A Fast Convolution Method for Implementing Single-Porosity Finite/Infinite Aquifer Models for Water-Influx Calculations
- W.F. Leung (Gulf R and D Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- September 1986
- Document Type
- Journal Paper
- 490 - 510
- 1986. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 4.3.4 Scale, 5.5.8 History Matching, 5.6.9 Production Forecasting, 4.1.5 Processing Equipment, 5.5 Reservoir Simulation
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Summary. Existing aquifer models are either computationally inefficient or inaccurate for reservoir water-influx calculations. A new fast convolution method (FCM) that does not require lengthy data manipulation is used to implement four different single-porosity finite and infinite aquifer models for unsteady-state water-influx calculations. Numerical examples on circular and linear aquifers, together with a step-by-step calculation, are used to demonstrate the accuracy and efficiency of the models. These numerical studies show that the FCM approach is as much as 20% more accurate than the widely used methods for both finite and infinite aquifers. The models can be implemented readily in numerical reservoir simulators, as well as in hand computations.
Water influx is an important natural mechanism for primary recovery of hydrocarbons. It is normally primary recovery of hydrocarbons. It is normally calculated with mathematical aquifer models during both the history-matching phase and the production-forecast phase. Steady-state, transient, pseudosteady-state (pSS), and pot aquifer models have been used to calculate water influx into reservoirs. However, these models are either computationally inefficient or inaccurate; therefore, they are not suitable for either simple material-balance calculations or for more complex reservoir simulations. Because of these difficulties, four different unsteady-state single-porosity aquifer models have been developed. They are the PSS model, modified pseudosteady-state model (MPSS), transient model, and infinite-aquifer model. The first three models, listed in increasing order of accuracy, are applicable for finite aquifers, whereas the last model pertains to infinite-acting aquifers. Water influx can be calculated accurately for time-dependent reservoir boundary pressure with these four models. The common denominator for these models is a new FCM that does not require lengthy manipulation of past boundary-pressure data as in the superposition method. The computational effort associated with the present models is comparable to existing present models is comparable to existing superposition-free methods, such as the Carter and Tracy and Fetkovitch methods. Moreover, the models implemented with the FCM are significantly more accurate than existing methods. Several examples of both circular and linear aquifers illustrate the utility and accuracy of the models.
The steady-state model is applicable for a reservoir and aquifer of such a large extent that water influx does not affect the pressure state of either system. The size of reservoirs and aquifers is usually limited. In practice, unsteady-state water influx in which the pressure at the reservoir or the aquifer changes with time is pressure at the reservoir or the aquifer changes with time is frequently encountered. The literature discusses three types of unsteady-state models used to model water influx from finite and infinite-acting aquifers. They are described briefly below.
Pot Aquifer Model-Finite Aquifers. The aquifer Pot Aquifer Model-Finite Aquifers. The aquifer pressure is assumed to be at instant equilibrium with the pressure is assumed to be at instant equilibrium with the boundary pressure in this model. This restrictive assumption is valid only if fluid transmissibility between the aquifer and reservoir is very large and the size of the aquifer is very small compared with the reservoir.
Fetkovitch's PSS Model-Finite Aquifers. Fetkovitch proposed the PSS aquifer model to explain the late-time proposed the PSS aquifer model to explain the late-time transient effect of water influx. The development of the model is largely empirical, and there is no set guideline to determine when the model is valid. This aspect will be discussed later.
Transient Model-Finite and Infinite Aquifers. This model is based on the solution of the differential equations for unsteady-state water influx for linear and circular aquifers. Therefore, the transient model is more rigorous in treatment of unsteady-state behavior than other models. One disadvantage of this model, however, is that the analytical solution for aquifers of arbitrary shape is generally not available. A second disadvantage of the model is that the general solution for time-dependent reservoir boundary pressure is expressed by superposition. The superposition procedure requires both the past boundary-pressure history and the influx solution at different effective periods of time. The calculation algorithm becomes lengthy and tedious as the number of timesteps increases.
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