Application of Buckley-Leverett Displacement Theory to Noncommunicating Layered Systems
Authors: Snyder R.W. , Ramey H.J. 
Published: December 2007

Abstract
This paper presents the results of applying the Buckley-Leverett' displacement theory to petroleum reservoirs consisting of a finite number of layers. The layers are assumed to communicate only in the wellbores, and the reservoir may be represented as a linear system. Most previous investigations of this nature were limited by assumptions and by inconsistent calculation techniques. This study improves on previous work by applying the Buckley-Leverett displacement theory to a noncommunicating layered system where permeability, porosity, initial saturation, residual saturation and relative permeability vary from layer to layer in a logical and consistent manner. Gravity and capillary-pressure effects are neglected. A modification of the Higgins-Leighton calculation method was used in this study. Waterflood predictions were made with all properties varying, and then with only permeability varying using several mobility ratios. These results were compared with the Stiles and Dykstra-Parsons predictions. It is shown that the latter methods generally give poor values for the breakthrough recovery and pessimistic predictions for the performance after breakthrough. Similar results were obtained for a gas-displacement case.

Introduction
Field experience with immiscible displacement usually shows constant producing conditions until breakthrough of the displacing fluid. Then oil production continues at increasing displacing-to-displaced fluid ratios until the economic limit is reached. Three different ideal mechanisms are known that will produce this behavior:
- relative permeability effects as described by Buckley-Leverett frontal advance theory,
- vertical stratification as considered by Stiles, Dykstra and Parsons and others and
- different path lengths involved in areal (two-dimensional) flow between wells as described by Dyes et al. Without question, a combination of these factors modified by formation heterogeneity and other known and unknown factors actually does control the behavior of real systems, This paper presents results of an investigation of certain factors that should affect performance but which have received little attention to date.
In 1944, Law demonstrated that porosity and permeability are often found to have normal and logarithmic-normal distributions, respectively, throughout cored intervals in natural formations. This led to the concept of the noncommunicating, multilayered reservoir model for immiscible displacement. This model assumes that the reservoir is composed of a number of layers that communicate only at the wellbores. Each layer is individually homogeneous, but may be different from every other layer.
Stiles presented one of the earliest applications of this model to waterflood performance. In addition, Stiles assumed that the initial saturations and relative permeabilities were the same for each layer, porosity was the same. displacement was piston-like, fluids were incompressible and injection into each layer was proportional to that layer's permeability capacity (permeability-thickness product). The last assumption would be true if the mobility ratio for the displacement were unity." Dykstra and Parson's used the same model as Stiles, but rigorously included mobility ratios other than unity for piston-like displacement. Dykstra and Parsons used their general result to produce charts for log-normal permeability distributions between layers. Similarly, Muskat published analytical solutions for linear and exponential permeability distributionsIn 1959, Roberts described a scheme for calculating water-drive performance for the noncommunicating, layered reservoir model which considered two-phase flow in the displaced region. Roberts used the same model and assumed that the injection rate into a layer was proportional to that layer's permeability capacity, but that flood front locations could be evaluated from the Dykstra-Parsons results. These assumptions are inconsistent, and a material balance cannot be maintained except for a mobility ratio of unity. At the same time, Kufus and Lynch coupled Buckley-Leverett displacement theory with the layered model to provide an improvement of the Dykstra-Parsons method that was consistent. In 1960, Higgins and Leighton presented a numerical method for calculating waterflood performance also considering two-phase flow in the displaced region. The result was used to investigate variation in absolute permeability and oil viscosity. An excellent, detailed history of using the noncommunicating, layered reservoir model was presented by Nielsen. The preceding techniques (and many related ones) were similar in that differences in initial saturations, residual saturations and relative permeabilities from layer to layer were neglected. It is well known that the irreducible water saturation is an important function of absolute permeability.