Waterflooding Performance of Communicating Stratified Reservoirs With Log-Normal Permeability Distribution
- Noaman El-Khatib (King Saud U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- December 1999
- Document Type
- Journal Paper
- 542 - 549
- 1999. Society of Petroleum Engineers
- 5.4.1 Waterflooding, 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements, 4.1.2 Separation and Treating, 6.5.2 Water use, produced water discharge and disposal, 5.6.4 Drillstem/Well Testing, 5.5 Reservoir Simulation, 1.2.3 Rock properties, 4.3.4 Scale, 5.3.4 Reduction of Residual Oil Saturation, 1.6.9 Coring, Fishing, 5.7.2 Recovery Factors
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An analytical solution is developed for waterflooding performance of layered reservoirs with a log-normal permeability distribution with complete crossflow between layers. The permeability distribution is characterized by the Dykstra-Parsons (DP) variation coefficient VDP or the standard deviation of the distribution sk. The performance is expressed in terms of vertical coverage as function of the producing water-oil ratio. Also an expression for the dimensionless time (pore volumes of injected water) at a given water-oil ratio is derived. Expressions are also derived for pseudorelative permeability functions and fractional flow curves that can be used in reservoir simulation. Correlation charts are also presented to enable graphical determination of the performance. The variables are combined in such a way that a single chart is constructed for the entire range of water-oil ratio, mobility ratio and permeability variation.
Analogy to the Buckley-Leverett (BL) multiple-valued saturation profile is found to occur at low mobility ratios (M<1) where a multiple-valued displacement front is formed. A procedure similar to the BL discontinuity is suggested to handle this situation. Successive layers with different permeabilities are allowed to move with the same velocity resulting in a single-valued profile with a discontinuity. No such behavior is observed for mobility ratios greater than unity. A criterion for the minimum mobility ratio at which this behavior occurs is presented as a function of the variation coefficient VDP.
Waterflooding is still the recovery process responsible for most of the oil production by secondary recovery. Water injected into the reservoir displaces almost all of the oil except the residual oil saturation from the portions of the reservoir contacted or swept by water. The fraction of oil displaced from a contacted volume is known as the displacement efficiency and depends on the relative permeability characteristics of the rock as well as the viscosities of the displacing and displaced fluids. The extent to which a reservoir is swept by a displacing fluid is separated into areal and vertical sweep efficiencies. The areal sweep efficiency accounts for the nonlinearity of the flow patterns between injection and production wells. The vertical sweep efficiency or coverage is caused by the heterogeneity of the reservoir, i.e., variation of horizontal permeability in the vertical direction. The displacing fluid tends to move faster in zones with higher permeabilities, resulting in earlier breakthrough into producing wells. Both areal and vertical sweep efficiencies are highly dependent on the mobility ratio of the displacement process and depend on the volume of the injected fluid expressed in pore volumes. The vertical sweep efficiency, however, is mainly dependent on the permeability distribution in the producing layer. Because of the variation in the depositional environments, reservoir rocks usually exhibit random variations in their petrophysical properties. Porosity is usually found to have a normal distribution, while the permeability has a log-normal distribution. The log-normal distribution of permeability is characterized by two parameters: the mean permeability Km and the standard deviation sk The standard deviation sk can also be expressed in terms of the DP variation coefficient VDP. It may also be related to the Lorenz coefficient L.
The methods available in the literature to predict the waterflooding performance of stratified reservoirs can be grouped into two categories depending on the assumption of communication or no communication between the different layers. The method of DP1 is the basis for performance prediction in noncommunicating stratified reservoirs. In addition to the basic equations presented in their work, they also presented correlations of the vertical coverage for log-normal permeability distributions in terms of mobility ratio and permeability variation coefficient at different values of the water-oil ratio. Also presented in their paper is a correlation of actual recovery factor vs. vertical coverage, initial water saturation, and water-oil ratio. This correlation was based on experimental runs performed on core plugs with permeability distributions determined by measuring the permeability at different locations on the core with a minipermeameter. Johnson2 later on combined the theoretical charts based on DP equations with the experimental correlation chart into a group of correlation charts from which the recovery factor at given values of water-oil ratio can be calculated directly without first computing the vertical coverage. Mobarek3 found discrepancies between results obtained by this method and results obtained using a numerical model.
Muskat4 presented analytical solution for waterflooding performance of stratified systems with linear and exponential permeability distributions. Reznik et al.5 derived expressions for the variation of pressure drop or injection rate as function of injection time for the DP model.
Prediction of waterflooding performance for communicating reservoirs was presented by Hiatt.6 This model assumes instantaneous crossflow between layers to keep the pressure gradient the same in all layers at any distance. Warren and Casgrove7 applied the Hiatt model to a system with log-normal permeability distribution and normal porosity distribution. Their method is semigraphical, semianalytical since they obtain values from plots of permeability and formation capacity distributions on probability graphs. Hearn8 used the same model of Hiatt to develop expressions for pseudorelative permeabilities that can be used in numerical reservoir simulation to reduce a three-dimensional model to a two-dimensional areal model with average (pseudo) functions for the vertical direction. El-Khatib9 extended the work of Hiatt to account for variable rock properties other than the absolute permeability. He also presented equations for the variation of the injectivity ratio with injection time and compared performance of communicating and noncommunicating systems.
Since it is widely accepted that the permeability in reservoir rocks exhibits a log-normal distribution, the objective of this work is to present a solution in a closed form for the waterflooding performance of stratified reservoirs with such permeability distributions. This would be the limiting case for a stratified system composed of a very large number of layers. In such a case, it is reasonable to assume complete communication between the layers since it is highly unrealistic to assume such large number of layers to be separated by an equal number of thin insulating strata.
Assumptions and Definitions
The following assumptions are made:
The sysstem is linear, horizontal and of constant thickness.
The flow is isothermal, incompressible and obeys Darcy's law.
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