determining petrophysical properties of facies using a ... · determining petrophysical properties...

DETERMINING PETROPHYSICAL PROPERTIES

OF FACIES USING A HIERACHICAL HISTORY

MATCHING METHOD

A REPORT SUBMITTED TO THE DEPERTMENT

OF PETROLEUM ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

BySatomi SuzukiAugust 2003

ii

Abstract

This report outlines the master’s degree research carried out to develop a history

matching method which perturbs a geostatistically modeled permeability field. The goal

of the research is to propose a methodology that can preserve geological information

about permeability heterogeneity, which is strongly controlled by facies in many cases.

The work is an extension of probability-perturbation-based history matching methods

proposed by Caers1 and Kim and Caers2. In their work, the novel methodology was

developed to perturb facies realization with facies having known permeability in order to

obtain history match preserving underlying geological concepts. We extend their history-

matching scheme to the cases where the within-facies permeability is not known.

The methods investigated in this work are one-dimensional and multi-dimensional

optimization techniques which perturb the histogram of permeability within each facies.

The strength of this approach lies in a preservation of facies shape, while determining the

petorphysical properties of each facies through a history matching. The applicability of

the methodology was investigated using 2D and 3D synthetic reservoir models, which

comprise sand and mud.

iii

Acknowledgement

I would like to express my sincere gratitude to my adviser, Prof. Jef Caers, for his

advice and encouragement. This research was carried out under his precise and valuable

instruction and he provided me a great guidance to complete the research. I also would

like to show appreciation to my senior students in SCRF, Junrai Kim and Todd Hoffman.

Junrai Kim provided me his source code and the facies perturbation part of my computer

program was coded based on it. Todd Hoffman provided me the reference reservoir

models used in Chapter 5.

The financial support for my study was provided by Japan Oil Engineering, Co., Ltd.

and Japan National Oil Corporation.

iv

Contents

Abstract ............................................................................................................................... ii

Acknowledgement ............................................................................................................. iii

Contents ............................................................................................................................. iv

List of Figures ..................................................................................................................... v

1. Introduction..................................................................................................................... 1

2. Review of Previous Works ............................................................................................. 3

2.1. Multiple-point Geostatistics ..................................................................................... 3

2.2. History Matching under Training-image-based Geological Constraints.................. 4

2.3. Inclusion of Uncertainty in Facies Related Parameters............................................ 7

3. Method for Perturbing Permeability ............................................................................... 8

4. Synthetic Reservoir Application Using One-dimensional Optimization...................... 10

4.1. Model Description .................................................................................................. 10

4.2. Selection of Production Data.................................................................................. 11

4.3. Hierarchical Perturbations ...................................................................................... 12

4.4 Results ..................................................................................................................... 13

4.5. Order of Parameter Perturbation............................................................................. 15

4.6. Discussion............................................................................................................... 15

5. Synthetic Reservoir Application Using Multi-dimensional Optimization ................... 16

5.1. Model Description .................................................................................................. 16

5.2. Sensitivity of Objective Function to Parameters .................................................... 19

5.3. Results and Discussion ........................................................................................... 21

6. Conclusions................................................................................................................... 23

References......................................................................................................................... 25

v

List of Figures

Fig. 1 Permeability maps of reference and initial models

Fig. 2 Histogram of permeability, reference model vs. initial model

Fig. 3 Comparison of simulated water cut

Fig. 4 Comparison of simulated BHSP

Fig. 5 Permeability maps of additional models

Fig. 6 Comparison of water saturation distribution at simulation end, additional models

Fig. 7 Result of water cut matching, perturbing facies only

Fig. 8 Perturbation of facies realization during water cut matching, perturbing facies

only

Fig. 9 Comparison of water saturation distribution at simulation end, perturbing facies

only

Fig. 10 Result of water cut & BHSP matching, perturbing facies only

Fig. 11 Perturbation of facies realization during water cut & BHSP matching, perturbing

facies only

Fig. 12 Comparison of water saturation distribution at simulation end, perturbing facies

only

Fig. 13 Comparison of pressure distribution at simulation end, perturbing facies only

Fig. 14 Result of history matching, Case1

Fig. 15 Perturbation of facies and permeability realization during history matching,

Case1

Fig. 16 Water saturation distribution at simulation end, Case1

Fig. 17 Pressure distribution at simulation end, Case1

Fig. 18 Histogram of permeability, history matched model vs. reference model, Case1



Case2



vi




Case3






Case4




Fig. 34 Comparison of history matched permeability realizations, original vs. reversed

order, Case2

Fig. 35 Comparison of history matched permeability realizations, original vs. reversed

order, Case3

Fig. 36 Horizontal slice of training image used for facies simulation, Reference model A

Fig. 37 3D view of facies realization, Reference model A

Fig. 38 Horizontal slice of permeability realization, Reference model A

Fig. 39 Well location, Reference models A & B

Fig. 40 Horizontal slice of facies realization, Reference model B

Fig. 41 Horizontal slice of permeability realization, Reference model B

Fig. 42 Horizontal slice of porosity realization, Reference model B

Fig. 43 Horizontal slice of permeability realization, Initial model

Fig. 44 Histogram of permeability, Initial model vs. Reference models A & B

Fig. 45 Comparison of simulated water cut, Reference model A vs. initial model

Fig. 46 Comparison of simulated BHSP, Reference model A vs. initial model

Fig. 47 Comparison of simulated water cut, Reference model B vs. initial model

Fig. 48 Comparison of simulated BHSP, Reference model B vs. initial model

Fig. 49 Sensitivity of objective function to matching parameters

vii

Fig. 50 Result of history matching, Case A

Fig. 51 Horizontal slice of permeability realization, History matched model, Case A

Fig. 52 Histogram of permeability, Reference model vs. history matched model, Case A

Fig. 53 Optimization performance during history matching process, Case A

Fig. 54 Result of history matching, Case B

Fig. 55 Horizontal slice of permeability realization, History matched model, Case B

Fig. 56 Histogram of permeability, Reference model vs. history matched model, Case B

Fig. 57 Optimization performance during history matching process, Case B

1

Chapter 1

1. Introduction

Geostatistical reservoir modeling has been widely used for reservoir characterization

for its ability to describe detailed petrophysical heterogeneity, its suitability for

integrating various types of data from different sources, and its capability to produce

multiple reservoir description related to reservoir uncertainty. Recently, multiple-point

geostatistics3 has been successfully applied to capture complicated geological features,

introducing the concept of a “geological training image”. The inclusion of dynamic data

into geostatistical reservoir modeling has also been extensively pursued since the

reproduction of past production behavior is crucial for reservoir models to be predictive.

However, little attention has been paid to the preservation of geological features in

history matching process despite the increased capability of geostatistical techniques for

integrating geologic and seismic information. The limitation of traditional approaches

that rely on the perturbation of individual gridblock permeability is the lack of geological

control on the history matching process. Without constraints, a history matched

permeability field can easily lose consistency with geology and hence lose predictive

power.

Caers1 proposed a probability perturbation method as a novel approach for history

matching, focusing on imposing geological constraints on dynamic data integration. The

key to his approach is the perturbation of the probability model that generates facies

realizations, rather than the direct perturbation of gridblock properties, to obtain a history

match. The history matched facies realization keeps consistency with geological models

since the probability perturbation is constrained by geology through a variogram or a

training image. Later, his work was extended by Kim and Caers2 to cover the geological

uncertainty in net-to-gross ratio or facies anisotropy directions.

All of the examples presented in Caers1 rely on the assumption that the petrophysical

properties of each facies (i.e. porosity and permeability) are known and constant for each

facies. This paper presents a methodology for perturbing the petrophysical properties per

2

facies in addition to the facies perturbation. The proposed approach perturbs a facies

realization and the within-facies permeability in a hierarchical manner. A facies

realization is perturbed using the probability perturbation method1. The within-facies

permeability is optimized to a obtain history match by perturbing the histogram of

permeability. The advantage of this approach is that facies control on permeability

heterogeneity is easily preserved without losing geological plausibility.

This paper consists of six chapters. Chapter 2 outlines the review of previous works,

i.e. multiple-point geostatistics and probability perturbation method. Chapter 3 presents

the proposed method used for perturbing within-facies permeability field. The results of

two-dimensional and three-dimensional synthetic reservoir applications are discussed in

Chapters 4 and 5, respectively. Chapter 6 summarizes the conclusions derived from the

application results.

3

Chapter 2

2. Review of Previous Works

2.1. Multiple-point Geostatistics

The limitation of geostatistical techniques based on the variogram, essentially a two-

point statistic, is that it cannot capture complicated geological patterns such as

meandering channels. Variograms only account for correlation between any two data

locations to characterize spatial continuity, despite the fact that the feature of local spatial

continuity is strongly related to the pattern of data events especially in complex geology.

Multiple-point geostatistics3 advocates the use of a “training image”, a reservoir

analog of sorts, that depicts the desired geological pattern or heterogeneity without the

need of being constrained to any specific reservoir data. Since the role of a training image

is only to provide the information about spatial continuity, or the information about

pattern of geological feature, it can be a purely conceptual image inferred through

geological interpretation. The multiple-point geostatistics approach accounts for the

multiple-point pattern when deriving conditional probability distributions in sequential

simulation. Unlike kriging-based simulation techniques such as sequential Gaussian

simulation, the local conditional probability is directly borrowed from a “geological

training image” which depicts geological patterns, such as meandering channels, as a

conceptual realization. Using a training image with sufficient volume in space, the

conditional probabilities are scanned from the training image and stored in a training

image database, termed “search tree”5. In the sequential simulation scheme, the

conditional probability can be retrieved at any unsampled location in accordance with the

configuration of the available conditioning data and is used to draw a simulated value.

The local conditional probability can be conditioned to soft information, such as

seismic data, as well as hard data events. The following relation is applied for conditional

probabilities based on the assumption of conditional independence6:

4

a

c

b

x = (1)

where, )(

)(1

AP

APa

−= )|(

)|(1

BAP

BAPb

−= )|(

)|(1

CAP

CAPc

−= ),|(

),|(1

CBAP

CBAPx

−=

P(A) is global probability of facies A, P(A|B) is probability of facies A at any

location conditioned to hard data events B, and P(A|C) is probability of facies A

conditioned to secondary data C and obtained through some form of well-to-seismic

calibration. P(A|B,C) is probability of facies A jointly conditioned to hard data events (B)

and secondary data (C). Accordingly, P(A|B,C) is obtained from Eq. (1) as:

bca

a

xCBAP

+=

+=

1

1),|( (2)

2.2. History Matching under Training-image-based Geological

Constraints

Consider some binary geological realization i(0)(u). The notation i(l)(u) denotes the

simulated facies indicator at location u obtained in the lth iteration of the history

matching process . Recall that the indicator random function I (u) is defined as;

=other wise 0,

locationat occurs A facies if 1, )I(

uu (3)

The probability perturbation method1 history matches a facies realization by

perturbing the local facies probability P(A|B) using a training image as geological

constraints. By perturbing probabilities rather than grid properties, geological consistency

can be maintained. The probability P(A|B) is perturbed using another probability P(A|D)

which is defined as follows.

5

This probability P(A|D) depends on the production data through a free parameter rD

as follows:

P(A|D) = (1 - rD) i(0)(u) + rD P(A) (4)

i(0)(u) is facies indicator at location u in the initial realization, P(A) is net-to gross

ratio, and rD is a perturbation parameter between [0,1]. In order to create a perturbation of

P(A|B), the probability P(A|B) and P(A|D) are combined to form P(A|B,D), using Eq. (2).

The probability P(A|B,D) can be considered as a perturbation of P(A|B) and is used in

sequential simulation or draw a new i(l)(u). Note that P(A|B,D) depends on rD. If rD=0,

P(A|D) remains the same as i(0)(u), thus a realization generated using P(A|B,D) is the

same as the current realization i(0)(u). When rD=1, P(A|D) is equal to P(A), which means

the realization generated with P(A|B,D) is equi-probable with the current realization.

Accordingly, the facies realization i(0)(u) is perturbed between the current realization

i(0)(u) and another equi-probable realization through the parameter rD. The magnitude of

perturbation is determined by finding the value of rD that provides the best history match.

The flowchart of the algorithm is presented below.

6

Outer iteration: Generate initial facies realization, i(0)(u)

Generate geological training image

Define P(A|D) using rD: Eq. (4)

Inner iteration: Using one-parameter optimization technique, find rD

that obtain the best history match as follows:

Perturb P(A|B,D) using P(A|D): Eq. (2)

Set new rD to reduce objective function

Perform flow simulation

Generate facies realization, irD(l)(u) using P(A|B,D)

YESObjective function

minimized ?

NO

YES

History Matched ?

NO

Update facies realization, i(l)(u) = irD_opt (l)(u)

Change random number seed

YES

END

7

2.3. Inclusion of Uncertainty in Facies Related Parameters

The strength of the probability perturbation method is the sure-fire preservation of

underlying geological concepts during a history matching process. However, since the

geological concept imposed as constraints itself has uncertainty, history matching might

be difficult if the geological training image is used without accounting for its uncertainty.

In order to overcome this problem, the probability perturbation method can be extended

to account for the uncertainty in facies direction or net-to-gross of the training image. The

fundamental idea is to rotate the training image or change its net-to-gross as well as

perturbing the facies distribution to obtain history matching of dynamic data2.

In Kim and Caers2, the perturbation of facies anisotropy direction, ∆θ, is taken to be

linearly related to the facies perturbation parameter, rD, in Eq. (4) as:

max

min

min

1θθ ∆

−−

=∆D

DD

r

rr(5)

where the minimum / maximum limits of rD and ∆θ are defined based on geological

plausibility. Using this relation, the problem of jointly perturbing the parameters rD and

∆θ is reduced to a one-dimensional optimization problem. This direct link of facies

anisotropy direction to the perturbation parameter rD enables to apply a simple one-

parameter optimization technique for jointly finding the optimum perturbation of the

parameter rD and the facies related parameters, reducing computational burden of history

matching process.

Similarly, in order to perturb the net-to-gross ratio of facies, net-to-gross, p, is related

to the facies perturbation parameter, rD, as:

minminmax

min

min )(1

pppr

rrp

D

DD +−−−

= (6)

where the minimum / maximum limits of rD and p are defined by an engineer.

8

Chapter 3

3. Method for Perturbing Permeability

The method used by Caers1 relies on the fact that the permeability is constant and

known for each facies. To match both petrophysical properties and facies, we propose the

following hierarchical approach. First, we generate a facies realization and, for each

facies, a within-facies permeability realization. Using a simple cookie cut technique, we

can paste the permeability realizations into the facies model. History matching now

proceeds as follows:

1) Keeping the permeability per facies fixed, perform one outer iteration of Caers’

method for history matching facies.

2) Keeping the facies realization from step 1 fixed, change the permeability

realization for each facies using the method described below.

3) Iterate steps 1 and 2 while a history matching is achieved.

To modify the permeability realization, we simply modify the permeability

histogram. This can be motivated by the fact that the contrast between mud and channel

permeability is more important than the heterogeneity of within channel permeability.

The permeability heterogeneity is modeled assuming a log-normal distribution:

k logk log µ )y(σ )k( log += uu

10 )k( )k( log uu = (7)

where k(u) is grid permeability, µlogk,i and σlogk,i are mean and standard deviation of

logarithm of permeability within facies i, and y(u) is the outcome of spatially correlated

Gaussian random function Y(u), u = (x, y, z) ∈ D.

9

In step 2, we perturb the mean and standard deviation of the logarithm of

permeability, µlogk,i and σlogk,i, to obtain a history match of the production data. The

advantage of this method is that history matched permeability field preserves geological

consistency since we optimize the within facies permeability histogram under the

constrains of a facies geometry rather than perturbing individual grid block permeability.

One way to perturb mean and standard deviation of permeability in individual facies

is a sequential perturbation of the parameters (mean and variance of each facies) using a

one-dimensional optimization technique, i.e. Brent method4. The other way is to apply a

multi-dimensional optimization method, i.e. Levenberg-Marquardt method4, and perturb

the parameters jointly. In this work, both approaches are evaluated through synthetic

reservoir applications. The problems are restricted to two facies, sand and shale.

Each optimization process, step 1 and 2, includes an inner iteration to find the optimal

distribution parameter. The objective function to be minimized is defined as the weighted

summation of deviation between historical and simulated production data. This objective

function is examined at the end of each inner iteration to see whether it is reduced from

the previous parameter optimization. The optimized parameter is retained only when the

objective function is reduced. The spatially correlated Gaussian random function for

generating within-facies permeability field is updated in each outer iteration by changing

the random number seed. The variogram of Gaussian random number field is fixed

throughout the history matching. In order to preserve the plausibility of permeability

distribution for each facies, the perturbation of µlogk,i and σlogk,i are limited between

minimum and maximum values.

10

Chapter 4

4. Synthetic Reservoir Application Using One-dimensional

Optimization

4.1. Model Description

The proposed method is evaluated using a synthetic reservoir. The sequential

perturbation of the parameters using a one-dimensional optimization technique is applied

to obtain the history match of production data in this section. As a sample problem, a

five-spot-pattern water injection was simulated within a 2D horizontal reservoir model

which comprises sand and shale bodies. Fig.1 shows the permeability maps of the

reference and initial models. The reference model is supposed to be a “true reservoir

description” which provides a production history. The initial model is supposed to be a

“wrong initial guess”. The average and standard deviation of permeability in each model

is specified as below.

Reference Model Initial ModelAverage

permeabilityStandarddeviation

Averagepermeability

Standarddeviation

Sand3600 mD

(µlogk = 3.4)2930 mD

(σlogk = 0.4)490 mD

(µlogk = 2.6)266 mD

(σlogk = 0.3)

Shale5 mD

(µlogk = 0.7)1.2 mD

(σlogk = 0.1)1 mD

(µlogk = 0.0)0.3 mD

(σlogk = 0.15)

The histograms of permeability in sand and shale are compared between the reference

model and the initial model in Fig. 2. Both sand and shale permeability is higher in the

reference model than in the initial model. Also, the larger variation of permeability is

modeled in the reference model compared to the initial model (Fig. 2). The variogram of

the reference and initial model is the same and not changed during history matching.

11

Figs. 3~4 compare the simulated water cut and bottom-hole pressure (BHSP) of the

producer obtained from the reference model and the initial model. Simulation runs were

also made for two alternative initial models to study the impact of a wrong initial model.

1. An initial model is created with the same facies geometry as the reference but

with a “wrong” within-facies permeability as depicted in Fig. 5.

2. An initial model is created with the same permeability histogram per facies as

the reference but with a “wrong” facies geometry as depicted in Fig. 5.

The results are also included in Figs. 3~4. The comparison of water saturation

distribution at the end of the simulation is illustrated in Fig.6. As shown in Figs. 3~4, the

simulation result of the initial model exhibits lower water cut and lower BHSP compared

to the reference model due to “wrong” facies direction and lower permeability. The lower

water cut is mainly due to the “wrong” facies as shown by the result of the initial facies +

reference permeability case (i.e. “wrong facies” case, Fig. 3). The simulated water cut of

the initial permeability + reference facies case (i.e. “wrong permeability” case) is higher

than the reference model since low permeability partly blocks the flow path of water (Fig.

6). However, the cases with lower permeability result in the lower simulated BHSP

compared to the reference model (Fig. 4).

4.2. Selection of Production Data

Before applying the proposed hierarchical history matching method, an attempt was

made to match historical production behavior by perturbing only facies (the method of

Caers), hence freezing the permeability within each facies to a “wrong” histogram. Fig.7

depicts the result of water cut matching obtained by perturbing facies only. First, only the

water cut was used as data to be matched. Fig. 8 shows the permeability realizations

obtained during the optimization process in comparison with the reference permeability

model. Fig. 9 compares the water saturation distribution at the end of simulation among

the initial model, the optimized model, and the reference model. As shown in the figure, a

good match of water cut was obtained by facies perturbation although permeability

12

remained the same as that in the initial model. This can be explained by the fact that

water cut provides information about the preferential flow path of water, which is mostly

dependent on facies connectivity (Fig. 9).

Next, bottom-hole pressure (BHSP) was included in the matching process to assess

whether it provides additional information for within-facies permeability. Fig. 10 depicts

the water cut and BHSP matching result obtained by optimizing facies only. The

permeability realizations generated during the optimizing process are presented in Fig.

11. The water saturation and pressure distribution at the end of the simulation are

illustrated in Figs. 12~13. As shown in Fig. 10, BHSP is not sufficiently matched by

perturbing facies only since the pressure difference between producer and injector is

ultimately controlled by permeability (Fig.13).

4.3. Hierarchical Perturbations

The following four cases were considered to evaluate the applicability of the method

and to investigate appropriate choice of optimizing parameter;

Case 1: Optimize µlogk and σlogk of sand and shale

Case 2: Optimize µlogk of sand only

Case 3: Optimize µlogk of sand and shale

Case 4: Optimize µk and σk of sand and shale

where µlogk and σlogk are the mean and standard deviation of logarithm of permeability,

while µk and σk are the mean and standard deviation of permeability. In all cases, facies is

perturbed first, then for a fixed facies geometry the histograms of permeability within

each facies is perturbed by perturbing either/or mean and variance. To perturb the

histograms, the mean is perturbed first, then the variance. Cases 1~3 assume a log-normal

distribution for permeability whereas Case 4 assumes a normal distribution for

permeability. The facies realizations are also perturbed in all cases. The effect of the

13

order of parameter perturbation (i.e. which is the first to optimize – facies or

permeability?) was also investigated as described later in this section.

4.4 Results

Case 1

The histograms of permeability were optimized by perturbing µlogk and σlogk of

individual facies in this case. Fig. 14 depicts the history matching result obtained by

optimizing µlogk and σlogk of sand and shale. Fig. 15 shows the order of each parameter

perturbation during the optimizing process. Fig. 16~17 illustrates the water saturation and

pressure distribution at the simulation end corresponding to the realizations depicted in

Fig. 15. Fig. 18 compares the histograms of permeability in sand and shale between the

history matched model and the reference model. As shown in Fig. 14, a sufficient history

match was obtained both for water saturation and BHSP. Most of the history matching

was completed by perturbing facies, µlogk and σlogk of sand in the first outer iteration. The

perturbation of shale permeability showed little effect on the history match. As exhibited

in Fig. 15, change in σlogk generated significant fluctuation in the sand permeability

realizations. The resulting history matched model is quite different from the reference

model. It appears that modifying σlogk generated strong heterogeneity into the model,

hence enabling a history match (Fig. 16). The probability distribution of the history

matched sand permeability exhibits a much higher variance compared to that of the

reference model (Fig. 18).

Case 2

This case is designed to eliminate the effect associated with the change in σlogk of

sand. Only the facies and µlogk of sand are perturbed in this case. The result of history

matching, the optimized permeability realizations, the water saturation and pressure

distribution at the end of simulation, and the comparison of histograms of permeability

are depicted in Figs. 19~23. A good history match was obtained as shown in Fig. 19.

14

Moreover, as shown in Fig. 20, the history matched facies realization seems to be more

accurately representing the nature of facies distribution of the reference model. The

probability distribution of sand permeability in the reference model was also reasonably

reproduced in the history matched realization as depicted in Fig. 23. Note that increasing

µlogk increases the variance of permeability as well as mean since it has the same effect as

multiplying permeability.

Case 3

This case is designed to examine the effect of perturbing µlogk of shale in addition to

the perturbation of µlogk of sand in Case 2. Facies and µlogk of sand and shale were

optimized. The result of history matching, the optimized permeability realizations, the

water saturation and pressure distribution at the end of simulation, and the comparison of

histograms of permeability are depicted in Figs. 24~28. As shown in Fig. 24, the effect of

perturbing µlogk of shale on the history match is quite marginal.

Case 4

In this case, a normal distribution was assumed for the probability density function of

within facies permeability. The mean and standard deviation of permeability were used as

optimization parameters. The result of history matching, the optimized permeability

realizations, the water saturation and pressure distribution at the end of simulation, and

the comparison of histograms of permeability are illustrated in Figs. 29~33. As depicted

in Fig. 29, a good match of water cut and BHSP was obtained only by changing facies

and µk of sand. The σk of sand permeability and the parameters of shale permeability did

not reduce the objective function. The optimized facies distribution successfully

predicted the feature of facies in the reference model (Fig. 30). However, the history

matched sand permeability was too homogeneous compared to the reference model (Figs.

30 and 33), due to the use of a normal distribution. Nevertheless, the distribution

assumption (normal vs. lognormal) does not seem to affect the history match. The within-

facies mean permeability is most consequential.

15

4.5. Order of Parameter Perturbation

The effect of the order of parameter perturbation was examined based on Case 2 (i.e.

perturb µlogk of sand only) and Case3 (i.e. perturb µlogk of sand and shale), which are the

most successful cases in Section 4.4. In Case2, facies and µlogk of sand were perturbed in

the reversed order, i.e. optimize µlogk of sand first and optimize the facies realization next.

The history matched permeability realization is compared to the result of the original

perturbation order in Fig. 34. In Case3, µlogk of sand and µlogk of shale were perturbed in

the reversed order, executing facies perturbation first as in the original order. The result is

compared to the original perturbation order as shown in Fig. 35.

In this particular case, Case2 obtained better reproduction of faces geometry and sand

permeability of the reference when facies was perturbed first. In Case3, although the

history matched sand permeability is not sensitive to the order of the perturbation of µlogk

of sand and µlogk of shale, the facies geometry is better reproduced when perturbing µlogk

of shale first.

4.6. Discussion

The synthetic case study showed that the optimized facies and permeability

realization is strongly affected by the parameter choice. The optimization by perturbing

facies and µlogk, with an assumption of log normal distribution for permeability, provided

the best result among the cases. Perturbation of σlogk generated excessive fluctuation of

the permeability realization and obscured the impact of facies distribution on production

performance behavior. The normal distribution assumption for permeability distribution

produced extremely homogeneous permeability realization yet an adequate match to the

production data is achieved. It therefore appears that production data is not much

sensitive to the within-facies small-scale heterogeneity.

The result of the sequential optimization of facies and permeability using a single-

parameter optimization algorithm is sensitive to the order of the parameter perturbation.

The perturbation of various µlogk associated with each facies can be coupled using a

multi-dimensional optimization algorithm to reduce the concern about perturbation order.

16

Chapter 5

5. Synthetic Reservoir Application Using Multi-dimensional

Optimization

5.1. Model Description

The multi-dimensional optimization method was applied to perturb the histogram of

permeability using synthetic reservoir models. The two types of three-dimensional fluvial

type reservoir models, comprising mud and channel sand, are built as reference reservoirs

based on a sample problem presented in the paper of Hoffman and Caers7.

Reference Model A

The Reference Model A is a 3D horizontal reservoir with the top depth of 5000 ft.

The model consists of 50*65*15 grid blocks. The size of grid blocks is 200 ft in

horizontal and 30 ft in vertical. The facies realization comprising mud and meandering

channel sand is generated using the training image depicted in Fig.36. Fig.37 illustrates

the 3D view of the facies distribution. The permeability field within mud and channel

was simulated using Gaussian simulation and cookie-cut technique as shown in Fig.38.

The mean and standard deviation of permeability were specified as below:

Reference Model AAverage


Mud8.8 mD

(µlogk = 0.8)8.0 mD

(σlogk = 0.3)

Channel Sand485 mD

(µlogk = 2.7)213 mD

(σlogk = 0.2)

The constant porosity of 0.07 and 0.29 were set for sand and channel, respectively.

17

Six producers and three injectors were placed as depicted in Fig.39. Water-flooding

performance was simulated for 3000 days under the surface production/injection rate

control. Production and injection rates are 5000 STB/D/WELL and 12000

BBL/D/WELL, respectively. The reservoir was modeled as an undersaturated oil

reservoir with the initial pressure of 2500 psia. The formation volume factor and viscosity

of oil are 1.0 STB/RB and 1.0 cp, while those of water are 1.05 STB/RB and 0.325 cp.

The fluid compressibility / viscosibility are assumed negligible. Relative permeability

curve is specified with the irreducible water saturation of 0.15 and the residual oil

saturation of 0.30.

Reference Model B

The Reference Model B was directly borrowed from the sample problem of Hoffman

and Caers7. The model was constructed based on the Stanford V reservoir8 and consists

of three sets of layers which have channel sands with different shape and meandering

direction from layer to layer. Fig.40 shows the typical horizontal slices of facies

realization in individual layers. Permeability and porosity within mud and channel sand

are populated using Gaussian simulation and cookie-cut technique as depicted in

Figs.41~42. The mean and standard deviation of permeability are as below:

Reference Model BAverage


Mud0.07 mD

(µlogk = -1.6)0.12 mD

(σlogk = 0.5)

Channel Sand448 mD

(µlogk = 2.6)214 mD

(σlogk = 0.3)

Model B has a uniform the top depth of 2000 ft and the same reservoir size as Model

A. The model consists of 100*130*30 grid blocks with the grid block size of 100 ft in

horizontal and 15 ft in vertical.

18

Six producers and three injectors were placed as in Model A. Water-flooding

performance was simulated for 3600 days. Producers were controlled by the total liquid

production rate of 6400 STB/D/WELL. Minimum BHFP of 100 psia was specified as a

constraint for producers. Water was injected with 12000 BBL/D/WELL using surface

rate control. The reservoir is an undersaturated oil reservoir with the initial pressure of

1000 psia. The same fluid properties and relative permeability curves as in Model A were

specified.

The facies and permeability realizations of the initial model for history matching was

generated based on the Reference Model A by changing the random number seed for

simulating facies, the mean and standard deviation of permeability, and the Gaussian

random function of permeability field. The training image for facies simulation and the

variogram for permeability modeling were fixed to those used for the Reference Model

A. The grid coordinate was specified to the same as Model A. Fig.43 shows the

horizontal slices of permeability realization of the initial model. The mean and standard

deviation of permeability in each facies were specified as below:

Initial ModelAverage


Mud1.5 mD

(µlogk = 0.1)1.0 mD

(σlogk = 0.3)

Channel Sand34 mD

(µlogk = 1.5)14 mD

(σlogk = 0.2)

Fig.44 compares the histograms of permeability within the individual facies among the

Reference Models A, B, and the initial model. The mean permeability of the initial model

was specified as lower than that of the reference models as shown in the figure.

The two cases of history matching were performed starting from the same initial

geological model but using the different reference production data as follows:

19

Case A: Match pressure and water cut to the reference production data of the

Reference Model A

Case B: Match pressure and water cut to the reference production data of the

Reference Model B

Case A is designed assuming that the facies distribution and the permeability

heterogeneity are unknown, however, the geological pattern of the facies represented by a

training image and the variogram of permeability are known. On the other hand, Case B

is performed under the condition that the whole information about facies and

permeability heterogeneity, including the variation of channel shape/direction with

layers, are unknown. The operational condition of the wells and the reservoir parameters

other than facies, permeability and porosity were specified to be consistent between the

initial and reference models.

Both of the history matches were executed by perturbing facies distribution and

permeability histograms. Based on the conclusion derived in the one-dimensional

optimization case, only the mean of logarithm of permeability (µlogk) was perturbed in

this application with the fixed standard deviation of logarithm of permeability (σlogk)

when modifying the histograms of permeability. Figs.45~48 compare the simulated water

cut and bottom-hole shut-in pressure between the initial model and the Reference Models

A and B.

5.2. Sensitivity of Objective Function to Parameters

The Levenberg-Marquardt method4 was applied for perturbing the histograms of

permeability to obtain a history match. The parameters to be optimized are µlogk of mud

and µlogk of channel sand, hence a 2D optimization needs to be performed. The objective

function, O, to be minimized was defined as follows:

2211 OwOwO += (8)

where,

20

21

1 ,1

,,1,,11 ∑

=

−=

N

i i

iobsisim ddO

σ

22

1 ,2

,,2,,22 ∑

=

−=

N

i i

iobsisim ddO

σ

Subscripts 1 and 2 denote water cut and bottom-hole shut-in pressure, respectively. N

is the number of observed data point, dobs is observed the value (or simulated value of the

reference model), dsim is the simulated value of the current model, and σi is the

measurement error of data point i. The weighting factors w1 and w2 were determined so

that the term w1O1 equals w2O2 when evaluated on the initial model, in order to avoid

over-emphasizing one of the dynamic data, unless this is the explicit objective.

The derivatives of the objective function with regard to µlogk of mud and µlogk of

channel sand are calculated numerically. The same objective function was used for both

of facies perturbation and permeability perturbation.

Before making any actual history matches, the sensitivity of the objective function to

µlogk of mud and µlogk of channel was investigated using the Reference Model A as a

reference model. 100 cases of sensitivity simulation runs were made by changing µlogk of

mud and channel sand individually, and the error functions (O1 and O2 in Eq.8) were

calculated for each case. Fig.49 shows the contours of the error function of water cut (O1)

and that of bottom-hole shut-in pressure (O2) on the µlogk of mud vs. µlogk of channel

plane. The “true” minimum (i.e. µlogk of mud and µlogk of channel of the Reference Model

A) is also shown in the figure. As depicted in the figure, the minimum of the error

function of water cut extends along the straight line on the µlogk of mud vs. µlogk of

channel plane. This observation indicates that fractional flow behavior is controlled by

the degree of the contrast of permeability between facies rather than the magnitude of

permeability itself. The error function of bottom-hole pressure, on the contrary, shows the

favorable configuration for solving a minimization problem with the single minimum at

the “true” minimum location. This result of the sensitivity analysis coincides with the

21

observation in Chapter 4, that is, during the history matching process, water cut mainly

provides the information about facies distribution whereas bottom-hole pressure informs

on the magnitude of permeability within facies.

5.3. Results and Discussion

Case A

Fig. 50 shows the history matched production performance with comparison to the

reference data of the Reference Model A. The horizontal slices of the history matched

permeability field are depicted in Fig. 51. Fig. 52 compares the histogram of permeability

between the history matched model and the reference model. Change in the objective

function and the matching parameters during the inner iterations are plotted against the

iteration count and presented in Fig. 53. The figure also plots the contribution of water

cut and that of bottom-hole pressure to the objective function to observe the optimization

behavior.

As depicted in Fig. 50, the satisfactory matching of water cut and bottom-hole

pressure was obtained by the proposed method. Production history was matched with one

outer iteration in this case (Fig. 53). The histogram of history matched permeability

exhibited good agreement with the permeability histogram of the reference model

especially in channel sand (Fig. 52). The feature of permeability and facies of the

reference model was reasonably captured by the history matched model as illustrated in

Fig. 51. As depicted in Fig. 53, the perturbation of facies almost exclusively reduced the

error function due to water cut and showed negligible effect on pressure matching. The

perturbation of permeability mainly decreased the error function due to bottom-hole

pressure and provided little contribution to the matching of water cut.

Case B

Fig. 54 shows the history matched production performance obtained using the

production data from the Reference Model B. The horizontal slices of the history

22

matched permeability field are depicted in Fig. 55. The comparison of the histogram of

permeability between the history matched model and the reference model is presented in

Fig.56. Change in the objective function and the matching parameters during the inner

iterations are plotted against iteration count and shown in Fig. 57.

As expected, matching production data was more difficult in Case B than in Case A

due to the condition that the training image of facies is unknown. The optimization

process required eight outer iterations to obtain sufficient history match shown in Fig.54.

As shown in Fig.57, the water cut was matched in the first two outer iterations by mainly

perturbing facies, then the pressure was matched by perturbing the permeability

histograms in the subsequent outer iterations. As depicted in Fig.55, the history matched

geological model exhibited meandering channel sands mostly in the upper part of the

reservoir (i.e. Layer 3) where the channel direction of the reference model (Fig.40) shows

good agreement with that of the training image used for facies simulation (Fig.36). It

appears that the optimization process avoided to generate extended channels in the layers

where the “wrong” training image was used (i.e. Layers 1&2). The histogram of channel

sand permeability was well reproduced by the history matching as depicted in Fig.56.

The change in the mud permeability due to the history matching was marginal because of

the small impact of mud permeability on simulated production performance.

23

Chapter 6

6. Conclusions

1. A history matching method to determine the petrophysical properties of facies was

investigated as an extension of probability perturbation method1. The hierarchical

perturbation of facies and permeability was implemented. Along with the facies

perturbation, the histogram of permeability specific to individual facies is perturbed

to obtain a history match by optimizing the mean and standard deviation of the

logarithm of permeability under a log-normal distribution assumption. The advantage

of this approach is the sure-fire preservation of facies shape yet determining within

facies permeability.

2. In order to perturb the permeability histogram, two methods were tested, 1) the

sequential perturbation of parameters using an one-dimensional optimization

technique, and 2) the joint perturbation of parameters using a multi-dimensional

optimization technique. The parameter choice for optimization was also investigated.

The quality of the prediction of facies and permeability field is extremely deteriorated

if we include the perturbation of σlogk in the history matching scheme. Since

production data provides little information about the magnitude of small-scale

heterogeneity, the resulting perturbations of σlogk appeared unconstrained and

arbitrary, generating unrealistic fluctuations in the permeability field. The one-

dimensional optimization case worked well when only facies and µlogk are perturbed.

However, the result was sensitive to the order of the perturbation of parameters. A

multi-dimensional optimization method appeared to be the most efficient since no

hierarchical order is needed and since the production data is most sensitive to the

relative difference between mud and channel permeability.

3. During history matching, water cut data mainly provided the information about facies

distribution, while pressure data provided the information about µlogk of each facies.

24

This observation was clearly confirmed through the behavior of objective function

during iteration.

25

References

1. Caers, J., 2003. History matching under training-image based geological model

constraints. paper SPE74716, SPE Journal, in press.

2. Kim, J., and Caers, J., 2002. Training image-based history matching under unknown

facies anisotropy distribution. SCRF report.

3. Caers, J. and Zhang, T., 2002. Multiple-point geostatistics: a quantitative vehicle for

integrating geologic analogs into multiple reservoir models. SCRF report.

4. Press, W. et al., 1988. Numerical recipes in C. Cambrige university press.

5. Guardian, F. and Srivastava, S., 1993. Multivariate Geostatistics: beyond Bivariate

moments. Geostatistics-Troia, p 133-144, Kluwer Academic Publications.

6. Strebelle, S., 2000. Sequential simulation drawing structure from training images.

Ph.D dissertation, Stanford University, Stanford, USA.

7. Hoffman, T.B and Caers, J., 2003. History matching using the regional probability

perturbation method. SCRF report 16.

8. Mao, S. and Journel A.G, Generation of a reference petrophysical/seismic data set:

the Stanford V reservoir. SCRF report.

26

REFERENCE INITIAL

LOG(K) AVE. STD.SAND 3.4 0.4SHALE 0.7 0.1

LOG(K) AVE. STD.SAND 2.6 0.3SHALE 0.0 0.15

K AVE. STD.SAND 3600 2930SHALE 5 1.2

K AVE. STD.SAND 490 266SHALE 1 0.3

Fig.1 Permeability maps of reference and initial models

27

Fig.2 Histogram of permeability, Reference model vs. initial model

28

IMPACT OF FACIES & PERM ON SIMULATED WATER CUT (M=1.5)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

REF

INIT PERM + REF FACIES

REF PERM + INIT FACIES

INIT

PROD: 1500 BOPD INJ: 3000 BPDPROD: 2000 BOPD, INJ: 3000 BPD

IMPACT OF FACIES & PERM ON SIMULATED BHSP

1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

REF

INIT PERM + REF FACIES

REF PERM + INIT FACIES

INIT

PROD: 2000 BOPD, INJ: 3000 BPDPROD: 1500 BOPD INJ: 3000 BPD

Fig.4 Comparison of simulated BHSP

Fig.3 Comparison of simulated water cut

29

INIT FACIES + REF PERM

REFERENCE INITIAL

REF FACIES + INIT PERM

Fig.5 Permeability maps of additional models

30

INIT FACIES + REF PERM

REFERENCE INITIAL

REF FACIES + INIT PERM

Sw @ 360 DAYS15% 70%

Fig.6 Comparison of water saturation distribution at simulation end, Additional models

31

HISTORY MATCHING RESULT(W.C. Matching by Optimizing Facies)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY INITIAL

ITR 1 ITR 3

ITR 4

Fig.7 Result of water cut matching, perturbing facies only

32

Fig.8 Perturbation of facies realization during water cut matching, perturbing facies only

Iteration #1

Iteration #2 Iteration #3

33

REFERENCE MODEL

ITERATION #4(FACIES OPTIMIZED)

INITIAL MODEL

Sw @ 360 DAYS15% 70%

Fig.9 Comparison of water saturation distribution at simulation end, perturbing facies only

34

HISTORY MATCHING RESULT(W.C. & BHSP Matching by Optimizing Facies Only)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY INITIAL

ITR 1 ITR 2

ITR 3 ITR 4

ITR 5 ITR 6

ITR 7

Surface Oil Rate Control

SIMULATED BHSP(W.C. & BHSP Matching by Optimizing Facies Only)

1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

HISTORY INITIALITR 1 ITR 2ITR 3 ITR 4ITR 5 ITR 6ITR 7


Fig.10 Result of water cut & BHSP matching, perturbing facies only

35

Fig.11 Perturbation of facies realization during water cut & BHSP matching, perturbing facies only (1/2)

Iteration #1


Iteration #5Iteration #4

36

Fig.11 Perturbation of facies realization during water cut & BHSP matching, perturbing facies only (2/2)


37

REFERENCE MODEL

REALIZATION #7(FACIES OPTIMIZED)

INITIAL MODEL

Sw @ 360 DAYS15% 70%

Fig.12 Comparison of water saturation distribution at simulation end, perturbing facies only

38

REFERENCE MODEL

REALIZATION #7(FACIES OPTIMIZED)

INITIAL MODEL

Pressure @ 360 DAYS1500 psia 3500 psia

Fig.13 Comparison of pressure distribution at simulation end, perturbing facies only

39

HISTORY MATCHING RESULT(W.C. & BHSP Matching by Optimizing Facies & Perm.)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY

INITIAL

ITR 1 (Facies)

ITR 1 (sand ave. logK)

ITR 1 (sand std. logK)

ITR 1 (shale ave. logK)

ITR 1 (shale std. logK)

ITR 2(sand ave. logK)


ITR 4 (Facies)



SIMULATED BHSP(W.C. & BHSP Matching by Optimizing & Perm.)

1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

HISTORYINITIALITR 1 (Facies)ITR 1 (sand ave. logK)ITR 1 (sand std. logK)ITR 1 (shale ave. logK)ITR 1 (shale std. logK)ITR 2(sand ave. logK)ITR 2 (shale ave. logK)ITR 4 (Facies)ITR 5 (shale ave. logK)


Fig.14 Result of history matching, Case 1

40

Fig.15 Perturbation of facies and permeability realization during history matching, Case 1 (1/2)

41


42

Sw @ 360 DAYS15% 70%

INITIAL MODEL

Fig.16 Water saturation distribution at simulation end, Case 1 (1/2)

43

Sw @ 360 DAYS15% 70%

REFERENCE MODEL


44

INITIAL MODEL


Fig.17 Pressure distribution at simulation end, Case 1 (1/2)

45

REFERENCE MODEL



46

Fig.18 Histogram of permeability, History matched model vs. Reference model, Case1

47


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY

INITIAL

ITR 1 (Facies)


ITR 2 (Facies)




1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

HISTORY

INITIAL

ITR 1 (Facies)


ITR 2 (Facies)




48

Fig.20 Perturbation of facies and permeability realization during history matching, Case 2

49

REFERENCE MODEL

INITIAL MODEL

Sw @ 360 DAYS15% 70%

Fig.21 Water saturation distribution at simulation end, Case 2

50


INITIAL MODEL

REFERENCE MODEL

Fig.22 Pressure distribution at simulation end, Case 2

51


52


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY

INITIAL

ITR 1 (Facies)



ITR 2 (Facies)


ITR 3 (Facies)




1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

HISTORY

INITIAL

ITR 1 (Facies)



ITR 2 (Facies)


ITR 3 (Facies)




53


54


55

Sw @ 360 DAYS15% 70%

INITIAL MODEL


56

Sw @ 360 DAYS15% 70%

REFERENCE MODEL


57

INITIAL MODEL



58

REFERENCE MODEL



59


60


0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 60 120 180 240 300 360

TIME (DAYS)

WA

TE

R C

UT

(F

RA

C.)

HISTORY

INITIAL

ITR 1 (Facies)

ITR 1 (sand ave. K)

ITR 2 (Facies)

ITR 2 (sand ave. K)

ITR 2 (shale ave. K)



1000

1500

2000

2500

3000

3500

0 60 120 180 240 300 360

TIME (DAYS)

BH

SP

(P

SIA

)

HISTORY

INITIAL

ITR 1 (Facies)

ITR 1 (sand ave. K)

ITR 2 (Facies)

ITR 2 (sand ave. K)

ITR 2 (shale ave. K)



61

Fig.30 Perturbation of facies and permeability realization during history matching, Case 4

62

Sw @ 360 DAYS15% 70%

INITIAL MODEL

Fig.31 Water saturation distribution at simulation end, Case 4

63

INITIAL MODEL


Fig.32 Pressure distribution at simulation end, Case 4

64


65

Fig.34 Comparison of history matched permeability realizations, Original vs. reversed order, Case 2

Case 2 : Original Order Case 2 : Reversed Order

1. Perturb facies2. Perturb µlogk of sand3. Iterate 1 & 2

1. Perturb µlogk of sand2. Perturb facies3. Iterate 1 & 2

66Fig.35 Comparison of history matched permeability realizations, Original vs. reversed order, Case 3

Case 3 : Original Order Case 3 : Reversed Order

1. Perturb facies2. Perturb µlogk of sand3. Perturb µlogk of shale4. Iterate 1 - 3

1. Perturb facies2. Perturb µlogk of shale3. Perturb µlogk of sand4. Iterate 1 - 3

67

ChannelMud

Fig.37 3D view of facies realization, Reference model A

XY

Z

X

Y

Fig.36 Horizontal slice of training image used for facies simulation, Reference model A

* The labels of X, Y& Z coordinates showthe number of gridblocks.

68

X

Y

Fig.38 Horizontal slice of permeability realization, Reference model A

* The labels of X, Y & Z coordinates show the number of grid blocks.

69

P3

I2

P1I6

I7P4

P5

P9

P8

Producers (6 wells):

Injectors (3 wells):

P1, P3, P4, P5, P8, P9

I2, I6, I7

Fig.39 Well location, Reference models A & B

X

Y

70Fig.40 Horizontal slice of facies realization, Reference model B

Layer 1

Layer 3

Layer 2ChannelSand

Mud

* The labels of X, Y& Z coordinatesshow the number ofgrid blocks.

71

Fig.41 Horizontal slice of permeability realization, Reference model B

X

Y


72

Fig.42 Horizontal slice of porosity realization, Reference model B

X

Y


73

X

Y

Fig.43 Horizontal slice of permeability realization, Initial model


74

ReferenceModel A

Mud Channel

InitialModel

Mud Channel

AVE. STD.LOG(K) 0.8 0.3K 8.8 8.0

AVE. STD.LOG(K) 2.7 0.2K 485 213

AVE. STD.LOG(K) 0.1 0.3K 1.5 1.0

AVE. STD.LOG(K) 1.5 0.2K 34 14

Fig.44 Histogram of permeability, Initial model vs. Reference models A & B

ReferenceModel B

Mud Channel

AVE. STD.LOG(K) -1.6 0.5K 0.07 0.1

AVE. STD.LOG(K) 2.6 0.3K 448 214

75

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

Time (Days)

Wat

er C

ut

(fra

c.)

P1P3P4P5P8P9P1P3P4P5P8P9

Well

Reference

Initial

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000

Time (Days)

BH

SP

(p

sia)


Fig.46 Comparison of simulated BHSP, Reference model A vs. initial model

Fig.45 Comparison of simulated water cut, Reference model A vs. initial model

Well

Reference

Initial

76

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

Time (Days)

Wat

er C

ut

(fra

c.)


0

1000

2000

3000

4000

5000

6000

0 500 1000 1500 2000 2500 3000 3500

Time (Days)

BH

SP

(p

sia)


Fig.47 Comparison of simulated water cut, Reference model B vs. initial model

Fig.48 Comparison of simulated BHSP, Reference model B vs. initial model

Reference

Initial

Reference

Initial

Well

Well

77

2.42.2 2.6 2.8 3.0 3.2

0.2

0.6

1.0

1.4

1.8

Objective Function Contour : Fractional Flow

Mean Log(K) : Channel

Mean Log(K) Mud

Reference Model

2.42.2 2.6 2.8 3.0 3.2

0.2

0.6

1.0

1.4

1.8

Objective Function Contour : BHSP

Mean Log(K) : Channel

Mean Log(K)Mud

Reference Model

Fig.49 Sensitivity of objective function to matching parameters

78

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

Time (Days)

Wat

er C

ut

(fra

c.)


0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000

Time (Days)

BH

SP

(p

sia)


Well

Reference

History Matched

Well

Reference

History Matched

Fig.50 Result of history matching, Case A

79

X

Y

Fig.51 Horizontal slice of permeability realization, History matched model, Case A


80

Reference Model A

Mud Channel

History Matched Model

Mud Channel

AVE. STD.LOG(K) 0.8 0.3K 8.8 8.0

AVE. STD.LOG(K) 2.7 0.2K 485 213

AVE. STD.LOG(K) 0.6 0.3K 4.6 3.1

AVE. STD.LOG(K) 2.7 0.2K 537 212

Fig.52 Histogram of permeability, Reference model vs. history matched model, Case A

81

Iteration #1: Facies Perturbation

100

1000

10000

100000

0 1 2 3 4 5 6 7 8# of iteration

Ob

ject

ive

Func

tio

n

objective function contribution of W.C.contribution of BHSP

Iteration #1: Perm Perturbation

100

1000

10000

100000

0 1 2 3 4 5 6 7 8# of iteration

Obj

ecti

ve F

unct

ion

objective function contribution of W.C.

contribution of BHSP


0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8# of iteration

Par

amte

r

rD


0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8# of iteration

Par

amet

er

logk1_ave logk0_ave

Fig.53 Optimization performance during history matching process, Case A

82

Fig.54 Result of history matching, Case B

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

Time (Days)

Wat

er C

ut

(fra

c.)


0

200

400

600

800

1000

1200

0 500 1000 1500 2000 2500 3000 3500

Time (Days)

BH

SP

(p

sia)


Reference

History Matched

Reference

History Matched

Well

Well

83

Fig.55 Horizontal slice of permeability realization, History matched model, Case B

X

Y


84

Fig.56 Histogram of permeability, Reference model vs. history matched model, Case B

Reference Model B

Mud Channel

History Matched Model

Mud Channel

AVE. STD.LOG(K) -1.6 0.5K 0.07 0.1

AVE. STD.LOG(K) 2.6 0.3K 448 214

AVE. STD.LOG(K) 0.5 0.3K 4.7 3.2

AVE. STD.LOG(K) 2.5 0.2K 411 221

85

Fig.57 Optimization performance during history matching process, Case B (1/2)


10000

100000

1000000

0 1 2 3 4 5 6 7 8 9 10# of iteration

Obj

ecti

ve F

unct

ion



0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8 9 10 11 12# of iteration

Par

amte

r

rD


0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8 9 10# of iteration

Par

amte

r

rD


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8 9# of iteration

Par

amet

er

logk1_ave logk0_ave


10000

100000

1000000

0 1 2 3 4 5 6 7 8 9 10 11 12# of iteration

Obj

ecti

ve F

unct

ion



100

1000

10000

100000

0 1 2 3 4 5 6 7 8 9# of iteration

Obj

ecti

ve F

unct

ion


86

Fig.57 Optimization performance during history matching process, Case B (2/2)


10

100

1000

10000

0 1 2 3 4 5 6 7 8 9# of iteration

Obj

ecti

ve F

unct

ion



-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8 9# of iteration

Par

amet

er

logk1_ave logk0_ave


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1 2 3 4 5 6 7 8 9 10 11# of iteration

Par

amet

erlogk1_ave logk0_ave


100

1000

10000

100000

0 1 2 3 4 5 6 7 8 9 10 11# of iteration

Obj

ecti

ve F

unct

ion


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