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0Accepted Manuscript
Application of artificial intelligence to forecast hydrocarbon production from shales Palash Panja, Raul Velasco, Manas Pathak, Milind Deo
PII: S2405-6561(17)30114-1
DOI: 10.1016/j.petlm.2017.11.003
Reference: PETLM 175
To appear in: Petroleum
Received Date: 15 June 2017 Revised Date: 22 September 2017 Accepted Date: 22 November 2017
Please cite this article as: P. Panja, R. Velasco, M. Pathak, M. Deo, Application of artificial intelligence to forecast hydrocarbon production from shales, Petroleum (2017), doi: 10.1016/j.petlm.2017.11.003.
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Application of Artificial Intelligence to Forecast Hydrocarbon Production from Shales
Palash Panja1 *, Raul Velasco1, Manas Pathak2, Milind Deo2
1Energy & Geoscience Institute
432 Wakara Way, Suite 300, Salt Lake City, UT 84108
2Department of Chemical Engineering, University of Utah 50 Central Campus Dr., Salt Lake City, UT 84112
*ppanja@egi.utah.edu Corresponding author
Abstract
Artificial intelligence (AI) methods and applications have recently gained a great deal of attention in many areas, including fields of mathematics, neuroscience, economics, engineering, linguistics, gaming, and many others. This is due to the surge of innovative and sophisticated AI techniques applications to highly complex problems as well as the powerful new developments in high speed computing. Various applications of AI in everyday life include machine learning, pattern recognition, robotics, data processing and analysis, etc. The oil and gas industry is not behind either, in fact, AI techniques have recently been applied to estimate PVT properties, optimize production, predict recoverable hydrocarbons, optimize well placement using pattern recognition, optimize hydraulic fracture design, and to aid in reservoir characterization efforts. In this study, three different AI models are trained and used to forecast hydrocarbon production from hydraulically fractured wells. Two vastly used artificial intelligence methods, namely the Least Square Support Vector Machine (LSSVM) and the Artificial Neural Networks (ANN), are compared to a traditional curve fitting method known as Response Surface Model (RSM) using second order polynomial equations to determine production from shales. The objective of this work is to further explore the potential of AI in the oil and gas industry. Eight parameters are considered as input factors to build the model: reservoir permeability, initial dissolved gas-oil
ratio, rock compressibility, gas relative permeability, slope of gas oil ratio, initial reservoir pressure, flowing bottom hole pressure, and hydraulic fracture spacing. The range of values used for these parameters resemble real field scenarios from prolific shale plays such as the Eagle Ford, Bakken, and the Niobrara in the United States. Production data consists of oil recovery factor and produced gas-oil ratio (GOR) generated from a generic hydraulically fractured reservoir model using a commercial simulator. The Box-Behnken experiment design was used to minimize the number of simulations for this study. Five time-based models (for production periods of 90 days, 1 year, 5 years, 10 years, and 15 years) and one rate-based model (when oil rate drops to 5 bbl/day/fracture) were considered. Particle Swarm Optimization (PSO) routine is used in all three surrogate models to obtain the associated model parameters. Models were trained using 80% of all data generated through simulation while 20% was used for testing of the models. All models were evaluated by measuring the goodness of fit through the coefficient of determination (R ) and the Normalized Root Mean Square Error (NRMSE). Results show that RSM and LSSVM have very accurate oil recovery forecasting capabilities while LSSVM shows the best performance for complex GOR behavior. Furthermore, all surrogate models are shown to serve as reliable proxy reservoir models useful for fast fluid recovery forecasts and sensitivity analyses.
Keywords: Surrogate models; LSSVM; ANN; Oil recovery; Artificial intelligence; Unconventional reservoirs 1. Introduction
Surrogate models are particularly useful for quick predictions given a range of input parameters. These models are used to forecast oil production and perform sensitivity and uncertainty analyses. Polynomial equations and other non-linear equations known as response surface models (RSM) have been popularized for their simple mathematical structure and for easier
implementation. Recently, artificial intelligence applications have gained the interest of engineers and scientists due to their unconventional ways of connecting input data to output. RSM coupled with a proper design of experiments [1] was proven to be an efficient and fast proxy model for forecasting production performance and analyzing uncertainties [2]. Oil rate and water cut results were also predicted using RSM [3]. Response surface models are widely applied to various aspects of reservoir engineering including estimating initial hydrocarbon uncertainty, [4] production uncertainty [5-10], finding an optimal scheme for well placement [7, 11-14], history matching [13, 15, 16], and determining the dew point of water in natural gas processing unit [17]. Field cases have been studied using pattern recognition techniques [18] to determine pressure and production variation according to well locations.
Even though researchers have developed numerical, analytical, and semi-analytical techniques to understand the physics underlying the production from hydraulically fractured tight formations [19-22], many of these systems grow ¿in complexity rendering most of these methods inapplicable. The AI approach on the other hand is very useful when dealing with highly complex systems. At the cost of understanding the physical mechanisms taking place in tight formations, AI helps us analyze and forecast hydrocarbon production and assess performance. In this study, two of the most common AI techniques namely, ANN and LSSVM as well a second order polynomial RSM are used to predict oil and gas-oil ratio production from hydraulically fractured low permeability reservoir. The comparison of these three methods in terms of performance and accuracy is also discussed. The application of ANN started before LSSVM in the early 90's, data from well tests were already being interpreted using ANN [23, 24]. Rock characteristics such as lithology were determined from well logs using fuzzy neural networks [25]. Reservoir heterogeneity with respect to porosity, permeability, and oil saturations
were characterized from geophysical well logs such as gamma ray, bulk density, deep induction, etc. using ANN [26]. Thermodynamic properties from reservoir fluids such as bubble point pressures and formation volume factors at the bubble point have been predicted from four inputs: solution GOR, reservoir temperature, oil gravity, and gas density using ANN, SVM and nonlinear regression [27]. Similarly, crude oil viscosity and solution GOR as functions of pressure have been determined from 12 variables including compositions of oil, bubble point pressure, bubble point viscosity [28], etc. using ANN. Calculations of gas condensate dew point pressures were also made using gas composition, temperature, and heavy fraction properties [29-31] and condensate to gas ratio [32]. Results predicted by ANN for asphaltene precipitation [33] showed promising results compared to experimental studies [34]. Oil rates have also been measured in the pipe line using ANN for varying pressures and temperatures [35]. Various applications of LSSVM include porosity and permeability determination [36-39], water conning in horizontal wells [40, 41], well placement [40], gas-oil relative permeability curves [42], phase equilibrium calculations of hydrates [43], oil flow rate predictions [44], and temperature-pressure relationship in natural gas production and processing [45]. Wide applications of artificial intelligence in improved oil recovery were recently described by researchers [46-49]. Other applications include the description of CO2 solubility [50] and calcium carbonate [51] in brine sequestration processes.
Eight important parameters are considered as input data that include geological parameters (initial reservoir pressure, rock compressibility, and permeability), operational parameter (bottom hole pressure), completion parameters (fracture spacing), rock-fluid properties (Corey gas relative permeability exponent), and fluid properties (initial solution gas-oil ratio and the linear slope of solution gas-oil ratio versus pressure) which are selected from a previous study [52]
where a mechanistic study revealed these parameters to be highly significant. Six models (5 time-based models for production after 90 days, 1 year, 5 years, 10 years, and 15 years and one rate-based model when oil rate drops to 5 bbl/day/fracture) of oil recovery and produced GOR are developed for each surrogate model (RSM, ANN and LSSVM). Data is generated from a generic reservoir model with one vertical hydraulic fracture placed in the middle of the reservoir using a commercial reservoir simulator. The mathematical formulations and workflow to create these surrogate models are discussed in this article. The results obtained for all models are compared using error analyses in terms of coefficient of determination (R2) and normalized root mean square error (NRMSE). 2. Reservoir Model
Unconventional reservoirs such as shales and other tight formations are very complex in terms of possible natural fracture presence and heterogeneity. However, it is possible to build a homogeneous reservoir model using average properties if the variation is not very large. Typically, wells are drilled vertically and then directed horizontally for 1 to 2 miles, where as many as 100 vertical hydraulic fractures are induced to generate high conductive flow paths to the wellbore. Simulating an entire reservoir model that consists of 100 hydraulic fractures is very computationally expensive. Hence, only a small representative portion of the reservoir is simulated where production from a single vertical fracture is considered. The reservoir properties are assumed to be homogeneous as listed in the Table 1.
Table 1: Simulation parameters
Reservoir top depth (ft) Reservoir thickness (ft)
200 750 0.05 200
Reservoir width (ft) Fracture width (ft) Fracture height (ft)
Fracture half-length (ft) 375
Fracture orientation Parallel to YZ plane
Reservoir porosity (%) 5
Initial water saturation (%) 16
The number of unique input parameter combinations could lead to an enormous number of experiments or simulations. The Box-Behnken method [53] is chosen in this study to keep the required number of simulations to a minimum. This simulation design is also suitable for second order response surface models. Using the Box-Behnken design, 114 simulations are modeled for eight input parameters in three levels (minimum, medium and maximum) as shown in Table 2. The values of all input parameters are converted to a -1, 0, and 1 scale using a linear relationship, except for matrix permeability and rock compressibility (where logarithmic values are used instead).
Table 2: Input parameters and their values
Variable Symbol Minimum Medium Maximum
(-1) (0) (+1)
1 Matrix Permeability (nD) X1 10 225 5000
2 Gas Rel. Permeability Exponent, ng X2 1 2 3
3 Rock Compressibility (1/psi) X3 4x10-6 4x10-5 4x10-4
4 dRs/dp (SCF/STB/psi) X4 0.50 0.65 0.80
5 Initial Gas Oil Ratio, Rsi (scf/stb) X5 800 1900 3000
6 Initial Pressure, Pi (psi) X6 4000 5250 6500
7 BHP (psi) X7 500 1000 1500
8 Fracture Spacing (ft) X8 60 180 300
Apart from the 114 simulation results that were used to train the models, 30 additional simulations were ran to test the models. Therefore, the training data is comprised of approximately 80% of the total data set (114 out of 144) while the testing portion is comprised of approximately 20% (30 out of 144). The list of simulations used to train and test the models can be found in the Appendix (Tables A.3 and A.4). IMEXTM from the Computer Modeling Group,
Calgary, Canada was used to conduct all black-oil simulations. The minimum number of simulation grid blocks necessary to obtain accurate results and avoid convergence issues was used as prescribed by Panja et al. [54]. 3. Surrogate Model
As mentioned earlier, three types of surrogate models - a Response Surface Model (RSM), a Least Square Support Vector Machine (LSSVM) model, and an Artificial Neural Networks (ANN) model - were developed and compared in this study. Simulation results in terms of oil recovery and gas oil ratio (GOR) were obtained in two ways: by recording oil recovery and GOR after certain production times and when the oil rate dropped to 5 bbl/day/fracture. In other words, five time-based models and one rate-based model were developed as summarized below:
• Time based model: Models for oil recovery and GOR at 90 days, 1 year, 5 years, 10 years and 15 years.
• Rate based model: Models for oil recovery and GOR when oil rate drops to 5 bbl/day/fracture.
All unknown parameters in the surrogate models (RSM, LSSVM and ANN) are obtained using an optimization routine known as Particle Swarm Optimization (PSO) using Matlab (MathWorks® Inc.). The same optimization routine was used for all surrogate models to eliminate any performance bias. Sometimes, unacceptable physical values such as negative oil recovery factors or gas oil ratios are obtained using surrogate models. To avoid this pitfall, logarithms of the outcomes (recovery factors and gas-oil ratio) are used to build the models. A simplified schematic of methodology used to develop the surrogate model is shown in Figure 1.
Figure 1: Surrogate model development schematic All unknown parameters are listed in Table 3. These parameters are discussed in more detail in the upcoming sections.
Table 3: Number of parameters determined in each surrogate model
Method Number of parameters Optimized parameter
RSM 1 intercept 44 coefficients all
LSSVM 1 Bias term (b) 1 Regularization parameter (y) 1 Kernel parameter (a) 92 Support values (ai) Regularization parameter (y) Kernel parameter (a)
ANN 15 Biases ( 14 hidden layer +1 output layer) 126 Weights (8X14 for hidden layer+14 for output layer) All
The first two models (i.e. RSM and LSSVM) are discussed in detail in a previous article [55].
Therefore, these two models are intentionally discussed in brief here and the reader is referred to the referenced article for more details.
3.1. Response Surface Model (RSM)
The response surface model is the most common method used in many branches of engineering. Basically, an algebraic equation is fitted to develop a relationship between input and output data. During equation fitting with training data, the parameters (coefficients, intercepts etc.) are determined through an optimization routine to minimize error. A second order polynomial equation is chosen in this study. The equation is defined as:
For 8 input variables, there are 8 interaction coefficients, ak, 36 second order interaction coefficients, aij, and one intercept, a0, as shown in equation 1. A workflow to develop surrogate models (RSM and LSSVM) is shown in Figure 2.
i=1 j=i
Figure 2: Workflow used to develop RSM and LSSVM. Modified from Panja et al. [55] As part of the development of a model, validation is performed using test data to assess robustness. An accepted error margin is set for the surrogate model. In this fashion, surrogate models are continuously improved unless the error reaches its acceptance limit. 3.2. Least Square Support Vector Machine (LSSVM)
The Support Vector Machine (SVM) is usually used for classification and regression analysis. A modified form of SVM, namely the least square support vector machine (LSSVM) is used in this
study. LSSVM is close to SVM formulation but solves a linear system instead of a quadratic programming (QP) problem. It has been widely applied in various fields because it is easier to implement, speedy solution convergence, etc. On the other hand, LSSVM has the inherent nature of overfitting to minimize error. Various combinations of data training and testing sets such as 90-10 (%), 85-15(%), 80-20(%), and 70-30(%) were tried. Eventually, a data set with 80% used for training and 20% used for testing yielded the best prediction capabilities in this study. The same combination was used for the other two surrogate models (RSM and ANN). The input and output relationship in LSSVM is given by Equation 2:
y t = wT ^(^j) + b w/ ere Xj e Rv an d yj e R .........................( 2 )
The final form of LSSVM is given by Equation 3: rO 1 ••• 1
1 1 i)+-Y
a i = yi .......( 3 )
ajv. ( ) Jn. ( )
(N+1) X (V+1)
K(x,xi) is known as the kernel function which is chosen a priori. The Radial Basis Function (RBF) kernel is used in this study as shown in Equation 4
K^Xj'Xy) = e xp (--j
Radial Basis Function: Where,
xi: Input vector of ith data b: Bias term
y: Regularization parameter a: Kernel parameter ai: Support values
It is evident from equations 2, 3, and 4 that if the regularization parameter, y, and the kernel
parameter, a , are provided, the bias term and all support values can be determined from a linear
relationship. This is accomplished by using an optimization technique where initial y and a
values are guessed and iteratively improved as described in figure 2. For the optimization part, the training data is further divided into LSSVM training data (80%) and optimization data (20%) as shown in Figure 3
All Data ( 144)
«= 80% (114)
Training Data » 20% (30)
ft 80% (91) * 20 % (23) Test Data
Support Values: a. Y><?
Figure 3: Division of total data set into LSSVM training, optimization, and test data LSSVM training is over once all parameters in the Table 3 are found. At this point, the model can be applied for any unknown input vector using the RBF kernel as shown in Equation 5
V ( ll*-*ill2\ y(x) = ^ ai exp (--—2-) + b ................( 5 )
i=1 ^ ' 3.3. Artificial Neural Networks (ANNs)
The Artificial Neural Networks (ANNs) algorithm was developed based on human learning processes through brain and nerve networks. This is a connectionist technique where input and output are linked through neurons. The most common feed forward architecture consists of one input layer, one or more hidden layers, and one output layer as shown in Figure 4.
ANN Biases
Matrix Permeability
¡Gas Rel. Perm, Expoi Rock Compressibility y
Rock Compressibility
Slope of GOR
Initial GOR
Initial Pressure
Bottom Hole Pressure
Input nodes
Hidden nodes
Output nodes
Figure 4: Basic structure of ANN with input, hidden, and output layers. Links between input and output are established through the internal computations in the hidden layers. The complexity and non-linearity of the model are increased by increasing the number of hidden layers where the individual components of a layer are known as nodes. In this study, there are nine input nodes (eight input parameters as listed in Table 2 and one bias) contained in one hidden layer. A weight was given to each connection for every node and a bias term was added to each hidden and output node. Bias and weight values used in this study are summarized for one output in Table 3. In the process of training the ANN model, all weights and biases are determined by minimizing the error between the predicted output and the training output via activation function at each the node. All output data is normalized as shown in Equation 6:
Yj — Minimum ( Yf)
K' YlOVTYlClliZ6d / \
M ax imum ( Yj) — M inimum ( Yj) Computations in hidden nodes and output nodes are shown in Figure 5.
Figure 5: Input-to-output structure and calculations inside (a) hidden and (b) output nodes As shown in figure 5 computation consists of two calculations: summation and transformation through activation functions where activation functions may be linear or non-linear. In this study, sigmoid transfer function was used as shown in Equation 7
The sensitivity of the model to the number of hidden nodes (neurons) was also investigated. As described earlier, the non-linearity relationship between input and output data increases with the number of hidden nodes. However, increasing non-linearity doesn't always guarantee higher prediction accuracy. To find out the optimum number of hidden nodes (neurons), a sensitivity study was conducted on the training and testing data for oil recovery and gas oil ratio after 5 years of production as shown in Figure 6.
1 + exp(-x)
Number of Neurons in the First Layer Number of Neurons in the First Layer
(a) (b)
Figure 6: Coefficients of determination using different number of hidden nodes for (a) Oil
recovery and (b) gas oil ratio at 5 years.
2 . 2 It is evident from figure 6 that the R is close to unity for training data. On the other hand, the R
value for the test data increases initially with the number of neurons for oil recovery and gas oil
ratio. A maximum R value can be clearly identified at 14 neurons for the case of oil recovery. Therefore, 14 hidden neurons are used in this study. The ANN parameters used in study are summarized below:
• Number of neurons in the first layer (hidden layer) = 14
• Number of neurons in the second layer (output layer) = 1
• Number of weights = Number of neurons in the first layer * Number of input + Number of neurons in the first layer * Number of neurons in the second layer=126
• Number of biases = Number of neurons in the first layer + Number of neurons in the second layer = 15
The unknown parameters in the ANN structure are summarized in Table 3. During training of the ANN, these 126 weights and 15 biases are determined using an optimization routine, namely the Particle Swarm Optimization (PSO) which is discussed in the upcoming sections. 3.4. Goodness of Fit
There are various error measuring tools used in every branch of science and engineering. Their uses are mostly dependent on the model and purpose of the system. During the fitting portion of the model (training the model) the Mean Square Error (MSE) is set as the objective function to determine the optimized model parameters using PSO. As the minimum value of MSE is the indication of a good match between experimental or simulated values and modeled values, MSE is minimized during optimization in PSO. The MSE is calculated between experimental or simulated values and modeled values as shown in Equation 8.
. 2i=i(Yobs,i — Ymodel,i)
MSE =... ....................( 8)
Yobs and Ymodel are the simulated and modeled values respectively, n is the number of data sets. In this study, the Normalized Root Mean Square Error (NRMSE) and the coefficient of determination (R2) are adopted to measure the discrepancy between simulated data and model data. The NRMSE is used over MSE to compare various models (time based and rate based models) in the same scale. The coefficient of determination, R2, is defined as shown in Equation
R2 =1............................(9)
¿¿tot
Where,
SSres = !£= i( Yobs,i - Ymodei,i) , the residual sum of squares
SStot = 1.?= i( Yobs - Ymodei,i) , the total sum of squares
Y0bs = i ^ofc s,i , the mean of observed values The NRMSE is defined in Equation 10,
NR MSE = --—..............................( 1 0 )
*obs,max 'obs.min
Where Yobs,max is the maximum value and Yobs,min is the minimum value of the observed data.
The value of R varies from 0 to 1. R values close to unity and small NRMSE values are indication of a good fit.
4. Optimization Routine: Particle Swarm Optimization
Inspired by the motion of bird swarms, the Particle Swarm Optimization (PSO) routine was developed by Eberhart and Kennedy [56]. In this method, each potential solution is treated as particle. Each particle is characterized by its position and velocity. The position of a particle is defined in a hyperspace whose dimension is equal to the number of unknown parameters being optimized as shown in Table 3. For example, in the case of ANN, particles fly in a 141-dimensional hyperspace. Several particles are initially defined in hyperspace where they iteratively change their position to determine the optimum position. Fitness of a particle is determined by a fitness function such as the MSE. This algorithm is similar to the method followed by a bird groups searching for food in a vast area.
Two solutions, pbest and gbest, at any iteration during execution of the algorithm are tracked. The local best or pbest is defined as the best position of a particle in the hyperspace as determined by the fitness value. The global best or gbest is the overall best value by any particle so far in the population. At each iteration step, the velocity is updated first and then position. Accelerating the particle towards its pbest and gbest by updating velocity is done by two separate random numbers (random 1 and random 2) as shown in Equation 11
ffc+j = • VfcH" C1 ' randomt ■ (Pbesi - Pk) + c2 ■ random2 (gbest - p'k)
( 1 1)
Inertia Component Cognitive Component Social Component
Weighted current velocity Influence of particle Influence of entire population
The cognitive components guide the local search from its local best (pbest) and the social
component is responsible for global search depending on the population best (gbest) [57]. In
equation 11, v\+1 is the velocity in the next iteration step which is partially preserved from the
current velocity by an inertia weight, w ^range 0.4 to 0.9). The acceleration coefficients (C1 and C2) for cognitive and social components are chosen by trial and error. The position of a particle for the next iteration step is updated by the following equation:
Pk+1 =Pk+ vk+1 • At ..............................(12)
The values of wi, C1, C2, and other parameters used in this study are given in Table 4.
Table 4: Particle swarm optimization parameters in various surrogate models
Surrogate Model Particle Swarm Optimization
Number of Particles Maximum
Name Parameters Optimized for a single parameter C1 C2 w Iteration
RSM 1 intercept 44 coefficients 100 2 2 0.6 1000
LSSVM 1 Regularization parameter (y) 1 Kernel parameter (a) 100 2 2 0.6 1000
ANN 15 Biases 126 Weights 100 2 2 0.6 1000
Initial position and velocity of each particle is randomly distributed. After the initialization of positions and velocities of all particles, fitness is calculated. In subsequent steps, positions and velocities are updated iteratively by the local best and the global best parameters as summarized in the flowchart shown in in Figure 7 [58].
Figure 7: Particle Swarm Optimization flow chart. Modified from Ahmadi et al. [58] The entire flowchart can be divided into four parts, namely, initialization, fitness evaluation, condition check, and updates of velocity and position. Acceptance of any particle as potential solution is determined by its fitness value which is calculated in each iteration step. As described earlier, one local best, pbest, and one global best, gbest, are recorded during each iteration. The number of iterations is only limited by time and computational constraints; hence the maximum iteration number is defined by the user.
5. Results and Discussion
Five time-based models (90 days, 1 year, 5 years, 10 years and 15 years) and one rate-based model (5 bbl/day/fracture) were trained using RSM, LSSVM and ANN. The objective of this study is to compare performance of three surrogate models. Production performance in terms of oil recovery and gas-oil ratio are compared with simulation data. Since all time-based models behave similarly, only two time-based models (one early production model after 90 days and a long production model after 10 years) along with one rate-based model are discussed here. The fitness of a model with training data and test data are both discussed in this section. Once the model is trained, it is tested against an unknown data set (i.e., test data) to check for robustness of forecasting capabilities. As discussed earlier, the fitness is determined by two measures, the coefficient of determination (R ) and normalized root mean square error (NRMSE) which are used here to compare different surrogate models. 5.1. Training
In this section, model fitness as compared with training data is evaluated and discussed. It is important to assess individual models to check for overfitting. Three surrogate models (RSM, LSSVM, and ANN) for oil recovery and gas oil ratio after 90 days, 10 years of production and at a terminal rate (5 bbl/day/fracture) are shown in Figures 8.
Gas Oil Ratio (SCF/STB) at 90 days ji
Training Data \ • js* •
- RSM • ANN • LSSVM
1000 2000 3000 SIMULATION
Oi Recovery (%) at 10 yrs. 1
Training Data
• ANN • LSSVM
| 4500
§ 3000
1500 0
Gas Oil Ratio (SCF/STB) at 10 yrs.
Training Data
• • # •
•t / • • •
.. RSM • ANN • LSSVM
20 30 SIMULATION
3000 4500 SIMULATION
6000 7500
Figure 8: Comparison of RSM, ANN and LSSVM models using training data for (a) Oil
recovery after 90 days (b), Oil recovery after 10 years (c), Oil recovery after oil rate drops to 5 bbl/day/fracture (d), Gas Oil Ratio after 90 days (e), Gas Oil Ratio after 10 years (f), and Gas Oil Ratio after oil rate drops to 5 bbl/day/fracture
Typical high model fitness as compared with training data can be observed for all cases in Figure 8. However, the capture of the production behavior using surrogate models without apprehending the underlying physics is a great challenge. It is difficult to model the GOR from low permeability reservoirs [59] due to its complex behavior. Mainly at higher value of GOR (obtained from 10 years model), flow becomes boundary dominated. On the other hand, lower GOR (obtained from 90 days and 1 year models) occurs when flow is at the transient linear
regime. Overall, both oil recovery and GOR from surrogate models are in good agreement with
simulation results. The errors are calculated in terms of R and NRMSE as listed in Tables A.1 and A.2. For visual comparison, R2 and NRMSE for RSM, ANN and LSSVM models of oil recovery are shown in Figure 9a and b respectively.
90d-4ri lyr, 5yrï, IQyri, 15 vre. Rate ÇQtfjys 1 yr, S yrS, IQyn. ISyre, Rate
bated b4icd
(a) (b)
Figure 9: Fitness of RSM, ANN and LSSVM models for of oil recovery for training data (a) Coefficient of determination, R2 and (b) NRMSE
R2 and NRMSE for oil recovery using RSM, LSSVM and ANN for all time- and rate-based models are greater than 0.95 and less than 6% respectively. These values are evidence of well-trained models.
5.2. Testing
Some models may have the tendency to overfit with training data and consequently fail to predict unseen test data with high accuracy. In this study, 20 percent of all data was used to check the forecasting capabilities of all developed models. Results for oil recovery and GOR after 90 days, 10 years production, and at terminal oil rate (5 bbl/day/fracture) are shown in Figure 10.
Oil Recovery (X) at 90 days
Test Oata •
' \It • # «
«/ s it
/* rfRSM • ANN • LSSVM
0 0.6 1.2 1.8 2.4 3. SIMULATION
Oil Recovery (%) at 10 yrt.
Test Data / •
• • / " » * • i •• t »
• /« / 8
■■■J f *
• RSM • ANN • LSSVM
Gas Oil Ratio (SCF/STB) at 90 days Test Data
1000 2000 3000 SIMULATION
1 3000
g 2000
Gas Oil Ratio (SCF/STB) at 10 yrs. • • * • «
Test Data T* <
• V -jfi'
Y J • JBf /
• RSM • ANN • LSSVM
10 15 SIMULATION
2000 3000 SIMULATION
4000 5000
Oil Recovery [%) at oil rate 5STB/day/frac.
Test Data 1 • :
• • • • i. • •
• / RSM • ANN • LSSVM
S 4000
Gas Oil Ratio (SCF/STB) at oil rate 5STB/day/frac. /
Test Data •
y* f / / • •
¿f. ' * ' * . • •
• RSM • ANN • LSSVM
18 27 SIMULATION
4000 SIMULATION
(c) (f)
Figure 10: Comparison of RSM, ANN and LSSVM models using test data for (a) Oil recovery
after 90 days, (b) Oil recovery after 10 years, (c) Oil recovery after oil rate drops to 5 bbl/day/fracture, (d), Gas Oil Ratio after 90 days (e), Gas Oil Ratio after 10 years (f), and Gas Oil Ratio after the oil rate drops to 5 bbl/day/fracture
Although all models had high fitness compared with training data, they showed relatively lower fitness when compared with test data. Considering the fact that test data was not accounted for
during training, the models show promising forecasting capabilities without significant
•2 • 2 aberrations. R and NRMSE are calculated as listed in Tables A.1 and A.2. The R and NRMSE
values for RSM, ANN, and LSSVM oil recovery models are shown in Figures 11 a and b respectively. 1.0
90 dsvs 1 VT.
Syr:* 10 yrs. lSvrs. Rate based
Figure 11: Fitness of RSM, ANN and LSSVM oil recovery models for testing data: (a) Coefficient of determination, R2 and (b) NRMSE
Except for a few cases, the forecast accuracy for all models are within decent ranges. As shown in the figures above, RSM shows higher accuracy predicting oil recovery followed by LSSVM. As shown in Figures 10 and 11, AI tools have the potential to predict oil recoveries and fluid ratios given a small training data set. Large amounts of completion, geological, and production data can indeed be used to train more robust models and complement current conventional tools to evaluate the potential of tight oil reservoirs. The cost, however, is that AI skips the physical description and understanding of the multiphase production mechanisms in tight formations. This cost may not be too high to pay since the current conventional understanding of these systems may not be sufficiently developed yet. In fact, researchers have recently reported on the discrepancies between conventional thinking and fluids under nanoconfinement in tight formations [60, 61].
6. Conclusion
Artificial intelligence tools aimed to predict oil recovery and gas-oil ratio from hydraulically fractured tight formations can be successfully developed using simulation information as a training framework. In this study, three models were developed based on RSM, ANN, and LSSVM to predict recovery from wells producing under time-based (90 days, 1 year, 5 years, 10 years, and 15 years) and rate-based constraints (5 bbl/day/fracture). Eight key factors, namely, matrix permeability, gas relative permeability exponent, rock compressibility, initial gas-oil ratio, slope of solution gas-oil ratio versus pressure, initial pressure, flowing bottom-hole pressure, and fracture spacing were considered as input parameters for all cases. After all models were trained with the same database, they were used to predict production for different
scenarios. Using simulation as a comparison basis, all models were evaluated in terms of their oil recovery and producing gas-oil ratio predictive capabilities. It was found that RSM and LSSVM have better predictive capabilities for oil recovery than ANN. In addition, LSSVM
exhibits the highest accuracy with respect to gas-oil ratio prediction.
Field-scale modeling and simulation of hydraulically fractured ultra-low permeability reservoirs lead to very expensive computational overhead in commercial simulators. Surrogate reservoir models, on the other hand, are useful for quick oil production forecast and assessment. Additionally, these models can be used for risk and uncertainty analysis. Overall, artificial intelligence applications such as LSSVM have promising applications in various aspects of production and reservoir engineering.
Nomenclature
Symbol Description Units
y Regularization Parameter -
a Kernel Parameter -
a; Support Values Unit of Output
Mean Of Observed Values Unit of Output
ao The Intercept Of The Surrogate Model Unit of Output
AI Artificial Intelligence -
aii Coefficient Of 2nd Order Interaction Of Inputs -
ak Coefficient Of Independent Input -
ANN Artificial Neural Networks -
b Bias Term Unit of Output
BHP Bottom Hole Pressure psi
Cl Acceleration Coefficient For Cognitive Components
C2 Acceleration Coefficient For Social Components
DOE Design Of Experiments
dRs/dp Slope Of Gas/Oil Ratio In PVT (SCF/STB)/psi
gbest Population's Best Particle's Position
GOR Gas/Oil Ratio SCF/STB
LSSVM Least Square Support Vector Machine -
MSE Mean Square Error Unit of Output
...........ng.............................................................. Exponent Of Relative Permeability Curve For Gas -
NRMSE Normalized Root Mean Square Error
pbest Particle's Best Position
Pi Initial Reservoir Pressure psi
PSO Particle Swarm Optimization -
PVT Pressure-Volume-Temperature -
..........R2..................................................... Coefficient Of Determination -
Rsi Initial Gas/Oil Ratio SCF/STB
RSM Response Surface Model -
SSres Residual Sum Of Squares Unit of Output
SStot Total Sum Of Squares Unit of Output
SVM Support Vector Machine -
vk Velocity Of Particle -
wi Inertia Weight -
Ymodel.i Modeled Value Unit of Output
Yobs,i Observed Data Unit of Output
Yobs,max The Maximum Value Of Observed Data Unit of Output
Yobs,min The Minimum Value Of Observed Data Unit of Output
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Appendix A: Supplementary Information
Table A.1: Coefficient of determination (R2) of RSM, LSSVM and ANN for all models
Output Training Data Test Data
Model RSM LSSVM ANN RSM LSSVM ANN
Oil 90 days 0.99 0.99 0.96 0.69 0.52 0.51
1 year 0.98 0.99 0.98 0.78 0.69 0.53
5 years 0.99 0.99 0.99 0.63 0.81 0.60
Recovery 10 years 0.99 0.99 0.98 0.91 0.9 0.72
15 years 0.99 0.99 0.99 0.97 0.93 0.84
Rate Based 0.98 0.98 0.99 0.57 0.54 0.48
90 days 0.98 0.99 0.95 0.92 0.84 0.80
1 year 0.98 0.98 0.96 0.93 0.91 0.90
Gas Oil 5 years 0.98 0.98 0.97 0.41 0.73 0.30
Ratio 10 years 0.88 0.92 0.93 0.76 0.73 0.46
15 years 0.83 0.77 0.84 0.79 0.75 0.32
Rate Based 0.84 0.88 0.92 0.68 0.45 0.43
Table A.2: Normalized Root Mean Square Error (NRMSE) of RSM, LSSVM and ANN for all
models
Output Training Data Test Data
Model RSM LSSVM ANN RSM LSSVM ANN
90 days 1.9 1.9 3.5 16.5 20.3 20.7
1 year 2.4 2.3 2.5 12.4 14.7 18.1
Oil 5 years 2.0 1.9 2.1 16.1 11.5 16.7
Recovery 10 years 1.9 1.7 2.6 7.9 8.5 14
15 years 2.7 2.4 2.1 4.9 7.3 10.8
Rate Based 3.5 3.3 2.4 20.7 21.2 22.6
90 days 2.6 2.0 4.6 8.7 11.8 13.3
1 year 3.3 3.3 4.2 7.9 9.3 9.7
Gas Oil 5 years 3.0 3.1 3.7 24.0 16.1 26.2
Ratio 10 years 5.7 4.6 4.3 16.1 17.2 24.3
15 years 5.8 6.8 5.6 14.4 15.5 25.7
Rate Based 5.2 4.5 3.8 14.1 18.4 18.8
Ol Recovery (H) at 1 yr. /
y • RSM • ANN • LSSVM
SIMULATION
mi £ o s
5 2000
Gas Oil Ratio (SCF/ST8) at 1 yr. r1
? '<#1*!
- RSM • ANN • LSSVM
1000 2000 3000 4000 SIMULATION
i ¡20
Ol Recovery (%) at IS yrs. /
••
J* V • RSM • ANN • LSSVM
20 30 SIMULATION
5 6000
Gas Oil Ratio (SCF/STB) at IS yrs.
• • • • •
• • • •
• RSM • ANN • LSSVM
6000 SIMULATION
(c) (f)
Figure A.1.: Data training comparison of RSM, ANN and LSSVM models for (a) Oil recovery after 1 year (b) Oil recovery after 5 years (c) Oil recovery after 15 years (d) Gas Oil Ratio after 1 year (e) Gas Oil Ratio after 5 years (f) Gas Oil Ratio after 15 years
Table A.3: List of simulations using Box-Behnken DOE used to train surrogate models
Sr. No. Km (nD) Vn? Cf (1/psi) dRs/dp ((SCF/STB)/psi) Rsi (SCF/STB) Pi (Psi) Pwf (Psi) Xf (ft.)
1 10 1 4.00E-05 0.65 1900 5250 1000 180
2 10 3 4.00E-05 0.65 1900 5250 1000 180
3 5000 1 4.00E-05 0.65 1900 5250 1000 180
4 5000 3 4.00E-05 0.65 1900 5250 1000 180
5 10 2 4.00E-06 0.65 1900 5250 1000 180
6 10 2 4.00E-04 0.65 1900 5250 1000 180
7 5000 2 4.00E-06 0.65 1900 5250 1000 180
8 5000 2 4.00E-04 0.65 1900 5250 1000 180
9 10 2 4.00E-05 0.5 1900 5250 1000 180
10.......... 10 2 4.00E-05 0.8 1900 5250 1000 180
11 5000 2 4.00E-05 0.5 1900 5250 1000 180
12 5000 2 4.00E-05 0.8 1900 5250 1000 180
13 10 2 4.00E-05 0.65 800 5250 1000 180
14 10 2 4.00E-05 0.65 3000 5250 1000 180
15 5000 2 4.00E-05 0.65 800 5250 1000 180
16 5000 2 4.00E-05 0.65 3000 5250 1000 180
17 10 2 4.00E-05 0.65 1900 4000 1000 180
18 10 2 4.00E-05 0.65 1900 6500 1000 180
19 5000 2 4.00E-05 0.65 1900 4000 1000 180
20 5000 2 4.00E-05 0.65 1900 6500 1000 180
21 10 2 4.00E-05 0.65 1900 5250 500 180
22 10 2 4.00E-05 0.65 1900 5250 1500 180
23 5000 2 4.00E-05 0.65 1900 5250 500 180
24 5000 2 4.00E-05 0.65 1900 5250 1500 180
25 10 2 4.00E-05 0.65 1900 5250 1000 60
26 10 2 4.00E-05 0.65 1900 5250 1000 300
27 5000 2 4.00E-05 0.65 1900 5250 1000 60
28 5000 2 4.00E-05 0.65 1900 5250 1000 300
29 225 1 4.00E-06 0.65 1900 5250 1000 180
30 225 1 4.00E-04 0.65 1900 5250 1000 180
31 225 3 4.00E-06 0.65 1900 5250 1000 180
32 225 3 4.00E-04 V 0.65 1900 5250 1000 180
33 225 1 4.00E-05 0.5 1900 5250 1000 180
34 225 1 4.00E-05 0.8 1900 5250 1000 180
35 225 3 4.00E-05 0.5 1900 5250 1000 180
36 225 3 4.00E-05 0.8 1900 5250 1000 180
37 225 1 <4.00E-05 0.65 800 5250 1000 180
38 225 1 4.00E-05 0.65 3000 5250 1000 180
39 225 3 4.00E-05 0.65 800 5250 1000 180
40 225 3 4.00E-05 0.65 3000 5250 1000 180
41 225 1 4.00E-05 0.65 1900 4000 1000 180
42 225 1 4.00E-05 0.65 1900 6500 1000 180
43 225 3 4.00E-05 0.65 1900 4000 1000 180
44 225 3 4.00E-05 0.65 1900 6500 1000 180
45 225 1 4.00E-05 0.65 1900 5250 500 180
46 225 1 4.00E-05 0.65 1900 5250 1500 180
47 225 3 4.00E-05 0.65 1900 5250 500 180
48 225 3 4.00E-05 0.65 1900 5250 1500 180
49 225 1 4.00E-05 0.65 1900 5250 1000 60
50 225 1 4.00E-05 0.65 1900 5250 1000 300
51.......... 225 3 4.00E-05 0.65 1900 5250 1000 60
52 225 3 4.00E-05 0.65 1900 5250 1000 300
53 225 2 4.00E-06 0.5 1900 5250 1000 180
54 225 2 4.00E-06 0.8 1900 5250 1000 180
55 225 2 4.00E-04 0.5 1900 5250 1000 180
56 225 2 4.00E-04 0.8 1900 5250 1000 180
57 225 2 4.00E-06 0.65 800 5250 1000 180
58 225 2 4.00E-06 0.65 3000 5250 1000 180
59 225 2 4.00E-04 0.65 800 5250 1000 180
60 225 2 4.00E-04 0.65 3000 5250 1000 180
61 225 2 4.00E-06 0.65 1900 4000 1000 180
62 225 2 4.00E-06 0.65 1900 6500 1000 180
63 225 2 4.00E-04 0.65 1900 4000 1000 180
64 225 2 4.00E-04 0.65 1900 6500 1000 180
65 225 2 4.00E-06 0.65 1900 5250 500 180
66 225 2 4.00E-06 0.65 1900 5250 1500 180
67 225 2 4.00E-04 0.65 1900 5250 500 180
68 225 2 4.00E-04 0.65 1900 5250 1500 180
69 225 2 4.00E-06 0.65 1900 5250 1000 60
70 225 2 4.00E-06 0.65 1900 5250 1000 300
71 225 2 4.00E-04 0.65 1900 5250 1000 60
72 225 2 4.00E-04 0.65 1900 5250 1000 300
73 225 2 4.00E-05 K 0.5 800 5250 1000 180
74 225 2 4.00E-05 0.5 3000 5250 1000 180
75 225 2 4.00E-05 0.8 800 5250 1000 180
76 225 2 4.00E-05 0.8 3000 5250 1000 180
77 225 2 4.00E-05 0.5 1900 4000 1000 180
78 225 2 <4.00E-05 0.5 1900 6500 1000 180
79 225 2 4.00E-05 0.8 1900 4000 1000 180
80 225 2 4.00E-05 0.8 1900 6500 1000 180
81 225 2 4.00E-05 0.5 1900 5250 500 180
82 225 2 4.00E-05 0.5 1900 5250 1500 180
83 225 2 4.00E-05 0.8 1900 5250 500 180
84 225 2 4.00E-05 0.8 1900 5250 1500 180
85 225 2 4.00E-05 0.5 1900 5250 1000 60
86 225 2 4.00E-05 0.5 1900 5250 1000 300
87 225 2 4.00E-05 0.8 1900 5250 1000 60
88 225 2 4.00E-05 0.8 1900 5250 1000 300
89 225 2 4.00E-05 0.65 800 4000 1000 180
90 225 2 4.00E-05 0.65 800 6500 1000 180
91 225 2 4.00E-05 0.65 3000 4000 1000 180
92 ■■■ 225 2 4.00E-05 0.65 3000 6500 1000 180
93 225 2 4.00E-05 0.65 800 5250 500 180
94 225 2 4.00E-05 0.65 800 5250 1500 180
95 225 2 4.00E-05 0.65 3000 5250 500 180
96 225 2 4.00E-05 0.65 3000 5250 1500 180
97 225 2 4.00E-05 0.65 800 5250 1000 60
98 225 2 4.00E-05 0.65 800 5250 1000 300
99 225 2 4.00E-05 0.65 3000 5250 1000 60
100 225 2 4.00E-05 0.65 3000 5250 1000 300
101 225 2 4.00E-05 0.65 1900 4000 500 180
102 225 2 4.00E-05 0.65 1900 4000 1500 180
103 225 2 4.00E-05 0.65 1900 6500 500 180
104 225 2 4.00E-05 0.65 1900 6500 1500 180
105 225 2 4.00E-05 0.65 1900 4000 1000 60
106 225 2 4.00E-05 0.65 1900 4000 1000 300
107 225 2 4.00E-05 0.65 1900 6500 1000 60
108 225 2 4.00E-05 0.65 1900 6500 1000 300
109 225 2 4.00E-05 0.65 1900 5250 500 60
110 225 2 4.00E-05 0.65 1900 5250 500 300
111 225 2 4.00E-05 0.65 1900 5250 1500 60
112 225 2 4.00E-05 0.65 1900 5250 1500 300
113 225 2 4.00E-05 0.65 1900 5250 1000 180
114 225 2 4.00E-05 0.65 1900 5250 1000 180
Table A.4: List of simulations performed using random input parameter values used to test the
surrogate models
Sr. Km ng Cf dRs/dp Rsi Pi Pwf Xf
No. (nD) - ) (1/psi) ((SCF/STB)/psi) (SCF/STB) (psi) (psi) (ft.)
1 1496 ^ 1.17 1.0E-05 0.62 2069 5738 743 300
2 361 1.27 3.8E-05 0.67 1456 4170 1417 180
3 31 1.35 8.0E-06 0.58 1625 4637 769 60
4 44 1.78 2.0E-05 0.59 1724 4560 1266 60
5 2475 2.66 1.9E-05 0.69 2408 5670 689 300
6 12 2.61 1.4E-05 0.58 2820 6111 787 60
7 210 1.12 2.0E-05 0.75 1775 4861 591 180
8 28 1.80 1.9E-05 0.79 2969 5951 1076 60
9 4390 2.05 6.0E-06 0.72 2531 5688 1183 300
10 840 1.83 5.4E-06 0.60 1460 4017 1047 300
11 225 2.31 4.0E-05 0.68 2106 5505 926 180
12 187 2.26 5.9E-06 0.53 1689 4967 1144 180
13 14 1.58 4.3E-06 0.77 2779 6290 1148 60
14 694 1.86 1.5E-05 0.76 1539 4003 1179 300
15 13 1.03 3.0E-05 0.75 1273 5156 1136 60
16 16 2.97 1.9E-05 0.58 1413 5061 1445 60
17 256 1.33 6.2E-06 0.68 1323 5152 709 180
18 18 1.21 9.4E-06 0.51 2281 5925 y~09 60
19 1618 1.74 1.2E-05 0.63 1552 4806 736 300
20 1612 1.40 3.8E-05 0.59 2527 5962 619 300
21 892 1.98 5.7E-06 0.55 1566 5178 1107 300
22 25 1.68 2.9E-05 0.55 1900 4089 950 60
23 604 2.90 1.8E-05 0.63 1614 4440 959 300
24 251 2.84 9.5E-06 0.53 2944 5804 1162 180
25 4237 1.11 6.2E-06 0.68 1710 5184 1270 300
26 565 2.48 1.1E-05 0.64 1966 4382 850 300
27 1448 1.54 1.2E-05 0.71 1393 4853 1162 300
28 168 1.85 5.3E-06 0.71 2676 5518 916 180
29 147 2.10 1.6E-05 0.69 1431 4479 1342 180
30 1692 2.89 6.7E-06 0.51 2672 5846 1333 300
