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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<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^

A mathemathematical model is developed for performance prediction of waterflooding performance in stratified reservoirs using the Buckley-Leverett displacement mechanism. A modified definition of the mobility ratio is untroduced to account for the saturation variation behind the displacement front. Using this modified mobility ratio, the Dykstra-Parsons equations can be used to estimate the relative locations of the displacement fronts in different layers at the time of water breakthrough at a given layer. For layers after water breakthrough, expressions for the flow rate and water throughput are derived in terms of integral equations that are solved by iteration. The Buckley-Leverett and Welge tangent method is used to obtain the outlet and average saturations in each layer. These saturations are used to obtain the fractional oil recovery and water cut of each layer. Summation over all layers yields the performance of the total system. Expressions for the injectivity ratio are also derived. Solutions for stratified systems with log normal permeability distribution were obtained and compared with those for the piston-like displacement (Dykstra-Parsons). The effects of viscosity ratio and the Dykstra-Parsons permeability variation coefficient (V DP) on the performance is investigated. The Introduction of pseudo relative permeability functions is discussed.

Capillary pressure curves are represented by the equation Pc=a/ (Sw- Swi)b. It is shown that for the Leverett J-function to produce a single correlation, the tortuosity. the irreducible water saturation and the saturation exponent b must be the same for the different formations. A modified capillary pressure function J* is introduced that incorporates the tortuosity and irreducible water saturation in its definition. The modified function is correlated with the normalized saturation SD so that all curves will have their vertical asymptotes at a single point SD=0 The modified function represents a significant improvement over the original one since it is now sufficient to have the value of the saturation exponent b be the same for two samples to get the same correlation. This was verified by analyzing data from the literature and comparing the two correlations. A modified linear regression by means of weighted least squares method is used to fit the experimental capillary pressure data. Data sets from different sources were analyzed. It was found that a good single correlation can be obtained for samples from the same formation but the correlations were different for different formations. Introduction The capillary phenomena occurs in porous media when two or more immiscible fluids are present in the pore space. Due to the interfacial energy of the interface between the two phases. a difference in the pressure across the interface results and causes a curvature of the interface. The capillary forces causes retention of fluids in the pore space against the gravity forces. Immiscible fluids segregate due to gravity if placed in large containers such as tanks and pipes of large diameters. In porous rocks, however, the dense fluid (water) can be found at higher elevations above the oil-water contact. P. 547

Summary. The line-source solution of the diffusivity equation is differentiated and rearranged so that a linear relation between the variables is obtained. A plot of log tp' vs. 1/t yields a straight line whose intercept and slope are used to estimate the transmissivity, k h/mu, and the storativity, h phi ct, respectively. The method is extended to handle two-rate tests, including buildup tests, and can he used for the analysis of the combined data of the two periods. Introduction In interference test analysis, the semilog plotting techniques are inadequate because of the invalidity of the logarithmic approximation of the exponential integral function at large times. Usually type-curve matching is used. Recently, analysis methods based on the pressure derivative, p', were introduced. Tiab and Kumar used the maximum value of p' and the time at that point to estimate the transmissivity and storativity of the reservoir. Bourdet et al. introduced type-curve matching methods that involve both pressure-drop and pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-derivative methods to variable-rate testing pressure-derivative methods to variable-rate testing using a superposition time function. In this work, a derivative method that yields a straight-line plot is introduced. More details can be found in Ref. 4. Method The derivative of the pressure, with respect to time at a radial distance Ar from an active well as obtained from the line-source solution, is (1) Multiplying Eq. 1 by -t and taking logarithms of both sides, we get (2) The constants A and b are related to the transmissivity, T, and storativity, S, by the following equations: (3) and (4) It is clear from Eq. 2 that a plot of tp' vs. 1/t on a semilog graph gives a straight line. The intercept and slope can be used to estimate the transmissivity and storativity of the reservoir. The intercept, A, is read directly on the logarithmic scale as the value of tp' at 1/t=0. The slope in cycles/hr would be -b/2.303. Extension to Two-Rate Tests If the rate at the active well is changed from q1 to q2 at time t, it can be shown that (5) where (6) Eq. 5 is similar to Eq. 2 and yields a straight line if Delta t(p'+C)q1/(q1 -q2) is plotted vs. 1/At on a semilog plot. In this format, the slope and intercept of the straight line are the same as those of Eq. 2. This means that data points from the first rate region (q = q1) calculated according to Eq. 2 can be combined with data points from the second rate region (q = q2) calculated according to Eq. 5 and analyzed together to obtain the values of T and S that fit the data points in the two regions. In pressure-buildup testing, q2 --0 and a semilog plot of Delta t(p'+ C) vs. 1/Delta t would give a straight line with the same slope and intercept as those of Eqs. 2 and 5. Because C is not known in advance and depends on A and b in addition to At, an iterative procedure must be used in the analysis. An initial value of C = 0 may be used, and the constants A and b are estimated either graphically or by linear regression. The values of A and b obtained are used to update C according to Eq. 6. The iteration is continued until successive values of A and b become constant within a prescribed limit. The final values of A and b are then used to estimate the transmissivity and storativity from Eqs. 3 and 4, respectively. Illustrative Example The developed method is applied to the interference test data of a gas well reported by Ramey et al. Table 1 shows the test data, calculations, and final results. The term is approximated by Deltap/Delta 1n at the geometric average time . The method of least squares was used to find the constants A and b. A graphic presentation of the data is shown in Fig. 1. Type-curve matching presentation of the data is shown in Fig. 1. Type-curve matching results reported by Ramey et al. are also shown in Table 1. Comparison with results obtained by this method indicates a difference of about 3% in the transmissivity and 0.6% in the storativity, indicating the accuracy of the proposed method. Conclusions A new approach for interference test analysis is introduced. A semilog plot of tp vs. 1/t gives a straight line from its intercept, and slope reservoir parameters can be estimated. The method can also be applied to two-rate interference tests for which an iterative procedure is used. Data points from the two regions may be analyzed procedure is used. Data points from the two regions may be analyzed separately or combined. SPEFE P. 609