Summary.
This paper presents a new water influx model that differs from traditional approaches in that it includes the effect of vertical flow at the reservoir/aquifer interface. The results are presented in the form of dimensionless groups, which makes the model readily applicable to a wide range of systems. The paper concludes with a sample calculation showing how the predictions of this new model can be significantly different from those of conventional radial flow models.
Introduction
Petroleum reservoirs are often in contact with an aquifer that Petroleum reservoirs are often in contact with an aquifer that provides pressure support through water influx. Thus, the prediction provides pressure support through water influx. Thus, the prediction of reservoir behavior usually requires an accurate model of the aquifer. Reservoir/aquifer systems are commonly classified on the basis of flow geometry as either edgewater or bottomwater drive. For edgewater drive, the most rigorous aquifer influx model developed to date is that of van Everdingen and Hurst, 1 which is essentially a solution to the radial diffusivity equation. Although the assumptions made in deriving this model are not strictly valid for bottomwater drive systems, water influx in this case can sometimes be closely approximated by radial flow. Therefore, because the results are often quite adequate and for lack of a better model, it has been common practice to apply the van Everdingen and Hurst method to both bottomwater and edgewater systems. Coats has developed a model that takes into account vertical flow effects and has shown these effects to be fairly significant. The model as presented, however, has two principal limitations:the solution given applies to the "terminal-rate" case, which allows the user to calculate pressure from a known influx rate rather than the reverse, andthe solution is applicable only to infinite aquifers.
This paper is essentially an extension of the work of Coats. The bottomwater model presented here is a solution to the "terminal-pressure" case and applies to both finite and infinite aquifers. The results are presented in the form of dimensionless groups that are tabulated in a manner similar to that of van Everdingen and Hurst. The paper concludes with a sample calculation that illustrates the use of this new model and shows that predictions of the bottom-water model can be significantly different from those of the radial flow model. The calculation of water influx is important in a number of reservoir engineering applications, such as material-balance studies and the design of pressure maintenance schemes. The fact that a large percentage of reservoirs have adjoining aquifers means that percentage of reservoirs have adjoining aquifers means that development of an accurate aquifer model is critical to proper understanding of reservoir behavior. It is not surprising, therefore, that considerable research effort has been devoted to this subject. During the past 50 years, a large number of models describing water encroachment have emerged, and the majority of these have been subject to a great deal of modification. In these models, the reservoir is typically visualized as a right cylinder surrounded by a series of concentric cylinders representing the aquifer. Most of the models, such as me steady-state model of Schilthuis or the finite-aquifer, pseudosteady-state model of Fetkovich, are applicable to only a limited range of flow conditions or reservoir/ aquifer geometries. The model that possesses the most general applicability is the unsteady-state model of van Everdingen and Hurst. In fact, this model is a solution to the radial diffusivity equation and as such is valid for all flow regimes, provided, of course that the flow geometry is actually radial.
The radial flow geometry assumed by van Everdingen and Hurst is best understood by means of an illustration. Fig. 1 shows, in cross section, a reservoir subject to edgewater drive and the idealized radial flow model that represents this reservoir/aquifer system. The flow vectors in this case are horizontal, and water encroachment occurs across a cylindrical plane encircling the reservoir. This situation can be compared to the bottomwater drive system shown in Fig. 2. In this case, the flow vectors have a significant vertical component, and water encroachment occurs across a horizontal circular plane representing the oil/water contact. Thus, a rigorous bottomwater influx model must take into account vertical flow, and as will be shown below, the effect of vertical flow becomes increasingly more pronounced as the ratio of aquifer thickness, h, to reservoir radius, rR, becomes larger. The discussion below provides a detailed treatment of the bottomwater flow model depicted in Fig. 2. The diffusivity equation governing flow for this system is reduced to dimensionless form by introducing dimensionless variables. The resultant equation is then solved with a numerical simulator, and as in the work of van Everdingen and Hurst, the results are presented in the form of tables of dimensionless influx, WD, vs. dimensionless time, tD. Included in the discussion is an example calculation for a hypothetical bottom-water drive reservoir. This example illustrates the use of the new model and, by calculating influx with both the radial and bottom-water models, clearly shows that ignoring vertical flow can result in very significant error.
Discussion
Basic Equations. The partial-differential equation governing flow of a slightly compressible fluid in a system such as that shown in Fig. I is the well-known radial diffusivity equation:
..........................................(1)
For the bottomwater flow model depicted in Fig. 2, an additional term is added to this equation:
..........................................(2)
where Fk is the ratio of vertical to horizontal permeability. There are an infinite number of solutions to Eq. 2, representing all possible reservoir/aquifer configurations.
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