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CONTENTS

1. INTRODUCTION

1.1 Averaging

1.2 Effective, equivalent and pseudo permeability

1.3 When to Use Upscaling

2. SINGLE-PHASE UPSCALING

2.1 Introduction

2.2 Analytical Scale-up Methods

2.3 Tensor Permeability

2.4 Numerical Methods

2.5 Examples

2.6 Simulation with Full Permeability Tensors

2.7 Awkward Cases

2.8 Summary of Single-Phase Scale-up

3. TWO-PHASE UPSCALING

3.1 Introduction

3.2 Balance Of Forces

3.3 Steady-State Methods

3.4 Dynamic Methods

3.5 Additional Upscaling Methods

3.6 Summary of Two-Phase Scale-up

4. PRACTICAL APPLICATIONS

4.1 Introduction

4.2 Upscaling Stages

4.3 Recent Approaches

4.4 SPE Upscaling Study

4.5 Additional Topics

4.6 Upscaling Packages

4.7 Summary

6. REFERENCES

7 Permeability Upscaling

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2

Learning Objectives

After reading through this Chapter, the Student should be able to do

the following:

• Dene effective permeability, and understand when the denition applies.

• Recognise the difference between effective and equivalent permeability, and

pseudo relative permeability.

• Know how to calculate the effective permeability for simple models

analytically.

• Realise that permeability is a tensor property, and be able to distinguish

between the terms of the tensor (kxx, kxy, etc.).

• Understand numerical upscaling for single-phase flow, and know when to

apply different boundary conditions.

• Know how to carry out steady-state upscaling for two-phase ow.

• Become familiar with conventional two-phase dynamic upscaling

techniques, particularly the Kyte and Berry and Stone methods.

• Be aware of a range of other two-phase upscaling techniques.

• Judge when upscaling from the core-scale to the geological model cell is

required.

• Identify which upscaling methods are suitable for different stages of

upscaling

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Institute of Petroleum Engineering, Heriot-Watt University 3


1. INTRODUCTION

Reservoir modelling frequently involves generating multi-million cell models, which

are too large for carrying out ow simulations using conventional techniques. The

numbers of cells must therefore be reduced by “upscaling” (Figure 1). This is what most

engineers refer to when they talk about upscaling. However, as shown in this Chapter,

upscaling may be used to calculate effective properties at a variety of scales.

Geological model

Full-field model

Some quantities, such as porosity and water saturation, are easy to upscale, because

they may be averaged arithmetically. However, other quantities - notably permeability

- are much more difcult to upscale.

1.1 AVERAGINGThis simple example shows how to average porosity and permeability. Suppose we

have 10 grid blocks of size 1 cm3, 4 of which have a porosity of 0.15, and 6 of which

have a porosity of 0.20, as shown in Figure 2.

φ = 0.15 φ = 0.20

Sw = 0.40S

w = 0.50

The average porosity, φ , is given by:

φ = = × + ×

=total pore volume

total volume

4 0 15 6 0 20

100 18

. .. . (1)

When averaging the water saturation, we need to take the porosity into account. In theprevious example, suppose the water saturation was 0.5 in the blocks with porosity of

0.15, and 0.4 in the blocks with porosity 0.2, then the average water saturation is:

Figure 2.

Example for averaging porosity and water

saturation

Figure 1.

Upscaling to reduce

the size of a geological

model for full-eld ow

simulation. The model on

the left contains 20 million

grid cells and the one on

the right contains about

300,000 cells.

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4

S

total amount of water

total pore volumew = =

× × + × ×

× + ×

=+

= =

4 0 15 0 5 6 0 20 0 4

4 0 15 6 0 20

0 3 0 48

1 8

0 78

1 80 433

. . . .

. .

. .

.

.

.. . (2)

1.2 Effective, Equivalent And Pseudo PermeabilityWe cannot, in general, average the permeability in such a simple way, because the

effect on ow of a block of rock depends on the direction in which uid is owing

through the rock, and on the surrounding rock. Because of these problems, we

give two different denitions for single-phase permeability upscaling, and another

denition for two-phase ow.

Note that, when we upscale permeability, we are assuming the Darcy’s Law holds at

the coarse-scale as well as the ne-scale.

1.2.1 Effective PermeabilityIn scale-up, we wish to replace a ne-scale heterogeneous permeability distribution,

with a single effective permeability value. This process is called homogenisation, and

has been studied by researchers in a variety of elds. The true effective permeability

of a medium is an intrinsic property of that medium, and does not depend on the

boundary conditions. (Boundary conditions are the pressures, pressure gradients

and ows surrounding the model.) However, a true effective permeability may only

be calculated in limited cases. Figure 3 shows two examples, one stochastic and the

other deterministic, where the calculation of effective permeability is valid. In each

case, it is assumed that a permeability pattern, of length-scale a, extends over a large

region of size c. The condition for being able to calculate the effective permeability

of a region of size, b, is that a

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1.2.2 Equivalent Permeability

If we do not have a wide separation of distance scales, then the “averaged” permeabilityis not strictly speaking an effective permeability. It will depend on the boundary

conditions, and should be called the equivalent permeability. In reservoir simulation,

it is often the case that we are only calculating the equivalent permeability. However,

people usually refer to it as the effective permeability.

1.2.3 Pseudo permeabilityThere is a third type of permeability “average”, and this is the pseudo permeability.

When we are dealing with two-phase ow, we need to scale-up the relative permeabilities

as well as the absolute permeability. (See section 3.) Pseudo relative permeabilities

(pseudos) are tables of numbers, which allow us to reproduce a ne-scale simulation

on a coarse grid. They depend on many factors, such as the injection rate, and

the direction. In addition, pseudos may be calculated so that they compensate for

numerical dispersion.

Oil

Water

Oil

Water

Fine

Coarse

Time

R e c o v e r y

I P

Scale-up using pseudos is not a rigorous process, and is subject to error, unless carried

out carefully. There are several methods, each of which may perform well in some

situations and badly in others.

Figure 4

Example of a permeability

distribution where only an

equivalent permeability may

be calculated

Figure 5

A ne- and a coarse-scale

simulation of a waterood.

The pseudo relative

permeabilities are

calculated so that the

recovery at the coarse-scale

matches that for the ne-

scale

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6

Before describing how to calculate effective, equivalent and pseudo permeabilities,

we outline different situations where upscaling may be necessary.

1.3 When To Use UpscalingMost reservoirs are modelled using, what is commonly termed a “ne-scale geological

model”. This is a stochastic model with grid cells of size approximately 50 m in the

horizontal directions, and about 0.5 m in the vertical. There are typically about 107

such cells in a full eld model. These cells must be reduced in number to about 104 for

full-eld simulation. However, each of the grid cells in the geological model is likely

to be heterogeneous, containing, for example, sedimentary structures. Petrophysical

data (permeabilities, relative permeabilities, and capillary pressures) are acquired from

core plugs, which are only a few cm long. When small-scale structure is present,

petrophysical data should be upscaled before being applied to the grid blocks of thegeological model.

For convenience, we split upscaling into two separate stages (see gure 6). Stage

1 is upscaling from the smallest scale at which we may treat the rock as a porous

medium (rather than a network of pores), up to the scale of the stochastic geological

model, i.e. from the mm - cm scale to the m - Dm scale. Stage 2 is upscaling from

the stochastic geological model to the full-eld simulation model. (Note that each

stage could involve several levels of upscaling.

1.3.1 Stage 1 UpscalingThis level of upscaling is frequently ignored by engineers, and “rock” relative

permeability curves are applied directly to the geological model. However, work

carried out at Heriot-Watt University has demonstrated that small-scale structures,

such as sedimentary lamination may have a signicant effect on oil recovery (Corbett

et al., 1992; Ringrose et al., 1993; Huang et al., 1995). For example, in a wateroodof a water-wet rock, water is imbibed into the low permeability laminae, and oil

becomes trapped in the high permeability laminae.

Figure 6

Two separate stages of

upscaling. (Geological

model taken from “TenthSPE Comparative Solution

Project: A Comparison of

Upscaling Techniques”, by

Christie and Blunt, 2001)

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The Geopseudo Method is an approach, where upscaling is carried out in stages, using

geologically signicant length-scales. Models of typical sedimentary structures arecreated and permeability values are assigned to the laminae (from probe permeameter

measurements, or by analysing core plug data). Relative permeabilities and capillary

pressure curves are also assigned to each lamina-type (by history matching SCAL

experiments on core plugs). Flow simulations are carried out to calculate the effective

single-phase permeability and the two-phase pseudo parameters. Additional stages

of modelling and upscaling may be required - e.g. upscaling from beds to bed-sets

or facies associations.

In the nest-scale model, the grid cells may be a mm cube, or less. If we upscale to

blocks of 50 m x 50 m x 0.5 m, we are upscaling by a factor of at least 5 x 104 in the

horizontal directions and 500 in the vertical.

Low Perm High Perm

Individual Rel. Perm Curves

Pseudo Rel. Perm CurvesEffective Perm

1.3.2 Stage 2 UpscalingWhen engineers talk about upscaling, they usually refer to reducing the number of

cells in the geological model from about 107 to about 104, in order to carry out full-

eld simulation. Note that this is equivalent to upscaling by a factor of about 10 in

each direction, so the scale-up factor for Stage 2 upscaling is much smaller than for

Stage 1 upscaling. Once again two-phase (or even three-phase!) upscaling should be

carried where necessary, although frequently, only single-phase upscaling is used.

2. SINGLE-PHASE UPSCALING

2.1 IntroductionSingle-phase upscaling is relatively easy, compared with two-phase scale-up. We shall

assume in this section that we have steady-state, linear ow, with no sources or sinks

in the region of interest. In most cases, we shall also assume that we have widely

separated distance scales, so that we are calculating true effective permeabilities. We

start with simple cases, where the permeabilities may be averaged.

Figure 7

Illustration of the

Geopseudo Method

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8

2.2Analytical Scale-up Methods

2.2.1 Flow Parallel to Uniform Layers

∆

x

P1 P2

k i, tiQi

Consider a set of (innite) parallel layers of thickness, ti and permeability k

i, where i

= 1, 2, .. n (the number of layers). The effective permeability of these layers is given

by the arithmetic average , ka.

k k

t k

t eff a

i i

i

n

i

i

n= = =

=

∑

∑

1

1

. (3)

(Equation (3) may be proved by applying a xed pressure gradient along each

layer.)

Example 1

x

z

t1 = 3 mm, k 1 = 10 mD

t2 = 5 mm, k 2 = 100 mD

1

2

Suppose we have two layers as shown in Figure 8. The effective permeability for

ow in the x-direction is given by Equation (3), and is:

k a

=× + ×

+

= =3 10 5 100

3 5

530

866 25. mD

Figure 8

Along-layer ow

Figure 9

A simple, two-layer example

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2.2.2 Flow Across Uniform Layers

∆x

Q

k i, ti∆Pi

For ow perpendicular to the layers, the effective permeability is given by the

harmonic average, kh:

k k

t

t

k

eff h

i

i

n

i

ii

n= = =

=

∑

∑

1

1

. (4)

(Equation (4) may be proved by assuming a constant ow rate through each layer.)

Example 2

Equation (4) may be used to calculate the effective permeability for ow across the

two layers in the model shown in Figure 9, i.e. ow in the z-direction.

k h

=+

+

= =3 5

3 10 5 100

8

0 3522 86

.. mD.

From Examples 1 and 2, we see that the permeability is different in different directions.

In reservoirs with approximately horizontal layers, the arithmetic average may be

used for calculating the effective permeability in the horizontal direction, and the

harmonic average may be used for calculating the effective permeability in the

vertical direction.

2.2.3 Flow through Correlated Random FieldsFigure 11 shows an example of a correlated random permeability distribution.

(“Correlated” means that areas of high or low permeability tend to be clustered, so

that the spatial distribution is smoother that a totally random one.) The “correlation

length” is approximately the size of patches of high or low permeability. The longer

the correlation length, the longer will be the range of the semi-variogram for the

permeability distribution.

Assuming that we are averaging over many correlation lengths (i.e. we have a large

separation of distance scales), permeability should be isotropic (same in the x-, y- andz-directions). The effective permeability for a random permeability distribution is

proportional to the geometric average, which is given by:

Figure 10

Across-layer ow

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10

k k

ng

ii

n

= ( )

=∑exp ln1 (5)

where i = 1, 2, .. n is the number of cells in the distribution.

Correlation Length

The results given below have been derived theoretically for log-normal distributions,

with a standard deviation of σY, where Y = ln(k). The results depend on the number

of dimensions:

k eff = k g 1 −σY 2

2( ) in1 D

k eff = k g in 2 D

k eff = k g 1 +σY 2

6( ) in 3 D

(6)

These formulae are approximate, and assume σY is small (< 0.5). (Gutjahr et al,

1978.) The 1D result is an approximation of the harmonic average. Note that the

results do not depend on the correlation length of the eld, provided it is much

smaller that the system size.

Also note that ka > k

g > k

h, and the effective permeability always lies between the

two extremes: ka and k

h.

Example 3

Figure 11

A correlated, random

permeability distribution

Figure 12

A random arrangement of

the permeabilities in the

simple example

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Suppose that the permeability values in the simple example are jumbled up, so that

there are 75 small cells of 10 mD, and 125 small cells of 100 mD. See Figure 12.The effective permeability of this model will be:

pseudos - solid linesave. rock curves - dashed lines

Water Saturation

R e l a t i v e P e r m e a b i l i t y

0 0.2 0.4 0.6 0.8

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

keff = k g = 1075× log(10 )+125× log(100)

200

= 1075+ 250200

= 101.625

= 42.17mD.

2.2.4 Additional Averaging MethodsTo increase the accuracy of permeability averaging, some engineers use power

averaging. The power average is dened as:

k

k

n p

i

i

n

=

=∑ α

α

1

1/

(7)

where α is the power.

Also, sometimes, engineers use a combination of the arithmetic and harmonic averages,

e.g. they take the arithmetic average of the permeabilities in each column and them

calculate the harmonic average of the columns.

2.3 Tensor PermeabilityNow suppose that we have layers which are tilted at an angle to the horizontal, as

in Figure 13.

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12

∆P

net flow in x-dir

net flow in z-dir

x

z

A pressure gradient has been applied in the x-direction. This will obviously give rise to

a ow in the x-direction. The uid takes a path through the medium, so that it expends

a minimum amount of energy. There will be a component of ow up the high perm,

and only a small amount of ow across the low perm laminae, as shown. This gives

rise to a net ow in the z-direction, or crossow. Here, the term crossow is used

to describe ow perpendicular to the applied pressure gradient. When calculating

the effective permeability of this model, we need to take this crossow into account.

This may be done using a tensor effective permeability, k , where:

k

k k k

k k k k k k

xx xy xz

yx yy yz

zx zy zz

=

(8)

The rst index applies to the ow direction, and the second to the direction of the

pressure gradient. For example kxy

gives the ow in the x-direction caused by a

pressure gradient in the y-direction. The terms kxx

, kyy

, kzz

are known as the diagonal

terms. These are the terms which are usually considered - the horizontal and vertical

permeabilities, kh and k

v, respectively. The other terms, which describe the crossow,

are the off-diagonal terms.

With tensor permeabilities, Darcy’s Law becomes:

u = − ⋅ ∇k

µ P, (9)

where u is the Darcy velocity (vector) and P is pressure (scalar).

u k P

xk

P

yk

P

zxx xy xz x

= − ∂

∂ +

∂∂

+ ∂

∂

1

µ (10)

u k P

x k P

y k P

zy yx yy yz= −

∂∂ +

∂∂ +

∂∂

1

µ (11)

Figure 13

Crossow due to tilted

layers. White represents

high perm and dark

represents low perm

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u k

P

x k

P

y k

P

zz zx zy zz= −

∂

∂ +

∂

∂ +

∂

∂

1

µ (12)

In the model with horizontal layers (Figure 9), there is no crossow, so the tensor

is diagonal:

k =

66 25 0

0 22 86

.

.

or in 3D:

k =

66 25 0 0

0 66 25 0

0 0 22 86

.

.

.

In the random model (Figure 11), there will be little net crossow, so the effective

permeability tensor will be approximately:

k =

42 17 0

0 42 17

.

.

2.3.1 Flow Through Tilted Layers

θ

x

z

x’

z’

If the layered model in the previous example is tilted, the effective permeabilities in

the x’ and z’ directions will be the same as before (i.e. the arithmetic and harmonic

averages, respectively). However, in the x-z co-ordinate system, the effective

permeability should be represented by a full tensor. The terms of the tensor may becalculated from the arithmetic and harmonic averages, as follows:

Figure 14

Layers tilted at an angle of θ

to the horizontal

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14

k

k k k k

k k k k

a h a h

a h a h=

+ −( )−( ) +

cos sin sin cos

sin cos sin cos

2 2

2 2

θ θ θ θ

θ θ θ θ (13)

This formula is obtained by rotating the co-ordinate axes through an angle θ. (Youare not required to know the proof). This example is in 2D, so only the k

xx, k

xz, k

zx

and kzz

are shown. Further rotations may be carried out around the x’ or z’ axes to

obtain a full 3D tensor.

Note that:

• The tensor is symmetric (kxz

= kzx

).

• Depending on the sign of θ, the off-diagonal terms may be positive ornegative.

Example 4

Suppose the example in Figure 9 is rotated by 30º, and calculate the effective

permeability tensor.

cos230 = 0.75, sin230 = 0.25, sin30.cos30 = 0.433.

From before, ka = 66.25 mD, and k

h = 22.86 mD.

kxx

= 66.25 x 0.75 + 22.86 x 0.25 = 55.40 mD,

kxz

= (66.25 - 22.86) x 0.433 = 18.79 mD,

kzz

= 66.25 x 0.25 + 22.86 x 0.75 = 33.71 mD.

k =

55 40 18 79

18 79 33 71

. .

. .

2.4 Numerical MethodsIn general, the permeability distribution will not be simple enough for us to be able to

calculate the effective permeability analytically, and we will have to perform a numerical

simulation. We can use a nite difference method to calculate the pressures.

2.4.1 Recap on Flow SimulationThe continuity equation tells us that there is no net accumulation or loss of uid

within a grid block:

qxin

+ qzin

= qxout

+ qzout

(14)

(We are assuming incompressible rock and uids, here.)

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q xin q xout

q zin

q zout

i,j i+1,ji-1,j

i,j+1

i,j-1

x

z

Darcy’s law is used to express the ows in terms of the pressures and permeabilities.

For example, if the grid block in Figure 14 is of length Δx and height Δz and width

Δy, and if it is centred on location (i,j), then:

qk z y P P

x xin

x i j i j i j = −

−( )− −, / , , ,1 2 1∆ ∆∆µ

(15)

where kx,i-1/2,j

is the harmonic average of the permeabilities in the x-direction in blocks

(i-1,j) and (i,j). (The harmonic average is used, because ow from one cell to the

next, is equivalent to ow across two layers. See Section 2.2.2) The other ows are

calculated in a similar manner.

It is useful to use the transmissibilities, Tx = kxΔzΔy/Δx and Tz = kzΔxΔy/Δz. We cantherefore derive the pressure equation:

T T T T P

T P T P

T P T P

x i j x i j z i j z i j i j

x i j i j x i j i j

z i j i j z i j i j

, / , , / , , , / , , / ,

, / , , , / , ,

, , / , , , / ,

− + − +

− − + +

− − + +

+ + +( )− −

− −

1 2 1 2 1 2 1 2

1 2 1 1 2 1

1 2 1 1 2 1

== 0

(16)

An equation is set up for each Pij, 1 = 1, 2, .. nx and j = 1, 2, .. nz. The transmissibilities

are known, and using the appropriate boundary conditions, we can solve this set of

linear equations to obtain the pressure in each grid block. The effective permeability isthan calculated from the total ow and the total pressure drop, as described below.

2.4.2 Boundary ConditionsBoundary conditions are required to specify what happens at the edges of the

model.

Figure 15

Recap on ow simulation

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16

(a) No-Flow Boundaries

no flow through the sides

no flow through the sides

P1 P2

The pressure is xed on two sides of the model, (four sides in 3D) and no ow is

allowed through the others sides of the model (see gure 16). This type of boundary

condition is suitable for models where there is little crossow: for example, models

with approximately horizontal layers, or a random distribution.

A diagonal effective permeability tensor may be calculated using the method outlined

below. See Figure 17.

on left face

Pressure= P2Pressure= P1

on right face

L

Area, AFlow Rate, Q

x

y

z

1 Solve the steady-state equation to give the pressures, Pij

, in each grid block.

2 Calculate the inter-block flows in the x-direction using Darcy’s Law.

(See Equation 15.)

3 Calculate the total flow, Q, by adding the individual flows between two y-z

planes. (Any two planes will do, because the total ow is constant.)

4 Calculate the effective permeability for flow in the x-direction, using

the equation:

Q

k A P P

L

eff x

=

−, ( )1 2

µ (17)

Repeat the calculation for ow in the y- and z-directions, to obtain keff,y

and keff,z

.

Figure 17

The calculation of effective

permeability with no-ow

boundary conditions

Figure 16

No-ow boundary

conditions

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(b) Periodic Boundary ConditionsPeriodic boundary conditions are useful for calculating the effective permeabilities

of crossbeds or other sedimentary structures, when using the Geopseudo approach.

(See Section 1.3.1.) We assume that there are innitely repeated geological structures

in each direction. The use of periodic boundary conditions ensures that we have an

innitely repeated pattern of ows and pressure gradients. In the example shown in

Figure 18, there is a net pressure gradient in the x-direction. The blocks are numbered

i = 1, 2, ..nx in the x-direction, and j = 1, 2, .. nz in the z-direction.

x

z

P(i,0) = P(i,nz)

P(i,nz+1) = P(i,1)

P(0,j) = P(nx,j)+∆P P(nx+1,j) = P(1,j)-∆

A full permeability tensor may be calculated using periodic boundary conditions.

When a pressure gradient is applied in the x-direction there will be ow in the x-

direction, and there may also be ow in the z-direction due to internal heterogeneity.

These ows can be used to calculate the kxx

and kzx

tensor terms. Then a pressure

gradient is applied in the z-direction to obtain the kzz

and kxz

terms. (In 3D, a pressure

gradient should also be applied in the y-direction.)

(c) Linear Pressure Boundary ConditionsThese are similar to the no-ow boundary conditions, but the pressure is varied

linearly along the sides of the model (instead of specifying no ow).

P1 P2

P1 P2

P1 P2

Figure 18 Periodic boundary

conditions

Figure 19

Linear pressure boundary

conditions

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18

Some engineers use no-ow boundary conditions for calculating the effective

permeability in the x-direction, and linear pressure boundary conditions for calculatingthe effective permeability in the z-direction (vertical direction). This is sometimes

useful when upscaling sand/shale sequences.

The effective permeability calculated using linear pressure boundary conditions is

always greater to or equal to the effective permeability calculated using periodic

boundary conditions, which is always greater to or equal to the effective permeability

calculated using no-ow boundary conditions.

(d) Jackets or Skins

To reduce the effect of boundary conditions when calculating the effective permeability,

some engineers perform the simulations on a larger grid than necessary. The extragrid blocks round the edges are referred to as a “jacket” or “skin”. See Figure 20.

k eff

calculated for

this block

boundary conditions applied

to outer edges of model

2.5 ExamplesThese examples were calculated using the Geopseudo Toolkit software package

developed at Heriot-Watt. In addition to demonstrating the scale-up methods, they

show the limitations of nite difference simulation. The nite difference method is

only valid when ΔP/Δx → dP/dx, i.e. when the model is very nely discretised.

2.5.1 Scale-up of Stochastic DistributionsIn 2D, the effective permeability for a correlated random distribution should be the

geometric average.

(a) Log-normal distribution, poorly correlated, with σ = 1Since this is a stochastic distribution, every realisation will be slightly different.

We only show one realisation here.

Figure 20

Example of a ow jacket

round a model. In this case

the jacket is 4 cells thick

Figure 21

Contour plot of a

poorly-correlated stochastic

distribution.

(A 40 x 40 grid was used.)

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The geometric average of the permeabilities (calculated by computer using Equation

5) was 151 mD, so the analytical results were:

k =

151 0

0 151

The effective permeability was also calculated numerically using both no-ow

boundary conditions (NOF) and periodic boundary conditions (PBC). (The

Geopseudo Toolkit does not use linear pressure boundary conditions.) The

numerical result was:

k kNOF PBC

=

=

132 0

0 137

131 1

1 137

The off-diagonal terms are negligible compared with the diagonal ones, so no-ow

boundaries are adequate in this case. The numerical result has underestimated the

analytical one, because the grid is too coarse.

(b) Log-normal distribution, well-correlated, with σ = 1.

In this case, the geometric average was calculated to be 119 mD, so theanalytical results was:

k =

119 0

0 119

The numerical results were:

k kNOF PBC

=

=

−

−

114 0

0 120

122 2

2 118,

Here the numerical results are closer to the analytical ones, because we have a

well-correlated grid, so the nite difference method is more accurate. Again, the

off-diagonal terms are negligible.

Figure 22

Contour plot of a well-

correlated permeability

distribution.

(An 80 x 80 grid was used.)

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20

2.5.2 Scale-up of a Structured Model

Figure 23 shows a model of alternating layers of 20 mD and 100 mD, of equalthickness, and inclined at an angle of 20.56° (= arctan(0.5)) to the horizontal.

The analytical solution may be calculated using Equation 13.

kk k k k

k k k k

a h a h

a h a h

= + −( )−( ) +

=

cos sin sin cos

sin cos sin cos

. .

. .

2 2

2 2

54 7 10 7

10 7 38 7

θ θ θ θ

θ θ θ θ

The model was gridded coarsely (8 x 4 blocks), and then more nely (24 x 12 blocks)

(Figure 24), and the effective permeability was calculated numerically, using both

no-ow and periodic boundary conditions.

The 8 x 4 grid gives:

k kNOF PBC=

=

44 9 0 0

0 0 36 6

47 8 6 1

6 1 36 9

. .

. . ,

. .

. .

The 24 x 12 grid gives:

k kNOF PBC

=

=

48 4 0 0

0 0 38 6

52 4 9 2

9 2 38 1

. .

. .,

. .

. .

A simulation with a 96 x 48 grid gave:

k kNOF PBC=

=

50 0 0 0

0 0 39 2

54 1 10 3

10 3 38 5

. .

. . ,

. .

. .

Figure 23

Model of tilted layers

Figure 24

Left: 8 x 4 grid. Right: 24

x 12 grid

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The 8 x 4 grid is obviously much too coarse, and underestimates the effective

permeability. The 24 x 12 grid, although it looks much better, still under-estimatesthe analytical results. A 96 x 48 grid gives a result which is very close to the analytical

answer. However, people rarely perform simulations with such a rened grid, because

they are too time-consuming.

We can see from these results that using no-ow boundaries gives a reasonable

approximation to the diagonal terms of the tensor, but produces zero off-diagonal

terms.

(Note that there are other methods of discretising the pressure equations, which

will give a better answer with fewer gridblocks. However, these methods are not

described here, because they are more complicated.)

2.6 Simulation With Full Permeability TensorsHaving calculated full effective permeability tensors using periodic boundary

conditions, we need special software to handle them at the larger scale. Conventional

nite difference simulators use a 5-point scheme in 2D and a 7-point scheme in 3D,

and only take diagonal tensors - e.g. ECLIPSE 100 allows you to specify PERMX,

PERMY and PERMZ. Simulation with full tensors is more complicated and more

time-consuming, but some packages allow the user to input full tensors. For example,

ECLIPSE now allows users to input PERMXY, etc.

In 2D, a 9-point scheme is required to take account of crossow. This means that

there are 9 terms in each of the pressure equations, as illustrated in Equation 18.

a P a P a P a P a P

a P a P a P a P

i j i j i j i j i j

i j i j i j i j

1 2 1 3 1 4 1 5 1

6 1 1 7 1 1 8 1 1 9 1 1 0

, , , , ,

, , , , .

− − − −

− − − − =

− + − +

− − − + + − + + (18)

The coefcients, ai, in Equation 18 depend on the transmissibilities between the

blocks. There are several different methods of discretisation which give slightly

different results. To extend this to 3D, we need either a 19-point scheme or a 27-point

scheme. See Figure 25. (The 19-point scheme leaves out the 8 corners of the cube.)

Obviously, it takes longer to solve equations with a larger number of terms.

i-1,j-1 i,j-1 i+1,j-1

i-1,j i,j i+1,j

i-1,j+1 i,j+1 i+1,j+1

x

z

a) b)

Figure 25

a) A 9-point scheme for 2D.

b) A 27-point scheme for 3D

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22

The examples in Section 2.5 show that sometimes the off-diagonal elements of the

permeability tensor (kxy, etc) are negligible, so the limitations of using a 5-point(2D) or a 7-point (3D) scheme are not serious. Figure 26 shows examples of models

where the off-diagonal terms of the effective permeability tensor will be zero, or very

small, while Figure 27 shows examples of models where the off-diagonal terms may

be signicant.

a)

c)

b)

a)

c)

d)

b)

Figure 26

(a) innite parallel layers,

with one co-ordinate axis

aligned with the layers;

(b) models with an axis

of symmetry along a

co-ordinate axis (2D cross-

beds viewed along paleo-

current direction);

(c) a correlated random

distribution, where the

correlation length is very

much larger, or very much

smaller than the coarse

block size

Figure 27

(a) layers tilted with respect

to the co-ordinate axes;

(b) crossbeds or ripples;

(c) correlated random

distributions where the

correlation length is similar

to the coarse block size;

(d) models with faults or

fractures at an angle to the

co-ordinate axes.

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In layered systems, the size of the off-diagonal term may be gauged from Equation

13 in Section 2.3.3:

k xz

= (k a-k

h) sinθcosθ (19)

This is a maximum for θ = 45o, and increases as (ka-k

h) increases. Therefore, full

permeability tensors become more important as the angle of the lamination or bedding

increases, and as the permeability contrast increases.

2.7 Awkward CasesIn most cases, upscaling will introduce some error. However, when the permeability

contrasts are large, the errors may also be very large. Examples of “awkward

cases” for upscaling are:

• Low perm shales in a high perm sand

• High perm channels in a low net/gross region

• High perm fractures in a low perm matrix

• Low perm faults in a high perm region

For example, consider the sand/shale model in Figure 28. The total model is split

into 3 coarse blocks. However, there is a shale lying across each coarse block. If

the shale has zero permeability, each coarse block will have zero permeability in

the z-direction (vertical). However, uid can ow through the model vertically, as

shown. When upscaling models such as this one, care has to be taken to make the

coarse blocks large enough to include a representative elementary volume (REV), sothat the upscaled permeabilities are reasonable. (See King et al, 1998 for a practical

approach to upscaling in turbidite reservoirs.)

In awkward cases, non-uniform scale-up may be used to increase the accuracy of the

simulation. This means that the coarse blocks are not all the same size. Durlofsky

et al. (1997) assign smaller coarse grid cells to regions where there are high ow

rates.

For cases with high permeability streaks, the effective medium boundary conditions

(EMBCs) improve the accuracy (Wallstrom et al., 2000). See Section 3.5.

Figure 28

Sand/shale model

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24

2.8 Summary Of Single-Phase Scale-Up

• In some simple permeability distributions, the effective permeability may be

calculated analytically

- Continuous parallel layers

- Correlated random elds

• Permeability is actually a tensor quantity

- 4 terms in 2D

- 9 terms in 3D

• In most cases, it is more accurate to perform a numerical simulation

• Several types of boundary conditions are commonly used

- No-ow

- Periodic - Linear pressure gradient

• Some commercial simulators now take full permeability tensors

• Figure 26 shows examples of where diagonal tensors are adequate. Figure 27

shows examples of where full tensors will probably be more accurate.

• The most awkward cases to upscale are those with high permeability contrasts.

3. TWO-PHASE UPSCALING

3.1 IntroductionTwo-phase scale-up is more complicated than single-phase scale-up. In addition to

scaling up the absolute permeability, we need to consider:

• Relative permeabilities and capillary pressure

• Well locations

• Injection rates

(Well locations should also be considered in connection with single-phase ow.

However, in Section 2, we ignored sources and sinks, for simplicity.)

In general, we have one uid displacing another, so the ood is not in steady-state.

We have to simulate ne-scale oods in representative parts of the reservoir to obtain

pseudo relative permeabilities. Sometimes, a ood may approach a steady state overa small region, in which case, the phase permeabilities may be considered effective

permeabilities (rather than pseudos) provided the condition of widely separated

distance scales holds, as discussed in Section 1.

When upscaling, we should use the phase permeabilities:

k f = k

absk

rf(20)

Where “f” stands for uid - oil, gas or water. Generally, we assume that both the absolute

and the relative permeabilities are homogeneous and isotropic at the smallest scale

(diagonal tensor with kxx

= kzz

). As we upscale, we may require full tensors for the

phase permeabilities, k o and k

w. (However, full two-phase tensors are rarely used.)

To obtain effective (or pseudo) relative permeabilities, the absolute permeability must

be scaled-up separately. Then terms of the phase permeability tensor are divided by

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the appropriate term of the absolute permeability tensor. For example:

krf,xz

= kf,xz

/kabs,xz

3.2 Balance Of ForcesThe paths which uids take through the reservoir depend on the forces acting on

them. There are three types of forces:

• Viscous,

• Gravity,

• Capillary.

(The viscous force is caused by the injection of uid.)

A ternary diagram (Figure 29) may be used to show the balance of forces. We may

estimate the balance by comparing the pressure gradients due to each force acting

on the uids in a model. (Actually the force balance depends on the location in the

model, and varies with time, but for choosing a scale-up method, it is sufcient to

make a rough estimate of the average force balance.)

viscous dominant

gravity negligible

forces equal

capillary

gravityviscous

If the injection rate is very high the viscous pressure drop will be large compared with

capillary pressure gradients. If, in addition, the densities of the uids are similar,

the ood may be considered to be viscous-dominated.

If the viscous and gravity forces are small, the ood will be capillary-dominated.

This is most likely to occur at the small scale, particularly where we have ne-scale

lamination with large permeability contrasts over the scales of mm-cm. If

the injection rate is very low, there may be time for the uids to come to

capillary equilibrium over small distances (< 1 m). This means that the ood

is approximately in a steady-state. We can take advantage of this when we are

upscaling. See Section 3.3.1.

Alternatively, if the capillary forces are negligible, and the injection rate is very

low, the ood may be gravity-dominated, and we may have vertical equilibrium. In

this case, the uids become segregated immediately during the ood. The verticalequilibrium assumption (VE) may be used to scale-up in the z-direction, reducing

the number of dimensions in the simulation from 3 to 2, or from 2 to 1.

Figure 29

Ternary diagram showing

the balance of forces

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26

3.3 Steady-State Methods

These are actually quasi steady-state methods. We assume that within a short intervalof time the zone of interest is in a steady-state, but we allow the uid saturation to

change gradually, so that a full range of saturation is obtained. At steady-state, the

water saturation does not change with time, i.e. ∂Sw / ∂t = 0, so the continuity equation

becomes:

∇ • =uf 0, (22)

where u is Darcy velocity, and f is uid. From Darcy’s law:

∇ • • ∇( ) =k Pf f

0. (23)

3.3.1 Capillary-EquilibriumAssume that the injection rate is very low, gravity forces are negligible, and

that the uids have come into capillary equilibrium over a small distance (< 1

m). This means that the saturation distribution is determined by the capillary

pressure curves.

The method is as follows:

1. Choose a Pc level.

2. Determine the water saturations, and then the rel perms.

3. Calculate the pore volume-weighted average water saturation

4. Calculate the phase permeabilities: ko = k

absk

ro, k

w = k

absk

rw.

5. Calculate the effective water phase permeability.

6. Calculate the effective oil phase permeability.

7. Calculate the effective relative permeabilities, krw

= kw/k

abs, etc.

8. Repeat the process with another value of Pc.

Steps 5 and 6 may be carried out analytically or numerically, depending on the

distribution. The upscaled phase permeability may be a full or a diagonal tensor.

Example 5

Consider a model with two layers of equal thickness, as shown in Figure 30. The

absolute permeabilities are 100 mD and 20 mD. Assume that the porosity in each layer

is equal to 0.2. The rel perm and Pc curves for each layer are shown in Figure 31.

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Using the arithmetic and harmonic averages (Section 2), the effective permeability

is:

k =

60 0 0 0

0 0 33 3

. .

. .

0

2

4

6

8

10

12

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

C a p P r e s s u r e

R e l P e r m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

lo

hi

hi

hi

lo

lo

Suppose we choose a capillary pressure of Pc = 0.45.

In the high perm layer: Sw = 0.34, k

rw = 0.0013, k

w = 0.13, k

ro = 0.5, k

o = 50.

In the low perm layer: Sw = 0.44, k

rw = 0.0016, k

w = 0.032, k

ro = 0.48, k

o = 9.6.

Since the layers are of equal width, the average saturation is Sw = 0.39.

Using the arithmetic and harmonic averages from Section 2:

k k

k k

w rw

o ro

=

=

=

=

0 081 0 000

0 000 0 051

0 00135 0 00000

0 00000 0 00154

29 8 0 0

0 0 16 1

0 50 0 00

0 00 0 48

. .

. .

. .

. .

. .

. .

. .

. .

Note that the kv/kh ratio ( = kzz

/kxx

) is different for oil and water. It is 0.63 for water,

but only 0.54 for oil.

Effective relative permeability curves may be derived by repeating this calculationfor a range of capillary pressure values.

Figure 30

Model with horizontal

layers

Figure 31

Rel perm and Pc curves for

each lamina

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28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

k rox

k roz

k rwz

k rwx

R e l P e r m

The capillary-equilibrium method is useful as a quick method for scaling up from

the lamina to the bed scale. However, it is only valid in cases where the ow rate

is very low.

3.3.2 Viscous-Dominated Steady-StateWhen the capillary pressure is negligible, the viscous-dominated steady-state method

may be used. It may seem strange to use a steady-state method when the ood is

certainly not in a steady state, because there will be a Buckley-Leverett shock front

moving through the model. However, the method works surprisingly well. In thiscase, it is assumed that the fractional ow of water, f

w = q

w/q

t, is constant throughout

the model. The method proceeds in a similar manner to the capillary-equilibrium

method, except different levels for f w are chosen.

Recently a new steady-state method for intermediate viscous/capillary ratios has been

developed (Stephen and Pickup, 2000).

3.4 Dynamic MethodsFor dynamic (or non steady-state methods), we need to perform a two-phase ow

simulation on a ne grid. There are basically two types of non steady-state scale-

up:

(a) Weighted Pressure Methods

There are various ways of averaging the pressure, but we shall concentrate on the

Kyte and Berry (1975) method.

(b) Total Mobility Methods

In these methods, we scale-up the total mobility, and use the average fractional ow

to calculate the pseudo relative permeabilities.

The total mobility is:

λ λ λ µ µ

t o w

ro

o

rw

w

k k= + = + . (24)

Figure 32

Effective rel perm curves

in the horizontal and

vertical directions (x- and z-directions)

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The fractional ow is the ow of water divided by the total ow:

f q

q q

q

qw

w

o w

w

t

=

+

= (25)

Again there are a number of variations of this method, but we concentrate on the

Stone (1991) method.

The “traditional” upscaling methods assume that permeability is a diagonal tensor.

More recently there have been attempts to calculate full two-phase tensors. The

Pickup and Sorbie (1996) method is described in the last subsection.

3.4.1 The Kyte and Berry MethodA simple version of the Kyte and Berry (1975) method is presented here, using the

grid shown in Figure 33.

i=1 2 3 4 5 6 7 8 9 10

j=1

2

3

4

5 ∆z

∆x

∆X ∆Z

The diagram shows two coarse grid blocks, each of which is made up of 5 x 5 neblocks. The equations below show how to calculate the pseudo rel perms and capillary

pressure for the left coarse block.

The rst step is to perform a ne-scale, two-phase simulation (in ECLIPSE), saving

the pressures and inter-block ows at specied intervals of time. The method proceeds

as follows:

1. Calculate the effective absolute permeability in the area shown in Figure 34.

Figure 33

Model used for describing

the Kyte and Berry Method.

The thickness of the model

in Δ y.

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30

Kyte and Berry approximate the effective permeability using the arithmetic average

in each column, and then taking the harmonic average of the columns. The area

between the two coarse blocks is used, for reasons explained below.

∆

∆

∆

∆ ∆

z k

x

z k

x x

i i

i

ij ij

ij i j j

=+( )+=

∑11

5

2, (26)

where Δxij and Δz

ij are the sizes of the ne blocks. (In this case, all the blocks are

of equal size.)

k

X

Z x

z k

I

i

i ii

=

=

∑

∆

∆ ∆

∆3

7 (27)

where Δx and Δz are the dimensions of the coarse blocks, andk I

is the requiredeffective absolute permeability.

The pseudos are then calculated, at certain times during the simulation. (These are

the times at which the restart les are written in the Eclipse simulation.)

2. Calculate the average water saturation:

S

S x z

x z

w

w ij ij ij ij

i j

ij ij ij i j

= ==

==

∑∑

∑∑

, φ

φ

∆ ∆

∆ ∆

1

5

1

5

1

5

1

5

(28)

where φij is the porosity.

3. Calculate the total ow of oil and water out of the left coarse block (Figure 35).

q q f f j j

==

∑ 51

5

, ,

(29)

where qf5,j

is the ow of uid “f” from ne block number (5,j).

Figure 34

The area used for

calculating the effective

absolute permeability

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Figure 35

Calculation of the total ow

4. Calculate the average phase pressures in the central column of each coarse block.

In this example, we use the ne blocks in columns 3 and 8, the shaded areas in

Figure 36.

I II

In the Kyte and Berry method, the pressures are weighted by the phase permeabilities

times the height of the cells (which in this case are all the same size). This is so thatmore weight is given to regions where there is greater ow. However, there is no

scientic justication for using this weighting. In the rst coarse block (numbered,

I), the average pressure is:

P

k k z P g D D

k k z

fI

j rf j f j f j

j

j rf j

j

=

− −( )=

=

∑

∑

3 3 3 3 3

1

5

3 3 3

1

5

∆

∆

ρ ( )

(30)

where D3j is the depth of cell (3,j) andD is the average depth of coarse cell I. The

term gρf (D3j -D) is to normalise the pressure to the grid block centre. The averagepressure for coarse block II is calculated in the same manner, but using column 8

instead of column 3. The pressure difference is then calculated as:

∆P P Pf fI fII= − (31)

5. The pseudo rel perms are then calculated using Darcy’s law. Firstly, calculate the

pseudo potential difference. (Potential is dened as Φ = P-ρgz, so that the ow rateis proportional to ∇Φ.)

∆Φ ∆ ∆f f f P g D= − ρ (32)

where ΔD is the depth difference between the two coarse grid centres. Then:

Figure 36

Averaging the phase

pressures

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32

k

q X

Zkrf f f

I I=

−µ ∆

∆ ∆Φ (33)

6. Calculate the pseudo capillary pressure using:

P P Pc oI wI= − (34)

The Eclipse PSEUDO package can be used for calculating Kyte and Berry pseu-

dos.

3.4.2 Discussion on Numerical DispersionOne advantage of pseudo-isation methods, such as that of Kyte and Berry is that they

can take account of numerical dispersion. When a simulation is carried out using a

larger grid, the front between the oil and water becomes more spread out. However,

the Kyte and Berry method counteracts this effect by calculating the ows on the

down-stream side of the coarse block, instead of the middle. This is illustrated by

a simple example of a homogeneous grid. Figure 37 shows an example of input rel

perm curves (“rock” curves).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

R e l

P e r m

If the water saturation is Sw = 0.5, the rock curves show that there is a small amount

of oil and water owing. However, when the average saturation, is 0.5 in the coarse

block, the distribution could be as shown in Figure 38.

coarse

block water

oil

Since the water has reached only half way across the coarse block, there should be

no water owing out of the right side. The Kyte and Berry method calculates the

pseudo rel perms using the ow on the down stream side of the coarse block, to

Figure 37

Example of “rock” curves

Figure 38

Example of the water

saturation in a coarse block

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prevent water breaking through too soon. The pseudo water rel perm curve is moved

to the right, relative to the rock curves, as shown in Figure 39.

rock

curves

pseudo kro

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

pseudo krw

R e l

P e r m

3.4.3 Disadvantages of the Kyte and Berry MethodThere are certain problems with the Kyte and Berry method.

• Negative rel perms are produced, if ∆Φ has the same sign as qf .

• Innite rel perms occur if ∆Φ f is zero.

• The method of averaging the pressures, using rel perm as a weighting function,

may cause errors when the fluids are separated due to gravity. For floods whichare gravity-dominated, the TW method works better (Section 3.5).

• Non-zero pseudo capillary pressure may be produced, even if there is no capillary

pressure in the ne-scale simulation. This is because a different weighting is used

for calculating the average pressure in each phase.

• The capillary pressure may be different in different directions, because only the

central column is used for averaging the pressures.

Because of the rst two disadvantages, i.e. negative, or innite rel perms, pseudos

obtained from packages like the PSEUDO must be vetted before using at the coarse

scale. Often “odd” values of relative permeability are set to zero.

The last two disadvantages may be overcome by using a pore volume weighted

average of the pressures over the entire coarse block. This method is also available

in ECLIPSE. The last two advantages may be overcome by using a pore volume

weighted average of the pressures over the entire coarse block. In the ECLIPSE

PSEUDO package, the capillary pressure is now calculated in this way. There is also

a Pore Volume Weighted (PVW) method, in which the rel perms are also calculated

from pressures which have been averaged using pore volume weighting, rather than

rel perm weighting.

Good reviews of various methods for calculating pseudos are presented in Barker

and Dupouy (1996) and Barker and Thibeau (1997).

Example 6 - A Simple Kyte & Berry Calculation

Figure 40 shows an 11 x 3 ne grid. The rst and last columns are for injection and

Figure 39

Example pseudo rel perm

curves

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34

production. The central 9 columns make up 3 coarse blocks, each of which is 3 x

3 ne blocks. The calculations below show how to derive the pseudo rel perms forthe central coarse block, at one specied time. We shall ignore gravity in this simple

simulation, so we have effectively a 1D model.

Assume that the blocks are homogeneous, with porosity = 0.2, and permeability kabs

= 100 mD = 0.1 D. Also assume that the viscosity of water is μw = 1 cP, and that

of oil is μo = 5 cP. The “rock” rel perms and capillary pressures were supplied by

tables. You only need to know kr and P

c for a few water saturation values, and these

are supplied below.

After simulating for 200 time steps, suppose the water saturation distribution is as

follows:

I II III 0 1 2 3 4 5 6 7 8 9 10

(inj) (prod)

Sw 0.59 0.56 0.54 0.53 0.51 0.47 0.41 0.32 0.28 0.27 0.27

1. Calculate the average water saturation in coarse block II:

Sw

= 0.46.

2. Calculate the total ow of each phase out of the central coarse block. The pressures,

which were calculated using a simulator, are as follows:

I II III

0 1 2 3 4 5 6 7 8 9 10

(inj) (prod)

Po 11.12 11.03 10.93 10.81 10.67 10.51 10.35 10.21 10.14 10.07 10.0 (atms)

Pc 0.15 0.16 0.17 0.18 0.19 0.21 0.27 0.53 2.72 4.25 4.25

(atms)

Pw 10.97 10.87 10.76 10.63 10.48 10.30 10.08 9.68 7.42 6.82 5.75

(atms)

The phase perms (in Darcies) are shown for each block, below. These were calculated

using the water saturations above, the rel perm tables, and the absolute perm, which

is 0.1 D.

Figure 40

Grid used for simple Kyte

and Berry example

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I II III

0 1 2 3 4 5 6 7 8 9 10 (inj) (prod) kw 0.012 0.009 0.008 0.006 0.005 0.003 0.001 0.000 0.000 0.000 0.000

(D)

ko 0.000 0.000 0.004 0.006 0.008 0.013 0.028 0.059 0.080 0.085 0.085

(D)

Use Darcy’s law to calculate the ows out coarse block II, i.e. ows out of column

6.

q k P

x

q q qf f

f

f f f

i

f = = = ×=

∑µ

∆

∆

,1

3

3

ΔPf is the pressure drop between ne blocks 6 and 7 (shown using dark shading in the

table), and Δx is the distance between the centres of these blocks, and so is 1 cm.

I II III

0 1 2 3 4 5 6 7 8 9 10

(inj) (prod) qw

0.0012

qo 0.0023

The units forq are cc/s.

3. Calculate the average pressures in the central columns of coarse blocks II and III.

Since the pressures are constant within each column, the averaging is trivial. The

pressures to use for calculating the pseudos are shown using the light shading in the

pressure table above.

The pressure drop between the centres of blocks II and III is therefore:

∆Pw = 10.30 - 7.42 = 2.88 atms.

∆Po = 10.51 - 10.14 = 0.37 atms.

4. Calculate the pseudo phase permeability using Darcy’s law again.

k

q x

A P f

f f =

µ ∆

∆

Here Δx is the distance between the centres of the coarse blocks II and III, and is 3cm, and A is the cross-sectional area of the coarse blocks, which is 3 cm2.

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36

I II III

0 1 2 3 4 5 6 7 8 9 10 (inj) (prod) kw

0.00042

(D)

ko 0.031

(D)

5. Calculate the rel perms by dividing by the absolute permeability (0.1 D).

krw

= 0.004 k

ro=

0.311

6. Finally calculate the average capillary pressure, which is the difference betweenthe average oil and average water pressure in block II.

Pc=P

o- P

w= 10.51 - 10.36 = 0.21 atms.

This calculation must be repeated at a number of different times in order to calculate

the pseudo rel perms and capillary pressures for a range of water saturations.

3.4.4 Example of K&B Pseudos for a Ripple ModelFigure 41 shows a 2D model of small-scale ripples. Pseudo rel perms were calculated

for a ripple using four different ow rates. There is a factor of 10 between each

rate. Rate 1 is the fastest. Figure 42 shows the resulting pseudos (from Pickup andStephen, 2000).

200 mD

10 mD

1 cm,

18 cells

3 cm, 54 cells

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

rate 1rate 2

rate 3

rate 4

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Water Saturation

rate 1rate 2

rate 3

rate 4



Figure 41

A model of ripples (based

on the Ardross Cliff, near

St. Monance, Fife)

Figure 42

Pseudo relative

permeabilities for the ripple

model, for different ow

rates (1: very fast ... 4: very

slow)

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Note the following:

1. At high rates, the pseudos are shifted to the right. This is to compensate for numeri-

cal dispersion.

2. At very low ow rates (rate 4), the ood is capillary-dominated, and the oil is

trapped. The pseudo oil rel perm goes to zero at Sw = 0.46.

3.4.5 The Stone MethodIn this method, the problem of how to average the pressure is avoided by upscaling

the total mobility, and using the fractional ow to calculate the pseudo rel perms.

As with the Kyte and Berry method, a ne-scale two-phase ow simulation is per-formed, and the inter-block ows are saved at specied times. We use the same

example as before to illustrate the method.

i=1 2 3 4 5 6 7 8 9 10

j=1

2

3

4

5 ∆z

∆x

At each time interval:

1. Calculate the average water saturation, as before:

S

S x z

x z

w

w ij ij ij ij

i j

ij ij iji j

= ==

==

∑∑

∑∑

, φ

φ

∆ ∆

∆ ∆

1

5

1

5

1

5

1

5 (35)

where φij is the porosity of each block.

2. Calculate the total (relative) mobilities in column 5.

λ λ λ µ µ

t o w

ro

o

rw

w

k k= + = + . (36)

3. Average the total mobilities as follows:

Figure 43

The Stone Method. The

thickness of the model is Δ y

and is set to one here, for

convenience.

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38

λ

λ

t

x j t j

j

x j

j

T

T

= =

=

∑

∑

5 5

1

5

5

1

5 , (37)

where Tx5j

is the x-direction transmissibility in block (5,j). Transmissibility is dened

as:

T k z

xx

x=

∆

∆ (38)

(It is assumed here that absolute permeability is directional, but relative permeabilityis isotropic. Also, this is a 2D model, and we are assuming that Δy = 1.)

4. Calculate the total ow out of the coarse block, for each phase:

q qf f j

j

==

∑ 51

5

(39)

Then calculate the total ow:

q q qt o w

= + (40)

The average fractional ow of water is then calculated:

f q

qw

w

t

= (41)

5. Finally, calculate the pseudos as follows:

k f k f rw w w t ro o w t= = −( )µ λ µ λ , 1 (42)

The Eclipse PSEUDO package has an option for the Stone Method.

3.4.6 Disadvantages of the Stone MethodAlthough the Stone Method avoids problems arising due to pressure averaging, there

are other problems:

• Capillary pressure and gravity are ignored.

• The calculation of the average total mobility is inadequate when there is a

signicant variation in the total mobility.

• This method performs badly when the model is tilted.

3.5 Additional Upscaling MethodsSince the aim of two-phase upscaling is to derive rel perms, which produce the sameresults for the course-scale simulation as the ne-scale simulation, some engineers

calculate the pseudo rel perms using history matching methods for representative

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sections of the eld. In this case, the pseudo rel perms are assumed to have a

particular shape (e.g. a power law), and the parameters are adjusted to produce thehistory match. See, for example, Johnson et al., 1982. This approach is sometimes

referred to as the regression method.

The following methods have been developed recently to overcome some of the

drawbacks of the Kyte and Berry and Stone methods. They are not available yet

in commercial simulators. This is not an exhaustive list - more methods are being

developed all the time.

(a) The Total Mobility Method

This method was developed by Christie et al (1995). It is similar to the Stone method,

but it handles gravity better.

(b) The Hewett and Archer (HA) Method

In this method, described by Hewett and Archer (1997), the pressures are not averaged,

the pressure in centre of the coarse grid block is used directly. This method can

handle both gravity and capillary pressure.

(c) The Transmissibility-Weighted (TW) Method

This method was developed by Nasir Darman at Heriot-Watt (Darman et al, 1999).

It is similar to the Kyte and Berry method, except transmissibility weighting is used

when calculating the average pressure. The method works better than the Kyte and

Berry method in cases where gravity effects are signicant (e.g. a gas ood) and it

also works better than the Stone method in tilted reservoirs.

Recent research (Darman, Pickup and Sorbie, 2001) shows that, although the HA

method and the TW method produce different pseudo rel perms, their effects are

equivalent, because they produce the same fractional ows.

(d) Upscaling with Effective Medium Boundary Conditions (EMBCs)

Usually in scale-up, a section of a whole model is removed and a ow simulation is

carried out on this section using simple BCs, such as constant pressure at the edges

and no-ow at the sides. This allows a large ux through any high perm streaks

- much higher than you get when the simulation is carried out on the whole model.

To overcome this problem, Wallstrom et al. (2000) use effective medium boundaryconditions (EMBCs). According to effective medium theory, if we have a circular

disk of high permeability, k, embedded in an innite medium of perm ko, then the

ux (Darcy velocity) owing inside the disk is:

u k

k k du

o

o=

+ −

( )1

(43)

where d is the dimension (2 for 2D, or 3 for 3D), and uo is the ow velocity for the

innite medium, in the absence of the disk.

In the method of Wallstrom et al (2000), Effective Medium Theory was used to set up

ow boundary conditions for the ne-scale simulations. For each coarse block, the

ux entering each ne-grid cell at the inlet face, or exiting from each ne-grid cell

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40

on the outlet face, is calculated using the theory. The method used for calculating

the pseudos was a variation of the Stone method.

(e) Two-Phase Tensor Upscaling

Using any of the above methods, you can get directional pseudo rel perms, but not full

tensors. It is possible to carry out full two-phase tensor upscaling (Pickup and Sorbie,

1996), and to simulate two-phase ow using full tensor permeabilities. However,

not many people use this method yet. It may be important to use two-phase tensor

methods in reservoir with large dip and in fractured or faulted reservoirs.

3.6 Summary Of Two-Phase Scale-UpScaling up two-phase ow is more complicated than single-phase scale-up. The

movement of uids through a reservoir depends on the balance of forces - viscous,

capillary and gravity. Calculations may be simplied if one of these forces is dominant.

For example, if the injection rate is very low, and if gravitational forces are negligible,

then the uids may come into capillary equilibrium over small distances. In this case

we have a quasi steady state, and we can calculate proper effective permeabilities for

each phase (assuming that we have widely separated distances scales).

Usually, the uids are not in a steady state, and we can only calculate pseudo functions,

rather than effective properties. There are many different methods for scaling up non

steady-state two-phase ow. They tend to fall into two categories:

• Pressure Averaging Methods,

• Total Mobility Methods.

These methods are not rigorous. They are numerical “recipes” which aim to reproduce

the ne-scale hydrocarbon recovery on a coarse-scale grid. Therefore, there is no

method which is guaranteed to give the “correct” answer.

Also, pseudos are case dependent. They depend on the injection rate and on the

direction of ow (i.e. they are history-dependent). Figure 42 showed how pseudos

vary for different ow rates.

4. PRACTICAL APPLICATIONS

4.1 IntroductionHaving described the theory behind various upscaling methods, we now outline

how to apply these methods in practice. We also present a few extra topics, such as

scale-up in the vicinity of wells, grouping of pseudos.

The choice of which method to use in often depends, in practice, on how much time

(money) and data are available. Figure 44 illustrates the range of methods from the

point of view of time taken to implement them.

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Upscale absolute

perms only

Simple

averaging

Numerical

simulation

Two-phase

upscaling

Steady

state

Numerical

simulation

Easy

Difficult

We should also examine the balance of forces (Section 3.2) to assess which method to

use. Figure 45 shows which methods are appropriate for different force balances. On

the whole, dynamic methods are more accurate, if you can get the boundary conditions

correct. However, if one force is dominant, you may be able to take shortcuts.

1

3 2

4

5

6

Capillary-

dominated

Gravity-

dominated

Viscous-

dominated

7

Key:

1. Capillary-dominated - use capillary equilibrium method.

2. Gravity dominated - Simulation may be in vertical equilibrium - uids

segregate under gravity. Eclipse has a vertical equilibrium option, which may

be used to convert a 3D model into a 2D one (or a 2D to a 1D).

3. Viscous-dominated - use viscous-dominated steady-state, or a dynamic

method like Kyte and Berry.

4. Capillary and gravity forces competing, negligible viscous force - can use

capillary-gravity equilibrium.

5. Both viscous and gravity forces important - the TW method works better

here than Kyte and Berry.

6. Viscous and capillary forces competing - could use Kyte and Berry here.

A new intermediate steady-state method has been developed (Stephen and

Pickup, 2000).7. All forces competing - probably should not take short-cuts in this region -

use a dynamic method, such as Kyte and Berry.

Figure 44

Upscaling methods, as a

function of ease of use

Figure 45

Balance of forces triangle

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42

4.2 Upscaling Stages

Different methods may be appropriate for different stages of upscaling.

4.2.1 Stage 1 - Geopseudo UpscalingThe rst stage of upscaling, which tends not to be given careful consideration, is

to calculate appropriate relative permeabilities and capillary pressure for the grid

cells of the stochastic geological model (with cells sizes of approximately 50 m in

the horizontal and 0.5 m in the vertical). As mentioned in section 1, the Geopseudo

Method (Corbett et al., 1992) may be used to upscale from the lamina-scale upwards

(Figure 46). Models of sedimentary structures are created, and simulations are carried

out to upscale from the lamina-scale to the bed-scale and, if necessary another step

is used to upscale to a set of beds. Corbett et al. (1992) applied the Kyte and Berry

Method to upscale hummocky cross-stratication in several stages. However, thesemodels were very simple and only in 2D. In more complex models, it is probably not

feasible to use the Kyte and Berry Method. One approach which has been adopted to

save time is the use of steady-state methods (Pickup et al. 2000) - either the capillary

equilibrium method or viscous-dominated steady-state.

Low Perm High Perm

Individual Rel. Perm Curves

Pseudo Rel. Perm CurvesEffective Perm

Example of Geoseudo Upscaling

These two examples are from Pickup et al., 2000. The rst case is a synthetic model

of a uvio-aeolian eld, and the second case is a model of a tidal deltaic reservoir.

Figure 46

Example of Geopseudo

upscaling

Figure 47

Scale-up stages used in

the uvio-aeolian example

- from the lamina-scale

models to the stochastic

geological model.

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Geopseudo upscaling maybe time-consuming and expensive. There is no point in

upscaling from the smallest scales, unless cores are available for the eld. Cores

must be studied to identify the sedimentary structures present, and probe permeability

measurements should be taken to populate the small-scale models. Additionally, and

very importantly reliable SCAL data is also required.

To decide whether or not it is worthwhile carrying out Geopseudo upscaling, you

need to examine the large-scale structure of the reservoir rst. If uid can ow

through massive, high permeability sandstone to the wells, it will avoid lower quality

rock which may contain laminae. In this case, Geopseudo upscaling is probably

unnecessary.

The following list of guidelines from Ringrose et al. (1999) is useful for assessing

whether Geopseudo upscaling may be necessary:

(1) Are immiscible uids owing?

(2) Are signicant small-scale heterogeneities present? Specically:

• Is the permeability contrast greater than 5:1?

• Is the layer thickness less than 20 cm?

• Is the mean permeability less than 500 mD?

(3) What is the large-scale structure of the reservoir? In many cases, large-scale

connectivity may be the dominant issue, in which case, small-scale structure may

have to be ignored. Use the Weber and van Geuns (1990) classication to describe

the large-scale structures:

• Layer cake reservoirs - small-scale structure will usually have primary

importance. • Jigsaw puzzle reservoirs - small-scale structure may be important.

• Labyrinth reservoirs - small-scale structure will usually be of secondary

importance.

Figure 48

The three models sizes used

in the tidal deltaic model:

a) a heterolithic sand

model (1m x 1m x 0.3 m),

b) the stochastic geological

(Storm), and c) the coarse-

scale model - 6 km x 2.2 km

x 0.25m.

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44

The Geopseudo method may require two stages as follows:

1. Lamina-scale (mm or less) → bed-scale (5 m x 5 m x 0.33 m)Use simple models of representative beds. In a waterood, capillary forces will

probably dominate here, so use the capillary equilibrium method. This method is

relatively fast, and is suitable for grids of up to a million blocks. (Dynamic methods,

such as the Kyte and Berry Method are valid here, but they are time consuming.)

2. Bed-scale (5 m x 5 m x 0.33 m) → geological model scale (50 m x 50 m x 1 m)This stage of upscaling takes account of the variation of beds within a genetic unit

- e.g. trends. Steady-state methods may be used for convenience. Since the model is

no longer capillary dominated at this scale, viscous-dominated upscaling may be used

(or intermediate steady-state upscaling - Stephen and Pickup, 2000). Again, dynamicmethods are more accurate here, but are time-consuming. Some people assume that,

once they have obtained the effective rel perms of a single bed, this can be applied

directly to the geological model - i.e. this level of upscaling is not necessary.

4.2.2 Stage 2 - “Conventional” UpscalingStage 2 upscaling means reducing the number of grid cells in the stochastic geological

model so that full-eld simulation may be carried out. Depending on the amount of

time available, engineers may only upscale absolute permeability. What’s more, they

may only carry out simple averaging, rather than use one of the numerical simulation

methods (Figure 44).

At the scale of the geological model, capillary forces are no longer important, but

gravity effects may be signicant. Refer back to Figure 45 for methods to use where

gravity is important.

If dynamic methods, such as that of Kyte and Berry, are to be used for upscaling from

a 3D geological model to a full-eld model, representative sections of the full model

must be simulated at the ne-scale (i.e. the geological model scale, in this case). We

assume here that the geological model is too large for two-phase ow simulation,

otherwise upscaling would not be necessary. However, the boundary conditions are

difcult to specify, since they depend on the location of the wells.

4.3 Recent ApproachesRecently, some methods have been developed to try to overcome the problem of

using appropriate boundary conditions for the ne-grid simulation. These methods

mostly involve streamline simulation.

In streamline simulation (e.g. Blunt et al., 1996), the pressure equations are solved

once (using total mobilities), and streamlines i.e. the paths along which uids will

move, are calculated. The uids are advanced along these paths to obtain the saturation

distribution at a later time. From time to time, the pressure equation is recalculated

to take account of varying mobilities as the ood progresses. This method is much

faster than conventional simulation, because the pressure equations are only solved

a few times during the simulation, instead of perhaps thousands of times. Althoughthis method is less accurate than conventional simulation, it is feasible to carry out

streamline simulations on the grids containing millions of blocks.

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Figure 49

Nested grid simulation

Streamlines may also be used along with upscaling methods - for example to speedup a ne-scale two-phase simulation.

4.3.1 Nested Grid SimulationThis method was developed recently by Gautier et al. (1999). The pressure equations

are solved on the coarse grid, and these are used to calculate the ows between each

coarse block (Figure 49). The pressure equations are then solved on the ne grid,

within each coarse block, using ow boundary conditions. In this way the velocity

eld is computed on the ne-grid. Streamline methods are used to move the saturations

forward in time. The pressure eld is updated periodically.

A number of other researchers have proposed similar methods, e.g. Guedes andSchiozer (2001).

Coarse-grid simulation

Fine-grid simulation

with flow BCs

4.4 SPE Upscaling StudyIn the recent SPE Comparative Solution Project, which focussed on upscaling

techniques (Christie and Blunt, 2001), two models were presented: a 2D model

with gas displacing oil, and a 3D waterood. We summarise the results of the 3D

waterood here, as

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