The Nepali Mathematical Sciences Report, Vol. 39, No. 1, 2022: 22-35 DOI: https://doi.org/10.3126/nmsr.v39i1.46915 COMPARISON OF FINITE DIFFERENCE SCHEMES FOR FLUID FLOW IN UNSATURATED POROUS MEDIUM (SOIL) RAMESH CHANDRA TIMSINA 1 AND KEDAR NATH UPRETY 2 1 Department of Mathematics, Patan Multiple Campus, Tribhuvan University, Kathmandu, Nepal 2 Central Department of Mathematics, Tribhuvan University, Kathmandu, Nepal Abstract: Water movement in unsaturated porous medium (soil) can be expressed by Richards equation with the mass conservation law and Darcy–Buckingham’s law. This equation can be expressed in three different forms as pressure head based, moisture con- tent based and mixed from. In this study, we solve one dimensional Richards Equation in mixed form numerically using finite difference method with various time–stepping schemes: Forward Euler, Backward Euler, Crank–Nicolson and a Stabilized Runge–Kutta–Legendre Super Time-Stepping and we compare their performances using Dirichlet boundary condi- tion on an isotropic homogeneous vertical soil column. Keywords: Finite Difference Methods, Richards Equation, Kirchhoff Transformation, Su- per Time–Stepping Schemes, Infiltration. 1. Introduction Water flow in unsaturated porous media (soil) is an important phenomena in ground- water hydrology. Water movement phenomena in unsaturated zone creates an emerging and realistic problems like contaminant transport, water–added transport of solutes and predicting water percolation in groundwater hydrology. In unsaturated zone the flow of water ascribable to capillary action and gravitational potential and is assumed to obey the classical Richards Equation [1]. Richards Equation is the combine from of mass conserva- tion law and Darcy–Buckingham’s law [2]. Consequently, this equation has three different forms as pressure head, moisture content and mixed from depending on either moisture content θ and pressure head ψ as the dependent variable. The constitutive (experimental) relationship between θ = θ(z,t) and ψ = ψ(z,t) allows the conversion from one another. As shown in [1] the couple or mixed form of Richards Equation takes the following form. ∂θ ∂t −∇.K(ψ)∇ψ − ∂K ∂z = S (z,t), (1.1) where, θ is the volumetric moisture content, ψ is the pressure head K(ψ) is the unsaturated hydraulic conductivity, describes the behavior of water flow that can move through pore Received: April 13, 2022 Accepted: June 25, 2022 Published Online: June 30, 2022.