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# Vertical displacement in a waterflood

This section discusses the impact of vertical variations in permeability and the effect of gravity on simple 2D reservoir situations in which the areal effects are ignored. Gravity effects always are present because for any potential waterflood project, oil always is less dense than water, even more so after the gas is included that is dissolved in the oil at reservoir conditions.

## Overview

Three particular situations are discussed here:

• Stratified systems with noncommunicating layers for various mobility ratios
• Homogeneous systems with gravity (including dipping beds)
• Stratified systems with communicating layers and assumed vertical fluid equilibrium

The discussion below does not include the Pc effects on vertical saturation distributions. Through countercurrent imbibition, Pc effects help to counteract nonequilibrium water/oil saturation distributions. The mathematics of including Pc effects makes the problems too complicated for inclusion here. Standard numerical reservoir simulators—which are needed for a complete analysis of real reservoir situations—account for the effects of countercurrent imbibition caused by Pc effects, as well as for the water/oil density and viscosity differences that lead to injection-water gravity underrunning and the layer-by-layer permeability variations, with or without communicating layers.

Before presenting some of the technical-literature techniques for studying the vertical displacement characteristics of water/oil displacement, one must first define some measure of the vertical permeability variations. Dykstra and Parsons developed a method that is based on routine-core-analysis data. In that approach, the routine-core-analysis permeability data for the pay intervals are arranged in descending order, and the percent of the total number of values that exceeds each entry is calculated. The values then are presented as a log-probability plot (see Fig. 1). A reasonably straight line is drawn through the data, with the points in the 10-to-90% range being more heavily weighted. This straight line is a measure of the dispersion and the heterogeneity of the reservoir rock. What has come to be known as the "Dykstra-Parsons coefficient of permeability variation" V is defined as: ....................(1)

where k50 = median permeability value, md, and k84.1 = permeability at 84.1% probability (one standard deviation), md.

Fig. 2 shows the relationship of the Dykstra-Parsons V values to varying degrees of rock heterogeneity. Note that in Fig. 2, the V for most reservoirs ranges from 0.5 to 0.9.

## Stratified systems with noncommunicating layers

Over the years, several waterflood prediction methods have been proposed and published that account for the vertical variations in rock properties, particularly permeability. These simple methods assumed that every rock layer acts independently of all other rock layers (even at 1-ft increments in the reservoir) and that each rock layer is continuous from the injection well to the production well. These early methods were developed when the ability to make detailed, complicated engineering calculations was limited. They focused on how to account for:

• The effect of the vertical permeability variation with minimal consideration of the mobility ratio
• The effect of vertical permeability variation and mobility ratio, assuming constant pressure at the injection and production wells. ....................(2)

Stiles developed one of the earliest methods, for which only the permeability-thickness (kh) distribution of the vertical reservoir interval and the mobility ratio at endpoint conditions need to be known (see Eq. 2). The water/oil ratio (WOR) Fwo after water breakthrough as a function of the fraction of the total flow capacity C represented by layers having water breakthrough is defined as: ....................(3)

where Bo = the oil formation-volume factor, RB/STB.

A more sophisticated method that is widely used is that of Dykstra and Parsons. Their method is based on calculations for linear layered models and assumes no crossflow and the use of the results of more than 200 floodpot tests that were performed on more than 40 California oil-reservoir core samples. This Dykstra-and-Parsons method takes into account initial fluid saturations, mobility ratios, producing WORs, and fractional oil recoveries. The permeability variation was taken into account by use of V, as defined in Eq. 1 above. Figs. 3 through 6 plot the results of the Dykstra-and-Parsons technique as V vs. M for four WOR levels (1, 5, 25, and 100).

## Homogeneous reservoirs subject to gravity effects

In essentially all reservoirs, even those with close well spacings, the horizontal distance between an injector well and a producer well is very long relative to the vertical thickness of the reservoir pay interval. This means that gravity plays a major role in the water/oil-displacement process, given that the fluids can move vertically within the pay interval. For conceptual and calculation purposes, the limiting case is to assume that gravity forces dominate the water/oil-displacement process, that gravity segregation of the oil and water is complete, and that the system is in "vertical equilibrium." This means that vertically the gravity and capillary forces are in balance and that the vertical saturation distribution is governed by the Pc/Sw function.

The first and simplest homogeneous reservoir situation described here is a reservoir whose permeability is constant throughout the pay interval. Craig studied a set of scaled laboratory vertical models experimentally and developed a correlation between the sweep efficiency at breakthrough and the values of the scaling parameter: ....................(4)

where (Δp)h = pressure difference in the horizontal direction, psi; (Δp)V = pressure difference in the vertical direction, psi; ut = horizontal Darcy velocity, ft/D; kx = permeability in the x direction, darcies; and Δρ = water/oil density difference, lbm/ft3.

As Fig. 7 shows, the sweep efficiency as it is related to the scaling parameter is a strong function of mobility ratio. Fig. 8 compares the fractional flow of water for a homogeneous system with vertical equilibrium to the fractional flow of water calculated from the original laboratory water/oil relative permeability curves. The effect of the water moving along the base of the reservoir interval because of the gravity effects—but with the Pc/Sw curve controlling the vertical distribution of the water and oil saturations—is that the water breaks through earlier and the WOR rises more slowly.

Another reservoir situation that involves gravity effects is a homogeneous reservoir with dipping beds. If the rate of water injection in a waterflood is too low for vertical equilibrium to occur, there will be gravity-stabilized flow between the water and the oil. Dietz has derived a relationship to predict the critical velocity qc required to propagate a stable interface through a linear system in which gravity forces dominate, but in which pistonlike displacement occurs and Pc effects are neglected: ....................(5)

where ρo = oil density, lbm/ft3; ρw = water density, lbm/ft3; and α = dip angle, degree.

When the oil/water interface is stable, the velocities of oil and water are equal at every point in the interface. The interface is linear and will move at a constant velocity through the system as long as q < qc. The stable linear interface will not necessarily be flat; however, it will be stable with a slope β, as defined by Eq. 6: ....................(6)

where y = the position in y-coordinate system, ft, and G is dimensionless and defined by Eq. 7: ....................(7)

Figs. 8a and 8b depict gravity-stable situations for two different mobility ratios. Fig. 8c depicts the unstable situation for an unfavorable mobility ratio where the displacement rate is too high for the water and oil to maintain vertical equilibrium.

## Stratified systems with communicating layers and assumed vertical equilibrium

One of the systems that have been analyzed with simple calculations is that of water/oil displacement with vertical permeability variations and gravity effects, but with capillary pressure neglected. Dake explores this in his reservoir-engineering textbook. Dake’s illustrative example assumes a three-layer system. He assumes the permeability variation to be highest to lowest from top to bottom, and then compares those results with results from assuming the reverse, the layer with highest permeability variation then on the bottom. Fig. 9 lists the properties of the three layers, and Fig. 10 presents the averaged relative permeability curves. Fig. 11 shows the pseudocapillary pressure (a) and fractional-flow curves (b). Note that Fig. 11b includes smoothed and unsmoothed versions of the fractional-flow curve. The smoothed version is the curve that would be used for a Welge-type fractional-flow calculation.

Dake’s example shows that in waterflooding where gravity effects are significant, having the high-permeability layers at the top of the reservoir interval allows a much more efficient oil displacement than when the high-permeability layers are at the bottom of the reservoir interval. This is because gravity causes the water to slump, and when the lower-permeability layers are the base, the water must move more slowly than the oil from the injector to the producer.

So far, the discussions have highlighted the development of techniques for understanding and analyzing key aspects of oil/water displacement. These techniques predate modern computers; hence, they were developed to simplify the real reservoir problem sufficiently to allow various engineering calculations to be made.

Of course, the availability of modern computers and advanced numerical-reservoir-simulation software has rendered many of these simplifying assumptions unnecessary when quantifying waterflood-type water/oil displacements in real reservoirs. Nevertheless, these historic techniques have been discussed here to provide an understanding of the dynamics of the water/oil-displacement process and the primary variables that influence the recovery efficiency.

## Nomenclature

 A = cross-sectional area available for flow, ft2 Bo = oil formation-volume factor, RB/STB C = total flow capacity g = gravity constant G = value as defined by Eq. 7, dimensionless h = reservoir thickness, ft i = fluid phase i k = absolute permeability, darcies k50 = median permeability value, md k84.1 = permeability at 1 standard deviation above mean value, md L = length, ft M = mobility ratio Pc = capillary pressure, psia x = position in x -coordinate system, ft y = position in y-coordinate system, ft α = reservoir dip angle, degrees β = slope, degrees (Δp)h = pressure difference in horizontal direction, psi (Δp)V = pressure difference in vertical direction, psi Δρ = water/oil density difference, lbm/ft3 or g/cm3 μi = viscosity of fluid phase i, cp μo = oil viscosity, cp μw = water viscosity, cp ρo = oil density, lbm/ft3 or g/cm3 ρw = water density, lbm/ft3 or g/cm3 fwf = fractional flow of water at flood front Fwo = water/oil ratio