Turkish Journal of Computer and Mathematics Education Vol.12 No.13 (2021), 6373 – 6382 6373 Research Article B-Spline Collocation Solution of One Dimensional Nonlinear Differential Equation Arising in Homogeneous Porous Media Nilesh Sonara a , Dr.Dilip C Joshi b , Dr. Narendrasinh B Desai c a Research Scholar, Mathematics Department, VNSGU, Surat,Gujarat,India. nilesh.sonara2012@gmail.com b Professor, Mathematics Department, VNSGU, Surat,Gujarat,India. c Associate Professor, Department of Mathematics, ADIT, Vallabh V.Nagar,Gujarat,India. Article History: _________________________________________________________________________________________________________ Abstract: This paper investigates the Numerical solution nonlinear partial differential equation for one dimensional instability phenomenon known as Boussinesq's equation arising in a porous media in oil-water displacement treatment (instability). Its Numerical Solution has been acquired by utilizing B-Spline method with proper boundary and initial conditions. The Numerical solution of Boussinesq's equation using Spline method is very nearer to Exact Solution obtained by analytical method .It is surmise that when distance and time increases, its saturation of injected water is increases . Numerical solution and graphical illustration has been obtained by Matlab. Keywords: Water-flooding process, Instability, Immiscible displacement, Fluid flow, B-Spline Collocation Method. ________________________________________________________________________________________________________ 1. Introduction The fingering phenomenon occurs during the secondary recovery process arising in porous media, which is popular in different engineering fields such as soil mechanics, Agriculture, groundwater and hydrology, and petroleum engineering (Brailovsky, Babchin, Frankel, & Sivashinsky, 2006) , (Posadas, Quiroz, Crestana, & Vaz, 2009), (Tullis & Wright, 2007). This kind of phenomenon can also be seen in the oil recovery treatment that occurs in oil reservoirs. It is common to practice oil recovery technology to inject water into oil fields at specific locations in an attempt to drive oil into a production well. This stage of oil recovery is referred to as secondary recover. Figure 1: The fingering process between native oil and injected water flow through a porous medium is visualized in fig (1). Only the average cross-sectional area occupied by the fingers was measured in the statistical treatment of fingering, Neglecting the size and shape of individual fingers (Scheidegger A. , 1960). The statistical behaviour of fingering phenomenon in porous media was studied by (Scheidegger & Johnson, 1961), who used the method of characteristics. With the use of a perturbation solution, (Verma, 1969) investigated the stabilization of instabilities in oil-water displacement treatment through heterogeneous porous media with capillary pressure. The confluent hypergeometric function was used by (Patel D. M., 1998) to solve the problem. Using the advection-diffusion concept, (Patel D. M., 1998) has explained this problem. Many Researchers (Mehta & Joshi, M. S, 2009), (Pradhan, Mehta, & Patel, 2011), and (Patel & Desai, 2015) have explained analytical and numerical approaches to the fingering phenomenon arising in homogeneous porous media using various methods such as Rdtm method, Crank-Nicolson method, and Homotopy methods. Using the Spline method, we can obtain a numerical values of a