Solutions of Unsteady-State Radial Gas Flow
Authors: Carter R.D. 
Published: December 2007

Abstract
Numerical solutions are presented to some problems of unsteady-state radial flow of gas in which Darcy's law holds. These solutions are intended to aid in explaining the observed behavior of gas wells during drawdown tests. The variation of viscosity and Z-factor with pressure is accounted for in the solutions. Results are presented for three types of problems:unfractured wells producing at a constant rate,fractured wells producing at a constant rate andunfractured wells producing through a critical flow prover.
In the critical-flow-prover solutions, a correction of the calculated wellbore pressure to account for non-Darcy flow is made. A good correlation is shown between the solutions to the problem of Type 1 and the corresponding diffusivity-equation solution. Using this correlation, the diffusivity solution may be used to obtain gas-well solutions. The solutions to the problems of Type 2 illustrate the difference in drawdown-curve behavior to be expected between fractured and unfractured gas wells. Methods for estimating fracture radius and flow capacity are presented and verified from solutions of Type 2. The solutions to the problems of Type 3 illustrate the differences to be expected between critical-flow-prover drawdown tests and constant-rate tests. A equation is presented for correcting critical-flow-prover data to constant-rate conditions. A description of the methods used in obtaining the solutions appears in the Appendix.

Introduction
It is necessary for many engineering purposes, such as predicting future deliverability performance, to be able to relate bottom-hole flowing pressure, formation pressure and flow rate for conditions of stabilized flow in gas wells. This information is customarily given as a plot of (p -p ) vs q on logarithmic coordinates. It is desirable to be able to predict stabilized performance curves for gas wells based on data obtained from drawdown tests of short duration. Tests of long duration may be wasteful of gas, and low-permeability wells do not stabilize in a short-duration test. Although it is now commonly recognized that non-Darcy flow plays a significant part in gas-well behavior, it is believed that an understanding of transient behavior of systems obeying Darcy's law is still important in the interpretation of short-term test data, and the prediction of stabilized performance curves there from. In this connection, an equation for correcting Darcy flow solutions for non-Darcy flow effects has existed in the literature for some time. This equation is
......................................(1)
It seems likely that solutions of the type presented herein, combined with the non-Darcy flow correction of the type of Eq. 1 and the usual laminar-flow skin factor, could provide a model which would adequately explain transient gas-well behavior for times after the non-Darcy flow region (which should be confined to a small area surrounding the well) has been established. The solutions presented in this paper are for three types of problems:unfractured wells producing at a constant rate,fractured wells producing at a constant rate andunfractured wells producing through a critical flow prover.
In the problems of Types 1 and 2, no corrections for non-Darcy flow were made in the results presented, since it was reasoned that the constant-rate boundary condition would facilitate such a correction if it was desired later. For the problems of Type 3, however, a condition of varying flow rate required that a non-Darcy flow correction of the type of Eq. 1 be included. This was done. Based on the numerical results presented, a number of approximations and a correlation are presented which may be useful in analyzing gas-well test data.

TYPE 1 PROBLEMS-UNFRACTURED WELL, CONSTANT FLOW RATE
It is assumed that the system can be described as a finite cylindrical gas reservoir produced by a single cylindrical well located at the center of the reservoir. Flow is strictly radial. When the flow obeys Darcy's law, the governing differential equation is
..........(2)
where M(p) = (2,703 x 10(–6) x 520), and f(R) =.
Boundary and initial conditions are:
...................(3)
JPT
P. 549^