NUMERICAL SOLUTION OF ONE-DIMENSIONAL DISPERSION EQUATION IN HOMOGENEOUS POROUS MEDIUM BY MODIFIED FINITE ELEMENT METHOD

Tejas Sharma, Gargi Trivedi, Vishant Shah, Shrikant Pathak

Abstract


This article discusses the mathematical modeling of the longitudinal dispersion phenomenon in a homogeneous porous medium and its solution using the modified finite element method. Also,  the theorem about existence and uniqueness, and stability of nonlinear system that arose in the numerical scheme, by utilizing nonlinear functional analysis and the Banach fixed point theorem are proved. Finally,  illustrations are added to show the efficacy of the derived method.

Keywords


Burger's equation; porous medium; finite element scheme

Full Text:

PDF

References


Bear J. Dynamics of Fluids in Porous Media. Dover Publication, New York, 1972.

Fetter C., Boving T., Kreamer D. Contaminant Hydrogeology. Prentice Hall, New Jersey, 1999.

Lake L.W. Enhanced Oil Recovery. Prentice-Hall Inc., New Jersey, 1989.

Peacemen D.W. Fundamentals of Numerical Reservoir Simulation. Elsevier Scientific, New York, 1977.

Hillel D. Introduction to Environmental Soil Physics. Academic Press, San Diego, 2003. DOI: 10.1016/B978-0-12-348655-4.X5000-X

Schnoor J. Environmental Modeling: Fate and Transport of Pollutants in Water, Air, and Soil. John Wiley and Sons, New York, 1996.

Fick A. V. On liquid diffusion. Philosophical Magazine and Journal of Science, 1855, vol. 10, no. 63, pp.30-39. DOI: 10.1080/14786445508641925

Taylor G. Dispersion of Soluble Matter in Solvent Flowing Slowly Through a Tube. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 1953, vol. 219, no. 1137, pp. 186-203. DOI: 10.1098/rspa.1953.0139

Aris R. On the Dispersion of a Solute in a Fluid Flowing through a Tube. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 1956, vol. 235, no. 1200, pp. 67-77. DOI: 10.1098/rspa.1956.0065

Rosenberg D. Methods for the Numerical Solution of Partial Differential Equations. American Elsevier Publishing Company Inc., New York, 1979.

Chen Z., Huan G., Ma Y. Computational Methods for Multiphase Flows in Porous Media, Siam, Philadelphia, New York, 2006.

Borana R.N., Pradhan V.H., Mehta M.N., Numerical Solution of One-dimensional Dispersion Phenomenon in Homogeneous Porous Medium by Finite Difference Method. Proceedings of International Conference on Emerging Trends in Scientific Research, Surendranafar, Gujarat, 2015.

Courant R. Variational Methods for the Solution of Problems of Equilibrium and Vibrations. Bulletin of the American Mathematical Society, 1943, vol. 49, no. 1, pp. 1-23.

Clough R. The Finite Element Method in Plane Stress Analysis. Proceedings of the 2nd ASCE Conference on Electronic Computation, 1960, pp. 345-378.

Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 2005.

Reddy J.N. An Introduction to the Finite Element Method. McGraw-Hill Education, New York, 2006.

Williams F.A. Numerical Methods for Partial Differential Equations, Academic press Inc., Cambridge, 1992.

Smith G.D. Numerical Solution of Partial Differential Equation: Finite Difference Methods. Clarendon Press, Oxford, 1985.

Evans G., Blackledge J., Yardley P. Numerical Methods for Partial Differential Equations, Springer-Verlag, London, 2000.

Dabral V., Kapoor S., Dhawan S. Numerical Simulation of One-Dimensional Heat Equation: B-Spline Finite Element Method. Indian Journal of Computer Science and Engineering, 2011, vol. 2, no. 2, pp. 222-235.

Mohebbi A., Dehghan M. High-Order Compact Solution of the One-Dimensional Heat and Advection-Diffusion Equations. Applied Mathematical Modelling, 2010, vol. 34, no. 10, pp. 3071-3084. DOI: 10.1016/j.apm.2010.01.013

Dag I., Irk D., Saka B. A Numerical Solution of the Burgers’ Equation Using Cubic B-Splines. Applied Mathematics and Computation, 2005, vol. 163, no. 1, pp. 199-211.

Dehghan M. A Finite Difference Method for a Non-Local Boundary Value Problem for Two-Dimensional Heat Equation. Applied Mathematics and Computation, 2000, vol. 112, no. 1, pp. 133-142.

Gorguis A., Benny Chan W.K. Heat Equation and Its Comparative Solutions. Computers and Mathematics with Applications, 2008, vol. 55, vol. 12, pp. 2973-2980.

Gupta M.M., Manohar R.P., Stephenson J.W. High-order Difference Schemes for Two-Dimensional Elliptic Equation. Numerical Methods for Partial Differential Equations, 1985, vol. 1, pp. 71-80.

Bernard R.A., Wilhelm R.H. Turbulent Diffusion in Fixed Beds of Packed Solids. Chemical Engineering Progress, 1950, vol. 46, pp. 233-244.

Kovo A.S. Mathematical Modelling and Simulation of Dispersion in a Nonideal Plug Flow Reactor. Journal of Dispersion Science and Technology, 2008, vol. 29, pp. 1129-1134. DOI: 10.1080/01932690701817859

Ebach E., White R. Mixing of Fluids Flowing through Beds of Packed Solids. AIChE Journal, 1958, vol. 4, no. 2, pp. 161-169. DOI: 10.1002/aic.690040209

Hunt B. Dispersion Calculations in Nonuniform Seepage. Journal of Hydrology, 1978, vol. 36, no. 3-4, pp. 261-277. DOI: 10.1016/0022-1694(78)90148-8

Sharma T., Pathak S., Trivedi G. Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical Methods to Solve Linear Partial Differential Equations. Indian Journal of Science and Technology, 2024, vol. 17, no. 10, pp. 924-931. DOI:10.17485/IJST/v17i10.1776

Sharma T., Pathak S., Trivedi G.J., Sanghvi R. Flow Modelling in Porous Medium Applying Numerical Techniques: A Comparative Analysis. Recent Research Reviews Journal, 2023, vol. 2, no. 2, pp. 288-304. DOI: 10.36548/rrrj.2023.2.004

Shah V., Trivedi G.J., Sharma J., Sanghvi R. On Solution of Non-Instantaneous Impulsive Hilfer Fractional Integro-Differential Evolution System. Journals of the Polish Mathematical Society, 2023, vol. 51, no. 1, pp. 33-50. DOI:10.14708/ma.v51i1.7167.

Sharma T., Pathak Dr.S., Trivedi G. Numerical Modeling of Fluid Flow Through Porous Media: A Modified Crank-Nicolson Approach to Burgers’ Equation. Journal of Advanced Zoology, 2024, vol. 44, no. s8, pp. 363-371. DOI: 10.53555/jaz.v44is8.4098


Refbacks

  • There are currently no refbacks.


 Save