parameter identification of multistage fracturing horizontal well...

Research ArticleParameter Identification of Multistage Fracturing HorizontalWell Based on PSO-RBF Neural Network

Rongwang Yin12 Qingyu Li 1 Peichao Li3 and Detang Lu 1

1School of Engineering Science University of Science and Technology of China Hefei 230026 China2Department of Basic Teaching and Experiment Hefei University Hefei 230601 China3School of Mechanical and Automotive Engineering Shanghai University of Engineering Science Shanghai 201620 China

Correspondence should be addressed to Qingyu Li liqyustceducn and Detang Lu dtluustceducn

Received 30 April 2019 Revised 14 March 2020 Accepted 18 June 2020 Published 3 July 2020

Academic Editor Cristian Mateos

Copyright copy 2020 Rongwang Yin et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to more accurately identify multistage fracturing horizontal well (MFHW) parameters and address the heterogeneity ofreservoirs and the randomness of well-production data a new method based on the PSO-RBF neural network model is proposedFirst the GPU parallel program is used to calculate the bottomhole pressure of a multistage fracturing horizontal well Secondmost of the above pressure data are imported into the RBF neural network model for training In the training process theoptimization function of the global optimal solution of the PSO algorithm is employed to optimize the parameters of the RBFneural network and eventually the required PSO-RBF neural network model is establishedird the resulting neural network istested using the remaining data Finally a field case of a multistage fracturing horizontal well is studied by using the presentedPSO-RBF neural network modele results show that in most cases the proposedmodel performs better than other models withthe highest correlation coefficient the lowest mean and absolute error is proves that the PSO-RBF neural network model canbe applied effectively to horizontal well parameter identification e proposed model has great potential to improve theprediction accuracy of reservoir physical parameters

1 Introduction

Reservoir description is an important means of estimatingthe physical properties of a reservoir in the petroleum in-dustry ese reservoir physical parameters require bettermethods to improve the prediction accuracy to furtherenhance the subsequent success rate of exploration anddevelopment Multiphase flow in oil and gas fields usuallyrefers to the simultaneous flow of multiple fluids in a res-ervoir which generally occurs during oil and gas production[1 2] For the old oilfields oil and gas development hasexperienced stages such as self-spraying secondary andtertiary oil recovery which makes the process of the oil andgas production complex and challenging In order to dealwith this complexity a lot of studies have been done at homeand abroad and the reservoir numerical simulations havebeen introduced to estimate the multiphase flow charac-teristics of oil natural gas and water [3 4]

In the petroleum industry well-logging data have beencommonly used to predict formation parameters such aspermeability formation boundaries and fracture lengthPermeability is defined as a measure of the ability of a porousmedium to allow fluid to pass through it e concept ofpermeability is important for determining accurate reservoirdescription simulation and management erefore priorto any modeling or calculation the permeability of porousmedia must be determined e earliest method of per-meability prediction is the empirical correlation betweenpermeability and other petrophysical properties such asporosity and water saturation ese correlations have madesome successes in sandstone reservoirs However it is notapplicable to heterogeneous formations

In recent years the artificial neural network (ANN) hasbeen applied to the reservoir field solving many highlycomplex nonlinear problems [5] e ANN is considered tobe a nonlinear tool that can predict complex nonlinear

HindawiScientific ProgrammingVolume 2020 Article ID 6810903 11 pageshttpsdoiorg10115520206810903

system behavior well It is also used to solve many differenttypes of problems in petroleum engineering such as res-ervoir characterization permeability prediction verticalmultiphase flow downhole pressure prediction and dis-solved gas driven flow prediction [6ndash8]

e ANN is considered to be a method that can effec-tively replace traditional techniques of predicting perme-ability from well-logging data e disadvantage of the ANNapplication is the requirement to identify the structure of thenetwork In order to overcome the shortcoming in thisstudy an MFHW parameter identification model based onthe PSO-RBF (particle swarm optimization PSO radialbasis function RBF) neural network is proposed e mainpurpose is to seek a better prediction of reservoir propertiese parameters of the radial basis neural network aredesigned by using the particle swarm algorithm to obtainhigher prediction accuracy e parameter identificationmodel based on the PSO-RBF neural network has highlearning ability and flexibility when encountering complexproblems which makes the proposed model applicable tomost engineering practices

2 Review of Related Works

In the past few decades researchers have done a lot of workto study reservoir parameter estimation problems such aspermeability porosity and fracture length Taking the es-timation of permeability as an example there are currentlythree main methods core analysis well test analysis andartificial intelligence methods

e most accurate method of permeability prediction isto perform core analysis in the laboratory by applyingDarcyrsquos law Among the different techniques the perme-ability obtained from the core analysis is more efficient thanthat obtained by other methods and the permeability fromthe core analysis can be used to verify other estimationmodels However the core analysis process is time con-suming and difficult to use widely

Another method of obtaining reservoir permeability iswell test analysis e analytical solution is solved by usingthe porous flow equation and some rock properties such aspermeability and porosity can be then obtained inverselyfrom the pressure curve e data available after well testinghelp petroleum engineers calculate formation permeabilityskin factor and wellbore storage [9] Due to the cost oftesting well testing is not a good solution for any reservoirsthat have been developed

ANNs are increasingly being employed to predict res-ervoir properties based on geophysical well-logging data[10 11] Mohaghegh et al pointed out that neural networksare powerful tools for identifying the relationship betweenpermeability and well-logging data [12] In addition Ami-nian et alrsquos research shows that artificial neural networks canbe used to predict formation permeability even in highlyheterogeneous reservoirs [13 14] Based on the artificialneural network and particle swarm optimization algorithmAhmadi et al proposed a method for predicting horizontalwell productivity in a pseudosteady state and the experi-mental results achieved a high fitting accuracy (lt082) [15]

Zhang et al proposed a hybrid algorithm that combines aparticle swarm optimization algorithm with a back-propa-gation algorithm to train the weights of a feed-forwardneural network e results show that the hybrid algorithmis superior to the adaptive particle swarm optimization al-gorithm and backpropagation neural network (BPNN) inboth convergence speed and convergence accuracy [16]Based on the RBF and BPNN Chen developed two watertreatment models e model has been trained and checkedseparately from actual data from the water plant Comparedwith the model based on the BPNN the model based on theRBF neural network has the characteristics of better ap-proximation and faster convergence [17] Based on themodels of the BPNN and RBFNN (the radial basis functionneural network RBFNN) Yang et al proposed a model forcement decomposition and calcination process e resultsshow that the model based on the RBFNN can achieve veryhigh fitting results [18]

Saemi et al improved the predictive performance ofneural networks by using the GA to optimize the networkparameters of the ANN [19] In addition Kaydani et alrsquosfindings indicate that the GA and ANN network structuresdesigned by subregion can predict the permeability of aheterogeneous reservoir in Iran [20] Tahmasebi et alproposed four methods for predicting permeability of dif-ferent neural network structures and statistically comparedthe results obtained Finally a new method of the modularneural network (MNN) was obtained [21] By combiningcuckoo optimization algorithm (COA) particle swarmoptimization algorithm (PSO) imperialist competitive al-gorithm (ICA) and LevenbergndashMarquardt algorithm (LM)Kaydan et al proposed a new method to estimate thepermeability [22]

Rough set theory can be successfully applied to theprediction of permeability of porous media [23ndash28] Byestablishing a rough set model Ilkhchi et al successfullypredicted the permeability of an offshore gas field in Iran[29]

Support vector regression (SVR) based on the principleof structural risk minimization is a promising machinelearning method rough experimental comparisonsGholami et al found that the SVR method has faster speedand higher accuracy in predicting reservoir permeability[30]

3 Problem Formulation

31 Optimized Design After fracturing tight sandstonereservoirs the physical properties of the reservoirs arecharacterized by extremely low porosity a wide range ofpermeability changes and complex pore-permeability re-lationships erefore conventional production predictionmodels often fail to meet the requirements in terms ofprediction accuracy

Effective algorithms for complex nonlinear problems areessentially optimization problems e so-called optimiza-tion refers to the problem of seeking the maximum orminimum value of the given objective function according tothe change of design variables under the condition of

2 Scientific Programming

meeting some certain constraints It can be described by thefollowing equation

min f(x) ormaxf(x)

st gi(x) 0 i 1 2 hj(x)ge 0 or hj(x)le 0

j 1 2

(1)

where x isin S and S is the design parameter space that meetsthe limiting conditions called the solution space f(x)

f(x1 x2 xn) is the objective function to be optimizedx (x1 x2 xn)T x1 x2 xn is the optimization de-sign variable and minf(x) ormaxf(x) is the minimum ormaximum value of the objective functiongi(x) 0 i 1 2 are the equality constraints andhj(x)ge 0 or hj(x)le 0 denotes the inequality constraints

Taking MFHW as the studied object the PSO-RBF neuralnetwork was used to predict permeability and fracture half-length e optimization objective function is the minimumRMSE (root mean square error) value of permeability andhalf-length of fracture e input variables of the network arethe formation pressure the number of fractures stratumthickness the well storage constant the well storage skin andso on and the output variables are the RMSE values

32 Data Set e data set describes the characteristics andbehavior of the reservoirrsquos input and output parameters andis the training data set for the reservoir model It is obtainedusing static and dynamic data such as porosity permeabilitypressure and reservoir production values For a given sourceunit according to the line source superposition principlethe pressure distribution at any point (xD yD and zD) in theformation space shall be

pDi j xD yD zD tD( 1113857

1113946tD

0qDij(τ)Gx D xD τD( 1113857Gy D yD τD( 1113857GzD zD τD( 1113857dτ

(2)

e coefficient matrix is expressed as followsM11 M12 middot middot middot M1NMi

minus1

M21 M22 middot middot middot M2NMiminus1

⋮ ⋮ ⋮ ⋮ ⋮

MNMi1 MNMi2 middot middot middot MNMiNMi⋮

1 1 middot middot middot 1 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

qD11

qD12

⋮

qDN Mi

pDw f

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0

0

⋮

0

1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)

As for the definitions of various symbols appearing inequations (2) and (3) the reader can refer to our previouspaper [31] Note that the formation pressure data of MFHWare obtained by the GPU parallel calculation method pro-posed in the reference [31] All training data sets are dividedinto two parts before the PSO-RBF training begins One isthe input data set and the other is the output data set einput and output data sets are normalized within a specificrange Before the training begins a standard normalizationfunction is used to limit the range of the input and output

data sets between minus1 and 1 and the mathematical function isgiven in the following equation

y e2x minus 1e2x + 1

(4)

During PSO-RBF training the training data set is di-vided into three parts training validation and testingTraining data are used in the training and validation data arealso used during training However it should be mentionedthat validation data are used to check the network learninginstead of training the network

4 PSO-RBF Model

e advantage of ANNs over other conventional techniquesis the ability to perform complex and highly nonlinear tasksaccurately and swiftly In most previous works related toreservoir models the researchers used the BPNN to con-struct reservoir models However the BPNN has a problemof falling into local minimum during training time In mostcases the network does not reach the global minimum tofind the minimum error value e global optimizationability of the PSO algorithm makes the radial basis neuralnetwork model based on the PSO algorithm have no localminimum problem

41 PSO Algorithm Inspired by the movement of the birdsthe PSO algorithm was developed by Kennedy [32] In thisapproach every possible solution is considered as a particleEach particle is characterized by its position and velocitye position of the particle is defined in the hyperspace andits dimension is equal to the number of nonoptimized pa-rameters Several particles are initially defined in hyperspacewhich iteratively change their positions to determine the bestposition e fitness of a particle is determined by a fitnessfunction such as RMSE is algorithm is similar to howbirds search for food in a wide area

During the iterative execution of the algorithm thevalues of pbest and gbest are constantly updated pbest isdefined as the optimal position of the particle in the hy-perspace determined by the fitness value gbest is the overallbest position for all particles In each iteration step the speedis updated first and then the location is updated e par-ticles are accelerated to pbest and gbest by updating the speed

vik+1 wi middot v

ik + c1 middot r1 middot p

ibest minus p

ik1113872 1113873 + c2 middot r2 middot gbest minus p

ik1113872 1113873

(5)

where vik+1 is the speed of the next iteration wi is the inertia

weight vik is the current speed r1 and r2 are random

numbers and pik is the current position of the particle

Update the position of the particle using the followingequation

pik+1 p

ik + v

ik+1 middot Δt (6)

e initial position and velocity of each particle arerandomly distributed After initializing the position andvelocity of all particles the fitness is calculated In thesubsequent steps the position and velocity are iteratively

Scientific Programming 3

updated by the local best parameters and the global bestparameters as shown in Figure 1

e entire flowchart can be divided into four partsnamely initialization fitness calculation status check andupdating of speed and position

42 Radial Basis Function Neural Network RBFNN has asmall number of hidden layer neurons and are used todevelop networks with good generalization capabilities[33] RBFNN is considered to be a special type of artificialneural network because its architecture requires only ahidden layer which allows the input space to be repre-sented in new spaces of different hidden layer neuronsestructure is shown in Figure 2 In the training process ofRBFNN it appears as a linear model because all hiddenneuron centers and calculations are fixed e RBFNNhidden layer neurons perform a nonlinear transformationand map all inputs into the new input space e outputlayer is considered as a linear converter and is applied tothe new input space e performance of RBFNN can bedetermined by adjusting the center (width) of the hiddenlayer neurons and there is no specific formula to choosethe width of the radial basis function [34] RBFNN hasbeen widely used in system prediction pattern recogni-tion speech recognition and adaptive control It has alsobeen used to solve the problems of oil and gas fields egoil and gas ratios of reservoirs electromagnetics resis-tivity log data inversion log data prediction seismic

properties and nonlinear relationships between reservoirproperties and seismic properties [35ndash37]

e growth and overall structure of RBFNN is affectedby the RBF e RBFNN input is directly connected to eachof the basic functions and produces an output at is

ϕi exp minus(x minus u)T(x minus u)

σ21113890 1113891 (7)

where x represents the input data of the network u is thecenter of the radial basis function and σ is the width of theRBF (σ gt 0) First the hidden layer neurons are calculatedbased on the radius of the radial basis function and thentransmitted to the output layer Here the sum of the

Get the global bestvalue

Initialize the position and speedof all particles

Calculate the fitness value of theith particle

Replace local optimum

Is the current value better thanthe local best value

Is the current value better thanthe global best value

i le n

Replace global best value

The global best value meetsthe requirements

The maximum number ofiterations reached

i = i

+ 1

No

No

Yes

Yes

NoYes

Yes Yes

Upd

ate t

he p

ositi

on an

dsp

eed

of p

artic

les

No

No

Figure 1 Flowchart of PSO algorithm

x1

x2

x3

xn

Hidden layer Output layer

o1

0j

Input layer

middotmiddotmiddot

middotmiddotmiddot

Figure 2 RBFNN structure


products between the hidden layer neurons and the weightvectors is calculated to obtain the final network output ynthat is

yn 1113944m

m1wi middot ϕi (8)

43 PSO-RBF Neural Network Method e prediction ac-curacy of the RBFNNmainly depends on the central vector u

of the radial basis kernel function the radius σ of the radialbasis and the connection weight wi between the output layerand the hidden layer e traditional RBF neural networkuses local information based on parameter space to setparameters which results in the values of u σ and wi beinglocal optimal solutions rather than global optimal solutionsIn view of these defects in the RBFNN this work employs thePSO algorithm to optimize the traditional RBFNN whenidentifying MFHW parameters e RBFNN parametersoptimized by the PSO algorithm are global optimal pa-rameters thus avoiding the problem of low reliability ofRBFNN learning e specific flowchart of the PSO-RBFalgorithm is shown in Figure 3

PSO optimization RBFNN parameters are divided intotwo steps the first step is to determine the center value andwidth of the basis function and the second step is to de-termine the connection weight between the hidden layer andthe input layer In the optimization process the data ob-tained by GPU parallel computing are used to train andverify the network

e algorithm flow of the first step is as follows(1) Collection of reservoir model data samples (2)

cluster analysis of samples to generate center u and width σof basis functions (3) initialization of the particle swarmalgorithm using u and σ (4) calculation of each particle rootmean square error (RMSE) (5) updating local optimal so-lution and global optimal solution (6) updating particleposition and velocity (7) repeating steps (4ndash6) until theaccuracy requirement or number of iterations is reachedand (8) obtaining the center u and width σ of the basisfunction

e algorithm flow of the second step is as follows(1) Calculation of the output of the hidden layer (2)

initialization of the weight wi and reinitialization of theparticle group (3) calculating the cumulative error of eachparticle (4) updating the local optimal solution and theglobal optimal solution (5) adjusting the position and ve-locity of the particles (6) repeating steps (3ndash5) and (7)obtaining the parameter wi of the RBF neural network

5 A Field Case Study

51 Computation e bottomhole flow pressure is analyzedand calculated by using the measured data of multistagefracturing horizontal wells in an oilfield in western Chinae three-dimensional PEBI grid of fracturing of under-ground horizontal wells is shown in Figure 4 e reservoirstratum is homogeneous the fluid flow satisfies Darcyrsquos lawthe horizontal and vertical permeability of the fracture are

different and the horizontal well is fractured in a closedrectangular reservoir Using the GPU-based MFHW bot-tomhole pressure calculation method proposed in ourprevious work [31] the bottomhole flow pressure of thehorizontal well is calculated and the data set is obtained

52 Training Model e various parameter settings in theparticle swarm algorithm are described in detail in thesensitivity study section e RBFNN consists of 5 inputnodes and 1 output node e PSO optimization stepspresented in the previous section are used to optimize thetraining of hidden layer neurons 4 8 12 16 20 24 and 28respectively e test results are shown in Figure 5 As canbe seen from the figure when the number of hidden layerneurons is 16 the error is the lowest erefore an RBNnetwork structure of 5-16-1 is obtained

Collecting the reservoir data samples

Clustering generates the basis and width ofthe basis function and initializes the particle

swarm

Calculate RMSE

Meet therequirements

Particleswarm

optimization

Get thewidth and radius

Calculate the hiddenlayer output

Initialize the weightsand particle swarm

optimization

Calculate cumulativeerror


Obtain the networkparameters

Particleswarm

optimization

Figure 3 PSO-RBF-based algorithm flowchart


53 Prediction After obtaining the RBF neural networkparameters through the PSO algorithm the network needsto be tested To this end the remaining data are importedinto the optimized RBF neural network to obtain the pre-dicted values of the bottomhole permeability and the frac-ture length of MFHW

In order to further detect the performance of the net-work the PSO-RBF network structure proposed in the paperis compared with the PSO-BPNN and the SVR algorithme result is shown in Figure 6 e permeability predictionresults and errors of the three algorithms are shown inTable 1 As can be obtained from Table 1 the RMSE of thePSO-BPNN is 0332 the RMSE of the SVR algorithm is0308 and the RMSE of the PSO-RBF is 0178 e fracturelength is predicted by three algorithms and the result isshown in Figure 7e fracture length prediction results anderrors of the three algorithms are shown in Table 2 As can beobtained from Table 2 the RMSE of the PSO-BPNN is 3776the RMSE of the SVR algorithm is 3319 and the RMSE ofthe PSO-RBF is 2250 It can be seen from the comparisonresults that the RMSE of the PSO-RBF algorithm is much

lower than that of the other two algorithms showing that thealgorithm has higher prediction accuracy and betterperformance

6 Sensitivity Study

In the particle swarm optimization algorithm the pop-ulation number particle speed inertia weight learningfactor and random number all have certain effects on theperformance of the algorithm According to the studies athome and abroad the improvement of the PSO algorithmmainly focuses on the inertia weights and learning factors[38 39] e inertia weight plays a major role in theconvergence of the particle swarm algorithm e largerthe value of wi the stronger the global search ability andconversely the stronger the local search capability c1 andc2 are the maximum step sizes of the particles to adjust tothe individual optimal or group optimal direction

Figure 4 Schematic diagram of horizontal well fracturing PEBI grid

030

028

026

024

022

020

018

016

014

012

010

RMSE

0 4 8 12 16 20 24 28Hidden layer neurons

Figure 5 RMSE of different hidden layer neuronsMeasured valuePSO-BPNN algorithm PSO-RBF algorithm

SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

Figure 6 Prediction of permeability of different algorithms


Table 1 Permeability prediction results of different algorithms

Fracture number Measured value PSO-BPNN algorithm SVR algorithm PSO-RBN algorithm1 1549 1463 1294 16632 0011 0085 0094 00563 2837 2668 2762 27934 4671 5172 4854 47015 1437 1660 1687 15326 1452 1887 1854 15647 1818 2024 1687 19008 1756 2034 2114 19029 19500 19060 18965 1922510 17539 17837 17365 1738511 12632 13092 12394 1306512 11966 11674 12321 1221113 13389 13157 13256 1339514 5250 5638 5054 534115 5633 5159 6254 5406

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)

Measured valuePSO-BPNN algorithm PSO-RBF algorithm

SVR algorithm

Figure 7 Prediction of fracture length in different algorithms

Table 2 Prediction results of fracture lengths for different algorithms



respectively When the learning factor is small theparticles may linger away from the target area whilethe particles can quickly move toward the target area oreven exceed the target area when the learning factor islarge

e values of the inertia weight and learning factor in theparticle swarm algorithm can change during the optimi-zation process During the search process the inertia weightis set to change with the number of iterationse formula ofinertia weight is expressed by the following equation

Table 3 Particle swarm algorithm parameters

Algorithm Parameter name Value

Classic PSO

Inertia weight ωmax 12ωmin 04Learning factor c1 c2 2

Position and speed range [minus1 1]

Maximum number ofiterations and population

sizeimax 800N 50

ImprovedPSO

Inertia weight ωi ωmax minus i times (ωmax minus ωmin) (imax) ωmax 12ωmin 04

Learning factor c1 c1start + (i times (c1end minus c1start))(imax)

c2 c2start + (i times (c2end minus c2start))(imax)c1start c2start 05c1end c2end 25



sizeimax 800N 50

Table 5 Fracture length prediction data of two algorithms

Fracture number Measured value Classic algorithm Improved algorithm1 234 230 2322 378 372 3763 286 291 2894 306 302 3045 276 273 2796 233 236 2307 257 261 2538 268 265 2699 254 250 25210 167 170 16611 149 145 14812 241 238 24313 239 236 24114 248 252 24615 305 302 303

Table 4 Permeability prediction data of two algorithms



ωi ωmax minus i timesωmax minus ωmin

imax (9)

where ωmax is the maximum value of inertia weight ωmin isthe minimum value of inertia weight i is the current numberof iterations and imax is the maximum number of iterationse calculation formula of the learning factor is shown in thefollowing equation

c1 c1start +i times c1end minus c1start1113872 1113873

imax


imax

(10)

where c1start and c1end are the initial and final values of thelearning factor c1 c2start and c2end are the initial and final

values of the learning factor c2 i is the current number ofiterations and imax is the maximum number of iterations

In order to illustrate the effect of parameter settings ofinertial weight and learning factor on the prediction resultsof the PSO-RBF model a classic particle swarm algorithmwith fixed parameter values and an improved particle swarmalgorithm with parameter value changing were used topredict the permeability and the half-length of the fracturerespectively e basic parameters of the two algorithms areshown in Table 3 e permeability prediction data andfracture length prediction data of the two algorithms areshown in Tables 4 and 5 respectively e prediction resultsof the two algorithms are shown in Figures 8ndash11 respec-tively From Table 4 it can be concluded that when makingpermeability predictions the RMSE of the classic algorithmis 0294 and the RMSE of the improved algorithm is 0178From Table 5 it can be concluded that when predicting the

25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)

Classic PSO algorithmMeasured value

Figure 8 Permeability prediction of classic algorithm

25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)

Improved PSO algorithmMeasured value

Figure 9 Permeability prediction of improved algorithm


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)

Figure 10 Fracture length prediction of classic algorithm


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)

Figure 11 Fracture length prediction of improved algorithm


fracture length the RMSE of the classic algorithm is 3823and the RMSE of the improved algorithm is 225 ereforethe improved particle swarm algorithm is applied to theneural network training process in this paper

7 Conclusions

In this study a neural network model of PSO-RBN wasproposed to predict the permeability and fracture length in aMFHW in western China A comparison of the performanceof the PSO-RBN algorithm and core permeability mea-surements shows that the presented model can adequatelyestimate reservoir permeability and fracture length despitethe highly nonlinear relationship between reservoir pa-rameters In addition the PSO-RBN algorithm and otheralgorithms are statistically compared in the prediction ofpermeability and fracture length e results depict that thepresented model has some advantages over the mentionedalgorithms erefore the proposed PSO-RBN model canprovide more accurate and efficient predictions to reservoirphysical parameters

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of the article

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (grant no 41672114) and NationalScience and Technology Major Project (grant no2017ZX05009005-002)

References

[1] P Pourafshary A Varavei K Sepehrnoori and A Podio ldquoAcompositional wellborereservoir simulator to model multi-phase flow and temperature distributionrdquo Journal of Petro-leum Science amp Engineering vol 69 no 1-2 pp 40ndash52 2009

[2] A Amini and A J Schleiss ldquoNumerical modeling of oil-watermultiphase flow contained by an oil spill barrierrdquo EngineeringApplications of Computational Fluid Mechanics vol 3 no 2pp 207ndash219 2009

[3] Y Jiang Techniques for Modeling Complex Reservoirs andAdvanced wells Stanford University Stanford CA USA2008

[4] A Roux J Corteville and M Bernicot ldquoWellsim and pepiteaccurate models of multiphase flow in oil wells and risersrdquo inProceedings of the International Meeting on PetroleumEngineering Tianjin China November 1988

[5] S Mohaghegh A Popa and S Ameri ldquoIntelligent systems candesign optimum fracturing jobsrdquo in Proceedings of the SPEEastern Regional Conference and Exhibition Pittsburgh PAUSA November 1999

[6] N V Queipo S Pintos N Rincon N Contreras andJ Colmenares ldquoSurrogate modeling-based optimization for

the integration of static and dynamic data into a reservoirdescriptionrdquo Journal of Petroleum Science and Engineeringvol 35 no 3-4 pp 167ndash181 2002

[7] N V Queipo J V Goicochea and S Pintos ldquoSurrogatemodeling-based optimization of SAGD processesrdquo Journal ofPetroleum Science and Engineering vol 35 no 1-2 pp 83ndash932002

[8] S DMohaghegh H H Hafez R Gaskari M Haajizadeh andM Kenawy ldquoUncertainty analysis of a giant oil field in themiddle east using surrogate reservoir modelrdquo in Proceedingsof the Abu Dhabi International Petroleum Exhibition andConference Abu Dhabi UAE November 2006

[9] AMohebbi R Kamalpour K Keyvanloo and A Sarrafi ldquoeprediction of permeability from well logging data based onreservoir zoning using artificial neural networks in one of anIranian heterogeneous oil reservoirrdquo Petroleum Science andTechnology vol 30 no 19 pp 1998ndash2007 2012

[10] M Jamialahmadi and F G Javadpour ldquoRelationship ofpermeability porosity and depth using an artificial neuralnetworkrdquo Journal of Petroleum Science and Engineeringvol 26 no 1ndash4 pp 235ndash239 2000

[11] F K Boadu ldquoRock properties and seismic attenuation neuralnetwork analysisrdquo Pure and Applied Geophysics vol 149no 3 pp 507ndash524 1997

[12] S Mohaghegh B Balan and S Ameri ldquoState-of-the-art inpermeability determination from well log data Part 2-veri-fiable accurate permeability predictions the touch-stone ofall modelsrdquo in Proceedings of the SPE Eastern RegionalMeeting Morgantown West Virginia September 1995

[13] K Aminian H I Bilgesu S Ameri and E Gil ldquoImproving thesimulation of waterflood performance with the use of neuralnetworksrdquo in Proceedings of the SPE Eastern RegionalMeetingCalgary Alberta Canada April 2000

[14] K Aminian B omas and H I Bilgesu ldquoPermeabilityDistribution predictionrdquo in Proceedings of the SPE EasternRegional Conference Canton Ohio October 2001

[15] M A Ahmadi R Soleimani M Lee T Kashiwao andA Bahadori ldquoDetermination of oil well production perfor-mance using artificial neural network (ANN) linked to theparticle swarm optimization (PSO) toolrdquo Petroleum vol 1no 2 pp 118ndash132 2015

[16] J-R Zhang J Zhang T-M Lok and M R Lyu ldquoA hybridparticle swarm optimization-back-propagation algorithm forfeedforward neural network trainingrdquo Applied Mathematicsand Computation vol 185 no 2 pp 1026ndash1037 2007

[17] X Chen ldquoDeformation measurement of the large flexiblesurface by improved RBFNN algorithm and BPNN algo-rithmrdquo in Proceedings of the International Symposium onNeural Networks pp 41ndash48 Berlin Heidelberg 2007

[18] B Yang H Lu and L Chen ldquoBPNN and RBFNN basedmodeling analysis and comparison for cement calcinationprocessrdquo in Proceedings of the ird International Workshopon Advanced Computational Intelligence pp 101ndash106 Suz-hou China August 2010

[19] M Saemi M Ahmadi and A Y Varjani ldquoDesign of neuralnetworks using genetic algorithm for the permeability esti-mation of the reservoirrdquo Journal of Petroleum Science andEngineering vol 59 no 1-2 pp 97ndash105 2007

[20] H Kaydani A Mohebbi and A Baghaie ldquoPermeabilityprediction based on reservoir zonation by a hybrid neuralgenetic algorithm in one of the Iranian heterogeneous oilreservoirsrdquo Journal of Petroleum Science and Engineeringvol 78 no 2 pp 497ndash504 2011


[21] P Tahmasebi and A Hezarkhani ldquoA fast and independentarchitecture of artificial neural network for permeabilitypredictionrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 118ndash126 2012

[22] H Kaydani and A Mohebbi ldquoA comparison study of usingoptimization algorithms and artificial neural networks forpredicting permeabilityrdquo Journal of Petroleum Science andEngineering vol 112 pp 17ndash23 2013

[23] S J Cuddy ldquoLitho-facies and permeability prediction fromelectrical logs using fuzzy logicrdquo SPE Reservoir Evaluation ampEngineering vol 3 no 4 pp 319ndash324 2000

[24] N Hambalek and R Gonzalez ldquoFuzzy logic applied to lith-ofacies and permeability forecasting case study sandstone ofnaricual formation El furrial field eastern Venezuela basinrdquoin Proceedings of the SPE Latin American and CaribbeanPetroleum Engineering Conference Port-of-Spain Trinidadand Tobago April 2003

[25] A A Taghavi ldquoImproved permeability estimation throughuse of fuzzy logic in a carbonate reservoir from southwestIranrdquo in Proceedings of the SPE Middle East Oil and Gas Showand Conference Kingdom of Bahrain March 2005

[26] J S Lim ldquoReservoir properties determination using fuzzylogic and neural networks from well data in offshore KoreardquoJournal of Petroleum Science and Engineering vol 49 no 3-4pp 182ndash192 2005

[27] I S Nashawi and A Malallah ldquoPermeability prediction fromwireline well logs using fuzzy logic and discriminant analysisrdquoin Proceedings of the SPE Asia Pacific Oil and Gas Conferenceand Exhibition Brisbane Australia October 2010

[28] S O Olatunji A Selamat and A Abdulraheem ldquoModelingthe permeability of carbonate reservoir using type-2 fuzzylogic systemsrdquo Computers in Industry vol 62 no 2pp 147ndash163 2011

[29] A K Ilkhchi M Rezaee and S A Moallemi ldquoA fuzzy logicapproach for estimation of permeability and rock type fromconventional well log data an example from the Kanganreservoir in the Iran offshore gas fieldrdquo Journal of Geophysicsand Engineering vol 3 no 4 pp 356ndash369 2006

[30] R Gholami A R Shahraki and M Jamali Paghaleh ldquoPre-diction of hydrocarbon reservoirs permeability using supportvector machinerdquo Mathematical Problems in Engineeringvol 2012 pp 1ndash18 2012

[31] R Yin Q Li P Li Y Guo Y An and D Lu ldquoGPU-basedcomputation of formation pressure for multistage hydrauli-cally fractured horizontal wells in tight oil and gas reservoirsrdquoMathematical Problems in Engineering vol 2018 pp 1ndash102018

[32] J Kennedy ldquoParticle swarm optimizationrdquo Encyclopedia ofMachine Learning and Data Mining Springer Berlin Ger-many pp 760ndash766 2010

[33] S D Mohaghegh J S Liu R Gaskari et al ldquoApplication ofsurrogate reservoir models (SRM) to an onshore green field inSaudi Arabia case studyrdquo in Proceedings of the North AfricaTechnical Conference and Exhibition Cairo Egypt February2012

[34] S Mohaghegh ldquoVirtual-intelligence applications in petro-leum engineering Part 1mdashartificial neural networksrdquo Journalof Petroleum Technology vol 52 no 09 pp 64ndash73 2000

[35] S Mohaghegh R Arefi S Ameri K Aminiand andR Nutter ldquoPetroleum reservoir characterization with the aidof artificial neural networksrdquo Journal of Petroleum Science andEngineering vol 16 no 4 pp 263ndash274 1996

[36] E A El-Sebakhy O Asparouhov A-A Abdulraheem et alldquoFunctional networks as a new data mining predictive

paradigm to predict permeability in a carbonate reservoirrdquoExpert Systems with Applications vol 39 no 12 pp 10359ndash10375 2012

[37] MMohammadpoor K Shahbazi F Torabi and A R QazviniFirouz ldquoA new methodology for prediction of bottomholeflowing pressure in vertical multiphase flow in Iranian oilfields using artificial neural networks (ANNs)rdquo in Proceedingsof the SPE Latin American and Caribbean Petroleum Engi-neering Conference Lima Peru December 2010

[38] A A A Esmin R A Coelho and S Matwin ldquoA review onparticle swarm optimization algorithm and its variants toclustering high-dimensional datardquo Artificial Intelligence Re-view vol 44 no 1 pp 23ndash45 2015

[39] F Van Den Bergh An Analysis of Particle Swarm optimizersPhD thesis University of Pretoria Pretoria South Africa2007


system behavior well It is also used to solve many differenttypes of problems in petroleum engineering such as res-ervoir characterization permeability prediction verticalmultiphase flow downhole pressure prediction and dis-solved gas driven flow prediction [6ndash8]

e ANN is considered to be a method that can effec-tively replace traditional techniques of predicting perme-ability from well-logging data e disadvantage of the ANNapplication is the requirement to identify the structure of thenetwork In order to overcome the shortcoming in thisstudy an MFHW parameter identification model based onthe PSO-RBF (particle swarm optimization PSO radialbasis function RBF) neural network is proposed e mainpurpose is to seek a better prediction of reservoir propertiese parameters of the radial basis neural network aredesigned by using the particle swarm algorithm to obtainhigher prediction accuracy e parameter identificationmodel based on the PSO-RBF neural network has highlearning ability and flexibility when encountering complexproblems which makes the proposed model applicable tomost engineering practices

2 Review of Related Works

In the past few decades researchers have done a lot of workto study reservoir parameter estimation problems such aspermeability porosity and fracture length Taking the es-timation of permeability as an example there are currentlythree main methods core analysis well test analysis andartificial intelligence methods

e most accurate method of permeability prediction isto perform core analysis in the laboratory by applyingDarcyrsquos law Among the different techniques the perme-ability obtained from the core analysis is more efficient thanthat obtained by other methods and the permeability fromthe core analysis can be used to verify other estimationmodels However the core analysis process is time con-suming and difficult to use widely

Another method of obtaining reservoir permeability iswell test analysis e analytical solution is solved by usingthe porous flow equation and some rock properties such aspermeability and porosity can be then obtained inverselyfrom the pressure curve e data available after well testinghelp petroleum engineers calculate formation permeabilityskin factor and wellbore storage [9] Due to the cost oftesting well testing is not a good solution for any reservoirsthat have been developed

ANNs are increasingly being employed to predict res-ervoir properties based on geophysical well-logging data[10 11] Mohaghegh et al pointed out that neural networksare powerful tools for identifying the relationship betweenpermeability and well-logging data [12] In addition Ami-nian et alrsquos research shows that artificial neural networks canbe used to predict formation permeability even in highlyheterogeneous reservoirs [13 14] Based on the artificialneural network and particle swarm optimization algorithmAhmadi et al proposed a method for predicting horizontalwell productivity in a pseudosteady state and the experi-mental results achieved a high fitting accuracy (lt082) [15]

Zhang et al proposed a hybrid algorithm that combines aparticle swarm optimization algorithm with a back-propa-gation algorithm to train the weights of a feed-forwardneural network e results show that the hybrid algorithmis superior to the adaptive particle swarm optimization al-gorithm and backpropagation neural network (BPNN) inboth convergence speed and convergence accuracy [16]Based on the RBF and BPNN Chen developed two watertreatment models e model has been trained and checkedseparately from actual data from the water plant Comparedwith the model based on the BPNN the model based on theRBF neural network has the characteristics of better ap-proximation and faster convergence [17] Based on themodels of the BPNN and RBFNN (the radial basis functionneural network RBFNN) Yang et al proposed a model forcement decomposition and calcination process e resultsshow that the model based on the RBFNN can achieve veryhigh fitting results [18]

Saemi et al improved the predictive performance ofneural networks by using the GA to optimize the networkparameters of the ANN [19] In addition Kaydani et alrsquosfindings indicate that the GA and ANN network structuresdesigned by subregion can predict the permeability of aheterogeneous reservoir in Iran [20] Tahmasebi et alproposed four methods for predicting permeability of dif-ferent neural network structures and statistically comparedthe results obtained Finally a new method of the modularneural network (MNN) was obtained [21] By combiningcuckoo optimization algorithm (COA) particle swarmoptimization algorithm (PSO) imperialist competitive al-gorithm (ICA) and LevenbergndashMarquardt algorithm (LM)Kaydan et al proposed a new method to estimate thepermeability [22]

Rough set theory can be successfully applied to theprediction of permeability of porous media [23ndash28] Byestablishing a rough set model Ilkhchi et al successfullypredicted the permeability of an offshore gas field in Iran[29]

Support vector regression (SVR) based on the principleof structural risk minimization is a promising machinelearning method rough experimental comparisonsGholami et al found that the SVR method has faster speedand higher accuracy in predicting reservoir permeability[30]

3 Problem Formulation

31 Optimized Design After fracturing tight sandstonereservoirs the physical properties of the reservoirs arecharacterized by extremely low porosity a wide range ofpermeability changes and complex pore-permeability re-lationships erefore conventional production predictionmodels often fail to meet the requirements in terms ofprediction accuracy

Effective algorithms for complex nonlinear problems areessentially optimization problems e so-called optimiza-tion refers to the problem of seeking the maximum orminimum value of the given objective function according tothe change of design variables under the condition of



min f(x) ormaxf(x)


j 1 2

(1)






1113946tD


(2)


minus1


⋮ ⋮ ⋮ ⋮ ⋮



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

qD11

qD12

⋮

qDN Mi

pDw f

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0

0

⋮

0

1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)




(4)


4 PSO-RBF Model




vik+1 wi middot v


ibest minus p


ik1113872 1113873

(5)





pik+1 p

ik + v

ik+1 middot Δt (6)









σ21113890 1113891 (7)








i le n




i = i

+ 1

No

No

Yes

Yes

NoYes

Yes Yes

Upd

ate t

he p

ositi

on an

dsp

eed

of p

artic

les

No

No


x1

x2

x3

xn


o1

0j

Input layer

middotmiddotmiddot

middotmiddotmiddot




yn 1113944m

m1wi middot ϕi (8)














swarm

Calculate RMSE


Particleswarm

optimization




optimization




Particleswarm

optimization






6 Sensitivity Study



030

028

026

024

022

020

018

016

014

012

010

RMSE



SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References













































min f(x) ormaxf(x)


j 1 2

(1)






1113946tD


(2)


minus1


⋮ ⋮ ⋮ ⋮ ⋮



⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

middot

qD11

qD12

⋮

qDN Mi

pDw f

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

0

0

⋮

0

1

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3)




(4)


4 PSO-RBF Model




vik+1 wi middot v


ibest minus p


ik1113872 1113873

(5)





pik+1 p

ik + v

ik+1 middot Δt (6)









σ21113890 1113891 (7)








i le n




i = i

+ 1

No

No

Yes

Yes

NoYes

Yes Yes

Upd

ate t

he p

ositi

on an

dsp

eed

of p

artic

les

No

No


x1

x2

x3

xn


o1

0j

Input layer

middotmiddotmiddot

middotmiddotmiddot




yn 1113944m

m1wi middot ϕi (8)














swarm

Calculate RMSE


Particleswarm

optimization




optimization




Particleswarm

optimization






6 Sensitivity Study



030

028

026

024

022

020

018

016

014

012

010

RMSE



SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References


















































σ21113890 1113891 (7)








i le n




i = i

+ 1

No

No

Yes

Yes

NoYes

Yes Yes

Upd

ate t

he p

ositi

on an

dsp

eed

of p

artic

les

No

No


x1

x2

x3

xn


o1

0j

Input layer

middotmiddotmiddot

middotmiddotmiddot




yn 1113944m

m1wi middot ϕi (8)














swarm

Calculate RMSE


Particleswarm

optimization




optimization




Particleswarm

optimization






6 Sensitivity Study



030

028

026

024

022

020

018

016

014

012

010

RMSE



SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References













































yn 1113944m

m1wi middot ϕi (8)














swarm

Calculate RMSE


Particleswarm

optimization




optimization




Particleswarm

optimization






6 Sensitivity Study



030

028

026

024

022

020

018

016

014

012

010

RMSE



SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References















































6 Sensitivity Study



030

028

026

024

022

020

018

016

014

012

010

RMSE



SVR algorithm

20

10

0

Perm

eabi

lity

(mD

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References














































0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400380360340320300280260240220200180160140120100

Frac

ture

leng

th (m

)


SVR algorithm









Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References
















































Classic PSO




sizeimax 800N 50

ImprovedPSO






sizeimax 800N 50







imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References













































imax (9)



imax


imax

(10)




25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)



25

20

15

10

5

00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Fracture number

Perm

eabi

lity

(mD

)




0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Fracture number

400

350

300

250

200

150

100

Frac

ture

leng

th (m

)




7 Conclusions


Data Availability




Acknowledgments


References













































7 Conclusions


Data Availability




Acknowledgments


References












































parameter identification of multistage fracturing horizontal well...

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