Estimation of subsurface petrophysical properties ...

Journal of Applied Geophysics 154 (2018) 93–107

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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo

Estimation of subsurface petrophysical properties using different stochastic algorithms in nonlinear regression analysis of pressure transients Zahra Arab Aboosadi a, Saeed Rooeentan b, Meisam Adibifard c,⁎ a b c

Young Researchers and Elite Club, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran Department of Petroleum Engineering, Fars Science and Research Branch University, Marvdasht, Iran Dave C. Swalm School of Chemical Engineering, Mississippi State University, Starkville, United States

a r t i c l e

i n f o

Article history: Received 10 January 2018 Received in revised form 16 April 2018 Accepted 23 April 2018 Available online 25 April 2018 Keywords: Pressure transient Nonlinear regression Petrophysical properties Stochastic optimization Homogeneous reservoirs Fractured reservoirs

a b s t r a c t Accurate characterization of the underground energy resources is crucial in rigorous prediction of their future behavior. Well testing is one of the main operations used in the oil and gas industry to characterize the underground hydrocarbon reservoirs. Among the various factors which affect the accuracy of the well testing, robustness of the optimization algorithm in nonlinear regression is of great important. Therefore, in this study, efficiency and computational time of four different population-based algorithms in solving the well testing regression problem are thoroughly investigated. The employed algorithms consist of a biological evolutionary algorithm, GA (Genetic Algorithm), two swarm-based algorithms, PSO (Particle Swarm Optimization) and FA (Fireflies Algorithm), and a social-based algorithm, ICA (Imperialist Competitive Algorithm). These algorithms have been applied on two different reservoir models including a homogenous infinite-acting, and a heterogenous fractured reservoir. Performances of the employed algorithms are then evaluated both statistically and graphically. The comparison study showed that FA fails to macth the data for both homogenous and heterogenous reservoirs. Although PSO, GA, and ICA come up with lower relative errors for the homogenous model, they still cannot accurately predict all the state variables for the fractured model. Based on relative error and residual plots, PSO and ICA outperform the other algorithms due to their localized searching capabilities. In detail, PSO and ICA end up with the R-squared values of 0.93 and 0.99 for the homogenous and heterogonous fractured models, respectively. Evolution of error over time unveiled that the indicated algorithms encounter problems in matching the transitional wellbore storage and infinite acting zones for the homogenous model; for the fractured model, however, most of the errors are distributed around the transitional matrix-fracture zone. The indicated stochastic algorithms were compared with a well-known derivative-based algorithm, namely LM (Levenberg-Marquardt), and sensitivity analysis showed that LM is very sensitive to the location of the initial point. On average, LM led to higher magnitudes of relative error for both homogenous and heterogenous reservoir models. The computational time for ICA was also lower than the other algorithms, indicating that ICA has the lowest relative error and computational time among the investigated algorithms. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Well testing is one of the reliable techniques used in the petroleum industry to estimate the reservoir's petrophysical parameters through analyzing the pressure transient data obtained from the well bottom hole. Robustness of the well testing, however, depends upon the accuracy of the utilized mathematical models, precision of the downhole pressure gauges, and the optimization technique used to match the field pressure data. The goal of well testing is to estimate the reservoir properties after matching the real field pressure data with the synthetic pressure ⁎ Corresponding author. E-mail address: [email protected]. (M. Adibifard).

https://doi.org/10.1016/j.jappgeo.2018.04.023 0926-9851/© 2017 Elsevier B.V. All rights reserved.

data belonging to a selected reservoir model. With this in mind, this study is aimed to investigate the applicability of four different meta-heuristic optimization algorithms in estimating the reservoir properties through nonlinear regression optimization. Well testing commences by a rate perturbation at the surface and monitoring the pressure-time data at the well bottom hole. Then, primary goal of the well testing is to recognize the reservoir model which honors the obtained pressure data. In the next phase, reservoir parameters are estimated by applying an appropriate nonlinear regression technique over the pressure data to further adjust the reservoir parameters of the selected model. Eventually, the total pressure history of the well is simulated to validate the results achieved through stages one and two. Among the indicated steps, estimation of reservoir

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Z. Arab Aboosadi et al. / Journal of Applied Geophysics 154 (2018) 93–107

Nomenclature pwd f(t) F s tD L−1 RMSE yEst. yAct. S CD k0 k1 c1 c2 wstart wend β0 α γ k C h rw Φ ct q μ B λ ω vi c1 c2 xi Pi Pg β0 γ α rij V Cn pn Ncol NCn dmov TCn ðimp;nÞ f cost ðcol;iÞ f cost xi+1 ∇f(xi) H λ I αm km kf

well pressure response in the Laplace medium time series function in the time domain time series function in the Laplace medium Laplace variable dimensionless time Laplace inverse operator Root Mean Squared Error estimated parameter real parameter skin factor dimensionless wellbore storage coefficient zero-degree modified Bessel function of the second kind first-degree modified Bessel function of the second kind social constant cognitive constant start value of the inertial weight end value of the inertial weight Initial attractiveness of the fireflies randomness parameter in the fireflies algorithm the light absorption coefficient reservoir permeability, md wellbore storage coefficient, bbl/psi reservoir thickness, ft. wellbore radius, ft. porosity total compressibility, psi−1 well flow rate, STBD oil viscosity, cp oil formation volume factor, Rbbl/STB interporosity flow coefficient fracture storativity ratio velocity vector in the PSO algorithm personal factor cognitive social factor position vector in the PSO and FA algorithms best solution for each particle in the PSO algorithm global optimum in the PSO algorithm firefly's brightness at the zero radius lights absorption coefficient in the FA algorithm the randomization parameter in the FA algorithm the Euclidean distance between each i–j pair of fireflies variables of each country in the ICA algorithm the Normalized cost functions for the imperialists in the ICA algorithm the normalized power of the n-th imperialist total number of all the colonies in the current population the number of colonies assigned to the n-th imperialist displacement vector for the colonies in the ICA algorithm total power of each empire within the ICA algorithm cost of the imperialist within the empire the magnitudes of cost function for the colonies in the empire update of the optimization variables using the LM algorithm gradient of the cost function evaluated at the previous iteration the Hessian matrix in the LM algorithm a scalar value used in the LM algorithm the identity matrix matrix shape factor matrix permeability fracture permeability.

parameters is of great importance since any miscalculation at this phase will lead to errors in the subsequent reservoir modeling and simulation studies; this will in turn influence the reservoir management studies. Therefore, opting a robust and sophisticated optimization algorithm for the well test analysis is necessary to ensure robust reservoir management studies. Although the first application of the computer-aided techniques in the well testing dates back to 1970’s, these techniques did not become practical at the oil industry until 1990's, when nonlinear regression well test analysis became a standard method and dozens of technical papers were disseminated (Dastan and Horne, 2010). Different authors have used different optimization techniques and cost functions to handle the nonlinear regression task for well test analysis. For example, Barua et al. (1985) used a non-weighted LS (Least Square) function as the cost function. It was concluded that the inclusion of the derivative data in the cost function will lead to higher convergence rate (Barua et al., 1985). El-Khatib (1987) used the LS technique to solve the nonlinear regression task in pulse test analysis and estimated the reservoir's transmissivity and storativity. To this end, he used a combination of NewtonRaphson and Regula-falsi method as the optimization algorithm. The developed method was tested by employing simulated data, and it was concluded that the introduced method converges rapidly. A perfect match was obtained between the estimated and assumed physical properties (El-Khatib, 1987). Nanba and Horne (1992) used a modification of the CF (Cholesky Factorization) method to estimate the reservoir parameters and concluded that the modified MG (Gauss-Cholesky) technique is more accurate than the other techniques such as GM (Gauss-Marquardt) and NB (Newton-Barua) (Nanba and Horne, 1992). Rosa and Horne (1995) employed a weighted cost functions during the optimization process; the weights depend upon the difference between the actual pressure and the synthetic data generated by a mathematical model. Their weighted cost function decreased the confidence interval while simultaneously lead to more accurate results, compared with other cost functions (Rosa and Horne, 1995). Employing LS and LAV (Least Absolute Value) objective functions in the well test analysis, Menekse et al. (1995) found out that LAV cost function is not sensitive to the outliers in the pressure data (Menekse et al., 1995). In 2000, Onur and Kuchuk adopted the maximum likelihood technique to conduct the nonlinear regression well test analysis by using the weighted test data. Their developed method did not require having prior knowledge about the error variance in order to calculate the weights of the pressure data. Applying this technique over several simulated and field data, they reached the conclusion that the new technique improves the accuracy of the estimated parameter, compared with the WLS (Weighted Least Square) methodology (Onur and Kuchuk, 2000). Using LS and TLS (Total Least Square) cost functions, which are based on the pressure and time differences, Dastan and Horne (2010) found out that TLS functions gives more accurate results in comparison with the former LS method, especially in the regions with noisy data (Dastan and Horne, 2010). Although there are enough numbers of research publications regarding the reservoir model's prediction by employing AI (Artificial Intelligence) techniques (Adibifard et al., 2014; Al-Kaabi and Lee, 1990; Alajmi and Ertekin, 2007; Allain and Houze, 1992; Athichanagorn and Horne, 1995; Ershaghi et al., 1993; Güyagüler et al., 2001; Kharrat and Razavi, 2008; Kumoluyi et al., 1995), there are limited resources addressing the estimation of the reservoir parameters by using the AI and data assimilation techniques in the nonlinear regression well test analysis (Adibifard et al., 2016; Bazargan and Adibifard, 2017). 1.1. Literature review on the derivative-free optimization The metaheuristic optimization algorithms, which rely upon the magnitudes of the objective function within the search domain instead of its derivative, have been successfully employed in different fields of the geophysics including, but not limited to, seismic modeling


(Aleardi, 2015; Haijun et al., 2017; Huang et al., 2013; Yang et al., 2017), geological model construction (Attwa et al., 2014), parameter estimation from petrophysical log data (Asoodeh and Bagheripour, 2012; Bagheripour and Asoodeh, 2013; Huang et al., 2001; Labani et al., 2010; Rajabi et al., 2010), Choke flow predictions (Ghorbani et al., 2017), and automated reservoir history match and uncertainty quantifications (Castellini et al., 2005; Hajizadeh et al., 2011, 2010; Sultan et al., 1994). Among the numerous free-derivative optimization algorithms that have been developed are GA (Genetic Algorithm), PSO (Particle Swarm Optimization), ACO (Ant Colony Optimization), FA (Fireflies Algorithm), COA (Cuckoo Optimization Algorithm), ICA (Imperialist Competitive Algorithm), SA (Simulated Annealing), and BCO (Bee Colony Optimization). These algorithms have been mostly adopted from the nature behavior of the different species. As an example, PSO algorithm emulate the forging behavior of the flocks of birds (Cui and Gao, 2012) while COA is developed based on the hatching behavior of the cuckoos (Rajabioun, 2011). Even though these algorithms have been basically developed to deal with the single-objective optimization problems, they can be modified or reformulated to handle the different types of problems involving multi-objective tasks. Given that, multi-objective algorithms such as NSGA-II (Non-dominated Sorting Genetic Algorithm II), MOPSO (Multi-Objective PSO), and f-MOPSO (Fuzzy Multi-Objective PSO) have been introduced as variants of MOEAs (Multi-Objective Evolutionary Algorithms) (Coello Coello, 2002; Deb et al., 2002; Rezaei et al., 2017). These algorithms are based on the pareto front generation and rank estimations to sort the objective functions in the multiobjective domain. In 2011, Gong et al. studied the optimization problems entailing uncertainties and hybrid indices which include both implicit and explicit indices. To efficiently deal with these types of problems, they transferred the uncertain optimization problem into couple of certain optimization issues and then developed a new EA (Evolutionary Algorithm) based on the GA and interactive GA to solve the system of precise optimization problems (Gong et al., 2011). In a similar study, Ji et al. (2011) proposed a large population EA in which a similar-based strategy was used to assign implicit indices to all the solutions in the population. The value of the user preference for each solution was then used to make a distinction between solutions with the same rank during the process of PO (Pareto Optimal). They used the new technique for the interior layout problem and their results showed that the newly introduced EA method outperforms the other algorithms (Ji et al., 2011). In a research study regarding uncertain MOPs (Multi-Objective Problems), Gong et al. (2013) introduced an IEA (Interactive Evolutionary Algorithm) which can be used in conjunction with a DM (Decision Maker) to solve the MOPs. They used a preference polyhedron to compare the solutions with the same rank and showed that their new technique is much more efficient that a posteriori method (Gong et al., 2013). Later, Gong et al. (2016) introduced a new set-based GA algorithm to handle the multi-objective problems with more than three cost functions in which at least one of the objective functions had uncertainties over its interval. To solve the designated IMaOPs (Interval Many-Objective Optimization Problems), they transformed the objective function into a new domain and then solved the problem with using the set-based GA (Gong et al., 2016). Interval programming is used to handle the optimization problems entailing uncertainties over the optimization variables. Contrary to the other techniques, the interval programming method solely utilizes the statistical information such as lower and upper bounds, and mid-point regarding the interval of the variables (Sun et al., 2018; Zaman et al., 2011). It is stated that interval programming is the most suitable technique to handle different types of uncertainties during the optimization process (Kreinovich, 2012). However, interval programming is not reliable when user does not possess enough knowledge about the interval analysis (Sun et al., 2018). With that in mind, most recently, Sun et al. (2018) proposed an ensemble framework to solve the interval

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programming problems even if user has little knowledge regarding the interval mathematics. Their proposed algorithm uses two different modules inducing reducing approaches to evaluate the solutions prior to the optimization and integrating approaches to combine the different set of dominance laws. They used two single-objective problems and four bi-objective tasks to verify the accuracy the developed algorithm. Despite requiring more computational time, the newly proposed ensemble framework proved to be efficient in dealing with the indicated problems (Sun et al., 2018). Because of the lack of the enough research studies regarding nonlinear regression analysis using stochastic algorithms, we have used four different meta-heuristic algorithms to estimate the reservoir parameters through nonlinear regression analysis. These derivative-free algorithms have shown to be effective in dealing the noisy data, where it is difficult to calculate the derivative of the cost function. The conventional Newton based optimization schemes require derivative of the objective function to handle the data; also, their results depend upon the initial point used to initiate the searching process. In this study, a biological evolutionary algorithm, namely GA (Genetic Algorithm), two swarm-based algorithms, namely PSO (Particle Swarm Optimization) and FA (Fireflies Algorithm), and a social-based evolutionary algorithm, namely ICA (Imperialist Competitive Algorithm), have been employed for the purpose of reservoir parameters' estimation. Interval programming is used to bound the optimization variables through defining a physical interval for the investigated petrophysical parameters in this study. The accuracy and computational time of the indicated algorithms have been compared statistically and graphically, and a comparison has been made between the employed population-based methods and the conventional LM (Levenberg-Marquardt) optimization scheme, which relies on the derivative of the cost function. 2. Developed algorithm Four different meta-heuristic optimization algorithms have been employed to address the nonlinear well test regression. To this end, the Stehfest numerical Laplace inverse algorithm is used to transfer the pressure solutions from the Laplace domain to the time domain. This algorithm was proposed by Stehfest (1970) and is straightforward for the engineering applications as it only requires determination of the following summation for the function F at the Laplace medium (Stehfest, 1970): f ðt Þ ≈

L ln ð2Þ X nlnð2Þ an F ; t n¼1 t

ð1Þ

where an coefficients are calculated by using the following equation:

L

an ¼ ð−1Þnþ2

minðn;2L Þ

X

k¼ðnþ1Þ =2

L=2

k ð2kÞ! ; L −k !k!ðk−1Þ!ðn−kÞ!ð2k−nÞ! 2

ð2Þ

The upper limit L defines the precision of the summation and there is an optimum value for L. In this paper, we have used L = 16 as it was the optimum value leading to the most accurate pressure predictions. In addition, since we need the derivative solution instead of the pressure solutions, the Okoye et al. (1991) method is used to generate the derivative data from the pressure solutions at the Laplace medium. This method is described in the following (Okoye et al., 1991): h i dp dpwd ¼ t Di L−1 fspwd ðsÞg ; t D wd ¼ t Di dt D i d ln ðt D Þ i

ð3Þ

The overall procedure utilized in this study is illustrated graphically at Fig. 1. The Stehfest Laplace inverse algorithm is used within the

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optimization algorithm to generate the derivative pressure data using the information provided by each particle in the optimization algorithm. Four different stochastic algorithms including GA (Genetic Algorithm), PSO (Particle Swarm Optimization), FA (Fireflies Algorithm), and ICA (Imperialist Competitive Algorithm) were applied over the simulated pressure data to estimate the reservoir parameters via the nonlinear regression analysis. Contrary to the conventional Newton like methods, the derivative-free algorithms, as the name stands for, do not need to calculate the derivative of the error function. Instead, they rely upon the generation of large number of random solutions, then, the generated random solutions are modified by using the sophisticated mathematical formulations which are usually inspired from nature rules. To compare the different individuals at each iteration, RMSE (Root Mean Square Error) over the derivative data, defined in the following, was used as the objective function for each optimization algorithm: RMSE ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N ∑ ðy −yAct: Þ2 ; N i Est:

ð4Þ

Table 1 Parameters of the GA algorithm used during the optimization process. Selection function Crossover function Mutation function Crossover fraction Elite count Generations

Roulette function Heuristic Adopt feasible 0.8 10% of the population 100

where N is the total number of the derivative points; yEst. and yAct. refer to the estimated and actual derivative data, respectively. 2.1. Genetic algorithm (GA) Genetic Algorithm (GA) was introduced first by John Holland in 1960s and then established by Holland and his students at the University of Michigan during 1960s and 1970s (Mitchell, 1998). This

Fig. 1. The main procedure used to estimate the reservoir properties by using derivative-free optimization algorithms.

Z. Arab Aboosadi et al. / Journal of Applied Geophysics 154 (2018) 93–107 Table 2 Parameters of the PSO algorithm used in this study for the nonlinear regression optimization. c1 c2 wstart wend Iterations

2 2 0.4 0.8 100

algorithm is in fact based on the Darwin's theory of natural selection. The GA algorithm encodes the optimization problem into strings with limited length. These strings, which represent the solutions in the search domain, are called chromosomes while each alphabet in the string is called a gene (Sastry et al., 2014) . GA is an evolution-based algorithm in which a population of the encoded chromosomes are updated successively by using different functions applied over the current chromosomes or individuals. Although there are many different variants of the GA, developed for different purposes, the main operations of a simple GA include selection, cross-over, and mutation. In the selection step, the best solutions are selected to generate the next generation or offsprings. Many selection functions such as roulette-wheel, stochastic universal, raking, and tournament selection have been suggested for this purpose (Sastry et al., 2014). In the cross-over or recombination phase, the selected candidates are combined in many different suggested ways to create a new solution in the encoded space. The generated chromosomes in the recombination phase would not be similar to any selected parent in the previous phase (Goldberg, 2002). In the final phase, to emulate the natural selection in the real world, a fraction of the solutions in the next generation are generated by utilizing the mutation function. In this function, one or several parts of the chosen chromosome are disturbed randomly. This helps the GA to keep searching nearby a solution (Sastry et al., 2014) while moving toward the probable optimal. Finally, the next generation is created by combing the recombined populations and mutated chromosomes. In some variants, however, it is possible to directly transfer a fraction of the best solutions to the next generation without any modification; these solutions are called elites of the population.

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Table 5 Fluid, rock and production data for the simulated homogeneous reservoir model. k, md C, bbl/psi S h, ft rw, ft Φ ct, psi−1 q, STBD μ, cp B, Rbbl/STB

50 7E−3 −2 50 0.3 0.3 4E−6 3000 1.2 1.05

used in different fields of engineering and hundreds of papers have been published in this area in less than a decade (Poli, 2007). In this algorithm, each particle in RN possesses two different attributes, a position vector and a velocity vector. The position vector xi represents the solution of the problem, and the velocity vector vi influences the convergence of the algorithm (Chun-man et al., 2012). Two other vectors also influence the movements of the particles in the searching space. The best solution obtained for each particle during its evolution is remembered by the algorithm as Pi; the best solution of the whole population also affects the current position of each particle and it is called as Pg. In this study we have utilized the inertia-based version of PSO developed by Shi and Eberhart (1998). This algorithm updates the velocity and position vectors using the following equations, respectively (Shi and Eberhart, 1998): vi ðt þ 1Þ ¼ vi ðt Þ þ c1 r 1i ðP i ðt Þ−xi ðt ÞÞ þ c2 r 2i P g ðt Þ−xi ðt Þ ;

ð5Þ

xi ðt þ 1Þ ¼ xi ðt Þ þ vi ðt þ 1Þ;

ð6Þ

The coefficients c1 and c2 at the velocity update formula refer, respectively, to the personal and cognitive social factors. r1i and r2i are also two random number defined in the range [0,1], and t refers to the current iteration. Pi is the best solution for each particle while Pgis the global best solution for the population.

2.2. Particle swarm optimization (PSO) 2.3. Fireflies algorithm (FA) The PSO algorithm is a kind of the swarm intelligent algorithms which is inspired from the social behavior of the birds and fishes foraging for foods (Cui and Gao, 2012). This algorithm has been successfully

A firefly emits short flashes of light which are produced during the bioluminescence process. The generated flashes are used either to fascinate the opposite gender or hunting a prey. In general, the intensity of

Table 3 Selected parameter values for the FA algorithm. β0 α γ Iterations

1 0.25 6 100

Table 4 Parameters of the ICA optimization algorithm used within the nonlinear regression process. Parameter

Assigned value

Decades Revolution rate Assimilation rate Number of initial empires Uniting threshold

100 0.1 1.4 25% of the population size 0.02

Fig. 2. Derivative plot at the log-log scale for the simulated homogeneous reservoir.

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The FA was developed in 2007 by Yang (2007) in the Cambridge University, and three different assumptions are used in developing this algorithm (Arora and Singh, 2013; Yang, 2009): • The fireflies in the population are unisexual, so all fireflies are attracted by all other fireflies • The attractiveness of the fireflies is proportional to their brightness which means fireflies with lower brightness move toward the ones with higher magnitudes of brightness • The concept of the fireflies' brightness depends upon the optimization task. To illustrate, for the maximization problems, the brightness is defined as the numerical value of the function while for the minimization tasks, the brightness is inversely proportional to the cost function

Fig. 3. Computation time versus population size for each optimization algorithm for the simulated homogenous reservoir model.

the produced lights determines how fast other fireflies move toward the particular firefly (Ali et al., 2014). The described process is the main process behind the Fireflies Algorithm (FA) used to solve the optimization process.

FA utilizes a population of fireflies, likewise any other populationbased algorithm, to obtain the best solution in the searching domain. In each iteration, the brightness of every i-th firefly is compared with any other j-th firefly, and it moves toward the firefly with higher magnitude of brightness using the following equation (Yang, 2009): 1 2 xi ¼ xi þ β0 e−γrij x j −xi þ α randn− ; 2

ð7Þ

where β0 is the firefly's brightness at the zero radius, γ is the lights absorption coefficient, randn is a random number which is selected from a uniform distribution at the range [0,1], and α is the

Fig. 4. RMSE over the derivative data plotted against the iterations for different population sizes (Homogenous model); a) GA algorithm, b) PSO algorithm, c) FA algorithm, and d) ICA algorithm.


99

Fig. 5. The derivative match for each optimization algorithm in the log-log plot, homogeneous reservoir model.

randomization parameter. The term rij represents the Euclidean distance between each i-j pair of fireflies and is defined as following: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u d X u 2 ð8Þ xi;k −x j;k ; r ij ¼ xi −x j ¼ t k¼1

2.4. Imperialist competitive algorithm (ICA) The Imperialist Competitive Algorithm or abbreviately ICA is developed based on the idea of the imperialist in which powerful countries influence other countries through expanding their power and political system. The purpose of imperialism was either to take advantage of the resources of the colonized country or to prevent the other imperialist to take control over the colonies (Atashpaz-Gargari and Lucas, 2007). The ICA algorithm was developed in 2007 by Atashpaz and Lucas, and it is one of the computational algorithms in nowadays systems (Biyanto et al., 2015). In this algorithm, each possible solution is assumed as a “country” with Nvar numbers of variables in the solution domain (Atashpaz-Gargari and Lucas, 2007): Country ¼ ½V 1 ; V 2 ; …; ; V Nvar ;

ð9Þ

After initialization of the countries, cost function will be evaluated for each country, and the whole population is divided into colonies and imperialists. Then, each imperialist is assigned a certain number of colonies. The number of specified colonies for each imperialist is determined by using the following equations (Atashpaz-Gargari and Lucas, 2007):

Fig. 6. Relative error over the pressure derivative data for the homogeneous reservoir model.

C n pn ¼ PN imp C i¼1

i

;

ð10Þ

100


Fig. 7. Regression plot and R-squared values for the derivative-free optimization schemes applied over a homogeneous reservoir model.

NC n ¼ roundðpn N col Þ;

ð11Þ

In the above equations, Cn and Ci are the normalized cost functions for the imperialists, pn is the normalized power of the n-th imperialist, Ncol is the total number of all the colonies in the current population, and NCn is the number of colonies assigned to the n-th imperialist. The colonies and their corresponding imperialist represent the n-th empire (Atashpaz-Gargari and Lucas, 2007). The ICA algorithm follows several major rules which are described in the following (Atashpaz-Gargari and Lucas, 2007; Biyanto et al., 2015; Talatahari et al., 2012): • Colonies are assimilated by the imperialist which means they move toward their imperialist as the algorithm evolves. This movement happens in direction of a vector in which the colony is at one side and imperialist is at the other end of the vector. The magnitude of the movement, dmov, is proportional to the distance between colony and imperialist: dmov U ð0; β dÞ;

• Total power of each empire is calculated through the following equation:

TC n ¼ f cost

ðimp;nÞ

PNCn þξ

i¼1

f cost NC n

ðcol;iÞ

;

ð13Þ

where TCn is the total cost for the n-th empire, f cost

ðimp;nÞ

is the cost of the ðcol;iÞ

imperialist within the empire, ξ is a number greater than zero, f cost are the magnitudes of cost function for the colonies in the empire, and NCn is the number of the total colonies. • If during the algorithm's evolution, the power of one of the colonies in the empire becomes greater than the power of the imperialist in the empire, the role of these two countries are exchanged. • Imperialist are always competing; as a result, the weakest empire is determined in each update, and it loses some of its colonies to the other empires.

ð12Þ

U refers to the uniform distribution, β is any number larger than one, and d is the distance between colony and imperialist. In the initial form of the ICA, also the movement vector is deviated for a small magnitude to look for the areas around the imperialist. • In addition to the assimilation, position of some countries is randomly changed by using a parameter called “revolution rate”. This process is called revolution, and it used to preserve randomness of the algorithm. This process is equivalent to the mutation in the GA algorithm.

Table 6 The relative error of the estimated parameter for the different optimization methods. Optimization algorithm

GA PSO FA ICA

Relative error, % k

C

S

12.24 12.26 26.99 12.29

13.96 13.89 27.05 13.93

35 34.89 68.42 34.88

Z. Arab Aboosadi et al. / Journal of Applied Geophysics 154 (2018) 93–107 Table 7 Fluid, rock and production data for the simulated dual porosity system. k, md C, bbl/psi S λ ω h, ft rw, ft Φ ct, psi−1 q, STBD μ, cp B, Rbbl/STB

101

Table 8 Range of the unknown variables used within the optimization procedure (Dual porosity model).

25 1E−3 4 4E−6 1E−2 80 0.2 0.25 5E−6 500 0.9 1.1

Parameter

Minimum

Maximum

k, md C, bbl/psi S λ ω

1 1E-4 -2 1E-8 1E-4

200 0.5 +10 1E-4 0.5

3. Results and discussion

2.5. Levenberg-Marquardt Algorithm (LM) The Levenberg-Marquardt algorithm (LM) is a derivative-based optimization algorithm which utilizes the derivative of the misfit function to search for the optimum variables in the solution space. In fact, LM can be considered as a hybrid algorithm combining the steepest descent and the Gauss-Newton optimization characteristics (Lourakis, 2005). It has become a standard algorithm in solving the non-linear least-square problems and is mostly used as the main optimization scheme, outperforming the gradient descent and conjugate gradient techniques (Lourakis and Argyros, 2004; Ranganathan, 2004). This algorithm is defined as following (Lourakis and Argyros, 2004):

To test accuracy of the employed algorithms, two simulated well test data belonging to homogenous and NFR (Naturally Fractured Reservoir) reservoirs were used in this study. Each simulated case is presented in a different section, and the corresponding statistical and graphical comparisons have been discussed separately for each model. After manipulating the parameters of the optimization algorithms and monitoring the error data, the optimum parameters for each algorithm were obtained, and these parameters are given in Tables 1 to 4 for each algorithm. For the complete description of the parameters used in this study, the interested readers might refer to (AtashpazGargari and Lucas, 2007), (Shi and Eberhart, 1999), (Goldberg and Holland, 1988), and (Yang and He, 2013). A brief description of each parameter, however, is provided in the nomenclature section at the end of this paper for the reader's information. 3.1. Homogenous reservoir

xiþ1 ¼ xi −ðH þ λIÞ−1 ∇f ðxi Þ;

ð14Þ

Where x refers to the optimization vector encompassing the optimization variables, H is the Hessian matrix determined at xi, ∇ is the gradient operator applied over the cost function f evaluated at the solution xi, I is the identity matrix, and λ is a scalar which its value is adjusted based on the LM algorithm. As a matter of fact, lower magnitudes of λ are selected when the error function goes down, indicating that the quadratic assumption on f(x) is reasonable. On the contrary, higher magnitudes of λ would be opted if the error function increases, which means we tend to rely on the quadratic approximate of the error function more than the previous iteration (Ranganathan, 2004). Obviously, the LM algorithm is an iterative technique which needs an initial point to update the solution reclusively. The selection of this initial point can influence the final solution greatly.

Fig. 8. The generated pressure derivative data for the simulated dual porosity model case.

The Laplace solution for a homogenous reservoir model is described using the following equation (Mavor and Cinco-Ley, 1979): pffiffi pffiffi pffiffi k0 s þ S sk1 s pffiffi pffiffi pffiffi ; pwd ðsÞ ¼ pffiffi pffiffi s sk1 s þ sC D k0 s þ S sk1 s

ð15Þ

Where s stands for the Laplace term, S is for the skin parameter, and CD is the dimensionless wellbore storage coefficient. Also, k0 is a zero degree modified Bessel function of the second kind, and k1 is the first degree modified Bessel function of the second kind. To test accuracy of the employed algorithms over the homogenous reservoirs, one simulated case was generated and used for this purpose. 3.1.1. The simulated homogenous model Opting the homogenous model and using the fluid, rock, and production data provided at the Table 5, the simulated derivative data

Fig. 9. RMSE versus iterations for four different optimization algorithms for the dual porosity fractured system.

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Fig. 10. The derivative match for different optimization algorithms over the dual porosity derivative data.

are plotted in Fig. 2. It is assumed that the test is long enough, so one can observe all different testing periods including initial wellbores storage and the late time infinite acting behavior. To investigate accuracy of the optimization algorithms with increases in the population size, the population size was increased from 20 to 100 and the RMSE was monitored against the iterations for each algorithm. In addition, computational time was measured for each population size and each algorithm. The computational time data are plotted in Fig. 3. As it is seen from Fig. 3, cost time increases almost linearly with increases in the population size. GA, PSO, and FA exhibit the same trend while the computational time for ICA is slightly lower than the other algorithms. The difference between the cost time of ICA and other algorithms becomes even more distinctive for larger population sizes. The RMSE data is plotted against iterations for different population sizes and for each optimization algorithm in Fig. 4. According to Fig. 4, the population size has almost no effect over the final response of the algorithm for GA, PSO, and ICA optimization algorithms. For these algorithms, the population size only influences the convergence rate.

For the FA algorithm, however, the population size not only influences the convergence of the algorithm, but it also affects the final results. For the FA algorithm, the optimum solution is obtained by using the population size of 100 in which the corresponding RMSE value is about 0.0822. For other algorithms, the lowest possible RMSE is about 0.0579. To graphically illustrate the best obtained match over the derivative data, the derivative match in a log-log plot is provided in Fig. 5. As it is seen from Fig. 5, despite for the FA algorithm, all other optimization

Table 9 The relative error over the estimated parameters of the dual porosity model for different optimization algorithms. Optimization algorithm

GA PSO FA ICA

Relative error, % k

C

S

λ

ω

3.96 0.74 27.23 1.06

3.19 1.04 5.30765 1.105653

16.23 0.14 111.00 1.29

16.25 8.35 928.42 8.49

193.24 159.75 4700.90 158.15


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Fig. 11. The overall and detailed error analysis for the derivative data belonging to the heterogenous reservoir model.

schemes accurately match the simulated derivative data. In fact, FA fails to precisely predict the hump in the wellbore storage zone. Two additional graphical illustrations including relative error and regression plot are provided in Figs. 6 and 7. Based on the evolution of the relative error data in Fig. 6, there is an excellent match between the predicted and real data till t = 0.02 h.

Then, a deviation from the real data is observed which lasts till the end of the test. Therefore, all employed algorithms successfully predict the data in the wellbore storage zone and fail to address the pressure response during the transitional wellbore storage and infinite acting behavior. Among the examined algorithms, FA exhibits the worst fitting characteristics by overestimating the derivative data in the transition

Fig. 12. The regression plot on the predicted derivative data for various stochastic optimization schemes.

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Z. Arab Aboosadi et al. / Journal of Applied Geophysics 154 (2018) 93–107 Table 10 Average relative error for the Levenberg-Marquardt algorithm for homogenous and heterogonous fractured models after using various initial points.

Fig. 13. Relative error data against the regression case obtained by the LM algorithm for the simulated homogenous model.

zone and underestimating the data in the infinite acting section. The relative error data for the remained algorithms approximately overlap each other, revealing a similar trend for the error data. The relative error data for all algorithms are almost located in the range [−30%, +30%], with the maximum overestimation of about +33% by the PSO algorithm and the maximum underestimation of about −31% by the FA algorithm. Regression plots and the corresponding R-squared values are provided in Fig. 7 for each optimization algorithm. The R-square, or coefficient of determination, represents the correlation existing between the real and predicted pressure data and is defined as following: PnS

2 p0Exp: ðiÞ−p0Model ðiÞ R ¼ 1− 2 ; PnS 0 0 i¼1 pExp: ðiÞ−pExp: i¼1

ð16Þ

where pExp.′ is the experimental or real derivative data point, pModel′ is the predicted pressure derivative record by the employed algorithm, and p0Exp: is the average of the real derivative data. Also, the upper limit of nS refers to the number of data points used for the regression plot. Based on Fig. 7, most of the algorithms come up with the R-squared values greater than 0.9 except for the FA algorithm which has the R2 =

Well-reservoir parameters

RE, % (homogenous model)

RE, % (dual porosity model)

k, md C, bbl/psi S λ ω

31.84 413.91 46.18 N/A N/A

42.73 47.68 72.32 920.47 595.80

0.8949. GA, PSO, and ICA have the same R2 value of 0.9331 which denotes they have equal capability in prediction of the real derivative data through the nonlinear regression analysis. Further statistical analyses of the results are provided in the Table 6. According to the Table 6, the highest error over all the estimated variables belong to the FA while for the remained algorithms the errors for permeability, wellbores storage, and skin factor are about 12, 14, and 35%, respectively. The increased error for the skin factor is due to the fact that derivative data are not a strong function of the skin factor, contrary to the permeability and wellbore storage coefficient. As a result, changing skin factor through the nonlinear regression may not tremendously influence the derivative data at the logarithmic plot. 3.2. Dual porosity reservoir For the second reservoir model, a dual porosity fractured model comprised of matrices and fractures was used. In this simplified model, only fractures communicate with the wellbore while matrices represent the source terms which transfer the stored fluid to the nearby fractures. The Laplace solution for the dual porosity model with the PSS interporosity flow is as following (Da Prat, 1990):

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi sf ðsÞ þ S sf ðsÞk1 sf ðsÞ k0 h pffiffiffiffiffiffiffiffiffiffi pwd ðsÞ ¼ npffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiio ; s sf ðsÞk1 sf ðsÞ þ sC D k0 sf ðsÞ þ S sf ðsÞk1 sf ðsÞ

ð17Þ where s stands for the Laplace variable while S represents the skin factor; CD is the dimensionless wellbore storage coefficient; k0 and k1 stands, respectively, for the zero degree and first degree modified Bessel functions of the second kind; f(s) is the Laplace function depending upon two important characteristics of the dual porosity systems, i.e. storativity ratio (ω) and interporosity flow coefficient (λ): f ðsÞ ¼

ωð1−ωÞs þ λ ; ð1−ωÞs þ λ

ð18Þ

The dimensionless parameters of λ and ω characterize the fractured reservoirs and make them distinct form the homogenous reservoirs. ω is called the storativity ratio and it is defined as following: ω¼

ðФct Þ f ðФct Þ f þ ðФct Þm

;

ð19Þ

In this equation, Ф is the reservoirs porosity, ct is the total compressibility factor, and indices f and m stand for the fracture and matrix, respectively. λ is the interporosity flow coefficient because it is a measure of the fluid flow connectivity between matrix and fracture. This parameter is related to the reservoir's petrophysical parameters through the following equation: Fig. 14. Relative error data against the regression case obtained by the LM algorithm for the simulated heterogenous fractured model.

λ ¼ α m r 2w

km ; kf

ð20Þ


where αm is the matrix shape factor, rw is the wellbore radius, km is the matrix permeability, and kf is the fracture permeability. One simulated test was generated by using a commercial well test software and was used to test accuracy of the developed stochastic technique over the heterogonous reservoir models. The dual porosity model is very important in the oil and gas industry because many fractured reservoirs within the world exhibit the same pressure drop as the dual porosity model. Therefore, it is very important to check applicability of the stochastic optimization algorithms over these types of reservoirs.

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magnitudes of the coefficient of determination are even higher than those obtained for the homogenous reservoir model. Therefore, it might be deduced that although the error over the estimated parameters is higher for the fractured reservoir, overall, there is a better match between the real and predicted data for the fractured model. In fact, the higher error over the estimated parameters for the heterogenous fractured model might be attributed to the very large magnitudes of the error data generated in the matrix-fracture transitional zone. 3.3. LM (Levenberg-Marquardt) algorithm implementation

3.2.1. The simulated dual-porosity model A commercial well test software was used to simulate the test data by using the fluid, rock, and production data provided at the Table 7. A quartz crystal pressure gauge was used to generate the pressure data, and the noise with magnitude of 0.01 psi was added to the simulated data to emulate the actual field data. The simulated derivative data are plotted in Fig. 8. A number of the unknown reservoir variables are increased to five for the dual porosity system, and the range of these variables in the optimization algorithms are given at the Table 8. Initially, the maximum iteration of 100 was used for the optimization, but it was observed that it is not sufficient for the convergence of the algorithms. Consequently, the maximum iteration was increased to 200 to ensure that the stochastic algorithms would not immaturely converge. RMSE versus iteration is plotted in Fig. 9 for all algorithms. From the error data, it is concluded that except for the FA algorithm, other algorithms end up almost with the same final RMSE. Among the different algorithms, PSO and ICA portray the rapid convergence while GA converges to its final solution at the iterations about 80. The final RMSE value for GA, PSO, FA, and ICA algorithms are, respectively, 0.0639, 0.0522, 0.2824, and 0.0522. This shows that PSO and ICA end up with the lowest magnitudes of RMSE. To illustrate the accuracy of each optimization algorithm, a graphical match is provided in Fig. 10 and the relative error data are presented in the Table 9. According to the Fig. 10 and Table 9, PSO and ICA end up with the lowest relative errors over all of the estimate reservoir parameters while the FA algorithm leads to the highest error over the estimated reservoir parameters. However, although error for most of the parameters is lower than 20% for GA, PSO, and ICA, the error is higher than 100% for the fracture storativity ratio, i.e. ω, for all employed algorithms. Despite the highest error over the fracture storativity ratio, these algorithms come up with a fair graphical match over the derivative data. FA algorithm, however, cannot accurately capture the U-shape in the derivative data and the resultant match is much more like the derivative response of a homogenous model. In addition to the utilized graphical and tabular analyses, the plots of the relative error and regression plots are used for further examinations of the results. The relative error values are plotted in Fig. 11 against the testing time. This plot demonstrates that except for the matrix-fracture transition zone, i.e. t = 0.1 to 1.0 h, a good match is obtained between the real and predicted derivative data. In the transition zone, PSO and ICA demonstrate the lowest possible error while FA shows a very high relative error. For better visualization, the y-axis is rescaled between −0.4 to 1.0. As it is seen from the detailed plot in Fig. 11, if we ignore the error data for the transition zone, then, the higher overestimation belongs to the ICA algorithm with the relative error of about +0.115 while the highest underestimation belongs to the FA with the error magnitude of −0.264. It shows that the highest possible error happens in the matrix-fracture transitional zone, and derivative-free algorithms fail to handle the noisy data in this region despite the successful match for the wellbore storage and the late-time infinite acting behavior. The regression plots for the dual porosity model are provided in Fig. 12 for each algorithm. Obviously, all off the four algorithms have very high R-squared values for the dual porosity model. The R2 value for all algorithms is greater than 0.98 which demonstrate existing a high correlation between the real and predicted derivative data. The

In this section, the well-known derivative-based LM algorithm has been used to interpret the simulated derivative data for both homogenous infinite acting and the heterogenous fractured model. Contrary to the aforementioned stochastic algorithms, LM demands the derivative of the cost function and an initial point to initialize the searching process. Therefore, selection of the initial point can greatly affect the final outcomes of the optimizations. With this in mind, we ran the LM for 10 different times and with ten different initial points. The relative error of the estimated reservoir parameters was calculated for each specific reservoir property for both of the examined models. The LM algorithm was applied over the simulated homogenous data with using different initial points. The final outcomes of the algorithm for each distinct initial guess were obtained and the corresponding RE (Relative Error) was calculated for each case. These RE data are plotted in Fig. 13 for the homogenous model. It seems that by altering the initial guess of the LM algorithm, the RE of the LM changes hugely; the minimum RE for all the estimated parameters is 0.15% and its maximum magnitude foes far beyond 100%. Among the estimated reservoir parameters for the homogenous model, the minimum and maximum RE belong to the wellbore storage coefficient. The C parameter has the minimum of 0.15% and maximum of 2864% which shows the huge impact of the initial point on the estimated reservoir parameters. Similarly, LM algorithm was applied over the simulated dual porosity data, and the RE data are plotted in Fig. 14 for various regression cases. Likewise, a large dependency to the initial point is observed for the dual porosity model. Among the estimated parameters, ω has the largest minimum error for all the regression cases; its RE lies between 51% and 900%, indicating inefficiency of the LM algorithm to estimate this crucial reservoir parameter. Since ω represents the storativity of the fractures with respect to the total matrix-fracture system, any miscalculation of ω will have huge effect on the reserve estimation of the reservoir. Similar to the homogenous model, the wellbore storage has the lowest RE of about 0.3% while λ has the highest RE magnitude which is more than 2000%. λ represents the fluid connectivity between matrix and fracture; therefore, erroneous estimations of λ will lead to inaccurate predictions of the production data. The average errors of all the runs for both reservoir models have been provided in Table 10. It seems that on average, the LM algorithm ends up with the maximum RE of about 400% for C and the minimum RE of about 32% for the reservoirs permeability, for the homogenous reservoir model. For the dual porosity model, however, the minimum and maximum of the RE data are 42% and 920%, respectively, which are extremely high compared with the best results achieved for the meta-heuristic algorithms (Tables 6 and 9). As a result, the ICA and PSO algorithms are more effective than the conventional LM algorithm as they do not need providing the initial point, and they also come up with lower magnitudes of RE. 4. Conclusion Based on the graphical and statistical analysis, it was concluded that GA, PSO, and ICA algorithms can successfully estimate the reservoir parameters for the homogenous model with relative errors between 12 and 35% for the estimated parameter; FA fails to accurately estimate the unknown reservoir variables. For the dual porosity heterogenous

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model, GA, PSO, and ICA end up with the relative errors between 0.74 and 16.25%, except for ω. Overall, for the complicated dual porosity model, PSO outperforms all other algorithms while ICA stands in the second place. Contrary to the mentioned algorithms, FA cannot handle the fractured reservoir model and ends up with large magnitudes of relative error. Lower magnitudes of RE for PSO and ICA might attribute to the fact that these algorithms utilize both local and global searching strategies. PSO gets benefit from the individuals local memory while ICA partitions the searching space into different empires with colonies moving toward the imperialist of each empire. This allows them to efficiently look for the best possible solution within the searching domain, especially when the number of unknown reservoir parameters increases, and the shape of the derivative curve becomes more complicated. Analyzing the residual and regression plots, it was unveiled that for the homogenous model, most of the residuals happen in the wellbore storage transitional and infinite acting regions; for the heterogonous fractured case, the matrix-fracture transition zone has the highest magnitudes of the relative error. In general, the coefficient of determination is greater for the fractured model compared with the homogenous model. Cost time data revealed that ICA algorithm is faster than the remained algorithms. Overall, it was observed that PSO and ICA algorithms outperform the other algorithms while ICA might look more favorable because of the smaller magnitude of the computational time. For comparison purposes, a Newton-like optimization algorithm, namely LM, was used also, and its sensitivity to the initial point was examined. It was observed that initial point greatly influences the final solution obtained by LM. Average RE data showed that LM algorithm ends up with RE values much greater than the RE values obtained for the stochastic algorithms. Overall, since the stochastic algorithms are not influenced by the initial point selection and do not require derivative calculations, these algorithms can be a better tool for the nonlinear well testing regression, provided that their internal parameters have been well optimized. References Adibifard, M., Tabatabaei-Nejad, S., Khodapanah, E., 2014. Artificial Neural Network (ANN) to estimate reservoir parameters in naturally fractured reservoirs using well test data. J. Pet. Sci. Eng. 122, 585–594. Adibifard, M., Bashiri, G., Roayaei, E., Emad, M.A., 2016. Using particle swarm optimization (PSO) algorithm in nonlinear regression well test analysis and its comparison with Levenberg-Marquardt algorithm. Int. J. Appl. Metaheuristic Comput. 7, 1–23. Alajmi, M.N., Ertekin, T., 2007. The development of an artificial neural network as a pressure transient analysis tool for applications in double-porosity reservoirs. Asia Pacific Oil and Gas Conference and Exhibition. Society of Petroleum Engineers. Aleardi, M., 2015. Seismic velocity estimation from well log data with genetic algorithms in comparison to neural networks and multilinear approaches. J. Appl. Geophys. 117, 13–22. Ali, N., Othman, M.A., Nor Husain, M., Misran, M.H., 2014. A review of firefly algorithm. J. Eng. Appl. Sci. 9, 1732–1736. Al-Kaabi, A., Lee, W., 1990. An artificial neural network approach to identify the well test interpretation model: applications. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Allain, O., Houze, O., 1992. A practical artificial intelligence application in well test interpretation. European Petroleum Computer Conference. Society of Petroleum Engineers. Arora, S., Singh, S., 2013. The firefly optimization algorithm: convergence analysis and parameter selection. Int. J. Comput. Appl. 69, 48–52. Asoodeh, M., Bagheripour, P., 2012. Prediction of compressional, shear, and Stoneley wave velocities from conventional well log data using a committee machine with intelligent systems. Rock Mech. Rock. Eng. 45, 45–63. Atashpaz-Gargari, E., Lucas, C., 2007. Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition, Evolutionary computation, 2007. CEC 2007. IEEE Congress on. IEEE, pp. 4661–4667. Athichanagorn, S., Horne, R.N., 1995. Automatic parameter estimation from well test data using artificial neural network. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Attwa, M., Akca, I., Basokur, A.T., Günther, T., 2014. Structure-based geoelectrical models derived from genetic algorithms: a case study for hydrogeological investigations along Elbe River coastal area, Germany. J. Appl. Geophys. 103, 57–70. Bagheripour, P., Asoodeh, M., 2013. Fuzzy ruling between core porosity and petrophysical logs: subtractive clustering vs. genetic algorithm–pattern search. J. Appl. Geophys. 99, 35–41.

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