Chapter 1: Pseudo Relative Permeability Curves

1
Chapter 1 Pseudo Relative Permeability Curves

• PETE 611
• R.A. Wattenbarger

2
Overview
1.1

• Introduction
• Driving Forces
• Gravity Segregation Homogeneous
• Stratified Reservoir High Rate (Hearn)
• Stratified Reservoir Low Rate
• Vertical Equilibrium
• Dynamic kr

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Introduction
1.2
The main purpose of this transformation is to
represent fine grid behavior using a coarser grid.
Reduce computer cost
Shorter computer time
Decrease complexity
Averaging the flow properties and the fluid
saturations in the vertical direction.
4
Introduction
1.3
Transformation of 3D model to 2D model
5
Introduction
1.4
with lab kr
6
Driving Forces
1.5
1)
2)
3)
The ratio of gravity forces to viscous forces is
known as Gravity Number
where
7
Gravity Segregation - Homogeneous
1.6
Assumptions
(a) Gravity forces dominate. (b) Viscous and
capillary forces are negligible. (c) If a
transition zone is present, it is negligible
compared to the thickness of the reservoir. (d)
There is a good vertical communication so that
the fluids are free to segregate vertically.
8
Gravity Segregation - Homogeneous
1.7
Application
kr curves for homogeneous gravity segregated case.
9
Stratified Reservoir High Rate (Hearn)
1.8
Order of breakthrough depends on
Assumptions
(a) Uniform layers. (b) Piston-like
displacement. (c) Oil and water phases
present. (d) Good vertical communication. (e)
Viscous forces dominate (ignore gravity, Pc
forces)
10
Stratified Reservoir High Rate (Hearn)
1.9
Calculation Procedures
11
Stratified Reservoir High Rate (Hearn)
1.10
12
Stratified Reservoir Low Rate
1.11
Hearns method and procedures are applied but in
vertical sequence.
gtgt
Viscous Forces
Gravity Forces
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Vertical Equilibrium
1.12
Factors that assist VE

• Low resistance to flow normal to the bedding
  planes.
• (2) Sands thin in the direction normal to the
  bedding planes.
• (3) Low areal rates of fluid movement.
• (4) if there is capillary pressure in transition
  zone, the capillary and gravity forces are
  balanced.

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Vertical Equilibrium
1.13
Geological equilibrium - Balancing capillary and
gravity forces
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Vertical Equilibrium Coats
1.14
Equations for calculations
The shape of the pseudo relative permeability and
capillary pressure curves are affected by

• Density difference (2) Dip angle
• (3) Sand thickness h (4) stratification
  (k(Z)and p(Z)).

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Vertical Equilibrium Coats
1.15
Coats paper. SPE 1961
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Vertical Equilibrium Coats
1.16
Coats paper. SPE 1961
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Vertical Equilibrium
1.17
Application
19
Dynamic Pseudo Relative Permeabilities
1.18
Description
20
Dynamic Pseudo Relative Permeabilities
1.19
Scaling from 2D to 1D
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Dynamic Pseudo Relative Permeabilities
1.20
a) Gravity Number
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Dynamic Pseudo Relative Permeabilities
1.21
b) Permeability Curves
23
Dynamic Pseudo Relative Permeabilities
1.22
c) Water Cut
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Dynamic Pseudo Relative Permeabilities
1.23
c) Water Cut
qw 1 STB/D
qw 3 STB/D
qw 5 STB/D
Best Match
25
Dynamic Pseudo Relative Permeabilities
1.24
Conclusions

• Dynamic pseudo relative permeabilities are an
  accurate approach to reduce the number of cells
  used to perform reservoir simulation.
• They work better in high-rate models, where
  viscosity forces are dominant over gravity
  forces.
• Curves appear to be independent of the injection
  flow rate.
• The use of these curves increases the capacity of
  performing sensitivity analysis since the
  computer time may result remarkably less.