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Buckley-Leverett Application

Technical Description and Manual

by
Paul Tijink

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Table of Contents

1. Introduction

THE BUCKLEY-LEVERETT THEORY

2. Fractional flow

3. Relative permeability curves

a) Corey relative permeability curves

b) LET relative permeability curves

4. The Buckley-Leverett Equation

5. Welge's method

6. Buckley-Leverett aspects and practical application

a) Time of water breakthrough

b) Recovery factor and water cut at time of water breakthrough

c) Recovery factor and water cut after time of water breakthrough

d) Mobility ratio

e) Gravity and production rate

f) Capillary pressure

g) Recovery factor versus pore volume injected

h) Displacement efficiency under relative permeability uncertainty

i) Critical rate estimation

GEOMETRY AND HETEROGENEITY

7. Reservoir geometry

a) Rectangular geometry

b) Trapezoidal geometry

c) Radial geometry

d) Coning geometry for vertical wells

e) Coning geometry for horizontal wells

8. Geometry and Gravitational Forces

9. Composite reservoirs

a) Radial composite reservoirs

THE BUCKLEY-LEVERETT APPLICATION

10. Input: Relative permeability curves

11. Input: Reservoir description and production

12. Output: Results obtained from Buckley-Leverett theory

13. Output: Graphs obtained from Buckley-Leverett theory

INFO

14. Nomenclature

15. Bibliography

16. Contact information

1. Introduction

This Technical Description and Manual gives a comprehensive description of the Buckley-Leverett theory and extensions of the theory to practical application areas. Furthermore, a description is given of the Buckley-Leverett tool, which is available from the MMbbls website at www.mmbbls.com.

The purpose of the Buckley-Leverett tool is to enable subsurface professionals to quickly compute the expected fractional flow behaviour over time of their mixed phase reservoirs. The tool is accompanied by a comprehensive reference manual which gives a detailed description of the concept of fractional flow, the Buckley-Leverett theory (Buckley & Leverett, 1942), the mathematical equations and the practical application of the theory within subsurface reservoir management.

The basic Buckley-Leverett equation describes the one-dimensional, frontal displacement of incompressible, immiscible water and oil, without mass transfer between the water and oil phases. In addition, Welge's method derives an expression for the position of the water front and the average water saturation value behind a propagating front. The derived equations can be used for the following purposes:

Obtain an estimate for the time of water breakthrough in a production well from a given location of water injection.a
Obtain an estimate of the sweep efficiency.
Obtain an estimate of water cut development after time of water breakthrough.
Obtain an estimate of the number of pore volumes that need to be injected in order to achieve a certain recovery factor.
Obtain an estimate of the range of water breakthrough times and the range of recovery factors and water cuts at the time of water breakthrough, under a range of relative permeability curves.
Obtain an estimate of a critical rate for an oil production well, by finding the balance between viscous forces and gravitational forces.

These estimates are based upon simplified physics and a 1D reservoir geometry and so can be very useful to obtain an understanding of the water flooding process at hand. The application areas are described in more detail in the chapter about THE BUCKLEY-LEVERETT THEORY. Traditionally, the Buckley-Leverett theory is described and applied by assuming a rectangular reservoir geometry or a radial reservoir geometry. In this Technical Description and Manual we have extended the Buckley-Leverett model to a trapezoidal reservoir geometry and coning geometries for vertical and horizontal wells. The latter extension can be used to model the production behavior of vertical and horizontal wells with bottom water coning after time of water breakthrough. Next to that, the Buckley-Leverett model has been extended to a composite reservoir model for rectangular and radial geometries with an arbitrary number of reservoir segments.

Furthermore, the chapter about THE BUCKLEY-LEVERETT APPLICATION describes the input and output parameters of the Buckley-Leverett tool related to the estimates described above.

A comprehensive description of various aspects of waterflooding can be found in (Forrest, 1971).

THE BUCKLEY-LEVERETT THEORY

2. Fractional flow

In this section we derive the fractional flow equation, which is based on a number of principles describing multi-phase flow, such as relative permeability and Darcy’s law. When two immiscible fluids simultaneously flow through a porous medium, they mutually hinder each other to flow due to capillary forces. In other words, the total flow capacity is reduced. This is expressed by the relative permeability curves of water and oil as a function of water saturation. The relative permeability curve of one fluid models how much the mobility of this fluid is reduced in the presence of a certain amount of the other fluid. In other words, it describes how well a fluid is transmitted in the presence of another fluid. Figure 1 (a) shows the typical shape of relative permeability curves in the case the rock is preferentially water-wet, Figure 1 (b) shows the typical shape of these curves for preferentially oil-wet rock.

Figure 1. Relative permeability curves for water-wet rock (a) and oil-wet rock (b).

In most cases relative permeability curves are non-linear. Furthermore, there exists a contrast in mobility between the displacing fluid and the fluid being displaced due to differences in fluid viscosity. As a result, even though the displacing fluid (water) is immiscible with the fluid being displaced (oil), the displacement does not take place as a piston-like process. Instead the saturations of the fluids change gradually, with varying fractions of the total stream consisting of water flow and oil flow. Hence, the usage of the concept of “fractional” flow, which was introduced by Leverett in 1941. Fractional flow of water expresses the fraction of water in the total flowing stream and is therefore also referred to as water cut. It is derived as follows. Consider Darcy's law for water and oil:

where α is the dipping angle of the one-dimensional reservoir. When we use the expression “one-dimensional reservoir”, we mean that the displacement process can be described by equations that are expressed in a single spatial variable x and in a time variable t. Furthermore, the geometry of the reservoir can be defined be specifying the cross-sectional area A(x) at location x along the length of the reservoir. As a result, the bulk volume of the reservoir between two locations x₁ and x₂ is then given by . However, for the time being, we restrict ourselves to a reservoir of constant cross-sectional area A. This is illustrated in Figure 2. The locations of the injector and the producer are also illustrated, such that a positive dipping angle means we are injecting down-dip in the reservoir.

Figure 2. One-dimensional reservoir with positive dipping angle.

Darcy’s law can be written as:

Subtracting Eq. ( 4 ) from Eq. ( 3 ) yields:

P_C = P_o – P_w∆ρ = ρ_w – ρ_o