee, P.C.Y. (1977) Flow in Fractured Porous Media. Water Resources Res., 13: 3, 558–566.
Edwards, A.L. (1969) TRUMP: A Computer Program for Transient and Steady State Temperature Distributions in Multidimensional Systems. National Technical Information Service, Springfield, VA., 265.
Ergatoudis, I., Irons, B.M. and Zienkiewicz, O.C. (1968) Curved Isoparametric, “Quadrilateral” Elements for Finite Element Analysis. Int. J. Solids Structures, 4, 31–42.
Finlayson, B.A. and Scriven, L.E. (1967) On the Search for Variational Principles. Int. Jour. Heat Transfer, 10, 799–821.
Fredlund, D.G. and N.R. Morganstern (1976) Constitutive Relationships for volume change in unsaturated soils, Canad. Geotech. Journ., 13, 261–276.
Galerkin, B.G. (1915) Rods and Plates. Series in Some Problems of Elastic Equilibrium of Rods and Plates. Vestn. Inzh. Tech., (USSR) Translation 63–18924, Clearinghouse, Federal Sci. Tech. Info., Springfield, VA.
Gray, W.G. and G.F. Pinder (1974) Galerkin approximation of the time derivative in the finite element analysis of groundwater flow, Water Resources Res., 10 (4), 821–828.
Gurtin, M.E. (1965) Variational Principles for Linear Initial Value Problems. Quarterly Applied Math, 22, 252–256.
Irons, B.M. (1970) A Frontal Solution Program for Finite Element Analysis. Int. J. Num. Meth. Eng., 2, 5–32.
Javandel, I. and Witherspoon, P.A. (1968) Application of the Finite Element Method to Transient Flow in Porous Media. Soc. Pet/ Eng. J., 8, 241–252.
Narashimhan, T.N. (1978) A Perspective on Numerical Analysis of the Diffusion Equation. Adv. in Water Resources, 1:3, 147–155.
Narasimhan, T.N. and Witherspoon, P.A. (1976) An Integrated Finite Difference Method for Analyzing Fluid Flow in Porous Media. Water Resources Res., 12, 57–64.
Narasimhan, T.N. and P.A. Witherspoon (1977) Numerical model for saturated-unsaturated flow in deformable porous media., 1. Theory, Water Resources Res., 13 (3), 657–664.
Narasimhan, T.N., A.L. Edwards and P.A. Witherspoon (1978) Numerical model for saturated-unsaturated flow in deformable porous media, 2. the algorithm, Water Resources Res., 14 (2), 255–261.
Narasimhan, T.N., Neuman, S.P. and Witherspoon, P.A. (1978) Finite Element Method for Subsurface Hydrology Using a Mixed Explicit-Implicit Scheme. Water Resources Res., 14: 5, 863–877.
Neuman, S.P. (19 73) Saturated-Unsaturated Seepage by Finite Elements. Proc. Am. Soc. Civil Eng., J. Hydraulics Div., 99:HY12, 2233–2250.
Neuman, S.P., Feddes, R.A. and Bresler, E. (1975) Finite Element Analysis of Two Dimensional Flow in Soils Considering Water Uptake by Roots: I Theory. Proc. Soil Sci. Soc. Am., 39: 2, 224–230.
Neuman, S.P. and Witherspoon, P.A. (19 71) Variational Principles for Fluid Flow in Porous Media. Am. Soc. Civil Eng., J. Eng. Mech. Div., 97, 359–379.
Pinder, G.F. and Frind, E.O. (1972) Aplication of Galerkin’s Procedure to Aquifer Analysis. Water Resources Res., 8: 1, 108–120.
Pinder, G.F., E.O. Frind and I.S. Papadopulos (1973) Func-tional coefficients in the analysis of groundwater flow, Water Resources Res., 9 (1), 222–226.
Remson, I., G.W. Hornberger and F.J. Molz (1971) Numerical Methods in Subsurface Hydrology, Wiley-Interscience.
Zienkiewicz, O.C. (19 71) The Finite Element Method in Engineering Science. McGraw Hill.
Zienkiewicz, O.C. and Cheung, Y.K. (1965) Finite Elements in the Solution of Field Problems. The Engineer, 507–510.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Martinus Nijhoff Publishers, Dordrecht
About this chapter
Cite this chapter
Narasimhan, T.N. (1984). Formulation of Numerical Equations. In: Bear, J., Corapcioglu, M.Y. (eds) Fundamentals of Transport Phenomena in Porous Media. NATO ASI Series, vol 82. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6175-3_16
Download citation
DOI: https://doi.org/10.1007/978-94-009-6175-3_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6177-7
Online ISBN: 978-94-009-6175-3
eBook Packages: Springer Book Archive