Waterflooding Performance of Communicating Stratified Reservoirs With Log-Normal Permeability Distribution
Authors: El-Khatib Noaman 
Published: December 2007

Abstract
Analytical solution is developed for waterflooding performance of layered reservoirs with a log-normal permeability distribution and assuming complete crossflow between layers. The permeability distribution is characterized by the Dykstra-Parsons variation coefficient VDP which is related to the standard deviation of the log-normal distribution k. The performance is expressed in terms of vertical converge as function of the producing water-oil ratio (W.O.R). Also an expression for the dimensionless time in terms of pore volumes of injected water at a given W.O.R. is derived. Dimensionless time and vertical converge can be obtained at any value of the W.O.R. for the corresponding mobility ratio by direct substitution in the derived equations which involve error functions and inverse error functions. The equations derived are presented as correlation charts to enable graphical determination of the performance. The variables are combined in such a way that a single chart for each of the dimensionless time and coverage is constructed for the entire range of W.O.R., mobility ratio and permeability variation.
The analytical solution is also used to derive expressions for pseudo relative permeability functions and fractional flow curves that may be used in reservoir simulation. A three-dimensional simulation can thus be reduced into two-dimensional (areal) simulation.
Analogy to the Buckley-Leverett multiple-valued saturation profile is found to occur at low mobility ratios (M&lt;1) where a multiple-valued displacement front is formed. A procedure similar to the B-L 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 distance profile with one or more discontinuities. 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.

Introduction
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 extend 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 k. P. 167^