An Analytical Extension of the Dykstra-Parsons Vertical Stratification Discrete Solution to a Continuous, Real-Time Basis (includes associated papers 13753 and 13830 and 14873 and 14695 )

Analytically exact and continuous solutions are developed for the space-time relationships of a linear water flood in a vertically stratified reservoir model. The solutions represent simple extensions of the analytical, but discrete, spatial relationships of Dykstra and Parsons to analytically continuous expressions. Explicit solutions for time are presented that permit the coupling of all instantaneous presented that permit the coupling of all instantaneous and cumulative performance parameters to a completely rational time basis. The continuous nature of the solutions permits unusual fluid behavior to be observed between successive bedbreak through points. Although the model assumes piston-like displacement, these novel phenomena do not appear to be artifacts of this limiting assumption. This work develops the concept of a bed property time that forms the basis for a generalized bed-ordering parameter. For the case of constant injection pressure, parameter. For the case of constant injection pressure, property time is shown to be identical to the real or property time is shown to be identical to the real or process time. For the common case of constant overall process time. For the common case of constant overall injection rate, the customary use of property time concepts to determine real or process time is shown to be completely erroneous, yielding values that are incorrect both in magnitude and in trend. A bed flood-front passing phenomenon is presented that allows the flood fronts of "slower" beds initially to lead those of "faster" beds if specified constraints are satisfied. It is shown that these constraints can be satisfied for moderate bed-fluid property variations. The analytical nature of the solutions provides greater insight into the controlling factors of such processes. The use of real time as a process parameter provides a more realistic basis for comparative performance between floods under the same or different injection conditions. The relationship between injected PV and time can be used to extend the linear model to approximate predictions for stratified, nonlinear, pattern floods. Introduction The first rational description of the saturation distribution created by immiscible displacement in homogeneous porous media was given by the classical work of Buckley and Leverett. The trailing zone relations developed by Welge greatly increased the use of the Buckley and Leverett relationships in water flood performance predictions and in the computation of relative performance predictions and in the computation of relative permeabilities from unsteady-state flows. These analyses did permeabilities from unsteady-state flows. These analyses did not, however, address the major problem of displacement in vertically stratified reservoirs. The problem of vertical stratification of the producingzones was first analyzed by Law using a set of horizontal, parallel, noncommunicating beds coupled only at the parallel, noncommunicating beds coupled only at the injection and production surfaces. This model remains the classic reservoir prototype for analysis of water flood performance in vertically stratified reservoirs. A stratified performance in vertically stratified reservoirs. A stratified reservoir with cross flow can be treated as a uniform system characterized by its average properties. Stiles used this model to predict water flood behaviour when the bed stratification was caused only by variations in the absolute and effective permeabilities. Further, the existence of a trailing zone was not accounted for, creating a piston-like displacement. The velocity of a bed flood front was assumed to be a function of absolute permeability only, which imposed a mobility ratio of unity on every bed. Unfortunately, the Stiles method was also used for mobility ratios other than unity by simple superposition. The results were completely irrational and alien to the model assumptions. The first rational inclusion of mobility ratios other than unity was presented in the work of Dykstra and Parsons. Although the piston-like displacement assumption is retained, this transport model is dynamically correct. Dykstra and Parsons also presented their results in a statistical fashion based on a log-normal permeability distribution. Unfortunately, these results were coupled to a severely restricted recovery correlation by Johnson, which greatly limited the universality of the original Dykstra and Parsons statistical work; this fact is implied by Craig. SPEJ P. 643