For quickly responding aquifers, the widely used step pressure (SP) formula for evaluating the water influx superposition integral is inaccurate as it is normally employed. Using the SP formula in the material-balance-with-water-influx equation causes it in turn to give inaccurate results. A totally general, linear pressure (LP) formula is introduced that more precisely evaluates the water influx superposition integral and leads to the following advantages when used in the material balance equation:
the, original gas or oil in place (OGIP/OOIP) and aquifer parameters can be determined earlier in a reservoir's life,
reservoir pressure can be predicted more accurately done the OGIP/OOIP and aquifer parameters are known, and
OGIP/OOIP and equivalent aquifer parameters can be determined for reservoirs with a variety of drive mechanisms including shale water influx and rock compressibility - without knowing what the actual drive mechanism is.
It is also pointed out that a computationally efficient linear pressure recursion (LPR) formula developed by Leung for reservoir simulators has the accuracy as the LP formula derived in this paper.
To find the most efficient and profitable production strategy for a partial water drive reservoir, two steps are required. First, it is necessary to determine the original-gas-place (OGIP) or original-oil-in-place (OOIP) and to accurately characterize water influx. Then. using this information as input to some form of a pressure prediction model, different production strategies must be evaluated to find the best One. In the second step, it is also necessary to accurately calculate water influx. The most general water influx calculation for mathematically linear aquifers* uses the superposition integral.
Van Everdingen and Hurst presented a formula that is commonly used to evaluate the superposition integral. In the work below, this formula will be called the step pressure (SP) formula because it is an exact evaluation of the superposition interval when the aquifer inner boundary pressure is a step function in each time interval. It is often believed that the SP expression is the moat accurate one available. It will be shown. however, that for instantaneously responding (i.e.. pot) aquifers or aquifers that respond moderately fast to pressure changes, this formula can substantially underpredict water influx unless a large number of time steps is urged. The largest errors in the water influx calculation occur whenever the aquifer inner boundary pressure is changing rapidly with time. The largest percent errors usually occur at early times.
A formula that is more accurate than the SP expression for pot or moderately fast aquifers is developed in this paper. It is called the linear pressure (LP) formula and is an exact evaluation of the superposition integral when the aquifer inner boundary pressure versus time curve is a series of straight line segments connecting the data points. The large errors in the water influx calculation occur when the aquifer inner boundary pressure versus time curve deviates the most from the straight line segments connecting the data Points. The largest serpent errors usually occur at early times. The LP expression can be used for any aquifer to which the superposition integral applies. Its computational efficiency and ease of application are about the same as the SP formula.
*Examples of mathematically nonlinear aquifers are those whose permeability or rock compressibility is a function of pressure.