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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Offshore Europe held in Aberdeen,

Scotland, U.K., 4-7 September 2007.

This paper was selected for presentation by an SPE Program Committee following review of

information contained in an abstract submitted by the author(s). Contents of the paper, as

presented, have not been reviewed by the Society of Petroleum Engineers and are subject to

correction by the author(s). The material, as presented, does not necessarily reflect any

position of the Society of Petroleum Engineers, its officers, or members. Papers presented at

SPE meetings are subject to publication review by Editorial Committees of the Society of

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for commercial purposes without the written consent of the Society of Petroleum Engineers is

prohibited. Permission to reproduce in print is restricted to an abstract of not more than

300 words; illustrations may not be copied. The abstract must contain conspicuous

acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.

Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

Determination of relative permeability data is required for

almost all calculations of fluid flow in petroleum reservoirs.

Water-oil relative permeability data play important roles in

characterizing the simultaneous two-phase flow in porous

rocks and predicting the performance of immiscible

displacement processes in oil reservoirs. They are used,

among other applications, for determining fluid distributions

and residual saturations, predicting future reservoir

performance, and estimating ultimate recovery. Undoubtedly,

these data are considered probably the most valuable

information required in reservoir simulation studies. Estimates

of relative permeability are generally obtained from laboratory

experiments with reservoir core samples. In the absence of the

laboratory measurement of relative permeability data,

empirical correlations are usually used to estimate relative

permeability data. Developing empirical correlations for

obtaining accurate estimates of relative permeability data

showed limited success, and proved difficult, especially for

carbonate reservoir rocks.

Artificial neural network (ANN) technology has proved

successful and useful in solving complex structured and

nonlinear problems. This paper presents a new modeling

technology to predict accurately water-oil relative

permeability using ANN. The ANN models of relative

permeability were developed using experimental data from

waterflood core tests samples collected from carbonate

reservoirs of giant Saudi Arabian oil fields. Three groups of

data sets were used for training, verification, and testing the

ANN models. Analysis of results of the testing data set show

excellent agreement with the experimental data of relative

permeability. In addition, error analyses show that the ANN

models developed in this study outperform all published

correlations.

The benefits of this work include meeting the increased

demand for conducting special core analysis, optimizing the

number of laboratory measurements, integrating into reservoir

simulation and reservoir management studies, and providing

significant cost savings on extensive lab work and substantial

required time.

Introduction

Artificial neural networks have seen an explosion of interest

over the past few years. They are powerful and useful tools for

solving practical problems in the petroleum industry

(Mohaghegh 2005; Al-Fattah and Startzman 2003).

Advantages of neural network techniques (Bishop 1995;

Fausett 1994; Haykin 1994; Patterson 1996) over conventional

techniques include the ability to address highly nonlinear

relationships, independence from assumptions about the

distribution of input or output variables, and the ability to

address either continuous or categorical data as either inputs or

outputs. In addition, neural networks are intuitively appealing

as they are based on crude low-level models of biological

systems. Neural networks, as in biological systems, simply

learn by examples. The neural network user provides

representative data and trains the neural networks to learn the

behavior of the data.

Design and Development of ANN Models

In regression problems, the objective is to estimate the value

of a continuous variable given the known input variables.

Regression problems can be solved using the following

network types: Multilayer Perceptrons (MLP), Radial Basis

Function (RBF), Generalized Regression Neural Network

(GRNN), and Linear. In this study, we experimented with the

first three types: MLP, RBF, and GRNN. The Linear model is

basically the conventional linear regression analysis. Since the

problem at hand is a regression type and because of its power

and advantages, we found GRNN performs the best for this

particular study. Hence, it is worth giving a brief description

of this neural network type. GRNN uses kernel-based

approximation to perform regression (Patterson 1996; Bishop

1995). It is one of the so called Bayesian networks. GRNN has

exactly four layers: the input layer, radial centers layer,

regression nodes layer, and an output layer as shown by Fig.

1. The input layer has equal number of nodes as input

variables. The radial layer nodes represent the centers of

clusters of known training data. This layer must be trained by

a clustering algorithm such as Sub-sampling, K-means, or

SPE 109018

Artificial-Intelligence Technology Predicts Relative Permeability of

Giant Carbonate Reservoirs

Saud M. Al-Fattah, SPE, and Hamad A. Al-Naim, SPE, Saudi Aramco

2 SPE 109018

Kohonen training. The regression layer, which contains linear

nodes, must have exactly one node more than the output layer.

There are two types of nodes: the first type of nodes calculates

the conditional regression for each output variable, whereas

the second type of nodes calculates the probability density.

The output layer performs a specialized function such that

each node simply divides the output of the associated first type

node by that of the second type node in the previous layer.

GRNNs can only be used for regression problems. A

GRNN trains almost instantly, but tends to be large and slow.

Although it is not necessary to have one radial neuron for each

training data point, the number still needs to be large. Like the

RBF network, the GRNN does not extrapolate.

GRNN has several advantages. It usually trains extremely

quickly, making the large number of evaluations required by

the input selection algorithm feasible; it is capable of

modeling nonlinear functions quite accurately; and it is

relatively sensitive to the inclusion of irrelevant input

variables. This is actually an advantage when trying to decide

whether input variables are required.

There are several important procedures that must be taken

into consideration during the design and development of an

ANN model. Fig. 2 is a flowchart illustrating the ANN

development strategies proposed and implemented in this

study.

Data Preparation

Data acquisition, preparation, and quality control are

considered the most important and most time consuming task,

Fig. 2. The number of data required for training a neural

network frequently presents difficulties. There are some

heuristic rules, which relate the number of data points needed

to the size of the network. The simplest of these indicates that

there should be ten times as many data points as connections

in the network. In fact, the number needed is also related to

the complexity of the underlying function which the network

is trying to model, and to the variance of the additive noise. As

the number of input variables increases, the number of input

data points required increases nonlinearly. Even a fairly small

number of input variables (perhaps fifty or less) require a huge

number of input data points. This problem is known as “the

curse of dimensionality.” If there is a larger, but still restricted,

data set, then it can be compensated to some extent by forming

an ensemble of networks, each network is trained using a

different resampling of the available data and then average

across the predictions of the networks in the ensemble.

Water-oil relative permeability measurements were

collected for all wells having special core analysis (SCAL) of

carbonate reservoirs in Saudi Arabian oil fields. These

reservoirs include Arab-D, Shuaibah, Arab-C, Arab-AB,

Fadhili, Upper Fadhili, Hanifa, and Hadriyah. The major fields

included in this study are the Ghawar field which is the largest

oil field in the world, Abqaiq, Shaybah, Qatif, Khurais, and

Berri. SCAL reports were thoroughly studied, and each

relative permeability curve was carefully screened, examined,

and checked for consistency and reliability. Hence, a large

database of water-oil relative permeability data for carbonate

reservoirs was created. All relative permeability experimental

data measurements were conducted using the unsteady state

method.

Developing ANN models for water-oil relative

permeability with easily obtainable input variables was one of

the objectives of this study. Initial water saturation, residual

oil saturation, porosity, well location and wettability are

considered the main input variables that significantly

contribute to the prediction of relative permeability data. We

made from these input variables several transformational

forms or functional links that are thought to play a role in

predicting the relative permeability. The initial water

saturation, residual oil saturation, and porosity of each well

can be easily obtained from either well logs or routine core

analysis. Wettability is an important input variable for

predicting the relative permeability data, thus to be included in

the pool of input variables. We found that not all wells with

relative permeability measurements have wettability data. For

those wells missing wettability data, we used Craig’s rule

(Craig 1971) to determine the wettability of each relative

permeability curve which is classified as oil-wet, water-wet, or

mixed wettability. It should be noted that Craig’s rule helps to

distinguish between strongly water wet and oil wet systems

based on relative permeability curves. If no information is

available on the wettability of a well, then it can be estimated

using offset wells data or sensitivity analysis can be

performed. The output of each network in this study is a single

variable either water or oil relative permeability.

Due to the variety of reservoir characteristics and using

data statistics, the database was divided into three categories

of reservoirs: Arab-D reservoir, Shuaibah reservoir, and all

other reservoirs having limited data. This necessitates the

development of six ANN models for predicting water and oil

relative permeability, resulting in two ANN models for each

reservoir category. The database of relative permeability that

is used in this study constitutes of a total of 3711 records or

cases. Table 1 presents the distribution of these data cases in

the three categories of reservoirs (Arab-D, Shuaibah, and the

Others).

Data Preprocessing

Data preprocessing is an important procedure in the

development of ANN models. All input and output variables

must be converted into numerical values to be introduced to

the network. Nominal values require special handling. Since

the wettability is a nominal input variable, it is converted into

a set of numerical values. That is oil-wet was represented as

{1,0,0}, mixed-wet as {0,1,0}, and water-wet as {0,0,1}. In

this study we applied two normalization algorithms:

mean/standard deviation, and minimax to ensure that the

network’s input and output will be in a sensible range (Al-

Fattah and Startzman 2003). The simplest normalization

function is the minimax, which finds the minimum and

maximum values of a variable in the data, and performs a

linear transformation using a shift and a scale factor to convert

the values into the target range, which is typically [0.0, 1.0].

After network execution, de-normalizing of the output follows

the reverse procedure: subtraction of the shift factor, followed

by division by the scale factor. The mean/standard deviation

technique is defined as the data mean subtracted from the

input variable value divided by the standard deviation. Both

methods have advantages in that they process the input and

SPE 109018 3

output variables without any loss of information and their

transform is mathematically reversible.

Input Selection and Dimensionality Reduction

One of the most difficult tasks in the design of the neural

network is the decision of which of the available variables to

use as inputs to the neural network. The only guaranteed

method to select the best input set is to train networks with all

possible input sets and all possible architectures, and to select

the best. Practically, this is impossible for any significant

number of candidate input variables. The problem is further

complicated when there are interdependencies or correlations

between some of the input variables, which means that any of

a number of subsets might be adequate.

To some extent, some neural network architectures can

actually learn to ignore useless variables. Other architectures

are adversely affected, and in all cases a larger number of

inputs imply that a larger number of training cases are

required to prevent over-learning. As a consequence, the

performance of a network can be improved by reducing the

number of input variables, even sometimes at the cost of

losing some input information. There are highly sophisticated

algorithms that determine the selection of input variables. The

following describes the input selection and dimensionality

reduction techniques that are used in this study. Table 2 lists

and defines all input variables that have been used in this

study.

Genetic Algorithm

A genetic algorithm is an optimization algorithm that can

search efficiently for binary strings by processing an initially

random population of strings using artificial mutation,

crossover and selection operators, in an analogy with the

process of natural selection (Goldberg 1989). It is applied in

this study to determine an optimal set of input variables that

contribute significantly to the performance of the neural

network. The method is used as part of the model building

process, where variables identified as the most relevant are

then used in a traditional model building stage of the analysis.

Genetic algorithm is a particularly effective technique for

combinatorial problems of this type, where a set of interrelated

yes/no decisions need to be made. For this study, it is used to

determine whether the input variable under evaluation is

significantly important or not. The genetic algorithm is

therefore a good alternative where there are large numbers of

variables (more than fifty, say), and also provides a valuable

second opinion for smaller numbers of variables. It is

particularly good at spotting interdependencies between

variables located close together on the masking strings.

Genetic algorithm can sometimes identify subsets of inputs

that are not discovered by other techniques. However, the

method is time consuming; it typically requires building and

testing many thousands of networks resulting in running the

program for a couple of days.

Forward and Backward Stepwise Algorithms

These algorithms (Hill and Lewicki 2006) are usually quicker

than the genetic algorithm if there are a reasonably small

number of variables. They are also equally effective if there

are not too many complex interdependencies between

variables. Forward and backward stepwise input selection

algorithms work by adding or removing variables one at a

time. Forward selection begins by locating the single input

variable that, on its own, best predicts the output variable. It

then checks for a second variable that when added to the first

it most improves the model, repeating this process until either

all variables have been selected, or no further improvement is

made. Backward stepwise feature selection is the reverse

process; it starts with a model including all variables, and then

removes them one at a time, at each stage finding the variable

that, when it is removed, least degrades the model.

Forward and backward selection methods each have their

advantages and disadvantages. Forward selection method is

generally faster. It may miss key variables if they are

interdependent or correlated. Backward selection method does

not suffer from this problem, but as it starts with the whole set

of variables, the initial evaluations are most time consuming.

Furthermore, the model can actually suffer purely from the

number of variables, making it difficult for the algorithm to

behave sensibly if there are a large number of variables,

especially if there are only a few weakly predictive ones in the

set. In contrast, because it selects only a few variables initially,

forward selection can succeed in this situation. Forward

selection is also much faster if there are few relevant variables,

as it will locate them at the beginning of its search, whereas

backwards selection will not whittle away the irrelevant ones

until the very end of its search.

In general, backward selection is to be preferred if there

are a small number of variables (say, twenty or less), and

forward selection may be better for larger numbers. All of the

above mentioned input selection algorithms evaluate feature

selection masks. These are used to select the input variables

for a new training set, and a GRNN is tested on this training

set.

Sensitivity Analysis

It is performed on the inputs to a neural network to indicate

which input variables are considered most important by that

particular neural network. Sensitivity analysis can be used

purely for informative purposes, or to perform input pruning,

which is removing excess neurons from input or hidden layers.

In general, input variables are not independent. Sensitivity

analysis gauges variables according to the deterioration on

modeling performance that occurs if that variable is not

available to the model. However, the interdependence between

variables means that no scheme of single ratings per variable

can ever reflect the subtlety of the true situation. In addition,

there may be interdependent variables that are useful only if

included as a set. If the entire set is included in a model, they

can be accorded significant sensitivity, but this does not reveal

the interdependency. Worse, if only part of the interdependent

set is included, their sensitivity will be zero, as they carry no

discernable information. In summary, precautions should be

exercised when drawing conclusions about the importance of

variables as sensitivity analysis does not rate the usefulness of

variables in modeling in a reliable or absolute manner.

Nonetheless, in practice sensitivity analysis is extremely

useful. If a number of models are studied, it is often possible

to identify variables that are always of high sensitivity, others

that are always of low sensitivity, and ambiguous variables

4 SPE 109018

that change ratings and probably carry mutually redundant

information.

Another common approach to dimensionality reduction is

the principle component analysis (Bishop 1995) which can be

represented in a linear network. It can often extract a very

small number of components from quite high-dimensional

original data and still retain the important structure.

Training, Verifying, and Testing

By exposing the network repeatedly to input data, the weights

and thresholds of the post-synaptic potential function are

adjusted using special training algorithms until the network

performs very well in correctly predicting the output. In this

study, the data are divided into three subsets: training set (50-

60% of data), verification or validation set (20-25% of data),

and testing test (20-25% of data), as presented in Table 1.

Typically, the training data subset is presented to the network

in several or even hundreds of iterations. Each presentation of

the training data to the network for adjustment of weights and

thresholds is referred to as an epoch. The procedure continues

until the overall error function has been sufficiently

minimized. The overall error is also computed for the second

subset of the data which is sometimes referred to as the

verification or validation data. The verification data acts as a

watchdog and takes no part in the adjustment of weights and

thresholds during training, but the networks’ performance is

continually checked against this subset as training continues.

The training is stopped when the error for the verification data

stops decreasing or starts to increase. Use of the verification

subset of data is important, because with unlimited training,

the neural network usually starts “overlearning” the training

data. Given no restrictions on training, a neural network may

describe the training data almost perfectly but may generalize

very poorly to new data. The use of verification subset to stop

training at a point when generalization potential is best is a

critical consideration in training neural networks. A third

subset of testing data is used to serve as an additional

independent check on the generalization capabilities of the

neural network, and as a blind test of the performance and

accuracy of the network. Several neural network architectures

and training algorithms have been attempted to achieve the

best results. The results were obtained using a hybrid approach

of genetic algorithms and neural network.

Results

All the six networks developed in this study were successfully

well trained, verified and checked for generalization. An

important measure of the network performance is the plot of

the root-mean-square error versus the number of iterations or

epochs. A well-trained network is characterized by decreasing

errors for both the training, and verification data sets as the

number of iterations increases (Al-Fattah and Startzman

2003). Statistical analysis used in this study to examine the

performance of a network are the output data standard

deviation, output error mean, output error standard deviation,

output absolute error mean, standard deviation ratio, and

Pearson-R correlation coefficient (Hill and Lewicki 2006).

The most significant parameter is the standard deviation (SD)

ratio that measures the performance of the neural network. It is

the best indicator of the goodness of a regression model and it

is defined as the ratio of the prediction error SD to the data

SD. One minus this regression ratio is sometimes referred to

as the explained variance of the model. The degree of

predictive accuracy needed varies from application to

application. Generally, an SD ratio of 0.3 or lower indicates a

very good regression performance network. Another important

parameter is the standard Pearson-R correlation coefficient

between the network’s prediction and the observed values. A

perfect prediction will have a correlation coefficient of 1.0. In

this study, we used the network verification data subset to

judge and compare the performance of a network among other

competing networks.

Due to its large size of data (70% of the database), most of

the results presented in this paper belong to the ANN models

developed for the Arab-D reservoir. Tables 3 and 4 present

statistical analysis of the ANN models for determining oil and

water relative permeability, respectively, for the Arab-D

reservoir. Both tables show that the Arab-D reservoir ANN

models for predicting oil and water relative permeability

achieved excellent results of accuracy by having low values of

SD ratios that are lower than 0.3 for all data subsets including

training, verification, and testing data set. Tables 3 and 4 also

show that a correlation coefficient of 99% was achieved for all

data subsets of the Arab-D reservoir model, indicating the

high accuracy of the ANN models for predicting the oil and

water relative permeability data.

Figs. 3 to 7 show that the results of ANN models are in

excellent agreement with the experimental data of oil and

water relative permeability. Crossplots of measured versus

predicted data of oil and water relative permeability are

presented in Figs. 8 and 9, respectively. The majority of the

data fall close to the perfect 45o straight line, indicating the

high degree of accuracy of the ANN models. Figs. 10 and 11

are histograms of residual errors of oil and water relative

permeability ANN models for the Arab-D reservoir.

Sensitivity analysis was performed on all input variables to

identify significant variables that are influential on the

network’s performance. Wettability was not found an

important input parameter for determining oil relative

permeability for all ANN models. On the other hand,

wettability was found to be the most influential input

parameter for determining water relative permeability. Figure

12 presents the most influential input variables that

significantly play an important role on the network’s outcome

for determining water relative permeability. Figure 12 shows

that the network performance is significantly get deteriorated

when removing wettability from the network by having the

highest error. It also shows that wettability is ranked first as

the most influential parameter on the network performance. To

study the effect of wettability on the network predictions, we

removed the wettability from the input variables and run the

network. Statistical analysis of the network performance is

presented in Table 5. The result of accuracy was badly

deteriorated indicating the significance of the wettability on

determining water relative permeability. Without using the

wettability as input, the ANN model has a correlation

coefficient of 79% for the verification subset and 51% for the

testing subset. Also, the SD ratios for the verification and

testing subsets are very high 0.6 and 0.9, respectively.

SPE 109018 5

This study differs from others’ work (Slipngarmlers et al.

2002) in that it used a large database of relative permeability

for giant carbonate reservoirs, it used less input variables such

that the developed ANN models use mainly six input variables

that can be easily obtained without performing additional

sophisticated experiments, and it achieved higher degree of

accuracy and performance. In addition, for the development of

the ANN models this study implemented several input

selection techniques, and used three data subsets (training,

verification, and testing) making sure that the network trained

very well avoiding the overlearning problem. Slipngarmlers et

al. (2002) used only two data subsets (training and testing),

and did not use input selection methods.

Comparison of ANN against Correlations

The newly-developed ANN models for predicting water-oil

relative permeability of carbonate reservoirs were validated

using data that were not utilized in the training of the ANN

models. This step was performed to examine the applicability

of the ANN models and to evaluate their accuracy against

correlations previously published in the literature. The new

ANN models were compared with published correlations of

Wyllie (1950); Pirson (1958); Naar et al. (1962); Jones (1978);

Land (1968); and Honarpour et al. (1986, 1982). Fig. 13

shows the results of comparison of the ANN model against

published correlations for predicting oil relative permeability

for one of the oil wells in the carbonate reservoir. The results

of comparison showed that the ANN models reproduced more

accurately the experimental relative-permeability data than the

published correlations. Although Honarpour’s (1986)

correlation gives the closest results to the experimental data

among other correlations, it does not honor the oil relative

permeability data at the initial water saturation by yielding a

value greater than one.

Fig. 14 presents a comparison of results of ANN models

against the correlations for predicting water relative

permeability data for an oil well in the Ghawar field. The

results clearly show the excellent agreement of the ANN

model with the experimental data and the high degree of

accuracy achieved by the ANN model compared to all

published correlations considered in this study.

Conclusion

In this study, we developed new prediction models for

determining water-oil relative permeability using artificial

neural network modeling technology for giant and complex

carbonate reservoirs. The ANN models were developed using

a hybrid approach of genetic algorithms and artificial neural

networks. The models were successfully trained, verified, and

tested using the GRNN algorithm. To the author’s knowledge,

this is the first study that uses this type of network, GRNN, in

the application of relative permeability determination.

Variable selection and dimensionality reduction techniques, a

critical procedure in the design and development of ANN

models, were presented and applied in this study.

Analysis of results of the blind testing data set of all ANN

models shows excellent agreement with the experimental data

of relative permeability. Results showed that the ANN models

outperform all published empirical equations by achieving

excellent performance and a high degree of accuracy.

Acknowledgement

The author would like to thank Saudi Aramco management for

their permission to publish this paper. Special thanks to

Ahmed A. Al-Moosa, and Fawzi Al-Matar, Saudi Aramco, for

the great support received during the course of this project.

Thanks are extended to the Petrophysics Unit of Saudi

Aramco’s EXPEC Advanced Research Center for providing

the data used in this work.

References

Al-Fattah, S.M., and Startzman, R.A. 2003. Neural Network

Approach Predicts U.S. Natural Gas Production. SPEPF 18

(2): 84-91. SPE-82411-PA. DOI: 10.2118/82411-PA.

Bishop, C. 1995. Neural Networks for Pattern Recognition.

Oxford: University Press.

Craig, F.F. 1971. The Reservoir Engineering Aspects of

Waterflooding. Richardson, TX: SPE Press.

Fausett, L. 1994. Fundamentals of Neural Networks. New

York: Prentice-Hall.

Goldberg, D.E. 1989. Genetic Algorithms. Reading, MA:

Addison Wesley.

Haykin, S. 1994. Neural Networks: A Comprehensive

Foundation. New York: Macmillan Publishing.

Hill, T. and Lewicki, P. 2006. Statistics Methods and

Applications. Tulsa, OK: StatSoft.

Honarpour, M., Koederitz, L., and Harvey, A.H. 1986.

Relative Permeability of Petroleum Reservoirs. Boca Raton:

CRC Press Inc.

Honarpour, M., Koederitz, L., and Harvey, A.H. 1982.

Empirical Equations for Estimating Two-Phase Relative

Permeability in Consolidated Rock. JPT 34 (12): 2905-2908.

SPE-9966-PA. DOI: 10.2118/9966-PA.

Jones, S.C. and Roszelle, W.O. 1978. Graphical Techniques

for Determining Relative Permeability from Displacement

Experiments. JPT 30 (5): 807-817. SPE-6045-PA. DOI:

10.2118/6045-PA.

Land, C.S. 1968. Calculation of Imbibition Relative

Permeability for Two- and Three-Phase Flow from Rock

Properties. SPEJ 8 (5): 149-56. SPE-1942-PA. DOI:

10.2118/1942-PA.

Mohaghegh, S.D. 2005. Recent Developments in Application

of Artificial Intelligence in Petroleum Engineering. JPT 57

(4): 86-91. SPE-89033-MS. DOI: 10.2118/89033-MS.

Naar, J., Wygal, R.J., Henderson, J.H. 1962. Imbibition

Relative Permeability in Unconsolidated Porous Media. SPEJ

2 (1): 254-58. SPE-213-PA. DOI: 10.2118/213-PA.

6 SPE 109018

Patterson, D. 1996. Artificial Neural Networks. Singapore:

Prentice Hall.

Pirson, S.J. 1958. Oil Reservoir Engineering. New York:

McGraw-Hill Book Co. Inc.

Slipngarmlers, N., Guler, B., Ertekin, T., and Grader, A.S.

2002. Development and Testing of Two-Phase Relative

Permeability Predictors Using Artificial Neural Networks.

SPEJ 7 (3): 299-308. SPE-79547-PA. DOI: 10.2118/79547-

PA.

Wyllie, M.R.J. 1950. Interrelationship between Wetting and

Nonwetting Phase Relative Permeability. Trans., AIME 192:

381-82.

Profile : GRNN 24:24-485-2-1:1 , Index = 10

Train Perf. = 0.076232 , Select Perf. = 0.102937 , Test Perf. = 0.137917

Fig. 1-Design of a Generalized Regression Neural Network

used in this study.

TABLE 1- Distribution of data records/cases for relative

permeability data sets.

Reservoir/Data set

Training

Verification

Testing

Total

Arab-D

1591

457

401

2449

Shuaibah

485

107

84

676

Others

390

93

103

586

TABLE 3- Statistical analysis of ANN model for Kro Arab-D

reservoir.

Training

Verification

Testing

Data S.D.

0.198159

0.133331

0.214694

Error Mean

-4.47E-05

0.002488

-0.000804

Error S.D.

0.019920

0.014860

0.032760

Abs. E. Mean

0.004571

0.005582

0.009307

S.D. Ratio

0.100502

0.111487

0.152606

Correlation-R

0.994949

0.993845

0.988549

TABLE 4- Statistical analysis of ANN model for Krw Arab-D

reservoir.

Training

Verification

Testing

Data S.D.

0.236330

0.225397

0.230525

Error Mean

-0.001382

-0.000521

-0.003320

Error S.D.

0.065039

0.065243

0.070861

Abs. E. Mean

0.041148

0.042435

0.044872

S.D. Ratio

0.275205

0.289457

0.307389

Correlation-R

0.965349

0.960761

0.956011

Fig. 2- Flowchart of procedure of ANN design and

development proposed in this study.

Input Layer Radial Layer

Regression

Layer

Output Layer

SPE 109018 7

TABLE 2- List of all input variables that have been tried in

this study.

Variable

Definition

w

S

Water saturation

wi

S

Irreducible water saturation

or

S

Residual oil saturation

Por

Porosity

Wet

Wettability (oil-wet, water-wet, mix-wet)

Xutm

X-coordinate of the well location

Yutm

Y-coordinate of the well location

on

S

or

orwSSS

1

1

wn

S

wi

wiwSSS

1

1o

S

orwi

orwSS SS

1

1

1w

S

orwi

wiwSS SS

1

2w

S

or

w

S

S

1

3w

S

wi

w

S

S

1

4w

S

wi

w

S

S

1

1

5

S

2

1

orwi

orwSS SS

6

S

or

S1

7

S

or

S

e1

8

S

1w

S

e

9

S

wiwSS

10

S

wiwSS

11

S

wiwwi SSS 1

12

S

orwor SSS 11

13

S

orwi

orwSS SS

1

Por1

wior SS 1

Por2

1

Scatterplot (Prediction (20 ) (Train) (FinalRelPerm_ARBD_noendpt_May29_2007) 6v*2443c)

Well=HRDH91.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Kro

Experimental

ANN Model

Fig. 3-Results of ANN model compared with experimental

data for oil-relative permeability.

Scatterplot (Prediction (20 ) (Train) (FinalRelPerm_ARBD_noendpt_May29_2007) 6v*2443c)

Well=HRDH89.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Kro

Experimental

ANN Model

Fig. 4-Results of ANN model compared with experimental

data for oil-relative permeability.

Scatterplot (Prediction (1 ) (FinalRelPerm_OTHERS_noendpt_May15_07) 6v*586c)

Well=KRSN55.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Krw

Experimental

ANN Model

Fig. 5-Results of ANN model compared with experimental

data for water-relative permeability.

8 SPE 109018

Scatterplot (Prediction (1 ) (FinalRelPerm_ARBD_noendpt_Jan30_2007) 7v*2449c)

Well=UTMN628.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Krw

Experimental

ANN Model

Fig. 6-Results of ANN model compared with experimental

data for water-relative permeability of Well U-628.

Scatterplot (Prediction (4 ) (FinalRelPerm_SHUB_noendpt_May27_07) 6v*676c)

Well=SHYB50.4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Sw

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Krw

Experimental

ANN Model

Fig. 7-Results of ANN model compared with experimental

data for water-relative permeability of Well SB-50.

Kro NN Model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Measured Kro

Predicted Kro

Fig. 8- Crossplot of ANN predicted and measured Kro for

Arab-D reservoirs.

Crossplot of Predicted Krw vs. Measured Krw

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Measured Krw

Predicted Krw

Fig. 9- Crossplot of ANN predicted and measured Krw for

Arab-D reservoirs.

Histogram of kro, Residual (T.20)

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

kro, Residual

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

No cases

Fig. 10- Histogram of Kro residual error for the Arab-D

reservoir model.

Histogram of krw, Residual (14)

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

krw, Residual

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

No cases

Fig. 11- Histogram of Krw residual error for the Arab-D

reservoir model.

SPE 109018 9

00.5 11.5 22.5

Wet

YUTM

XUTM

S10

Sor

Swi

por

S8

por2

Swn

Error Ratio

Fig. 12- Error ratio and ranking of the influence of input

variables on water relative permeability ANN model for Arab-

D reservoir.

TABLE 5- Statistical analysis of ANN model for Krw Arab-D

reservoir without the Wettability.

Training

Verification

Testing

Data S.D.

0.206744

0.270242

0.265920

Error Mean

-0.000831

-0.012651

-0.099167

Error S.D.

0.058208

0.166682

0.231691

Abs. E. Mean

0.036350

0.141533

0.179689

S.D. Ratio

0.281546

0.616790

0.871281

Correlation-R

0.963740

0.787427

0.508382

UTMN 1051

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation, fraction

Oil Relative Permeability, fraction

Kro_Ex

Wyllie

Pirson

Naar

Jones

Land

Honarpour

ANN

Fig. 13- Comparison of ANN model and correlations for

predicting Kro for one well in the Ghawar field.

SDGM 222

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation, fraction

Water Relative Permeability, fraction

Experimental

Wyllie

Pirson

Honarpour

ANN Model

Fig. 14- Comparison of ANN model and correlations for

predicting Krw for one well in the Ghawar field.