Abstract
A mathematical model is developed for waterfloodingperformance in linear stratified systems for both cases of noncommunicating layers with no crossflow and communicating layers with complete crossflow. The model accounts for variation of porosity and saturation inaddition to permeability of the different layers. The modelpredicts the fractional oil recovery, the water cut, the totalvolume injected, and the change in the total pressure drop, or the change in injection rate at the water breakthroughin the successive layers. A systematic procedure forordering of layers and performing calculations is outlined. Aprocedure for combining layers to avoid instability in the case of low mobility ratio is introduced. The developed model is applied to different examplesof stratified reservoirs. The effects of mobility ratio and crossflow between layers are discussed. The effects of variable porosity and fluid saturation are discussed also. It was found that crossflow between layers enhancesthe oil recovery for systems with favorable mobility ratios(lambda w/lambda o less than 1) and retards oil recovery for systems with unfavorable mobility ratios. It was found also that crossflow causes the effect of the mobility ratio on oil recovery to become more pronounced. The variation of porosity andfluid saturation with permeability is found to increase oilrecovery over that for the case of uniform porosity andsaturation for both favorable and unfavorable mobility ratios.
Introduction
Because of the variation in the depositional environments, oil-bearing formations usually exhibit random variationsin their petrophysical properties in both horizontal and vertical directions. Statistical as well as geological criteria usually are used to divide the pay zone betweenadjacent wells into a number of horizontal layers each with its own properties (k, phi, h, Swi, and Sor). Suchreservoirs usually are called "stratified," "layered,"or"heterogeneous" reservoirs. This variation in properties affects the performance of oil reservoirs during primary and secondary recovery processes. One of the significant factors influencingrecovery performance during waterflooding is thevariation of permeability in the vertical direction. In this case, the displacing fluid (water) tends to move faster in zones with higher permeabilities, causing earlier breakthrough of water into the producing wells and eventual by passing of some of the displaced fluid (oil). The various methods used for the prediction of waterflooding performance of stratified reservoirs differin the way the communication between the different layersis treated. Two ideal cases usually are used:completely noncommunicating layers andcommunicating layerswith complete crossflow.
For actual stratified Systems, however, the layers are partially connected in the vertical direction, and the performance of the system lies betweenthose of the two ideal cases. For the case of noncommunicating stratified layers, the methods of Stiles and Dykstra-Parsons usually areused. Stiles' method assumes unit mobility ratio for the displacement process when computing the recovery but accounts for the mobility ratio when computing the WOR, which results in contradictory formulas for the performance. The Dykstra-Parsons method and its modified version by Johnson use semiempirical correlations based on log-normal distribution of the layers' permeability. Muskat presented analytical expressions for the performance of reservoirs having linear and exponential permeability distributions. Two methods are available in the literature forestimating the performance of communicating systems with complete crossflow the method of Warren and Cosgrove and that of Hearn. Warren and Cosgrove's method requires a log-normal permeability distribution. Furthermore, it ignores the problem of ordering of layersfor low mobility ratio, which may cause physicallymeaningless results. The method of Hearn is intended to derive pseudorelative permeability functions for the stratified system to be used in reservoir simulation. Most of these methods assume that all layers have identical properties except permeability. Also, the time is notrelated explicitly to the performance. Furthermore, noneof these methods considers the variation in injection rateand total pressure drop as the displacement process progresses. Although these points can be treated numerically for a particular case using reservoir simulation methods, the objective of this work is to developan alytical expressions for waterflooding performance inidealized linear stratified systems that will consider the previously mentioned points.
Theoretical Analysis
Assumption and Definitions. For both the noncommunicating and communicating systems, these assumptions are made. 1. The system is linear and horizontal, and the flow is incompressible, isothermal, and obeys Darcy's law.
SPEJ
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