Least Square Support Vector Machine An E

SPE-180202-MS
Least Square Support Vector Machine: An Emerging Tool for Data Analysis
Palash Panja, Manas Pathak, Raul Velasco, and Milind Deo, University of Utah

Copyright 2016, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Low Perm Symposium held in Denver, Colorado, USA, 5– 6 May 2016.
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Abstract
Development of high speed computing leads to major advancements in every field of science and
engineering. Artificial intelligence (AI) method is emerging as new modern technology applied to
machine learning, pattern recognition, processing and understanding data, robotics etc. Its application in
oil and gas industry is new despite of the fact that it has huge potential to explore the knowledge regarding
reservoir characterization, PVT properties estimation, maximize productions, locating sweet spot using
pattern recognition, optimum design of fracturing job, calculation of recoverable hydrocarbon, well
placement etc. The main objective of this study is to put AI such as LSSVM in perspective from reservoir
engineering and encourage engineers and researchers to consider it as a valuable alternative tool in the

petroleum industry. Factors most affecting the production from fractured low permeability reservoirs such
as reservoir permeability, gas relative permeability exponent, rock compressibility, initial gas oil ratio,
slope of gas oil ratio in PVT, initial pressure, flowing bottom hole pressure and fracture spacing, are
studied. A wide range of values of each parameter based on real field data from Eagle Ford, Bakken and
Niobrara in the USA are assigned. Two different kinds of mathematical surrogate models, polynomial
response surface method (RSM) and least square support vector machine (LSSVM) are compared to seek
the better surrogate models in terms of predictability. Data are generated from a generic reservoir model
using commercial simulator. Various models of recovery factors and gas oil ratio are developed for
different times (after 90 days, 1 year, 5 years, 10 years, 15 years and 20 years) and for a minimum
economic rate (5 STB/ day). Multivariate regression was used to obtain coefficients for the second-order
polynomial response surface models using 80% of the simulated results (144). The LSSVM models
coupled with radial basis kernel function (RBF) are trained with 60% data. 20% of data is used to tune
the regularization parameter and kernel parameter using genetic algorithm (GA) optimization routine. Rest
20% data is utilized for testing the models’ predictability for future performance. Goodness of fit is
statistically measured by calculating coefficient of determination (R2), normalized root mean square error
(NRMSE) and average absolute relative error (AARE). LSSVM exhibits good predictability to forecast
the production such as oil recovery, gas recovery as surrogate models. The developed models can be used
with high accuracy to forecast the production of oil from ultra-low permeability reservoirs. Quick
sensitivity analysis of oil recovery to any parameter used in this study can be performed. The models are
also useful for uncertainty analysis of productions.

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Introduction
In the next five years the United States is expected to become the largest producer of oil in large part due
to the production of liquids from tight oil reservoirs such as the Bakken or shales like the Eagle Ford. It
is well recognized that ultra-low permeability reservoirs behave differently even when compared to low
permeability reservoirs. Importance of the petrophysical parameters can be assessed using parameter
sensitivity studies. Clear insight into the geologic and operational factors controlling production of
oil/condensate and gas from these reservoirs was developed in this study using surrogate models.
The application of response surface method was started in the early nineties to find the optimal
production and uncertainty associated with it (Damsleth et al., 1992, Egeland et al., 1992, Aanonsen et
al., 1995). Effectiveness of RSM is thoroughly investigated by comparing various design of experiments
(DOE) methods and different RSM (Yeten et al.,2005) and it is proved that response surface built through
appropriate DOE is an efficient and fast proxy model for forecasting production performance and
analyzing uncertainties (Amorim and Schiozer,2012). Amplitude factor and phase factors are adopted to
separate out the highly non-linear effects from the remaining effects to forecast the oil rate and water cut
(Li and Firedmann,2005). Among the artificial intelligence (AI) models, Artificial Neuron Networks

(ANN) and Least Square Support Vector Machine (LSSVM) have become the most effective methods
used in oil and gas industry. The RSM has been used for various purposes, these include estimating initial
hydrocarbon uncertainty (Peng and Gupta,2003), finding an optimal scheme for well placements (Guyaguler and Horne,2001, Manceau et al.,2001, Manceau et al.,2002, Landa and Güyagüler,2003, Carreras
et al.,2006), uncertainty in production and recovery performance (Dejean and Blanc,1999, Chewaroungroaj et al.,2000, Corre et al.,2000, Venkataraman,2000, Manceau et al.,2001, Mohaghegh,2006), history
matching (Landa and Güyagüler,2003, Yang et al.,2007, Slotte and Smorgrav,2008), and optimizing
production to flow through nano-pores (Sarma et al.,2005, Xie et al.,2013). The pressure and production
are studied using field cases applying surrogate reservoir models which are based on pattern recognition
techniques (Mohaghegh et al.,2012). AI was applied in unconventional fractured shale reservoirs (Dahaghi et al.,2012, Xie et al.,2013) and tight reservoirs (Khosravi et al.,2011, Khosravi et al.,2012).
Previous studies have shown that aquifer strength, fracture permeability and block height have a great
impact on oil recovery (Khosravi et al.,2011) for low permeable fractured reservoirs. LSSVM approach
is adopted more frequently over ANN for various purposes such as porosity and permeability determination(Ammal F. Al-anazi,2010, Fatai Adesina Anifowose,2010, Fatai Adesina Anifowose,2011, Mohammad-Ali Ahmadi,2014), well placement(Mohammad-Ali Ahmadi,2015), water conning (Mohammad
Ali Ahmadi,2014, Mohammad-Ali Ahmadi,2015) in horizontal wells, oil flow rate(Reza Gholgheysari
Gorjaei,2015) and gas oil relative permeability(Ahmadi,2015). Support vector machines were also used
to determine PVT properties such as bubble point pressure and formation volume factors of oil(E.
El-Sebakhy,2007, Shahin Rafiee-Taghanaki,2013), constant volume depletion (CVD) behavior (Milad
Arabloo,2014), dew point pressure(Mohammad Ali Ahmadi,2014, Arash Rabiei,2015), condensate to gas
ratio (CGR) (Mohammad Ali Ahmadi,2014)of gas condensate system.
In this work, second order polynomial response surface model and LSSVM are compared for oil
production from a hydraulically fractured low permeability reservoir. Eight important input factors are
considered based on previous study by authors(Palash Panja, 2015). A wide range of values of each

parameter based on real field data from Eagle Ford, Bakken and Niobrara in the USA are assigned to study
the impacts on production performance. The methodology to create surrogate reservoir models of the
production outcomes (oil recovery, gas recovery and produced gas oil ratio) are discussed in this paper.

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Methodology
Reservoir Model
A reservoir model with one horizontal well and a single vertical hydraulic fracture, located in the
middle of the reservoir, was constructed for this study. The shape of the reservoir is taken as box shoe.
The fracture height is considered the same as reservoir height. The reservoir dimensions in Y and Z
directions are kept constant, and boundaries in the X-direction are varied. The matrix permeability, initial
reservoir pressure and formation compressibility are varied. Fracture permeability, fracture width, fracture
orientation, matrix porosity, initial hydrocarbon saturation and the reservoir temperature are kept constant.
Parameters used in simulations are given in Table 1.
Table 1—Simulation parameters used in the study.
Reservoir Top (ft.)
Reservoir Thickness (ft.)

Reservoir Width (ft.)
Fracture Width (ft.)
Fracture Height (ft.)
Fracture Half length (ft.)
Fracture Orientation
Reservoir Porosity (%)
Initial Water Saturation (%)
Number of grids
Minimum Size of grid (ft.)
Maximum size of grid(ft.)

12000
200
750
0.05
Reservoir Height
Reservoir half width
Parallel to YZ plane
5
16

Variable depending on Fracture spacing
0.05 (X-direction),3(Y-direction),3(Z-direction),
6.5(X-direction), 110 (Y-direction), 20 (Z-direction),

Input parameters
Selecting the important input parameters requires prior knowledge and experience of simulations for
low permeability reservoirs. A mechanistic study(Palash Panja,2015) was conducted to choose the most
significant petrophysical inputs and operating parameters. Eight factors, namely matrix permeability, gas
relative permeability exponent, rock compressibility, initial gas oil ratio, slope of gas oil ratio in PVT,
initial pressure, flowing bottom hole pressure and fracture spacing, are selected in this study. Depending
on the type of DOE, each parameter may have different levels of values. The range of each variable is
chosen based on observed data variability in specific fields, on completion data and on how each well
operated as shown in Table 2.
Table 2—Field data from various Shale plays (Chipman et. al, 2011)
System/Unit/Zone
Bakken
o Elm Coulee (Middle Bakken)
o Parshall-Sanish (Middle Bakken)
o Elkhorn Ranch (Upper Bakken Shale)
Eagle Ford

o Karnes Trough (oil)
Niobrara
o Hereford Field area
o Powder River Basin
o North Park Basin

havg
(ft.)

␾avg
(%)

Km Range
(md)

Km,avg
(md)

Sw
(%)

Pore pressure
(psi)

GOR (SCF/STB)

11
16
10

6
6
2

0.04 to 0.50
.002 - .05
0.001 to 0.02

0.04
0.02

0.01

15 to 25
23
25

4800
6350
6500

500
500
500

150

10

.0011 to .0015

0.001

30

6000

500 to 1000

300
400
500

8
8
8

0.1
0.1
0.1

45
45
45

3500
4500
4000

⬍2000
⬍2000
⬍2000

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SPE-180202-MS

In this study, only gas relative permeability exponent (ng) is varied, hence water-oil relative permeabilities are fixed in all simulations.
Experimental Design
Many different designs of experiment techniques such as full factorial designs, fractional factorial
designs, central composite design. The Box-Behnken (Box and Behnken,1960) DOE was determined to
be best suited for this study to minimize the number of simulations for second order response surface
model (RSM). According to the Box-Behnken technique, 114 simulations are designed for eight input
parameters. The three absolute values of each input parameter except matrix permeability are scaled to
⫺1, 0 and ⫹1 using a linear relationship. Logarithmic values of matrix permeability are also scaled using
the same linear relationship. Ranges of all input parameters are summarized in Table 3.
Table 3—Ranges of input parameters chosen for the study
Variable
1
2
3
4
5
6
7
8

Matrix Permeability, (nD)
Gas Rel. Permeability Exponent, ng
Rock Compressibility, 1/psi
dRs/dp, (SCF/STB)/psi
Initial Gas Oil Ratio, Rsi, SCF/STB
Initial Pressure, Pi, psi
BHP, psi
Fracture Spacing, ft,

Symbol

Minimum

Maximum

X1
X2
X3
X4
X5
X6
X7
X8

10
1
4⫻10-6
0.50
800
4000
500
60

5000
3
4⫻10-5
0.80
3000
6500
1500
300

In addition to 114 simulations, 30 simulations are designed from randomly chosen values of each input
parameters. All input files are created for simulation using the combinations of independent variables
prescribed by the Box-Behnken method and randomized procedure. Different combinations of input
factors make each input file unique. The total number of input files is determined by the number of
independent variables and by the design of the experiment method.
Surrogate Model
RSM and LSSVM are developed using total 144 simulation results after dividing the data randomly
into training (80%) and test data (20%). We used the minimum number of grid blocks necessary to obtain
converged results as we observed the sensitivity (Panja et al.,2013) of other parameters. All simulations
were conducted using IMEX, a Computer Modeling Group Black-oil simulator. Two types of outcomes
are investigated in this study, a time based model and a rate based model.
In the time based model, outcomes (oil recovery, gas recovery and GOR) are recorded after certain
times such as 90 days, 1 year, 5 years, 10 years, 15 years and 20 years.
● In the rate based model, outcomes are obtained when the oil rate falls to 5 STB per day.
Response Surface Model: 2nd order Polynomial
●

A second order model with first order variable interactions was chosen. The functional relationship of
output with input variables are defined as:
(1)

Where,
Xi’s : the independent inputs
n : Total numbers of the independent inputs, n⫽8 for this study

SPE-180202-MS

5

a0 : the intercept
ak’s and aij’s : the coefficients
僐 : Error term, Target variable (absolute value) in the multivariate method to minimize.
To avoid non-negative values of outcomes, regression was performed using logarithms of outcomes.
The entire procedure of generating surrogate models (RSM and LSSVM) and its applications are shown
in Figure 1.

Figure 1—Workflow of the methodology to generate surrogate models

Coefficients for all models of each outcome are produced using multivariate regressions by fitting the
data in the second order model as displayed in Eq. 1 using Matlab (MathWorks® Inc.) programs where
the residual term was minimized. Forty-five coefficients are obtained from each model. The model is then
validated with the test data set to verify the robustness of the model. The surrogate models are then used
for further study like forecasting and sensitivity analysis.
Least Square Support Vector Machines
The support vector machine (SVM), a part of machine learning field is a supervised learning technique.
SVM is widely applied in classification and regression analysis. Many advantages of LSSVM over SVM

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SPE-180202-MS

have encouraged researchers to apply this method to various fields. A brief description of theory of
LSSVM is provided in Appendix B. Similar to equation 1, the following relationship is used in LSSVM
(B.1)
The final form of LSSVM in linear relationship is given by equation B.8 as
(B.8)

K(x,xi) is called kernel function. Various kernel functions are given in the Appendix B. Selection of
kernel function needs prior knowledge. Function can also be selected based on trial-error method. Radial
Basis Function kernel is a good default kernel which demonstrates satisfactory results. Radial Kernel
function is used in this study.
(B.9.3)

Regularization parameter, ␥ and kernel parameter, ␣2 in the LSSVM method are determined through
optimization technique such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and
Simulated Annealing (SA) by minimizing the objective function. In this study, mean square error (MSE)
(Equation A.2) between experimental (simulated) values and modeled values from LSSVM is considered
as objective function with genetic algorithm (GA) optimization routine.
(A.2)
yexp and ymodel are the experimental and modeled values respectively.
The workflow for LSSVM is shown in Figure 1. LSSVM models are trained using Matlab toolbox (K.
De Brabanter,2011) with 80% of training data, while 20% training data is used for optimization. The
training data is split into two sets unlike RSM (where all training data are used for regression) because
the predictability of the models is improved by avoiding overfitting of the models. The separate data set
(20%) is utilized to seek the optimum values of ␥ and ␣2. To check this hypothesis, initially all training
data were used to model LSSVM and to optimize ␥ and ␣2. The LSSVM models were almost perfectly
fitted with experimental data (R2⬎ 1) but the predictability with the test data was adversely affected (R2⬎
0.7). Splitting the training data into 80%-20% gave optimum predictability performance. Trained LSSVM
models (support values, i and bias term b) with optimized ␥ and ␣2 are used for forecast and sensitivity
analysis.

Results and Discussion
Oil recoveries, gas recoveries and gas oil ratio after 5 years, 10 years, 15 years, 20 years and when oil rate
reaches minimum economic rate of 5 STB/day are generated by simulations. These data are used to create
respective surrogate models, finally simulation and surrogate models (RSM and LSSVM) are cross plotted
for compared to see the fitness. Goodness of model fit is quantitatively measured by calculating
coefficient of determination, R2, Normalized Root Mean Square Error (NRMSE) and Average absolute
relative error (AARE) (see Appendix A).
Oil recoveries obtained from RSM and LSSVM are plotted in Figures 2a and b.

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Figure 2—Fitness of models of oil recovery from training data for (a) LSSVM and (b) RSM

Acceptance of surrogate models is self-explanatory as shown in Figures 2a and b; both RSM and
LSSVM models agree well with the simulation results. All recoveries fall well over the diagonal line.
There are few variations between the simulations and the model. The measurement of fitness in terms of
R2, NRMSE and AARE is provided in Table 4.

Table 4 —Fitness evaluation using statistical analysis of oil recovery models
Training Data: Oil Recovery
R2

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.975
0.990
0.994
0.996
0.992
0.987
0.931

0.978
0.987
0.992
0.995
0.991
0.987
0.941

2.8
1.9
1.4
1.3
1.9
2.7
7.1

2.7
2.2
1.6
1.4
2.0
2.7
6.8

0.073
0.048
0.035
0.027
0.032
0.036
0.127

0.070
0.053
0.038
0.031
0.031
0.033
0.117

The coefficient of determination, R2, values indicate a good correlation between surrogate models and
simulations. The coefficient of determination, R2, values are close to unity for time based models, and the
data for the rate based model is slight scatter around the diagonal line with reasonably good value of R2.
The predicted recoveries by model and the recoveries from simulation are in considerable agreement.
Deviation of models’ predictions are shown in Figure 3a & b in terms of relative error (see Appendix
A).

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Figure 3—Relative error distribution of oil recovery models using training data by (a) LSSVM (b) RSM

Larger deviations are observed at the lower recovery values i.e., recovery obtained at initial time such
as 90 days and 1 year. The economic rate of 5 STB/fracture/day also reaches very quickly even before first
year of production. Rate based models predicts results with maximum errors (approximately ⫹50% to
⫺80%). The errors in time based models are in the range of a30%. R2 and NRMSE for RSM and LSSVM
models of oil recovery are compared in Figure 4a and b respectively.

Figure 4 —Fitness evaluation of oil recovery models using (a) Co-efficient of determination, R2 and (b) NRMSE

It is evident that both RSM and LSSVM displayed a good fit with simulation results. The rate based
models have poor performance as shown in lower R2 values (~0.935) and higher NRMSE (~7%).
Testing of the Surrogate Model
Validation of model through predictability using blind data set (test set) is an important part of model
development. Surrogate models are generated based on the set of values of input factors as referred as
training data. Models should be tested with the other values of input factors within the range of study.

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9

Unsatisfactory fit of models will prescribe the modifications in original models. In this study, the test
(20% of all data) is kept aside to check model predictability. The modeled values are compared with
simulations results using test data to demonstrate the robustness of the surrogate models. Oil recovery, gas
recovery and gas oil ratio for time based models and rate based model are compared in Figures 5a and
b.

Figure 5—Predictability of models of oil recovery from test data for (a) LSSVM and (b) RSM

The overall fit of the models is satisfactory as indicated by R2 and NRMSE values (Table 5).
Models of long term recoveries (after 5 years) are more accurate (R2⬎0.97 and NRMSE⬍7%) than
the short term recovery models and rate based models. The relative error in oil recoveries are shown
in Figure 6a and b.

Table 5—Predictability evaluation using statistical analysis of gas oil ratio models
Test Data: Oil Recovery
2

R

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.918
0.960
0.976
0.984
0.980
0.973
0.935

0.955
0.971
0.976
0.983
0.982
0.978
0.931

7.8
5.5
4.0
3.1
3.4
4.0
6.5

5.8
4.7
3.9
3.2
3.3
3.6
6.4

0.12
0.09
0.08
0.06
0.05
0.06
0.24

0.09
0.08
0.07
0.06
0.05
0.05
0.23

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SPE-180202-MS

Figure 6 —Relative error distribution of oil recovery models using training data by (a) LSSVM (b) RSM

Predictability of the surrogate model is satisfactory except the rate based models where ⫺125 % to
⫹50 aberrations are observed in low oil recoveries. The quantitatively errors are given in Table 5.
Forecast and Sensitivity Analysis
Forecast and sensitivity analysis are two of many applications of surrogate models. These applications are
very rapid and easy to change the parameters. Any parameter or combination of parameters in the models
can be varied deterministically or probabilistically for sensitivity analysis. Forecast of recovery at any
time can be estimated from continuous recovery curves prepared by interpolating the available model data
(90 days to 20 years) for intermediate time periods.
Sensitivities of fracture spacing and initial gas oil ratio on recovery are studied here. All other
parameters except the parameter of interest are kept fixed as shown in Table 6.

Table 6 —Values of input parameters used in sensitivity studies
Variable

Sensitivity of Xf

Sensitivity of initial GOR

Matrix Permeability (nD)
Gas Rel. Permeability Exponent, ng
Rock Compressibility,1/psi
dRs/dp, (SCF/STB)/psi
Initial Gas Oil Ratio, Rsi, SCF/STB
Initial Pressure, Pi, psi
BHP, psi
Fracture Spacing, ft,

100
1.5
4* 10-5.5
0.55
1000
5000
500
60,120,180,240,300

100
1.5
4* 10-5.5
0.55
800,1200,1500,2000, 3000
6000
500
120

The values of input parameters are chosen within the range of study. The results of sensitivity analysis
are shown in Figure 8a and b.

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Figure 8 —Forecast of oil recovery and sensitivity of (a) Fracture Spacing and (b) Initial Gas Oil Ratio

No significant differences in oil recoveries predicted by LSSVM and RSM are observed. Only
marginal difference in oil recoveries from LSSVM and RSM is noticed for initial gas oil ratio of 3000
SCF/STB as shown in Figure 8b. Low fracture spacing improves oil recovery as shown in Figure 8a.
Major increase in recovery (approximately 8% at 20 years) is observed when fracture spacing is reduced
from 120 feet to 60 feet. In case of low fracture spacing, most of oil in the reservoir portion is exploited
and fracture interferes with the adjacent fractures. The stimulated volume is also reduced for low fracture
spacing reservoir hence smaller reservoir volume is used in recovery factor calculation. The change in oil
recovery is not significant (1-2% in 20 years) when fracture spacing increases beyond 180 feet. For
ultra-low permeable reservoirs, fracture spacing of 180 feet seems to be very high. Transient state flow
may persist for entire well life. Increasing fracture spacing increases the reservoir volume in calculation
of recovery factor thus, for same amount of oil recovered during moving boundary transient state flow;
recovery factor is less for higher fracture spacing. In Figure 6b, higher amount of oil is recovered with
higher initial gas oil ratio. Higher initial gas oil ratio provides energy in reservoir to sustain pressure for
long time. On the other hand, gas production is increased for higher initial gas oil ratio. Higher gas oil ratio
helps production of oil by improving the oil mobility with dissolved gas

Conclusion
Surrogate reservoir models can be used for a quick assessment of production performance from ultra-low
permeability reservoir like shales. Very few simulations are needed to generate the surrogate reservoir
models. The methodologies to create various surrogate models (RSM and LSSVM) are demonstrated.
Models are developed for different times (after 90 days, 1 year, 5 years, 10 years, 15 years and 20 years)
and for a minimum economic rate (5 STB/ day). Eight factors, namely, matrix permeability, gas relative
permeability exponent, rock compressibility, initial gas oil ratio, slope of gas oil ratio in PVT, initial
pressure, flowing bottom hole pressure and fracture spacing, are selected to study the impacts on oil
production from ultra-low permeable reservoirs. Both RSM and LSSVM exhibit good predictability to
forecast the production such as oil recovery, gas recovery as surrogate models. The accuracy of the models
is in the acceptance range. Thus the models for recoveries can be used confidently to predict the
production for any input parameters value within the given ranges of study. The surrogate models of
recoveries from ultra-low permeability reservoirs can be used confidently for any values of input factors
within the range of study. These models are also useful to study the risk associated with the production

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that will guide the development of a field. The support vector machine (SVM), a part of machine learning
field is a supervised learning technique. Many advantages of LSSVM over SVM have encouraged
researchers to apply this method to various fields. Least Square Support Vector Machine (LSSVM) is the
most effective method for predictability over other AI methods such as Artificial Neuron Networks
(ANN) because of its wide applicability to various aspects of reservoir engineering study.

Nomenclature
Symbol

Description

Units

a0
ak
aij
BHP
CGR
dRs/dp
GOR
km
n
ng
NRMSE
PDF
Pi
PVT
qo
Rsi
RMSE
R2
SSres
SStot
Xi
Ymodel,i
Yobs,i
Yobs,max
Yobs,min

The intercept of the surrogate model
Coefficient of independent input
Coefficient of 2nd Order Interaction of inputs
Bottom Hole Pressure
Condensate to Gas Ratio
Slope of gas/oil ratio in PVT
Gas/Oil Ratio
Matrix Permeability
Total numbers of independent inputs
Exponent of Relative Permeability Curve for Gas
Normalized Root Mean Square Error
Probability Density Function
Initial Reservoir Pressure
Pressure-Volume-Temperature
Oil Rate
Initial Gas/Oil Ratio
Root Mean Square Error
Coefficient of Determination
Residual Sum of Squares
Total Sum of Squares
Normalized Independent inputs
Modeled Value
Observed data
The Maximum value of Observed data
The Minimum value of Observed data
Mean of Observed Values

psi
STB/MMSCF
(SCF/STB)/psi
SCF/STB
nD
psi
STB/day
SCF/STB
Unit of Output
Unit of Output
Unit of Output
Unit of Output
Unit of Output
Unit of Output
Unit of Output
Unit of Output

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Appendix A
Goodness of Fit
There are many statistical tests to measure the accuracy or goodness of a modeled curve. R-squared (R2)
and normalized root mean squared error, relative error, and absolute average relative error are used in this
study to evaluate the regression models.

The coefficient of determination (R2):

The coefficient of determination, R2, indicates the overall accuracy of a regression. The R2, is defined

as:
(A1)
Where,
, the residual sum of squares
, the total sum of squares
, the mean of observed values
The values of R2 range from 0 to 1. The values closed to unity are interpreted as the better fit of the
model curve with observed data.

Normalized Root Mean Square Error (NRMSE):
(A.2)

The Root Mean Square Error (RMSE) (also known as the root mean square deviation, RMSD), is used
to measure the total residuals of modeled values and observed values.
The RMSE is defined as the square root of the mean squared error:
(A.3)
Where Yobs is observed values and Ymodel is modeled values.
Sometimes, it is difficult to analyze the error in terms of absolute values because different outcomes
vary in their absolute values, ranges and units. Non-dimensional forms of the RMSE are required to
compare RMSE for different units and outcomes. The RMSE is normalized by dividing with the range of
the observed data to get NRMSE (Normalized Root Mean Square Error) as follows:
(A.4)
Where,
Yobs,max is the maximum value of observed data.
Yobs,min is the minimum value of observed data.
The NRMSE can be expressed in term of percentage by multiplying by 100. The smaller percentage
values indicate the better fit of the model curve with observed data.

Average absolute relative error (AARE)
Relative error is measured for a particular point as
(A.5)

SPE-180202-MS

17

This the direct measurement of total relative errors. Absolute values of errors are generally used to
prevent the nullification of errors when adding positive and negative deviations. Another feature of AARE
is that it does not consider absolute value of errors but calculates error relative to the actual value. The
calculation method is shown in Equation A4
(A.6)

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SPE-180202-MS

Appendix B
Theory of Least Square Support Vector Machine
The LSSVM is developed by Suykens et al.(Johan A. K. Suykens,1999, Johan A. K. Suykens,2002) by
changing the inequality constraints in traditional support vector machines (SVM)(Corinna Cortes,1995).
The mathematical background is discussed in numerous places, a summary is given here. The regression
model from given training data (of size N) for LSSVM is given in equation B.1.

(B.1)
Where ␾(x) is the nonlinear mapping function and w and b are the weight vector and bias term,
respectively which are determined by linear regression formulated as equation B.2.

(B.2)
Subject to
(B.3)
where ␥ is the regularization parameter to prevent over fitting. The term ei is the model error.
To solve the restricted optimization problem given in equation B.2, first it is turned into unrestricted
by Lagrange multipliers (di), then optimal situation is found by partial derivatives based on the KarushKuhn-Tucker (KKT). Detailed calculations are given by Suyken et al.(Johan A. K. Suykens,1999, Johan
A. K. Suykens,2002, Amir Fayazi,2014, Mohammad Ali Ahmadi,2015).

(B.4)
Where,

Values of b and V are obtained after solving equation B.4

(B.5)

(B.6)
Final form of LSSVM i.e. equation B.1 after applying Mercer’s theorem is given as
(B.7)

where ⍀ij ⫽ ␸(xi)T ␸(xj) ⫽ K(xi, xj). Values of b and ␣ can be calculated by replacing ⍀ by K in
equation B.5 and B.6 respectively.

SPE-180202-MS

19

K(x,xi) is kernel function. Various kernel functions are discussed in next section. Linear relationship
given in equation B.4 can be described as
(B.8)

Kernel Function
Various kernel functions that satisfy Mercer’s condition are available such as linear, polynomial, radial
basis function (also known as Gaussian), exponential, Laplacian, hyperbolic tangent (also known as
sigmoid and Multilayer Perceptron (MLP)) etc.
(B.9.1)
(B.9.2)

(B.9.3)

(B.9.4)
Using RBF, equation 7 becomes
(B.10)

x and xi are vectors of size p (number of parameters)

Expanding equation 10
(B.11)

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SPE-180202-MS

Appendix C
Gas Recovery and Gas Oil Ratio

Figure C.1—Fitness of models of gas recovery from training data for (a) LSSVM and (b) RSM
Table C.1—Fitness evaluation using statistical analysis of gas recovery models
Training Data: Gas Recovery
R2

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.967
0.987
0.991
0.993
0.994
0.994
0.939

0.972
0.985
0.989
0.992
0.993
0.992
0.947

3.3
2.3
1.7
1.5
1.4
1.6
6.5

3.0
2.4
1.9
1.5
1.5
1.8
6.1

0.08
0.06
0.05
0.04
0.04
0.04
0.13

0.08
0.06
0.05
0.04
0.04
0.04
0.12

SPE-180202-MS

21

Figure C.2—Predictability of models of gas recovery from test data for (a) LSSVM and (b) RSM
Table C.2—Predictability evaluation using statistical analysis of gas recovery models
Test Data: Gas Recovery
R

2

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.883
0.929
0.956
0.961
0.968
0.969
0.928

0.939
0.955
0.960
0.964
0.969
0.969
0.924

8.8
7.7
5.8
5.3
4.8
4.7
7.1

6.4
6.1
5.5
5.1
4.7
4.7
7.2

0.13
0.11
0.10
0.09
0.08
0.08
0.24

0.10
0.09
0.09
0.08
0.08
0.07
0.24

Figure C.3—Fitness of models of gas oil ratio from training data for (a) LSSVM and (b) RSM

22

SPE-180202-MS

Table C.3—Fitness evaluation using statistical analysis of gas oil ratio models
Training Data: Gas Oil Ratio
R2

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.977
0.966
0.969
0.823
0.769
0.989
0.938

0.979
0.973
0.974
0.895
0.864
0.882
0.947

3.1
3.8
3.5
6.7
6.7
1.4
3.5

2.9
3.4
3.2
5.2
5.1
4.5
3.2

0.03
0.03
0.04
0.07
0.09
0.03
0.08

0.03
0.03
0.04
0.06
0.07
0.08
0.07

Figure C.3—Predictability of models of gas oil ratio from test data for (a) LSSVM and (b) RSM
Table C.4 —Predictability evaluation using statistical analysis of gas oil ratio models
Test Data: Gas Oil Ratio
2

R

NRMSE (%)

AARE

Models

LSSVM

RSM

LSSVM

RSM

LSSVM

RSM

90 days
1 year
5 years
10 years
15 years
20 years
Rate Based

0.959
0.974
0.890
0.872
0.843
0.621
0.880

0.961
0.943
0.863
0.819
0.746
0.694
0.860

4.9
4.2
9.1
10.0
10.7
16.9
9.5

4.8
6.2
10.2
12.0
13.6
15.2
10.2

0.05
0.05
0.07
0.09
0.10
0.14
0.09

0.05
0.06
0.07
0.11
0.13
0.14
0.10