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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.^{[1]}

## Contents

## Unsteady-state process

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:

and

where *x* = position in *x*-coordinate system, ft; *ρ*_{o} = oil density, lbm/ft^{3} or g/cm^{3}; *u*_{ox} = oil velocity in the *x* direction, ft/day; *t* = time, days; *S*_{o} = oil saturation, fraction PV; *ϕ* = porosity, fraction BV; *ρ*_{w} = water density, lbm/ft^{3} or g/cm^{3}; *u*_{wx} = water velocity in the x direction, ft/day; and *S*_{w} = water saturation, fraction.

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

and

where *q*_{o} = oil-production rate, B/D; *A* = cross-sectional area available for flow, ft^{2}; and *q*_{w} = 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:

and

where *f*_{o} = fractional flow of oil; *q*_{t} = the total production rate, B/D; and *f*_{w} = fractional flow of water.

Substituting **Eq. 6** into **Eq. 4** yields:

## 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.

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:

where *k*_{o} = permeability to oil, darcies; *g* = gravity constant; *α* = reservoir dip angle, degrees; and *k*_{w} = 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 *P*_{c} with position is very small and that the linear system is horizontal reduces **Eq. 10** to:

**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 *S*_{wf}. This saturation is equivalent to the saturation that Buckley and Leverett obtained through intuitive arguments.^{[3]} 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.^{[4]}

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 *S*_{wc} to the flood-front saturation *S*_{wf}. **Fig. 4** shows this saturation profile. The shock occurs because all saturations that are less than *S*_{wf} travel at the velocity of the flood front. Saturations that are greater than *S*_{wf} travel at velocities that are determined from **Eq. 9** by calculating the derivative of the fractional-flow curve at each *S*_{w} 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

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}Willhite, G.P. 1986.*Waterflooding*, Vol.**3**. Richardson, Texas: Textbook Series, SPE. - ↑ Craig Jr., F.F. 1971.
*The Reservoir Engineering Aspects of Waterflooding*, Vol.**3**. Richardson, Texas: Monograph Series, SPE. - ↑ Buckley, S.E. and Leverett, M.C. 1942. Mechanism of Fluid Displacement in Sands.
*Trans*., AIME**142**: 107–116. SPE-942107-G. - ↑ Terwilliger, P.L., Wilsey, L.E., Hall, H.N. et al. 1951. An Experimental and Theoretical Investigation of Gravity Drainage Performance.
*Trans*., AIME**192**: 285. http://dx.doi.org/10.2118/951285-G.

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## See also

Microscopic efficiency of waterflooding

Areal displacement in a waterflood