Non-Isothermic Gas Flow In a Porous Medium
Authors: Kaleta Andrzej , Marcak Henryk , Siemek Jakub 
Published: March 2010

Abstract
Solutions of the system of equations describing non-isothermic flow of real gases in porous media are given in the paper. Approximate solutions were obtained by the method of functional series for stationary flows, taking into account the thermal conductivity of the medium. Accurate solutions were obtained for stationary- flows without considering the thermal conductivity. Solutions are given for plane and plane-radial flows. Moreover, a method of solution is described which involves a consideration of the variation of the coefficient of gas compressibility, z, with pressure, p. Considering non-stationary non-isothermic flow, a method is given for solution with the use of the ‘minor parameter’, reducing the system of non-linear partial differential equations to the system of linear partial differential equations of the parabolic type which determine the coefficients of the solving series. The solutions may be used for the temperature testing of gas wells and production from condensate gas fields.

INTRODUCTION
THE FLOW OF GAS through a porous medium is determined univocally by the functions p/r, t/, T/r, t/,γ/r, t/ and v/r, t/, where p is pressure, T temperature, γ specific gravity, flow rate and r is the position vector. The distribution in time and space of the above-mentioned parameters may be obtained by solving a system of four equations:
(Equation Available In Full Paper)
On the whole, the flow of gas is defined in practice assuming that T = const, in a stationary state and with simple geometrical contours of the medium. There also exist several theoretical elaborations of the problems of non-stationary flow with the assumption that T = const.(2,3,4,5). The problem of non-isothermic flow has remained so far unsolved, with the exception of a few investigations. These were undertaken by Czekaluk(1,0) Czarny(4) and Gusein-Zade and Kolosovskaja(7). They referred mainly to the stationary flow of incompressible fluids, assuming that the stationary distribution of pressure in a porous medium is known, omitting the thermal conductivity of the medium. Un-dimensional flows (plane and plane-radial) were considered, with the only equation of energy being solved /3/. Thus, Gusein-Zade and Kolosovskaja(7), start from the equations of energy for linear and plane-radial flow under the form:
(Equation Available In Full Paper)
Equations /5/ and /6/ are directly obtained from equation /3/, assuming that A = O and ∂p / ∂t = O. Setting the original and boundary conditions in the form:
(Equation Available In Full Paper)
one seeks for a solution in the form of
(Equation Available In Full Paper)
As may be seen from the solutions obtained, the area of the temperature field is divided into area I, where x &lt; u °t, and area II, where x &gt; u °t; two different formulae are therefore justified. These considerations are substantial in the case of injecting a fluid into a well. Similarly in the case of plane-radial flow at the initial and boundary conditions in the form of
(Equation Available In Full Paper)
Solutions given in papers (1) and (4) are based on the same equations.