The Prediction of Gas-Well Performance Including the Effect of Non-Darcy Flow
Authors: Swift G.W. , Kiel O.G. 
Published: December 2007

Abstract
The concept of "a continuous succession of steady-states", which has been applied successfully by Aronofsky and Jenkins to obtain a solution for the nonlinear partial differential equation describing the transient Darcy flow of gas through porous media, is demonstrated to be equally valid for transient non-Darcy flow. A mathematical model, which numerically solves the partial differential equation, is used to check the validity of the succession of steady-states solution. Comparison of sand-face pressure histories computed by the two methods shows excellent agreement. The utility of the succession of steady-states solution in predicting performance of gas wells rests in the fact that no special computation equipment is required. The development of the succession of steady-states solution leads also to a practical method for determining and analyzing field test data. A method for taking gas-well test data under constant-rate conditions is presented. Experimental data obtained in the field by employing the constant-rate method are presented and analyzed in accordance with the succession of steady-states solution. Analysis of data in this fashion is demonstrated to give direct "in situ" information for reservoir permeability, porosity and turbulence or non-Darcy coefficient.

Introduction
The economics of gas production are dependent upon the transient behavior Of flow within the reservoir. For production from a finite reservoir, the transient flow behavior can be subdivided into two parts. At first, the transient caused by the movement of the pressure "wave" into the reservoir is of importance. Later in the production history, the pressure-wave movement ceases and the second transient stage of material depletion becomes controlling. For reservoirs of relatively high permeability, it can be shown that the pressure wave moves into the reservoir and stabilizes quite rapidly. In the case of relatively impermeable reservoirs, quite the opposite is true. Although it is theoretically possible to compute the production capability of a well from the properties of the reservoir as determined by static tests and core analyses, more reliable information is obtained by conducting flow tests on the well and thereby obtaining some measure of in situ formation properties. For gas wells, there are two basic types of tests in existence: the flow-after-flow method, and the isochronal method. Both of these techniques are tailored to obtain data that can be analyzed in accordance with the empirical performance equation:
...........................(1)
In addition, the isochronal method makes provision for the sluggish nature of pressure-wave movement in "tight" formations by requiring pressure build-up between flows and by stipulating that data obtained on successive flows be analyzed at equal elapsed flow times. It can be demonstrated that either test is valid for reservoirs of high permeability. Further, since it has been pointed out that the pressure wave stabilizes rapidly for reservoirs of this type, tests of relatively short duration will give stabilized information on the performance of a well. Further decline of sand-face pressure and/or production rate may be determined by employing material-balance techniques. Cullender points out that for relatively impermeable reservoirs the flow-after-flow method gives invalid results. (See Appendix C.) If the isochronal method of testing is used, there are two alternatives:the tests must be conducted for a sufficient length of time to obtain stabilized information (which may require months to accomplish); orsome method for extrapolating the results of short-term isochronal tests must be employed.
The first alternative is impracticable because of manpower, conservation and economic considerations. Recourse to the second alternative requires some assurance regarding the reliability of the extrapolation technique. Poettmann and Schilson present an empirical method for predicting stabilized performance. The present investigation was originally initiated to determine the reliability of this technique. To do this, a mathematical model was developed to simulate the Darcy and non-Darcy flow of gas through porous media. The model consisted of a finite-difference approximation of the nonlinear partial differential equation which was solved on an IBM 7090 computer. Long-term production histories were simulated by the model and compared against predictions obtained from the Poettmann-Schilson method. As the work progressed, it became apparent that a straightforward predictive equation could be developed by utilizing the concept of a succession of steady-states. As a result, the emphasis of the work was redirected to exploit the advantages of the new method.
JPT
P. 791^