In a gas reservoir, because of the assumptions inherent in the constant rate solution, it is not possible to accurately forecast deliverability during pseudo-steady state (after the reservoir has shown significant depletion). More specifically, the constant rate equation is not consistent with the tank model (material balance). The cause of the inconsistencies lies in the assumption of constant gas properties (ie. compressibility and viscosity). This paper deals with different attempts at modifying the constant rate solution to provide an approximate solution that is consistent with tank type depletion.
If possible, it is preferable to approach this type of solution by trying to avoid an iterative approach or a solution that requires the tank model (material balance). Once one begins to incorporate the tank model for pseudo steady state calculation, the calculation of the constant rate solution becomes redundant. Therefore, various procedures were attempted to develop a process that does not require a material balance calculation explicitly, and preferably no iterations.
This work also demonstrates that the constant rate solution and the constant pressure solutions are essentially equivalent during transient flow. However, there is a significant difference between them when boundary dominated flow is reached.
Although the quantity of reserves are obviously a very important part of determining the value of a new well, the length of time to produce these reserves is equally important. There is obviously much more value in reserves, if the reserves can be produced over a shorter period of time. Therefore, there is obvious value in preparing accurate gas deliverability forecasts. Currently, the following two methods are used to forecast gas deliverability:
Pseudo-steady state equation, and
Tank model approach (p/z material balance and Deliverability forecasting)
The pseudo-steady state solution is based on the "constant rate equation". For a given constant rate, the well's flowing pressure can be calculated as a function of time using the equation shown below. Equation (Available in full paper)
In practice, it is often more useful to forecast the rate at a fixed flowing pressure than to forecast the flowing pressure at a fixed gas rate (which is the basis of the pseudo-steady state solution). Since the gas rate will usually vary with time, the correct procedure is to use the equation below in conjunction with the principle of superposition to account for the rate variations. This can be a laborious procedure, and in many instances, a shortcut is taken to simplify the process. This simplification consists of "ignoring" the fact that the pseudo-steady state equation is a "constant rate" solution, and to simply invert the equation and use it to calculate the rate at a constant pressure. While there is some basis for this simplifying assumption(1,2) it can also lead to some erroneous answers. Equation (Available in full paper)
Figure 1 shows a comparison of the forecasts obtained by using the principle of superposition and that obtained by using the simplified procedure.