About one model of infiltration of oil and petroleum products into the ground during their spills

Abstract

The process of infiltration of oil and petroleum products into the ground when they spill onto the surface of the earth is considered. To construct a mathematical model of this process, the soil is represented as a solid body with a system of vertical cylindrical microtubules with the same diameter, and the infiltration of liquid into the soil is represented as the movement of a cylindrical liquid layer of variable height formed in the microtubule. It is assumed that liquid slides on the microtubule wall is according to Navier's law. First, formulas are proposed for determining the forces of inertia and viscous friction on the microtubule. Taking into account the acting forces, a mathematical model of the motion of a cylindrical liquid layer in a microtubule is constructed, which is a nonlinear ordinary differential equation of the second order. The resulting model is represented as a system of nonlinear ordinary differential equations of the first order with initial conditions. A discrete analogue of the latter problem is constructed using the finite difference method and a computational algorithm is proposed for the numerical solution of the resulting nonlinear system of difference equations. Numerical experiments were carried out on the basis of the proposed computational algorithm. Keywords: infiltration of oil and petroleum products into the soil; the model of ideal soil; sliding according to Navier's law; the finite difference method.