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# Macroscopic displacement efficiency of a linear waterflood

This page discusses the mathematical aspects of water/oil displacement for homogeneous linear systems. The presentation here is brief and does not include the intermediate steps of the mathematical derivation of the key equations. The details of these mathematical derivations are available in Willhite.

The displacement of oil by water from a porous and permeable rock is an unsteady-state process because the saturations change with time and distance from the injection point (see schematic diagram of Fig. 1). These saturation changes cause the relative permeability values and pressures to change as a function of time at each position in the rock. Fig. 2 illustrates the various stages of an oil/water displacement process in a homogeneous linear system.

The mathematical derivation of fluid-flow equations for porous media begins with the simple concept of a material-balance calculation: accumulation equals fluid in minus fluid out. This equation is written for the whole system and for each of the phases: water, oil, and gas. Eqs. 1 and 2 are the equations for the conservation of mass for a water/oil homogeneous linear system: ....................(1)

and ....................(2)

where x = position in x-coordinate system, ft; ρo = oil density, lbm/ft3 or g/cm3; uox = oil velocity in the x direction, ft/day; t = time, days; So = oil saturation, fraction PV; ϕ = porosity, fraction BV; ρw = water density, lbm/ft3 or g/cm3; uwx = water velocity in the x direction, ft/day; and Sw = water saturation, fraction.

Assuming that the oil and water are incompressible and that the porosity is constant, these equations become: ....................(3)

and ....................(4)

where qo = oil-production rate, B/D; A = cross-sectional area available for flow, ft2; and qw = water-production rate, B/D.

Next, the equations for fractional flow of oil and water are incorporated into these equations. The three fractional-flow equations are: ....................(5) ....................(6)

and ....................(7)

where fo = fractional flow of oil; qt = the total production rate, B/D; and fw = fractional flow of water.

Substituting Eq. 6 into Eq. 4 yields: ....................(8)

## Buckley-Leverett solution

Further mathematical manipulation of these equations obtains the Buckley-Leverett equation (Eq. 9), or frontal-advance equation. To derive this equation, it is assumed that the fractional flow of water is a function only of the water saturation and that there is no mass transfer between the oil and water phases. ....................(9)

This equation shows that in a linear displacement of water displacing oil, each water saturation moves through the rock at a velocity that is computed from the derivative of the fractional flow with respect to water saturation.

The general form of the fractional-flow equation for water is: ....................(10)

where ko = permeability to oil, darcies; g = gravity constant; α = reservoir dip angle, degrees; and kw = permeability to water, darcies. This equation includes terms for capillary pressure variation (as a function of saturation) in the linear direction and for the linear system possibly dipping at angle α.

Assuming that the gradient in Pc with position is very small and that the linear system is horizontal reduces Eq. 10 to: ....................(11)

Fig. 3 presents a typical fractional-flow curve that would be calculated from Eq. 11. This figure also shows a tangent to the fractional-flow curve that originates at the initial water saturation. The tangent point defines the "breakthrough" or "flood-front" saturation Swf. This saturation is equivalent to the saturation that Buckley and Leverett obtained through intuitive arguments. It subsequently was recognized that this tangent intersects the fractional-flow curve at the saturation that is common to the stabilized and the nonstabilized zones.

The frontal-advance equation (Eq. 9) cannot predict the saturation profile between the connate-water saturation and the breakthrough saturation. An approximation that was developed from the Buckley-Leverett solution considers the saturation change to be a step increase ("shock") from the connate-water saturation Swc to the flood-front saturation Swf. Fig. 4 shows this saturation profile. The shock occurs because all saturations that are less than Swf travel at the velocity of the flood front. Saturations that are greater than Swf travel at velocities that are determined from Eq. 9 by calculating the derivative of the fractional-flow curve at each Sw value.

That the Buckley-Leverett solution is reasonable has been experimentally verified. Fig. 5 compares experimental results with calculated values for two oils that have nearly a hundred-fold difference in viscosity.

Fig. 6 shows the viscosity ratio’s effect on the water fractional-flow behavior. The viscosity ratio is a key parameter; the efficiency of the linear displacement process of water displacing oil changes and is substantially different when the oil’s and the water’s viscosity is the same compared to when the oil’s viscosity is much higher than the water’s.

## Nomenclature

 A = cross-sectional area available for flow, ft2 fw = fractional flow of water = average fractional flow of water fwf = fractional flow of water at flood front Fwo = water/oil ratio Fpvg = fraction of displaceable pore volume that is gas saturated ko = permeability to oil, darcies ko = permeability to oil, darcies qw = water-production rate, B/D Pc = capillary pressure, psia qt = total production rate, B/D qw = water-production rate, B/D So = oil saturation, fraction PV Soi = initial oil saturation or (1 – Swc), fraction PV Sorw = residual oil saturation to waterflooding, fraction PV Sw = water saturation, fraction PV uwx = water Darcy velocity in the x direction, ft/day ρo = oil density, lbm/ft3 or g/cm3 ρw = water density, lbm/ft3 or g/cm3 ϕ = porosity, fraction BV μo = oil viscosity, cp μw = water viscosity, cp