ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ НЕИЗОТЕРМИЧЕКОЙ НЕУСТАНОВИВШЕЙСЯ ФИЛЬТРАЦИИ ГАЗА ПРИ НЕЛИНЕЙНОМ ЗАКОНЕ СОПРОТИВЛЕНИЯ

Abstract

This paper presents the development of a mathematical model of non-isothermal, unsteady flow of a real gas in deviation of Darcy's law and an algorithm for its numerical implementation. A formula for the flow velocity for non-linear binomial law of filtration was obtained. The formula includes a radial coordinate pressure derivative, thermo physical gas properties, and geophysical rock properties. There is also a partial differential equation and boundary condition at the well bottomhole for non-linear two-dimensional law of filtration. A method for simultaneous numerical integration for finite differences of differential equations shows an unsteady non-isothermic gas flow in the reservoir and energy conservation equation with initial and boundary conditions on the well bottom and the external drainage radius; functions of pressure and temperature at different time variances and their distribution along radial coordinates were calculated using a developed mathematical model. Graphs for these functions for linear and non-linear flow in reservoirs with different reservoir features for different well rates were built. Analysis of these graphs shows that taking into account the deviation of Darcy’s law of gas filtration in high permeable reservoirs for various flow rates does not make substantial corrections to the calculated values of pressure and temperature in comparison with the corresponding characteristics for flow. Calculations of bottomhole pressure - assuming that gas flow is steady - have calculation errors that are less than 0.2%, considered in the numerical implementation of task solution. With a low-permeable reservoir - when well production rates are low (for example, 169 thousand sm 3/day) - the bottomhole pressure decrease is small; therefore, a decrease of temperature at the bottomhole zone will also be small. The difference between the calculated values of these characteristics for linear and nonlinear filtering laws will also be negligible. Error of the value of bottomhole pressure calculated in assumptions that flow is steady does not exceed 0.7%. There is a deviation of Darcy's law in gas flow through the low-permeable reservoir at high well production flow rates (for example, 845 thousand sm 3/day), which leads to a substantial reduction of the reservoir pressure and temperature. Pressure stabilization for gas flow in the low-permeable reservoir does not depend on flow rate value. Error of value of bottomhole pressure, calculated in assumption that flow is steady exceeds 300%.