Single-Phase Fluid Flow in a Stratified Porous Medium With Crossflow(includes associated paper 12946 )

A new model is presented for a stratified reservoir. The model is used to study the influence of cross flow on pressure transient well tests and other single-phase flow problems. The reasons for single-phase cross flow in multilayer reservoirs are discussed. Solutions for linear and radial incompressible flow in a stratified reservoir with cross flow are presented; effects of reservoir parameters and cross flow on pressure are studied, and acriterion for considering cross flow between layers is suggested. Introduction Real reservoirs normally consist of many layers with different permeabilities. Frequently, thin low-permeabilitysilts or shales separate the layers. For simplicity, such reservoirs often are treated as a single uniform layer or asseveral independent layers. In reality, these layers influence each other through cross flow and cannot be treated so simply. In the early 1960's, several papers addressed the behavior and influence of single-phase fluid cross flow in multilayer reservoirs. These papers studied the unsteady flow behavior to explain transient well test results obtained in multilayer reservoirs with cross flow. From these papers it is clear that rigorous mathematical treatment of the single-phase cross flow problem in two-layer reservoirs is quite difficult-even under the highly idealized assumptions that each layer is homogeneous, that no low-permeability shale is between the layers, and so on. The problem is even more difficult if the reservoir has more than two layers. So, the problems really need to be simplified. A simplified model, called "semipermeable wall model," is suggested here to approximate the actual multilayer reservoir. In this model we ignore the pressure variation in the vertical direction in the differential equations and avoid the need for boundary conditions between layers, so the problem is greatly simplified mathematically. The purposes of this work areto establish fundamental equations for the semipermeable wall model,to discuss why single-phase cross flow occurs, andto study the flow in a multilayer system with cross flow to determine when to treat the multilayer system as a single uniform layer or as many independent layers and when neither of these simplifications applies. This paper gives some exact solutions for simple multilayer flow cases with cross flow. These examples are used to give a clear picture of the flow in a multilayer reservoir and to give criteria for deciding when we can treat the multilayer system as a single layer or as many independent layers. Semipermeable Wall Model and Fundamental Differential Equations. Reservoirs generally have horizontal dimensions much greater than their thickness between impermeable rocks at the top and bottom. If there is no low-permeability shale within a layer, the change of pressure is generally very small in the vertical direction. The pressure at the midway point in the vertical direction of the layer is a good representation of the average pressure in the layer. The vertical equilibrium (VE) concept is used widely in the petroleum literature. VE in each layer means that the vertical pressure drop is zero at all times and positions in each layer, so the pressure will be the same for all the points on any vertical line in each layer. Assuming VE implies perfect vertical communication, which is equivalent to assuming infinite vertical permeability. VE will be a good assumption for layers with effective length-to-thickness ratio of 10 or more. Since the pressure change is very small in the vertical direction in any layer, we can concentrate the vertical resistance to flow at the walls between the layers, and let the vertical resistance be zero within layers. Because the wall has concentrated vertical resistance, it is no longer an ordinary interface between layers. The pressures on opposite sides of the wall will differ by a finite amount. The resistances of the walls between layers should be taken such that they are equivalent to the actual vertical resistance of the reservoir. These imaginary walls, called "semipermeable walls," are a remedy for the assumption of infinite vertical permeability within layers. Five assumptions are used in the semipermeable wall model.The reservoir pore space is filled with a slightly compressible single-phase fluid.The reservoir is homogeneous in vertical direction in each layer.The thickness of each layer is constant.The reservoir consists of n layers. In each layer, the horizontal permeability is finite, but the vertical permeability is infinite.Gravity force is negligible. Fluid flowing through each semipermeable wall is assumed proportional to the local pressure difference across the wall and inversely proportional to viscosity of the fluid. Consider a two-layer model (see Fig. 1A). SPEJ P. 97^