Inertial and Slip Effects in Steady-State Radial Gas Flow Through Porous Media

Abstract

Steady horizontal radial gas flow through porous media may be described by means of the equation(1) provided there is no slip at the gas/solid interface and provided there is no slip at the gas/solid interface and q is taken to be positive in the r direction. If one assumes that the modified gas law holds and that viscosity, compressibility, and temperature remain constant, this equation may be integrated to yield(2) which is similar in form to that proposed by Tek et al. This equation may of course be reduced to the well known form employed for Darcy flow by deleting the second term on the right-hand side. However, it has been shown that neither slip nor inertial effects should be neglected when linear laboratory gas flow tests are used to evaluate rock properties. It has been observed that the same is true for laboratory data when taken in radial tests. Although Eq. 1, modified to allow for slip, may be integrated for linear flow, radial flow yields a nonlinear ordinary differential equation with no known solution. However, an approximate solution may be developed in the following manner: To allow for slip, Eq. 1 may be modified and written as(3) where(4) If the modified gas law applies and viscosity, compressibility, and temperature are assumed to remain constant at their average values, Eq. 2 may be converted to the form(5) Substitution of Eq. 4 into Eq. 5 yields(6) If one defines(7) and(8) Eq. 6 may be written as(9) or(10) As was stated earlier there is no known solution to this nonlinear ordinary differential equation. Although this equation may be attacked with numerical techniques to estimate rock properties using flow data, the procedure is involved and difficult. However, if the ratio b/p is sufficiently small, Eq. 9 may be approximated by deleting the term (1 + b/p) in the numerator, in which case(11) or(12) This equation may be integrated between the limits ; and; to yield (13) or(14) P. 1155