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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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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.
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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.
