Heriot-Watt University

Institute of Petroleum Engineering

Supervisors – Tony Peters - KERR- MCGEE (North Sea Ltd)

Gillian Pickup (Heriott-Watt)

MSc Petroleum Engineering

Project Report 2004/2005

Rohan Corlett

“Relative Permeability Upscaling

From Water-Oil Ratio Plot”

- 2 -

Declaration:

I Rohan Corlett confirm that this work submitted for assessment is my own and is

expressed in my own words. Any uses made within it of words of other authors in any

form (eg. Ideas, equations, figures text, tables, programs) are properly acknowledged at

the point of their use. A list of the references employed is included.

Signed…………………………………………………………..

Date………………………………………….

- 3 -

ACKNOWLEDGEMENTS

I would like to express my sincere thanks to my project supervisors, Tony Peters, Kerr

McGee North Sea U.K Ltd, Aberdeen, and Gillian Pickup, Heriot-Watt University for

their support, comments and knowledge throughout this project.

I would also like to thank Richard Todd, John Baillie, Sarah Brady, Kerr McGee North

Sea U.K Ltd and Jasper Schmidt, Horizon Energy Partners BV, for their involvement and

guidance.

- 4 -

SUMMARY

Dynamic upscaling especially in highly heterogeneous reservoir models is a very

challenging procedure and it is often hard to produce good history matches.

This report investigates the validity of a novel dynamic upscaling technique that only

requires knowledge of fluid properties (oil and water) and production history to generate

relative permeability pseudos that can be applied to full field simulation models to achieve

a history match.

The report

• Describes the background and method to the proposed technique.

• details the simulation study performed to investigate the validity of the technique

for a range of reservoir and fluid types

• Applies technique to north sea field

The results from the simulation studies showed good agreement with the fine scale models

indicating that this was a viable technique. The method was then applied to a full field

model which achieved excellent history matches to watercut.

- 5 -

TABLE OF CONTENTS

INTRODUCTION ............................................................................................... - 7 -

1 UPSCALING TECHNIQUE .................................................................... - 8 -

1.1 Background................................................................................................................................... - 8 -

1.1.1 – The Water-Oil Ratio ............................................................................................................... - 8 -

1.1.2 Background –Fractional Flow ................................................................................................. - 10 -

1.1.3 Background – Mobile Hydrocarbon ........................................................................................ - 10 -

1.1.4 Background – Corey Functions ............................................................................................... - 11 -

1.2 Upscaling Technique Method .................................................................................................... - 12 -

2 SIMULATION STUDY INVESTIGATION ............................................. - 15 -

2.1 Introduction ................................................................................................................................ - 15 -

2.2 Model boundary effects ............................................................................................................. - 15 -

2.3 Homogeneous Models ................................................................................................................ - 17 -

2.3.1 Fine Scale model ..................................................................................................................... - 17 -

2.3.2 Coarse grid model ................................................................................................................... - 20 -

2.3.3 Results ..................................................................................................................................... - 20 -

2.3.4 Pseudo Adjusting ..................................................................................................................... - 22 -

2.4 Varying Rock curves .................................................................................................................. - 28 -

2.5 Heterogeneous Models ............................................................................................................... - 30 -

2.5.1 Model with High Permeability Streak ..................................................................................... - 30 -

2.5.2 Heterogeneous model .............................................................................................................. - 32 -

2.5.3 Viscous model ......................................................................................................................... - 39 -

3 APPLICATION OF TECHNIQUE TO A NORTH SEA FIELD .............. - 42 -

4 CONCLUSIONS ................................................................................... - 46 -

5 References ................................................................................................................................... - 47 -

6 Appendix ..................................................................................................................................... - 48 -

- 6 -

TABLE OF FIGURES

Figure 1 - Typical Water-Oil Ratio Plot ........................................................................... - 8 -

Figure 2 - Examples of North Sea Field WOR Plots ........................................................ - 9 -

Figure 3 - Fraction Flow Curve Generated From WOR Plot ......................................... - 13 -

Figure 4 - Matching of fractional flow curves ................................................................ - 14 -

Figure 5 - Model showing different well locations ........................................................ - 16 -

Figure 6 - WOR plots for production wells in different locations ................................. - 16 -

Figure 7 - WOR for Homogeneous model ..................................................................... - 18 -

Figure 8 - Fractional flow curve matching for homogeneous model ............................. - 19 -

Figure 9 - Corey curves from Nw = 1.51 & No = 1.82, that were inputted in to Coarse

scale model ..................................................................................................................... - 19 -

Figure 10 - FOPT vs FWCT for coarse and fine models................................................ - 20 -

Figure 11 - Comparison of WWCT ................................................................................ - 21 -

Figure 12 – BHP match .................................................................................................. - 22 -

Figure 13a & b - Pseudo relative permeability curve and comparison of FOPT vs. FWCT . -

23 -

Figure 14 - Pseudo relative permeability curves where Krw &Kro are constant ........... - 24 -

Figure 15 - FWCT vs. FOPT comparison of fine and coarse with adjusted pseudos. ... - 25 -

Figure 16a & b - Pseudo curves with reduced water mobility and FWCT comparison for

COARSE Model with adjusted pseudo .......................................................................... - 26 -

Figure 17a & b - Pseudo curves and FWCT comparison for coarse with adjusted pseudo .. -

27 -

Figure 18 - WWCT comparision .................................................................................... - 28 -

Figure 19 - WOR plot and FOPT vs FWCT comparision .............................................. - 29 -

Figure 20 - WOR plot for heterogeneous model with high permeability streak ............ - 30 -

Figure 21 - Fw curve matching for high perm streak model .......................................... - 31 -

Figure 22 - FOPT vs. FWCT comparison for model with High permeability layer ...... - 31 -

Figure 23 - WOR plot for Hetrogeneous model ............................................................. - 32 -

Figure 24 - Satuartion profile showing coning ............................................................... - 33 -

Figure 25 - Saturation profiles after 13 & 14 years showing coning ............................. - 34 -

Figure 26 - WOR for model with Kv/Kh = 1 ................................................................. - 34 -

Figure 27 – WOR plot for Heterogeneous model........................................................... - 35 -

Figure 28 - Curve matching ............................................................................................ - 36 -

Figure 29 - FOPT vs. FWCT comparison for Heterogeneous model ............................. - 37 -

Figure 30 - Curve matching to flood front saturation, Swf. ........................................... - 38 -

Figure 31 - Simulation performance ............................................................................... - 38 -

Figure 32 – Comparison of WOR plot for models with high permeability in different

layers ............................................................................................................................... - 40 -

Figure 33 – Curve matching viscous model ................................................................... - 40 -

Figure 34 - FOPT vs FWCT comparison for viscous model .......................................... - 41 -

Figure 35 – WOR for North Sea Fulmar oil reservoir .................................................... - 43 -

Figure 36 - Curve match to WOR .................................................................................. - 43 -

Figure 37 – Relative permeability curves that were inputted into full field model ........ - 44 -

Figure 38 - watercut match for well ............................................................................... - 44 -

Figure 39 - Watercut match for well in same reservoir .................................................. - 45 -

- 7 -

Introduction

Reservoir simulation models are routinely used to predict the future performance of a

reservoir under different depletion and operating scenarios. To produce a reliable

characterisation of a reservoir, reservoir engineers and geoscientists need to build multi-

million geocellular models, which require a great deal of time and processing power for

flow simulations. The use of less complex models with a reduced number of cells is

therefore preferred, and is generally referred to as “upscaling”. Upscaling results in the

loss of important property information. Static reservoir properties, such as porosity, net-

to-gross, and initial water saturation are relatively straightforward to upscale. However,

the upscaling of dynamic properties such as absolute permeability (horizontal & vertical),

capillary pressure, and relative permeability tend to be more difficult. Dynamic upscaling

requires accurate reservoir description and fine scale simulation studies to generate

“pseudo” relative permeabilities that can be applied to full field models.

This report investigates the validity of a novel dynamic upscaling technique that only

requires knowledge of fluid properties (oil and water) and production history to generate

relative permeability pseudos that can be applied to full field simulation models to achieve

a history match. The technique is only applicable to undersaturated oilfields that have a

production history with a developing watercut.

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1 Upscaling Technique

This section describes the upscaling technique.

1.1 Background

1.1.1 – The Water-Oil Ratio

Many North Sea Oil fields exhibit a strong water-oil ratio (WOR) production trend. That is

a semi-log plot of:

Water Production Rate vs. Cumulative Oil Production [ 1]

Oil Production Rate

exhibits a linear trend that, in theory, can be extrapolated to an ultimate recovery for a

given watercut cutoff. Figure 1 is an example of a typical water-oil plot. In it, a linear

trend is observed for WOR > 0.4. From the extrapolation of the WOR trend an ultimate

recovery of around 4.5 MMbbls can calculated for a 95% watercut cutoff.

0.01

0.1

1

10

100

- 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 6,000,000

Cumulative Oil Production

Historical

Production

WOR

Extrapolation

WOR =19

95% watercut

WA

TE

R-O

IL R

atio

v/v

Ultimate Recovery

4.5 MMbbls

Figure 1 - Typical Water-Oil Ratio Plot

- 9 -

There is debate in the reservoir engineering community about the validity of the water-oil

ratio plot for predicting ultimate economic recovery from a reservoir [2]. Some of the

standard reservoir engineering reference books [3] [4] do not describe the water-oil ratio plot

as a tool for predicting ultimate recovery from a reservoir.

Example WOR plots for four actual North Sea Fields are shown in Figure 2 [5]. It can be

seen that linear trends are clearly discernible during periods of stable reservoir conditions.

Figure 2 - Examples of North Sea Field WOR Plots

0.01

0.10

1.00

10.00

100.00

Cum Oil

WO

R

WOR

0.0

0.1

1.0

10.0

100.0

Cum Oil bbl

WO

R

WOR

Consistent WI

Management

0.01

0.10

1.00

10.00

100.00

Cum Oil

WO

R

0

50000

100000

150000

200000

250000

300000

WOR

Field Injection

Late Time WOR

Upturn due to Low Water

injection

Small field, aquifer support

Auk WOR

0.001

0.01

0.1

1

10

100

Cum Oil

WO

R

WOR Monthly

Successful Infill Drilling campaign, WOR not valid, fractured reservoir ?

- 10 -

1.1.2 Background –Fractional Flow

The fractional flow of water (Fw), at any point in the reservoir is defined as:

Fw = QwBw / (QwBw + QoBo)…………………………………EQ. 1

Where (in field units)

, Qw = Water flow rate (stb/d)

Qo = Oil flow rate (stb/d)

Bw = Water formation volume factor (rb/stb)

Bo = Oil formation volume factor (rb/stb)

For displacement in a horizontal reservoir, and neglecting capillary pressure the fractional

flow equation can also be written as:

Fw = 1 / [1 + (μw/krw)*(kro/μo)]……………………………EQ. 2

Where, μo = Viscosity of oil

μw = Viscosity of water

krw = relative permeability to water

kro = relative permeability to oil

1.1.3 Background – Mobile Hydrocarbon

For a given porosity unit, the mobile hydrocarbon pore volume (HCPV) is defined as

(1-Swc-Soirr)…………………………………...………EQ. 3

Where

Swc = Initial water saturation

Soirr = irreducible oil saturation to water

- 11 -

1.1.4 Background – Corey Functions

Pore scale fractional flow information is usually measured during laboratory coreflooding

tests on cores taken from the reservoir. In the absence of such information, reservoir

engineers often generate relative permeability data (for use in reservoir simulation) using

Corey functions.

Corey functions are defined as:[6]

Relative permeability of Oil:

Kro =Kro (end point)* [(1-Sw-Soirr)/(1-Swc-Soirr)]No……………EQ. 4

Relative permeability of Water:

Krw = Krw (end point)*[(Sw-Swi)/(1-Swi-Soirr)]Nw……………EQ. 5

Where, Kro = Relative permeability of Oil

Krw = Relative permeability of Water

Krw (end point) = end point on water relative permeability curve

Kro (end point) = end point on oil relative permeability curve

No = Corey exponent for Oil

Nw = Corey exponent for water

The reservoir engineer uses different oil and water exponent to generate curves suitable for

the rock wettability and oil/water viscosity ratio.

- 12 -

1.2 Upscaling Technique Method

The upscaling technique simply converts field scale fractional flow to relative

permeability data that can be used in full field simulation models. It uses the base

assumption that the linear trend on the water oil ratio plot both extrapolates to the ultimate

recovery from the reservoir and accurately predicts the future fractional flow of the

reservoir. For the technique to be appropriate the WOR for the field must have exhibited a

linear trend (after initial rollover).

Workflow

FOPT = cumulative production

At connate water saturation oil is immobile, therefore:

When,

Sw = Swc, FOPT = 0

Then Sw=1-Soirr FOPT = Ultimate Recovery

Intermediate saturation values are determined for FOPT values from the following

approximation:

Sw = Swi + (FOPT/Ultimate Cumulative recovery)*(1-Swi-Soirr) …………EQ. 6

For each intermediate Sw value, the oil rate and water rates can be calculated directly from

the WOR plot. Hence a fractional flow curve can be generated from EQ. 1 (Figure 3) for

historic and future Sw.

- 13 -

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00Sw

Fw

Fractional flow curve

Swi

HCPV

FOPT0 Ultimate recovery

(1- Soirr)

Swf

Figure 3 - Fraction Flow Curve Generated From WOR Plot

The flood front saturation, Swf, is defined as the tangency point on the fractional flow

curve [4]. Relative permeabilities are then calculated for Saturation values between Swi

and 1-Soirr. Corey functions are then used to match the fractional flow for the saturation

range only when a linear trend on the WOR plot is observed (and predicted) [by

application of Equations 2,4,5]. Figure 4 shows the resultant match (note no match to the

FW prior to WOR linearity has been attempted)

- 14 -

Curve Matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Sw

Fw

0.1

1

10

100

1000

WO

R

Fw (from rel perm)

Fw from WOR

Series1

WOR

Start of linear WOR,

corresponds to the point at

which Fw curves are matched

Figure 4 - Matching of fractional flow curves

The relative permeabilities used to obtain the match are then inputted into the coarse scale

model.

This upscaling technique should produce good recovery and water cut matches from the

point that the fractional flow curves are matched.

- 15 -

2 Simulation Study Investigation

2.1 Introduction

The validity of this proposed upscaling technique was evaluated through a number of

simulations for different geological realisations. To investigate the technique a fine scale

model was run to produce the production data needed for the upscaling procedure. The

produced pseudos were then inputted into a coarse grid model with identical geometry as

the fine scale model and the performances were compared. The first tests that were carried

out used simple 3-dimensional homogeneous models. Further tests were performed where

parameters like heterogeneity, fluid viscosity and geometry were varied. This section

details the study.

2.2 Model boundary effects

As previously stated from real production data the WOR plot will tend towards a linear

straight line. To ensure the simulation models simulations truly reflected pore scale

fractional flow a study was performed to investigate effects that model boundaries might

have on production profiles (FW).

A fine scale simulation models was constructed. It was homogeneous and was run with an

injector / producer pair with injection controlled on a 100% voidage replacement basis.

Sensitivities investigating the effects that producer location has on the well’s WOR were

performed. (Figure 5 shows the position of each production relative to the injector). The

WOR plots for each well location are shown in Figure 6 It can clearly be observed that

when the producer is near the rear boundary of the model there is a large upturn in the

- 16 -

WOR plot. However, as the distance from the rear boundary to the producer is increased

the WOR plot tends more towards a straight line.

Figure 5 - Model showing different well locations

WOR Plots

0.1

1

10

100

1000

2000000 2500000 3000000 3500000 4000000 4500000 5000000 5500000 6000000

FOPT (STB)

Ln

WO

R

Well 1 Well 2

Well 3 Well 4

Well 5 Well 6

Figure 6 - WOR plots for production wells in different locations

From this it was concluded that the upturn on the WOR plots from the simulation is an

effect of the boundaries. When carrying out the simulation investigation only fine scale

Inj W 1 W 2 W 3 W 4 W 5 W 6

- 17 -

models that have a relatively straight line WOR plots were used (i.e. distances from

boundaries were adjusted to ensure correct pore scale fractional flow was being

represented in the well’s production).

2.3 Homogeneous Models

2.3.1 Fine Scale model

The first test was carried out using a simple 3-dimensional homogeneous model with 119

x 21 x 7 fine grid blocks. The properties of the fine grid blocks are shown in Table 1.

Property Value

Porosity, φ 0.2

Permeability 100mD

Viscosities, µo 0.48 cP

µw 0.312 cP

Formation factors, Bo 1.348 rb/bbl

Bw 1.05 rb/bbl

Grid Block sizes, dx 64.3 ft

dy 59.5 ft

dz 14.4 ft Table 1 - Fine scale model parameters

The rock relative permeabilities were derived from the Corey equation, where, values for

the Corey exponents of Nw = 2 and No = 2 were used. The end points on the relative

permeability curves were Krw = 0.362 and Kro = 0.65. The model used saturation values

of Swi = 0.35 and Swoirr = 0.25. Capillary pressure was zero.

Situated in the model were two wells: the producer producing 5000 rb/d and an injector

injecting at 100% voidage replacement. Both wells were completed vertically through the

entire reservoir interval. Further details of the fine scale simulation model are available in

Appendix A. The simulation was run for approximately 90 years in order to achieve a

water cut of 99%. The output data was used to produce a WOR for the fine scale

simulation, see Figure 7.

- 18 -

WOR Plot

y = 0.0006046256e0.0000014739x

0.1

1

10

100

1000

4000000 4500000 5000000 5500000 6000000 6500000 7000000 7500000 8000000 8500000 9000000

FOPT (STB)

WO

R

WOR

Extrapolation

Ultimate Recovery

8.4 MMbbls

WOR = 150

99% watercut

Equation of Extraploated line

Figure 7 - WOR for Homogeneous model

It was observed that there was a slight deviation from the straight line extrapolation due to

the effect of the model’s boundaries. The values at which the WOR plot began to deviate

from the straight line to the last point were discarded and new values of FOPT for values

of WOR were determined from the equation of the extrapolated straight line.

The fractional flow from the simulation was calculated and plotted along side the

fractional flow curve from the relative permeabilities. The fractional flow curve was

adjusted until the curves matched from the point that the WOR linear trend starts. Figure

8, shows the match that was achieved using Corey exponents of Nw = 1.51 and No = 1.83.

- 19 -

Curve Matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Sw

Fw

0.1

1

10

100

1000

WO

R

Fw (from rel perm)

Fw from WOR

WOR

Extraploation of WOR

Fw curves are matched to

start of linear WOR, with

Corey exponents of Nw = 1.51

& No = 1.83

Figure 8 - Fractional flow curve matching for homogeneous model

The subsequent Corey curves for water and oil are also shown in Figure 8. These are the

relative permeability curves that were inputted into the coarse grid model. The above plots

shows one of the potential limitations of the technique – when a straight line on WOR is

only achieved at high watercuts.. This limitation is perceived as not many North Sea

reservoirs are homogeneous and a WOR trend is usually observed at watercuts below 50%

(Figure 2).

Corey curves

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Sw

Kr

Psuedo Kro

Krw

Figure 9 - Corey curves from Nw = 1.51 & No = 1.82, that were inputted in to Coarse scale model

- 20 -

2.3.2 Coarse grid model

A coarse grid model with 17 x 3 x 1 grid blocks equating to a 7 x7 x 7 coarsening system

was built ensuring that it had the same STOIIP, reservoir properties and interwell

distances as the fine same model. The WOR generated relative permeabilities were

inputted into the coarse grid and the model run for the same number of years. The

performance of the coarse scale model was then compared to the fine scale model.

2.3.3 Results

2.3.3.1 Oil Production Total vs. Field Water Cut Total

The achieved cumulative production vs. watercut plot match is shown in Figure 10.

Performance Match

0.1

1

10

100

1000

2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT (STB)

WO

R

0

0.2

0.4

0.6

0.8

1

1.2

WOR

Fine

Coarse

Extrapolation

Start of match at

watercut of 78%

corresponds to start

of WOR linear trend

Figure 10 - FOPT vs FWCT for coarse and fine models

From the plot it can be seen that the coarse model shows a very good match for water cut

values above approximately 80%. As this upscaling technique is aimed at mature fields

- 21 -

that have high water cuts, this match is very satisfactory. When the WOR was plotted on

the same graph it could be seen that the WOR plot first begins to tend towards a linear

straight line at approximately WOR = 4, this corresponds to a FOPT value of 5900000

STB. This is the same FOPT value that the Fine and Coarse (FOPT vs FWCT) curves

begin to match. This shows that the extrapolated WOR data can be used to generate

accurate pseudos for a coarse homogeneous model of a high water cut field.

2.3.3.2 Watercut vs. Time

The watercut vs. time match is shown in Figure 11. it can be seen that there was early

water breakthrough in the coarse model due to numerical dispersion in the coarse grid

blocks. However the curves began to match at higher water cut values corresponding to

the point at which the WOR plot began to turn over towards a linear straight line.

FWCT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

WW

CT

Time (Days)

FINE

PSEUDO - COARSE

Figure 11 - Comparison of WWCT

2.3.3.3 Bottom Hole Pressure

The coarse and the fine scale bottom hole pressures do not match (Figure 12. However,

they do follow the same trend and shape. If the scale is refined to 20 days, see Figure 12,

- 22 -

it can be seen that the divergence occurs during the first 2 days. This was assumed to be

due to numerics because the fine scale cells better manage the pressure transients near the

producer.

BHP

4500

4700

4900

5100

5300

5500

5700

0 5000 10000 15000 20000 25000

BH

P (

ps

ia)

Time (Days)

FINE

PSEUDO - COARSE

BHP

4500

4700

4900

5100

5300

5500

5700

0 2 4 6 8 10 12 14 16 18 20

BH

P (

psia

)

Time (Days)

FINE

PSEUDO - COARSE

Figure 12 – BHP match

2.3.4 Pseudo Adjusting

To achieve a better match to the fine model at lower water cuts the relative permeabilities

for saturations less than the flood front saturation will have to be altered. A number of

different techniques were investigated to try to obtain a better match in the early time.

There is early water break through in the coarse model. This is due to numerical

- 23 -

dispersion; therefore, the main aim was to adjust the relative permeabilities to compensate

for the increasing numerical dispersion during upscaling. The first technique investigated

the effect of using straight line relative permeability curves for saturation values below

Swf ( see Figure 13a&bfor pseudo and resultant match)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT

FW

CT

Fine

Coarse - Straihtline Rel perms

Coarse

Figure 13a & b - Pseudo relative permeability curve and comparison of FOPT vs. FWCT

This was seen to promote early water breakthrough however the curves began to match at

a lower water cut values, FWCT = 0.55. It was therefore decided to increase the mobility

of the oil and at the same time reduce the mobility of the water to hopefully gain more

production before the water broke through to the producing well. This was achieved by

- 24 -

keeping Kro and Krw constant ie. Kro = 0.65 and Krw =0 for saturation values above Swi.

The pseudo curves are shown in Figure 14.

Figure 14 - Pseudo relative permeability curves where Krw &Kro are constant

The effect of this was investigated by incrementally increasing the saturation value by 1%

from Swi at which Kro and Krw were to remain constant. This was done to simulate a

piston-like displacement at low saturations. It was observed that the best match was

obtained when Kro and Krw were constant from Swi to Sw = 0.4 (see Figure 15). The

early water breakthrough had clearly been delayed and the coarse model curve now

matches the fine model curve at a FWCT of 40%.

- 25 -

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT

FW

CT

2/2 - Fine (119x21x7)

Coarse

Coarse - Kro & Krw constant to Sw=0.4

Figure 15 - FWCT vs. FOPT comparison of fine and coarse with adjusted pseudos.

The next technique was designed to reduce the mobility of the water so that more oil could

be produced before the water broke through. Kro remained a straight line and Krw was

reduced at different saturations, (the pseudo curves are shown in Figure 16a). The best

match was achieved when Sw = 0.5, Krw = 0.05 (see Figure 16b). Again the water

breakthrough had been delayed, however, not by as much as the previous technique, but its

curves also matched at approximately a FWCT = 40%.

- 26 -

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT

FW

CT

2/2 - Fine (119x21x7)

Coarse

Coarse - Krw reduced

Figure 16a & b - Pseudo curves with reduced water mobility and FWCT comparison for COARSE

Model with adjusted pseudo

The last technique used a combination of the two previous techniques. Kro and Krw

constant to Sw = 0.4 and at Sw = 0.5, Kro = 0.05 (see Figure 17a). This reduced the water

breakthrough further and the curves began to match at FWCT = 0.3.

- 27 -

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT

FW

CT

2/2 - Fine (119x21x7)

Coarse

Coarse - Combination

Figure 17a & b - Pseudo curves and FWCT comparison for coarse with adjusted pseudo

This investigation has shown that providing the pseudo-relative permeability curves generated

from the upscaling technique are maintained for saturations greater than Swf, then the rest of

the pseudo curves can be adjusted to gain a better match in the early time without affecting the

late time match.

Comparing the production well’s water cut for the fine model with the coarse model with the

adjusted pseudos it can be seen from Figure 18, that there is still some early water

breakthrough, however, the curves begin to match at reduced water cut values.

- 28 -

FWCT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

WW

CT

Time (Days)

FINE

COARSE - PSEUDO ADJUSTED

Figure 18 - WWCT comparision

2.4 Varying Rock curves

In the previous test Corey functions of Nw = 2 and No = 2 were used to generate the rock

curves that were applied to the fine scale homogeneous model. The next investigation

looked at the effect of using different Corey exponents and the ability to obtain a match.

For this investigation Corey exponents of Nw = 5 and No = 2 were used. These Corey

functions for water and oil are typical values for a water wet system. The model’s

geometry, geology and properties matched exactly the previous one. This model was run

for the same number of years to obtain a 99% water cut, and from the output data FOPT vs

log WOR was plotted. From Figure 19 it was observed that the WOR plot was similar to

the previous example. There was a slight deviation from the straight line at the end,

assumed to be due to boundary effects of the model. It was noted that the start of the

straight line was at a greater WOR, WOR = 5, than the previous example, where WOR =

3.5. From this it was inferred that if a match was produced, then, because of this the

performance plot of FOPT vs. FWCT for the fine and coarse models would begin to match

at a greater FWCT than the previous example.

- 29 -

A match to the fractional flow curves was achieved using Corey exponents of Nw = 1.70

and No = 1.81 (see Appendix B for curve matching plot).

The respective relative permeabilities produced were applied to the coarse scale model and

the performance plots compared.

Simulation Performance comparison

0.1

1

10

100

1000

3000000 4000000 5000000 6000000 7000000 8000000 9000000

FOPT (STB)

WO

R

0

0.2

0.4

0.6

0.8

1

1.2

FW

CT

WOR

Coarse

Fine

Extrapolation WOR plot

Match at 82%

watercut

Figure 19 - WOR plot and FOPT vs FWCT comparision

As previously anticipated the curves did match at a higher FWCT of Approximately 82%.

Again the FOPT where the two curves begin to match is the same as the start of the WOR plot

linear straight line. It was concluded that the value of the WOR where the linear straight-line

starts is related to the FWCT where the two curves match. The lower the WOR value that the

plot tends to a straight-line, the lower the FWCT the curves will begin to match. In practice,

the WOR plot straight line starts at a lower WOR than these previous two examples have

shown. Therefore, to prove this further, the next stage of the investigation was to produce

models where the turn over to the linear straight line could be observer at lower WOR

- 30 -

2.5 Heterogeneous Models

2.5.1 Model with High Permeability Streak

In order to try to reduce the WOR value at which the plot turns over to a straight line some

heterogeneity was introduced into the model. A high permeability horizontal layer of

1000mD was introduced to the centre of the fine scale model. This was done to promote

water breakthrough into the producer so that the linear WOR could be reached sooner.

The models rock curves were produced from Corey exponents of 5 and 2 for water and oil

respectively. The model had 199x49x7 fine grid blocks all other parameters remained the

same as the previous two models expect for this 1000mD layer. The WOR plot is shown

in Figure 20 below. The straight line linear WOR begins now at approximately WOR =

2.5.

Water Oil ratio

y = 0.02580927798767e0.00000035350136x

0.1

1

10

100

1000

50

00

00

0

70

00

00

0

90

00

00

0

11

00

00

00

13

00

00

00

15

00

00

00

17

00

00

00

19

00

00

00

21

00

00

00

23

00

00

00

25

00

00

00

FOPT

WO

R

WOR

Extrapolation of linearWOR

Equation of Extrapolated line

Ultimate Recovery

24MMbbls

WOR = 150

99% watercut

Figure 20 - WOR plot for heterogeneous model with high permeability streak

The fractional flow curves were matched to the start of the WOR linear trend using Corey

Exponents of Nw =0.96 and No = 2.16 (see Figure 21).

- 31 -

Curve Matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.40 0.50 0.60 0.70 0.80Sw

Fw

0.1

1

10

100

1000

WO

R

Fw (from relperm)

Fw from WOR

WOR

Figure 21 - Fw curve matching for high perm streak model

The performance plot of FOPT vs FWCT showed that the curves began to match at a

FWCT = 0.7, a better match in the early time could be achieved by manually adjusting the

pseudos, see section 2.3.4. Again this simulation study has shown that the lower the value

of WOR that the plot begins to tend to a straight line the lower the FWCT that the curves

begin to match.

Simulation Performance comparison

0.1

1

10

100

1000

0 5000000 10000000 15000000 20000000 25000000FOPT

WO

R

0

0.2

0.4

0.6

0.8

1

1.2

FW

CT

Fine - 1000mD high permstreak

Coarse

Match from 70%

watercut

Match better match in early

time could be achieved by

adjusting pseudos (see

homogeneous model)

Figure 22 - FOPT vs. FWCT comparison for model with High permeability layer

- 32 -

2.5.2 Heterogeneous model

The next simulation study introduced some more heterogeneity into the model to

determine if this had the effect of further lowering the start of the linear WOR plot. The

model’s parameters and geometry remained the same as the previous example but the

permeability within each layer was varied. The table below shows each layer’s

permeability.

Layer Number Permeability, mD

1 200

2 400

3 50

4 100

5 1000

6 300

7 200 Table 2 - Layer permeabilities

Figure 23, shows the WOR plot, it can be seen that the addition of this heterogeneity had

lowered the start of the linear straight line to approximately a WOR of 0.5.

0.1

1

10

100

1000

8000000 10000000 12000000 14000000 16000000 18000000 20000000 22000000 24000000

FOPT

WO

R

WOR - Kv/Kh = 0.1

Expon. (WOR - Kv/Kh = 0.1)

Figure 23 - WOR plot for Hetrogeneous model

Inflection point Deviation due to

boundary effects

- 33 -

It was observed that there was an inflection point on the WOR plot not long after the start

of the linear straight line and a deviation at the end of the plot. The deviation at the end is

assumed to be due to boundary effects, as this has been observed in other models. The

point of concern was the inflection near the start of the plot. To try to discover the cause

of this increase in FWCT, the models saturation profiles were investigated.

2.5.2.1 Coning

The saturation profiles clearly showed that there is a degree of coning in the model. This

coning occurred after approximately 10 years, from observing FWCT vs. Time this

corresponded to an increase in water cut. (see Appendix B).

Figure 24 - Satuartion profile showing coning

To further investigate the effect of coning and vertical connectivity on the WOR plot the

models vertical permeability was increased, Kv/Kh = 1. The saturation profiles in Figure

25 show a high degree of coning

Coning

- 34 -

Figure 25 - Saturation profiles after 13 & 14 years showing coning

The effect of this coning can clearly be seen on the WOR plot in Figure 26. Increasing

Kv/Kh in the model has resulted in the turn over to a linear straight line becoming more

gradual. The linear WOR was clearly visible then there was a large inflection on the plot.

This increase in FWCT occurred at the same time, approximately 13 years (see appendix

for FWCT vs Tme) when the physical coning was visible in the model’s saturation

profiles. Due to this large extent of coning it would be very hard to obtain a match with a

coarse pseudo.

WOR

0.1

1

10

100

8000000 10000000 12000000 14000000 16000000 18000000 20000000 22000000 24000000 26000000

FOPT

WO

R

WOR - Kv/Kh = 1

Extrapolation Linear WOR

Inflection point due to coning

Figure 26 - WOR for model with Kv/Kh = 1

Coning

- 35 -

In order to try to reduce the effect of the coning within in the model the vertical

permeability was reduced to Kv/Kh = 0.05. The WOR plot in Figure 27 showed that the

model was still being affected by coning and boundaries.

Water Oil ratio

y = 0.00083862308351e0.00000053500568x

0.1

1

10

100

1000

50

00

00

0

70

00

00

0

90

00

00

0

11

00

00

00

13

00

00

00

15

00

00

00

17

00

00

00

19

00

00

00

21

00

00

00

23

00

00

00

25

00

00

00

Cum Oil

Qw

/Qo

WOR -SIMULATION

Extrapolationof WOR

Linear trend

starting at

WOR >0.4

Ultimate

Recovery

22MMbbls

WOR = 150

99% watercut

Figure 27 – WOR plot for Heterogeneous model

The linear straight-line was extrapolated to a WOR = 150 to achieve a 99% watercut. The

resulting fractional flow curves were then matched from the start of the WOR linear trend.

- 36 -

Curve Matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Sw

Fw

0.1

1

10

100

1000

WO

R

Psuedo Krw

Psuedo Kro

Fw (from rel perm)

Fw from WOR

WOR Extrapolated

Figure 28 - Curve matching

The Corey Exponents used to achieve this match were Nw = 3.4 and No = 1.9. The

relative permeabilities were inputted into the coarse scale model. The permeability for the

coarse scale model was determined from the Harmonic average of the fine scale model

layer’s permeabilites.

Kh = SUM (ti) / SUM (ti/Ki) …………EQ. 7

Where, Kh = Harmonic average

ti = Layers thickness

Ki = Layers permeability

The Harmonic average permeability that was inputted into the coarse model was 185 mD.

- 37 -

Simulation performance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5000000 7000000 9000000 1.1E+07 1.3E+07 1.5E+07 1.7E+07 1.9E+07 2.1E+07 2.3E+07FOPT (STB)

FW

CT

5/2 - Fine Hetrogeneous

3.4/1.9 - Coarse

Early water

breakthrough in fine

model due to coning Misalignment in late

time due to

boundary effects in

coarse scale model

Figure 29 - FOPT vs. FWCT comparison for Heterogeneous model

The performance curves did not match in the early time because there was early water

break - through in the fine scale model due to the coning. However after the coning has

stopped the curves began to match at around 40% water cut. There was a misalignment

that occurred at approximately 80%, this was assumed to be boundary effects in the coarse

scale model.

The plot showed that the fine scale model had an earlier water breakthrough than the

coarse, therefore the mobility of the water in the coarse scale would have to be increased

to obtain a better match in the early time. It was decided to match only the curves to the

flood front saturation, Swf, on the fractional flow curve. This in effect would increase the

mobility of the water.

Figure 30 shows the fractional flow curves matched from the flood front saturation, Swf =

0.65. A match was obtained using Corey exponents of Nw = 1.85 and No = 1.7.

- 38 -

Curve Matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Sw

Fw

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00Fw (from relperm)Fw from WOR

Psuedo Krw

Psuedo Kro

Curves matched

to flood front

saturation,Swf

Figure 30 - Curve matching to flood front saturation, Swf.

Simulation Performance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5000000 10000000 15000000 20000000 25000000FOPT

FW

CT

5/2 - Fine Hetrogeneous

1.85/1.7 - Coarse Matched toshock front

Figure 31 - Simulation performance

- 39 -

The simulation performance curves in Figure 31 showed that the water breakthrough in

the coarse scale model was at approximately the same as the fine scale model. Matching

the fractional flow curve from Swf had compensated for the effect of the coning in the fine

scale model.

2.5.3 Viscous model

The next investigation looked at the effect of increasing the viscosity of the oil within the

model. The Model was the same as the previous example with the high permeability

streak in the middle except the viscosity of the oil was increased from μ = 0.48 cP to μ =

1.48. The WOR plot is shown in Figure 32. This model had a viscous-dominated flood;

the permeability heterogeneity dispersed the flood front, so that breakthrough occurred

earlier. There is a large deviation from the linear WOR this again is due to coning which

was observed on the saturation profiles. To try to omit this coning effect the high

permeability layer was place at the top of the model.

- 40 -

WOR

0.1

1

10

100

1000

0 5000 10000 15000 20000 25000

FOPT 10^3 (STB)

WO

R

WOR - High perm layer in middle

WOR - high perm layer at top

Deviation from linear

WOR due to coning

Figure 32 – Comparison of WOR plot for models with high permeability in different layers

The WOR plot showed that this did reduce the coning effect but did not get rid of it

completely. A small amount of coning downwards was observed in the saturation profile.

Placing the high permeability layer at the top of the model also had the effect of reducing

the start of the linear WOR.

Curve matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.40 0.50 0.60 0.70 0.80

Sw

Fw

0.1

1

10

100

1000

Fw (from relperm)

Fw from WOR

ExtraploatedWOR

Figure 33 – Curve matching viscous model

- 41 -

The fractional flow curve match in Figure 33 was achieved using corey exponents of Nw

= 1.95 and No = 1.55.

The performance curves below in Figure 34 showed that a good match was achieved in the

late time from about a water cut of 75%. The mismatch in the early time was again

because of the early water breakthrough in the fine scale model due to the coning.

Simulation Performance

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5000 10000 15000 20000 25000

FOPT 10^3

FW

CT

Fine - VISC

1.95/1.55 - Coarse

Mismatch due to coning

and boundary effects

Figure 34 - FOPT vs FWCT comparison for viscous model

- 42 -

3 Application of Technique to A North Sea Field

The upscaling technique was applied to production data from a well in a North Sea Fulmar

oil reservoir. The well’s water-oil ratio plot used is shown in Figure 35. The effects of

varying areal sweep patterns on the well’s WOR are clear. The WOR was extrapolated

from a linear trend during a period of sustained water injection. Figure 36 shows the match

to the WOR generated fractional flow. Figure 37 shows the generated relative

permeability curves that were inputted in the field full field simulation model. The

achieved watercut match for the well is shown in Figure 38. The watercut match of a well

that drained the same reservoir zone is shown in Figure 39. Both matches are good,

especially as no other adjustment to the model was made after application of the pseudo.

The early time matches could be improved by modifying the low water saturation parts of

the pseudo (as used in section 2.3.4 of this report to perfect the homogeneous model

match) or adjust other model parameters (e.g. irreducible oil saturation).

- 43 -

Water Oil Ratio

0.1

1

10

100

4000000

5000000

6000000

7000000

8000000

9000000

10000000

11000000

12000000

Cumultive Well production

Qw

/Qo

0

10000

20000

30000

40000

50000

60000

70000

80000Well WOR

Water Injection

Expon. (Extrap)

WOR Extrapolation from a period of

sustained water injection

Figure 35 – WOR for North Sea Fulmar oil reservoir

Curve matching

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

Sw

Fw

0.1

1

10

100

1000

Fw (from relperm)

Fw from WOR

Fw from WOR

WOR

Figure 36 - Curve match to WOR

- 44 -

0.00

0.20

0.40

0.60

0.80

1.00

- 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Sw

Kr

Psuedo Krw

Psuedo Kro

Fw (from rel perm)

Fw from WOR

Figure 37 – Relative permeability curves that were inputted into full field model

Figure 38 - watercut match for well

- 45 -

Figure 39 - Watercut match for well in same reservoir zone

- 46 -

4 Conclusions

• The WOR is a valid tool to predict future reservoir waterflood performance for

North Sea fields with developing watercuts

• Fine scale simulation studies have shown that when upscaling care must be taken

to eliminate boundary or near wellbore effects that are not true rock fractional flow

characteristics.

• Where a WOR trend occurs at high watercuts the approach shall only provide good

matches at the corresponding watercuts. In this instance improvements to the

match can be achieved by the manual introduction of a small shockfront to the

relative permeability curve.

• The investigated technique to generate pseudo relative permeability curves for use

in field models is shown to be valid and a non time consuming that does not

require detailed reservoir characterisation.

• The approach is valid for both heterogeneous and homogeneous reservoirs with

both favourable and unfavourable fluid mobility ratios.

- 47 -

5 References

[ 1] Willhite, G.P.W, “Waterflooding”, SPE textbook series VOL:3

[2] Miles, A, “Fractional Flow Approach to Performance Prediction in Mature Water-Flooded Fields”

DTI Oil & Gas Directorate, August 2002 (http://ior.rml.co.uk/issue1/articles/art-2.htm)

[3] Dake, L.P, “The Practice of Reservoir Engineering” (Revised Edition)

[4] Dake, L.P, “Fundamentals of Reservoir Engineering”

[5] Petroleum Production Reporting System, DTI Oil & Gas

http://www.og.dti.gov.uk/pprs/full_production.htm

[6] Stiles, J, “Using Special Core Analysis in Reservoir engineering – Relative Permeability &

Capillary Pressure”, Course notes, Dec 1994

[7] Darman, N.H, G.E Pickup and K.S Sorbie, “A Comparison of Two-phase Dynamic Upscaling

methods Based on Fluid Potentials”, Computational Geosciences 6: 5-27, 2002

[8] Azoug, Y, “The Performance of Pseudofunctions in the Upscaling Process”, SPE 80910, presented

at SPE Production and Operations Symposium, Oklahoma, March 2003

[9] Okano, H, “Quantification of Uncertainty in Relative Permeability for Coarse-Scale Reservoir

Simulation”, SPE 94140, Europec/EAGE Annnual Conference, Madrid, June 2005.

- 48 -

6 Appendix

- 49 -

Appendix A

Eclipse Input File – Fine Homogeneous Model

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-V2_2.DATA

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-- SIMPLE FINE SCALE MODEL

-- FOR UPSCALING STUDY

-- Model (119x21x7)

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

--

--

-- Authors: Rohan Corlett & Tony Peters

-- Date:25 July 2005

-- ECLIPSE: v2004

--

--

========================================================================

======

RUNSPEC

--

========================================================================

======

--

========================================================================

======

NOECHO

TITLE

Fine Scale Model

DIMENS

-- NX NY NZ

119 21 7 /

OIL

WATER

FIELD

TABDIMS

--Sat.Tabs PVT.Tabs Sat.Nodes Pres.Nodes FIP.Regs Rs.Nodes

Sat.EndPt.Tabs

1 1 50 50 15 25 3

/

EQLDIMS

--Equil.Regs Depth.Nodes Depth.Nodes Tracer.Tabs Depth.Nodes

1 10 10 0 0 /

REGDIMS

--FIP.Regs Sets.FIP.REGS

1 /

- 50 -

WELLDIMS

--No.Wells No.Cons.Per.Well No.Groups No.Wells.Per.Group

2 25 1 2 /

VFPPDIMS

-- MXMFLO MXMTHP MXMWFR MXMGFR MXMALQ NMMVFT

/

VFPIDIMS

-- MXSFLO MXSTHP NMSVFT

/

START

1 'JUL' 2005 /

NSTACK

25 /

UNIFOUT

UNIFIN

--NOSIM

-------------------------------------------------------

-- Set relative paths for INCLUDE files

GRID

--

MESSAGES

9* 1000 /

--PSEUDO

--

EQUALS

'TOPS' 10000 1 119 1 21 1 1 /

'DX' 64.3 1 119 1 21 1 7 /

'DY' 59.5 1 119 1 21 1 7 /

'DZ' 14.3 1 119 1 21 1 7 /

'PORO' 0.20 1 119 1 21 1 7 /

'NTG' 1.0 1 119 1 21 1 7 /

'PERMX' 100 1 119 1 21 1 7 /

/

COPY

'PERMX' 'PERMY' /

'PERMX' 'PERMZ' /

/

MULTIPLY

'PERMZ' 0.1 1 119 1 21 1 7 /

/

--

========================================================================

======

INIT

PROPS

--

========================================================================

======

--

========================================================================

======

SWOF

- 51 -

------------------------------------------------

-- Corey func nw=2 no=2 Krw=0.362 Kro=0.65

------------------------------------------------

-- Sw Krw Kro Pc

0.3500000 0.0000000 0.6500000 0.0000000

0.3600000 0.0002263 0.6179063 0.0000000

0.3700000 0.0009050 0.5866250 0.0000000

0.3800000 0.0020363 0.5561563 0.0000000

0.3900000 0.0036200 0.5265000 0.0000000

0.4000000 0.0056563 0.4976563 0.0000000

0.4100000 0.0081450 0.4696250 0.0000000

0.4200000 0.0110863 0.4424063 0.0000000

0.4300000 0.0144800 0.4160000 0.0000000

0.4400000 0.0183263 0.3904063 0.0000000

0.4500000 0.0226250 0.3656250 0.0000000

0.4600000 0.0273763 0.3416563 0.0000000

0.4700000 0.0325800 0.3185000 0.0000000

0.4800000 0.0382363 0.2961563 0.0000000

0.4900000 0.0443450 0.2746250 0.0000000

0.5000000 0.0509063 0.2539063 0.0000000

0.5100000 0.0579200 0.2340000 0.0000000

0.5250000 0.0692891 0.2056641 0.0000000

0.5480000 0.0886991 0.1657663 0.0000000

0.5500000 0.0905000 0.1625000 0.0000000

0.5600000 0.0997763 0.1466563 0.0000000

0.5700000 0.1095050 0.1316250 0.0000000

0.5800000 0.1196863 0.1174063 0.0000000

0.5900000 0.1303200 0.1040000 0.0000000

0.6000000 0.1414063 0.0914063 0.0000000

0.6100000 0.1529450 0.0796250 0.0000000

0.6200000 0.1649363 0.0686563 0.0000000

0.6300000 0.1773800 0.0585000 0.0000000

0.6400000 0.1902763 0.0491563 0.0000000

0.6500000 0.2036250 0.0406250 0.0000000

0.6600000 0.2174263 0.0329063 0.0000000

0.6700000 0.2316800 0.0260000 0.0000000

0.6800000 0.2463863 0.0199063 0.0000000

0.6900000 0.2615450 0.0146250 0.0000000

0.7000000 0.2771563 0.0101563 0.0000000

0.7100000 0.2932200 0.0065000 0.0000000

0.7200000 0.3097363 0.0036563 0.0000000

0.7300000 0.3267050 0.0016250 0.0000000

0.7400000 0.3441263 0.0004063 0.0000000

0.7500000 0.3620000 0.0000000 0.0000000

/

--

========================================================================

======

ROCK

-- pref Cr

-- psia 1/psi

5905 3.1E-06 /

PVTW

-- pref Bw Cw visw

-- psia rb/bbl 1/psi cp

5905 1.050 3.0E-06 0.312 /

PVDO

-- psia Bo viso

-- rb/bbl cP

2040 1.421 0.380

- 52 -

3000 1.392 0.405

4000 1.368 0.431

5000 1.354 0.459

5910 1.342 0.480

6500 1.334 0.49

/

GRAVITY

-- Surface Densities for Oil Water and Gas

-- API Water=1 Air=1

44.28 1.0474 1.0570 / Platform

RSCONST

-- GOR pref

-- Mcf/bbl psia

0.502 2040 /

--

========================================================================

======

REGIONS

--

========================================================================

======

SATNUM

17493*1 /

FIPNUM

17493*1 /

EQLNUM

17493*1 /

--

========================================================================

======

SOLUTION

--

========================================================================

======

EQUIL

--

--DATUM PRES WOC CAP.PRES GOC CAP.PRES RSVD RSVD EQUIL

--DEPTH DATUM FT @WOC FT @GOC TABLE TABLE ACC

11351 5910 11351 0 0 0 0 0 10

/

-------------------------------------------------------

RPTSOL

'RESTART=2' 'FIP=3' /

--

========================================================================

======

--

========================================================================

======

SUMMARY

SEPARATE

- 53 -

RPTONLY

FOPR

FWPR

FOPT

FWCT

--

WBHP

'W1' 'PR1' /

WWCT

'W1' 'PR1' /

FVPT

FVIT

WPI

'W1' 'PR1' /

--

--

--

========================================================================

======

SCHEDULE

--

========================================================================

======

-- Reporting

RPTRST

'BASIC=4' /

RPTSCHED

/

WELSPECS

'PR1' 'G1' 81 11 1* 'OIL' 7* /

'W1' 'G1' 11 11 1* 'WAT' 7* /

/

COMPDAT

'PR1' 1* 1* 1 7 'OPEN' 2* 0.508 3* 'Z' 1* /

'W1' 1* 1* 1 7 'OPEN' 2* 0.508 3* 'Z' 1* /

/

-- qo qw qg ql rb/d bhp

WCONPROD

'PR1' 'OPEN' 'RESV' 1* 1* 1* 1* 5000 2100 /

/

WPIMULT

'W1' 10 /

'PR1' 10 /

/

WCONINJ

'W1' 'WAT' 'OP' 'RESV' 1* 0 1.0 FVDG 10000 /

/

TSTEP

30*1

11*30

/

TSTEP

- 54 -

12*30

12*30

12*30

12*30

/

TSTEP

12*60

12*60

12*60

12*60

/

TSTEP

12*100

12*100

/

TSTEP

10*365

20*365

40*365

/

END

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-- END OF FILE

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

- 55 -

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-- V2_2_C_PS.DATA

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-- PSEUDO COARSE GRID CROSS-SECTIONAL MODEL

-- FOR UPSCALING STUDY (Nw = 1.16, No = 2.06)

--

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

--

-- Simple Coarse Grid

-- for use in pseudo upscaling study

--

-- Authors: Rohan Corlett & Tony Peters

-- Date: 26 Jul 2005

-- ECLIPSE: v2004

--

========================================================================

======

RUNSPEC

--

========================================================================

======

--

========================================================================

======

NOECHO

TITLE

Simple Coarse Model

DIMENS

-- NX NY NZ

17 3 1 /

OIL

WATER

FIELD

TABDIMS

--Sat.Tabs PVT.Tabs Sat.Nodes Pres.Nodes FIP.Regs Rs.Nodes

Sat.EndPt.Tabs

1 1 50 50 15 25 3

/

EQLDIMS

--Equil.Regs Depth.Nodes Depth.Nodes Tracer.Tabs Depth.Nodes

1 10 10 0 0 /

REGDIMS

--FIP.Regs Sets.FIP.REGS

1 /

WELLDIMS

--No.Wells No.Cons.Per.Well No.Groups No.Wells.Per.Group

2 25 1 2 /

VFPPDIMS

-- MXMFLO MXMTHP MXMWFR MXMGFR MXMALQ NMMVFT

- 56 -

/

VFPIDIMS

-- MXSFLO MXSTHP NMSVFT

/

START

1 'JUL' 2005 /

NSTACK

25 /

UNIFOUT

UNIFIN

--NOSIM

-------------------------------------------------------

-- Set relative paths for INCLUDE files

GRID

--

MESSAGES

9* 1000 /

--PSEUDO

--

EQUALS

'TOPS' 10000 1 17 1 3 1 1 /

'DX' 450 1 17 1 3 1 1 /

'DY' 417 1 17 1 3 1 1 /

'DZ' 100 1 17 1 3 1 1 /

'PORO' 0.20 1 17 1 3 1 1 /

'NTG' 1.0 1 17 1 3 1 1 /

'PERMX' 100 1 17 1 3 1 1 /

/

COPY

'PERMX' 'PERMY' /

'PERMX' 'PERMZ' /

/

MULTIPLY

'PERMZ' 0.1 1 17 1 3 1 1 /

/

--

========================================================================

======

INIT

PROPS

--

========================================================================

======

--

========================================================================

======-----------------------

------------------------------------------------

--

SWOF

------------------------------------------------

-- Corey func nw=1.51 no=1.83 Krw=0.362 Kro=0.65

- 57 -

------------------------------------------------

-- Sw Krw Kro Pc

0.3500000 0.0000000 0.6500000 0.0000000

0.3600000 0.0013856 0.6204967 0.0000000

0.3700000 0.0039428 0.5916183 0.0000000

0.3800000 0.0072691 0.5633673 0.0000000

0.3900000 0.0112196 0.5357467 0.0000000

0.4000000 0.0157105 0.5087591 0.0000000

0.4100000 0.0206849 0.4824077 0.0000000

0.4200000 0.0261010 0.4566954 0.0000000

0.4300000 0.0319266 0.4316254 0.0000000

0.4400000 0.0381353 0.4072010 0.0000000

0.4500000 0.0447057 0.3834256 0.0000000

0.4600000 0.0516194 0.3603027 0.0000000

0.4700000 0.0588608 0.3378360 0.0000000

0.4800000 0.0664160 0.3160292 0.0000000

0.4900000 0.0742729 0.2948864 0.0000000

0.5000000 0.0824208 0.2744118 0.0000000

0.5100000 0.0908500 0.2546096 0.0000000

0.5200000 0.0995516 0.2354844 0.0000000

0.5300000 0.1085176 0.2170411 0.0000000

0.5400000 0.1177407 0.1992845 0.0000000

0.5500000 0.1272143 0.1822201 0.0000000

0.5600000 0.1369319 0.1658535 0.0000000

0.5700000 0.1468879 0.1501905 0.0000000

0.5800000 0.1570769 0.1352375 0.0000000

0.5900000 0.1674938 0.1210013 0.0000000

0.6000000 0.1781340 0.1074890 0.0000000

0.6100000 0.1889929 0.0947085 0.0000000

0.6200000 0.2000664 0.0826681 0.0000000

0.6300000 0.2113506 0.0713769 0.0000000

0.6400000 0.2228417 0.0608450 0.0000000

0.6500000 0.2345362 0.0510834 0.0000000

0.6600000 0.2464307 0.0421042 0.0000000

0.6700000 0.2585220 0.0339213 0.0000000

0.6800000 0.2708072 0.0265505 0.0000000

0.6900000 0.2832832 0.0200097 0.0000000

0.7000000 0.2959473 0.0143206 0.0000000

0.7100000 0.3087969 0.0095095 0.0000000

0.7200000 0.3218293 0.0056095 0.0000000

0.7300000 0.3350422 0.0026659 0.0000000

0.7400000 0.3484332 0.0007473 0.0000000

0.7500000 0.3620000 0.0000000 0.0000000

/

--

========================================================================

======

ROCK

-- pref Cr

-- psia 1/psi

5905 3.1E-06 /

PVTW

-- pref Bw Cw visw

- 58 -

-- psia rb/bbl 1/psi cp

5905 1.050 3.0E-06 0.312 /

PVDO

-- psia Bo viso

-- rb/bbl cP

2040 1.421 0.380

3000 1.392 0.405

4000 1.368 0.431

5000 1.354 0.459

5910 1.342 0.480

6500 1.334 0.49

/

GRAVITY

-- Surface Densities for Oil Water and Gas

-- API Water=1 Air=1

44.28 1.0474 1.0570 / Platform

RSCONST

-- GOR pref

-- Mcf/bbl psia

0.502 2040 /

--

========================================================================

======

REGIONS

--

========================================================================

======

SATNUM

51*1 /

FIPNUM

51*1 /

EQLNUM

51*1 /

--

========================================================================

======

SOLUTION

--

========================================================================

======

EQUIL

--

--DATUM PRES WOC CAP.PRES GOC CAP.PRES RSVD RSVD EQUIL

--DEPTH DATUM FT @WOC FT @GOC TABLE TABLE ACC

11351 5910 11351 0 0 0 0 0 10

/

-------------------------------------------------------

RPTSOL

'RESTART=2' 'FIP=3' /

- 59 -

--

========================================================================

======

--

========================================================================

======

SUMMARY

SEPARATE

RPTONLY

FOPR

FWPR

FOPT

FWCT

--

WBHP

'W1' 'PR1' /

WWCT

'W1' 'PR1' /

FVPT

FVIT

FOIP

--

--

--

========================================================================

======

SCHEDULE

--

========================================================================

======

-- Reporting

RPTRST

'BASIC=4' /

RPTSCHED

/

WELSPECS

'PR1' 'G1' 12 2 1* 'OIL' 7* /

'W1' 'G1' 2 2 1* 'WAT' 7* /

/

COMPDAT

'PR1' 1* 1* 1 1 'OPEN' 2* 0.508 3* 'Z' 1* /

'W1' 1* 1* 1 1 'OPEN' 2* 0.508 3* 'Z' 1* /

/

-- qo qw qg ql rb/d bhp

WCONPROD

'PR1' 'OPEN' 'RESV' 1* 1* 1* 1* 5000 2100 /

/

WPIMULT

'W1' 10 /

'PR1' 10 /

/

WCONINJ

'W1' 'WAT' 'OP' 'RESV' 1* 0 1.0 FVDG 10000 /

- 60 -

/

TSTEP

30*1

11*30

/

TSTEP

12*30

12*30

12*30

12*30

/

TSTEP

12*60

12*60

12*60

12*60

/

TSTEP

12*100

12*100

/

TSTEP

10*365

40*365 /

END

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

-- END OF FILE

--

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++

- 61 -

Appendix B – curve matching

Curve Matching for Homogeneous Model

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

Sw

Fw

0.1

1

10

100

1000

Fw (from relperm)

Fw fromWOR

WOR -Extraploated

WOR -Extraploated

Expon.(WOR -Extraploated)

- 62 -

FWCT vs. TIME

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

FW

CT

Time (years)

FINE - Hetrogeneous model

Increasing watercut

due to coning

Coning in Heterogeneous model Kv/Kh = 0.1

FWCT vs. TIME

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

FW

CT

Time (years)

FINE - Hetrogeneous modelKv/Kh = 1

Increasing watercut

due to coning

Coning in Heterogeneous model Kv/Kh = 1