June 22-27, 2014
The present paper discusses the numerical solution of the Burgers’ equation arising in longitudinal dispersion phenomenon in fluid flow through porous media. In the porous medium pure water, salt water or contaminated water disperse in longitudinal direction gives rise to a non-linear partial differential equation in the form of Burgers’ equation. The equation is solved by using Crank-Nicolson finite difference scheme with appropriate initial and boundary conditions. The longitudinal dispersion phenomenon may be miscible or immiscible fluid flow through porous media. The problem of miscible displacement can be seen in the coastal areas, where fresh water beds are gradually displaced by sea water. Longitudinal dispersion phenomenon plays an important role to control salinity of the soil in western seashore region of the Gujarat state in India. To control salinity, the government of Gujarat has developed many check dams from where contaminated water diverted and poured to the soil of the farms, where the crops of cumin seed (jeera), fennel (saunf) and other grains are grown. In this region due to the infiltration of this infiltered water, free surface of sweet water table rises, consequently, saline seawater cannot cross the threshold in the nearby area of the seashore. In such a way, the dispersion of contaminated water plays key role to solve salinity problem. The immiscible dispersion also plays an important role in petroleum engineering during secondary oil recovery process, in which water or gas is injected in oil formatted area to drive the oil towards production well. An unconditionally stable Crank-Nicolson finite difference scheme has been employed to find the concentration C(X, T) of salty or contaminated water dispersion in uni-direction. The outcome is consistent with physical phenomenon of longitudinal dispersion in miscible fluid flow through porous media. It is concluded, that the concentration C(X, T) decreases as distance X as well as time T increases. The tables and graphs are developed by using MATLAB coding.
Ravi Borana, Vikas Pradhan, and Manoj Mehta, "Numerical Solution of Burgers' equation arising in Longitudinal Dispersion Phenomena in Fluid Flow through Porous Media by Crank-Nicolson Scheme" in "5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry", Prof. Kambiz Vafai, University of California, Riverside; Prof. Adrian Bejan, Duke University; Prof. Akira Nakayama, Shizuoka University; Prof. Oronzio Manca, Seconda Università degli Studi Napoli Eds, ECI Symposium Series, (2014). http://dc.engconfintl.org/porous_media_V/47