research article geomechanical properties of ...well logs such as gamma ray, available information...

Research ArticleGeomechanical Properties of Unconventional Shale Reservoirs

Mohammad O Eshkalak1 Shahab D Mohaghegh2 and Soodabeh Esmaili3

1Department of Petroleum and Geosystems Engineering The University of Texas at Austin Austin TX 78712 USA2Department of Petroleum and Natural Gas Engineering West Virginia University Morgantown WV 26506 USA3Occidental Petroleum Corporation Bakersfield CA 93311 USA

Correspondence should be addressed to Mohammad O Eshkalak eshkalakutexasedu

Received 12 July 2014 Accepted 12 October 2014 Published 3 December 2014

Academic Editor Andrea Franzetti

Copyright copy 2014 Mohammad O Eshkalak et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Production from unconventional reservoirs has gained an increased attention among operators in North America during pastyears and is believed to secure the energy demand for next decades Economic production fromunconventional reservoirs ismainlyattributed to realizing the complexities and key fundamentals of reservoir formation properties Geomechanical well logs (includingwell logs such as total minimum horizontal stress Poissonrsquos ratio and Young shear and bulkmodulus) are secured source to obtainthese substantial shale rock propertiesHowever running these geomechanical well logs for the entire asset is not a commonpracticethat is associated with the cost of obtaining these well logs In this study synthetic geomechanical well logs for a Marcellus shaleasset located in southern Pennsylvania are generated using data-driven modeling Full-field geomechanical distributions (mapand volumes) of this asset for five geomechanical properties are also created using general geostatistical methods coupled withdata-driven modeling The results showed that synthetic geomechanical well logs and real field logs fall into each other when theinput dataset has not seen the real field well logs Geomechanical distributions of the Marcellus shale improved significantly whenfull-field data is incorporated in the geostatistical calculations

1 Introduction

Shale gas reservoirs which are also called source rock reser-voirs (SRR) have some unique attributes that make hydraulicfracturing an essential option in order to commence aneconomic level of the natural gas production Unlike con-ventional gas reservoirs insufficient permeability ultra-lowporosity of shale rock and limited reservoir contact areabut vastly organic-rich formation cannot offer productionin a commercial value without stimulation processes Manystudies are conducted from shale pore-scale level to fieldscale reservoir simulations to improve the understandingof complex flow behavior that are developed and discussedthrough numerical analytical and semianalytical reservoirmodels for unconventional reservoirs [1ndash10] However inorder to predict the performance of a shale gas reservoirimplementing accurate shale rock properties is essential fordeveloping a geologic model for the entire asset Henceit is very critical to access more data while working onan unconventional reservoir In this study synthetic data

are generated using artificial intelligence and data miningtechniques (AIampDM)

Principal stress profile of an oil and gas reservoir dependshighly on the rock geomechanical properties Geomechanicalproperties of reservoir rock include Poissonrsquos ratio totalminimum horizontal stress and bulk Young and shearmodulus These properties play significant role in currentdevelopment plans of shale assets compared to conven-tional reservoirs that have established sufficient informationavailable Moreover having access to geomechanical datacan assist engineers and geoscientists during geomechanicalmodeling hydraulic fracture treatment design and reservoirsimulation in shale gas fields across the US and worldwideGeomechanical well logs are one of the sources that securesuch data Running geomechanical well logs (in all wellsin a shale asset) is not common practice among operatorsand the reason is attributed to the cost associated withrunning such logs In this paper data-driven models aredeveloped to accurately determine the Marcellus shale rockproperties

Hindawi Publishing CorporationJournal of Petroleum EngineeringVolume 2014 Article ID 961641 10 pageshttpdxdoiorg1011552014961641

2 Journal of Petroleum Engineering

Artificial intelligence and data mining have been usedwithin last 20 years in reservoir modeling and characteri-zation to perform analysis on formation of interest [11 12]Also some studies indicated that artificial neural network(ANN) is a powerful tool for pattern recognition and systemidentification such asmethodology developed byMohagheghet al in 1998 [13] to generate synthetic magnetic resonanceimaging (MRI) logs using conventional logs such as SP GRand resistivity Their methodology incorporated an artificialneural network as itsmain tool to generate the target variableThe synthetic magnetic resonance imaging logs were gener-ated with a high degree of accuracy even when the modeldeveloped used data not employed during model devel-opment Moreover Basheer and Najjar demonstrated thatANN is suitable to predict and classify soil compaction androcks characteristics as well as determining somemechanicalparameters such as Youngrsquos modulus Poissonrsquos ratio [14]They mainly investigated the neural network capability insolving geotechnical engineering problems and they provideda general view of some neural network application in theirfield of research

In this study it is demonstrated that AIampDM technologyis able to develop data-driven models for generating rockgeomechanical properties The overall work-flow includesdevelopment of synthetic geomechanical well logs from con-ventional logs such as gamma ray and bulk density that arecommonly available These data-driven models used around30 percent of data (coming from geomechanical logs) of theentire asset which were available to expand them for the restof the field with conventional logs but no geomechanical logsData-driven models have been validated with blind wellsBlind wells are wells with actual data which are selecteddue to different locations in the asset of 100 horizontal gaswells Moreover the logs generated from data-driven modelsare used to build an integrated field-wide geomechanicaldistribution (maps and volumes) for rock geomechanicalproperties In this work the ultimate purpose is meant topropose a technique in order to omit the necessity of runninggeomechanical logs for the entire asset once such logs areobtained for some portion of the asset to be used in thedata-driven models The number of well logs required to runthe data-driven models that leads to an accurate result isdetermined to be 30 percent of the wells in an unconventionalasset In this study just 30 out of 100 horizontal wells in theMarcellus shale asset in southern Pennsylvania own actualgeomechanical well logs that are provided by a major servicecompany

2 Methodology

In order to accomplish the objectives of this study a method-ology and thorough procedure are required to be definedThemethodology used to accomplish the objectives of this studyincludes four steps as indicated below A detailed descriptionof each step is explained afterwards

(a) data preparation

(b) data-driven model development using AIampDM

(c) validation of data-driven models(d) geomechanical property distribution

21 Step (a) Data Preparation Data preparation is the mostimportant step in developing data-driven models due to thefact that all the other steps are using the data prepared inthis step Data preparation involves checking the data foraccuracy entering data in a right format in a computer fileand developing and documenting a database structure thatintegrated the various properties used in the next stepsIn this study a dataset form the available well logs arerequired to be prepared that portrays specific property ofthe rock versus depth First production pay zone must beidentified from the conventional well logs along with thehorizontal wells trajectory After identifying the depth ofthe producing zones for Marcellus shale from conventionalwell logs such as gamma ray available information of eachindividual well is extracted from well logs in every oneor half a foot according to its log characteristics and toolsused Also in order for the models to understand thedifferences between production pay zones and while usingthese datasets it is essential to specify the contrast betweenpay zones (upper Marcellus Purcell lower Marcellus andOnondaga) and the adjacent rocks nonshale To account forthis contrast 50 feet depth of log data located above andbelow the pay zones of interest is also added to the maindataset

The prepared dataset contains rows and columns con-sisting of following data that are recorded versus depth thewells name the well coordinates the values for gamma ray(GR) bulk density (BD) sonic porosity bulk modulus (BM)shear modulus (SM) Youngrsquos modulus (YM) Poissonrsquos ratio(PR) and total minimum horizontal stress (TMHS) for eachhorizontal shale well It must be emphasized that not all thewells include geomechanical well logs thus geomechanicalvalues are only recorded for the wells that have such real data30 wells

22 Step (b) Data-Driven Models Development In this stepthe prepared dataset was processed using backpropagationalgorithm of neural network into two main parts

Part 1 (Conventional Models) In order to have consistentconventional logs data for all wells for the shale productionpay zone part 1 is defined and the conventional logs aregenerated for those wells that missed some conventional logsAs it is shown in Figure 1 the bulk density and sonic porosityfor 30 wells were produced by using two different data-driven models First model (neural network model 1) usedgamma ray depth location and well coordinates as an inputto develop training calibration and verification segmentsfor generating the bulk density of around 30 wells in theasset where bulk density and sonic data weremissing Secondmodel (neural network model 2) used also bulk density asinput beside inputs of first step to generate sonic porosity forthe part of asset without this property (around 30 wells alsoused in this part)

Journal of Petroleum Engineering 3

Inputs location gamma ray and

depth for 70 wells

Outputs bulk density and sonic

porosity for 70 wells

Training calibration and verification

Inputs locationgamma ray and

depth for 30 wells

Outputs bulkdensity and sonic


Apply Data-drivenmodels 1 and 2

Generate

Figure 1 Part 1mdashdata-driven model for conventional logs

Outputs geomechanical well logs

for 30 wells


Inputs conventional well logs for 70 new

wells


Generate Outputs geomechanicalwell logs for 70 new

wells

Inputs conventional well logs

for 30 wells

Figure 2 Part 2mdashdata-driven models for geomechanical logs

At the end of this step all existing wells in the asset havethe required conventional well log properties to be used inpart 2

Part 2 (Geomechanical Models) After completing the missingdata for conventional logs from part 1 for all horizontal wellsanother neural network structure is used to develop fivedifferent data-driven models (models 3 to 7) as shown inFigure 2 to generate the geomechanical well logs for all wellsAs it is shown in Figure 2 this step consists of five neuralnetwork models in which the inputs were completed in eachstep by using the generated geomechanical property in theprevious step In detail for each of these neural networksthe same conventional logs of 30 wells have been providedas the input and one geomechanical property was generatedat a time Then each generated geomechanical property wasused as an input for the next neural network model andthe process continued until all five geomechanical propertiesof interest were generated for the entire Marcellus shaleasset

All models are multilayer networks that are trained usinga backpropagation method of neural network technology Inorder to achieve the least error in backpropagation methoda different percentage of datasets are incorporated in themodels and finally it is concluded that considering 80 of

Longitude

Latit

ude

Wells with geomechanical logsBlind validation wellsWells with no geomechanical logs

Figure 3 Marcellus shale gas field southern Pennsylvania

data used in the training process and by considering 20 forthe calibration process and the remaining for the last stepverification (10 for each) the universal backpropagationerror is minimized A general backpropagation scheme andformulation is explained inAppendix In the next section twodifferent methods for error calculation 119904 are explained


Error Analysis The mean absolute percentage error (MAPE)also known as mean absolute percentage deviation is ameasure of accuracy of a method for constructing fitted timeseries values in statistics specifically in trend estimation Itusually expresses accuracy as a percentage and is defined by

MAPE = 100119899

119899

sum

119905=1

1003816100381610038161003816100381610038161003816

119860119905 minus 119865119905

119860119905

1003816100381610038161003816100381610038161003816

(1)

where 119860119905 is the actual value and 119865119905 is the forecast value

The difference between 119860119905 and 119865119905 is divided by theactual value 119860119905 again The absolute value in this calculationis summed for every fitted or forecasted point in time anddivided again by the number of fitted points 119899 Multiplyingby 100 makes it a percentage error Also 119877-squared is definedand determined in (2) for all three steps training calibrationand verification The higher the 119877-squared the closest theresults to the actual values

119877-squared = 1 minus Sum of squared distances between the actual and predicted valuesSum of squared distances between the actual and their mean values

(2)

In our study the highest achieved 119877-squared was around98 percent and the lower one in some cases around 89percent and in both situations the results presented arehighly acceptable A higher level of 119877-squared reflects inall three stages of training calibration and verification areliable correlation between actual and generated data It isalso important to mention that during the initial trainingof datasets the results obtained were with low 119877-squaredUnsuccessful behavior of models was understood because ofhaving some wells with log data for each 05 ft which is incontrast with the rest of the wells with every available 1 ftlog data Once piece of data of 05 ft turned to 1 ft whichis considered as discrepancy that misleads the data-drivenmodels the results came out properly and the data-drivenmodels showed rapid improvements Further second issuethat resulted in smoother behavior of the data-driven modelswas related to the removal of nonshaly thin intervals log datafrom the pay zones that exists within the upper MarcellusPurcell lower Marcellus and Onondaga Once these layerswere removed the models converged much faster with verylow backpropagation error

23 Step (c) Data-Driven Model Validation This step ex-plains a robustmethod to analyze the accuracy and validity ofthe data-driven modelrsquos results although the universal errorof all the models was very low according to previous sectionTo examine themodels validity thewell log data of somewells(which have both real conventional and geomechanical logs)was removed from the training dataset and it was attemptedto regenerate the geomechanical logs These removed wellsare so called blind wells Blind wells have been chosen fromdifferent location in the Marcellus shale asset under studyThen data-driven models used this new dataset to be trainedto generate geomechanical properties for blind wells Data-driven models number 3 to 7 was separately validated togenerate geomechanical properties In each step the gener-ated property compared and plotted against the actual valueswhich had been removed from main dataset The results ofthis step are explained in Results and Discussions section

Figures 4 through 8 demonstrate the actual well logs andgenerated logs for 5 blind wells that are chosen form different

location in the asset as illustrated in Figure 3 To compare theresults both actual and generated properties are plotted inthe same figure like an actual well log Properties such as bulkmodulus Youngmodulus Poissonrsquos ratio shearmodulus andtotal minimum horizontal stress are presented respectivelyBlue line shows the actual value and the red line is forgenerated values by data-driven models For well 1 to well4 there is a perfect match between blue and red lines thatshows the models have generated the exact actual dataThesewells are in proximity of wells with actual geomechanicalproperties according to their locations and depths As itwas expected results shown for these wells are accuratewhich demonstrate data-driven modelrsquos capability and accu-racy in generating geomechanical properties of Marcellusshale

24 Step (d) Geomechanical Property Distribution The firstobjective of this paper is accomplished in the previoussection and the geomechanical properties are generated for allexisting wells in the Marcellus shale asset To accomplish thisstep different geostatisticalmethods fromPetrel commercialsoftware are considered to create geomechanical propertydistribution for the Marcellus shale field Further geome-chanical well logs generated from the data-driven modelsare coupled with a commercial reservoir simulator in orderto create geomechanical distributions for properties such astotal minimum horizontal stress Poissonrsquos ratio and Youngshear and bulk modulus

Sequential Gaussian simulation (SGS) is finally usedto create distribution according to well locations for theentire field due to its very smooth and consistent sur-faces and distributions (maps) obtained compared to othermethods Two types of maps were created First map isonly incorporated with 30 wells which already had actualgeomechanical logs The second map is related to entirefield (70 wells with generated property and 30 wells withactual data) With comparing these two maps significantdifference between geomechanical property distributionwithand without having full-field data is observed as shown inFigures 9 through 13 Ten maps that show distribution of fiverock geomechanical properties in the Marcellus shale assetwere created


6220

6230

6240

6250

6260

6270

6280

Bulk modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Shear modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Total min hor stress

05

07

09

11

13

15

6220

6230

6240

6250

6260

6270

6280

Young modulus

1 2 3 46220

6230

6240

6250

6260

6270

6280

Poissonrsquos ratio

0 01

02

03

04

05

Well 1 Synthetic Well 1 Synthetic Well 1 Synthetic Well 1 Synthetic Well 1 Synthetic

Figure 4 Well 1 actual versus generated well logs

Table 1 Information used for databases for developing data-driven models

Well identifier Description Conventional well logs Geomechanical well logs Number of wellsCircle Blind validation wells Yes Yes 5Triangle Wells used for validation Yes Yes 25Diamond Wells with no geomechanical logs Yes No 70

3 Results and Discussions

The Marcellus shale under study consists of 100 multifrac-tured horizontal wells Figure 3 depicts the distribution ofexisting wells in the asset that is used in this study Table 1shows the information and number of wells that were used todevelop data-driven models in different steps as well as thevalidation purpose in step (c)

In this study a multilayer neural networks or multilayerperceptions are considered to develop the data-driven mod-els These networks are most suitable for pattern recognitionspecially in nonlinear problems neural network that have onehidden layer with different number of hidden neurons thatare selected based on the number of data records availableand the number of input parameters selected in each trainingprocess

The training process of the neural networks is conductedusing a backpropagation technique In the training processthe dataset is partitioned into three separate segments Thisis done in order to make sure that the neural network willnot be trapped in the memorization phase Moreover theintelligent partitioning process allows the network to adaptto new data once it is being trained The first segment whichincludes the majority of the data is used to train the modelIn order to prevent the memorizing and overtraining effectin the neural network training process a second segment ofthe data is taken for calibration that is blind to the neural

network and at each step of training process the networkis tested for this set If the updated network gives betterpredictions for the calibration set it will replace the previousneural network otherwise the previous network is selectedTraining will be continued once the error of predictions forboth the calibration and training dataset is satisfactory Thiswill be achieved only if the calibration and training partitionsare showing similar statistical characteristics Verificationpartition is the third and last segment used for the processthat is kept out of training and calibration process and isused only to test the precision of the neural networks Oncethe network is trained and calibrated then the final model isapplied to the verification set If the results are satisfactorythen the neural network is accepted as part of the entireprediction system [15 16]

Figures 4 to 8 show the actual well logs and generatedlogs for 5 blind wells shown as black circles in Figure 3 Tocompare the results both actual and generated properties areplotted in the same figure similar to an actual well log Inthese plots properties such as bulkmodulus YoungmodulusPoissonrsquos ratio shearmodulus and totalminimumhorizontalstress are presented respectively Blue line shows the actualvalue and the red line is for generated values by data-drivenmodels For well 1 to well 4 (Figures 5 6 and 7) there isperfect match between blue and red lines These wells arein proximity of wells with actual geomechanical propertiesaccording to their locations and depths As it was expected


6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio

0 02 04Well 2 Synthetic Well 2 Synthetic Well 2 Synthetic Well 2 Synthetic Well 2 Synthetic


6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05



results shown for these wells are accurate which demonstratedata-driven models capability in predicting geomechanicalproperties

For well 5 in Figure 8 the generated data is not in agree-ment with the actual logs and it is because of the location ofthe well that is far from (upper side of the asset in the fieldmdashFigure 2) the rest of wells in the asset that we have used for thetraining purposes This fact indicates that the models couldnot predict the behavior of outlier wells andmost importantlyemphasizes on the fact that data-driven modeling is perfectfor interpolations and not accurate for the extrapolation as itis in agreement with the neural network literature Moreover

it is found that the depth of producing pay zone of this well 5compared to other four blind wells is different (out of range)and it might be another reason related to the fact that modelscould not capture the behaviors very well

Figures 9 10 11 12 and 13 are showing distributions andmaps for the five geomechanical rock properties of interestin this study For each property there are two distributionsone that is generated by using the actual data and the secondthat considered the information of both generated and actualdata (full-field data) A comparison between maps for eachproperty demonstrates that more reasonable and accuratedistribution is achieved using more data for the asset


6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio



The sequential Gaussian simulation (SGS) algorithm wasused in order to generate these maps In the top distributionmap plus signs represent the wells with actual data whichhave been used in dataset for training calibration andverification during data-driven model development

4 Conclusions

In this study it is demonstrated that the data-drivenmodelingusing AIampDM technology is a reliable and robust tool toobtain accurate results for generating synthetic geomechani-cal logs for unconventional shale resources In simple terms

we used conventional well logs to generate field-wide geome-chanical properties and distribution maps of geomechanicalproperties for the entire asset Marcellus shale in southernPennsylvania

Five data-driven models were designed trained and val-idated to predict five geomechanical properties of interest forMarcellus shale unconventional reservoir First data miningissue in this study removing nonshaly intervals and adding 50feet contrast zone was successfully managed to lead a reliableprediction with least error calculated in backpropagationmethod Also second validation process the use of 5 blindwells was performed to show the robustness and accuracy of


425400375350325300275250225200

GeneralYoungrsquos modulus-30 wells-505 (U)


425400375350325300275250225200

Figure 9 Young modulus

Shear modulus (Mpxi)Shear modulus-20wells-909 (U)

220210200190180170160150140130


220210200190180170160150140130

Figure 10 Shear modulus

data-driven models for predicting Young modulus Poissonratio bulk modulus shear modulus and total minimumhorizontal stress

Geomechanical property distribution maps of the entireasset illustrated a significant difference between distributionswhen there are just a few available pieces of actual data ratherthan having access to the full-field data These syntheticgeomechanical logs and property distributions for Marcellusshale exhibit a great deal of assistance to better performingreservoir modeling characterization and the optimization ofhydraulic fracturing issues related to the current Marcellusshale development process Authors expect these models willconclude also accurate results in other unconventional shaleresources

Appendix

Backpropagation Method Formulation

We now derive the backpropagation technique for a gen-eral case The equations (A1) through (A8) show the

026

024

022

020

018

016

014

012

010

Poissonrsquos ratioPoissonrsquos ratio-30 wells-909 (U)

026

024

022

020

018

016

014

012

010


Figure 11 Poissonrsquos ratio

090

085

080

075

070

065

060

055

Principal stress (psi)Min horizontal stress-20 wells-909 (U)

090

085

080

075

070

065

060

055


Figure 12 Total minimum horizontal stress

300

280

260

240

220

200

180

160

140

Bulk modulus (Mpsi)Bulk modulus-90wells-909 (U)

300

280

260

240

220

200

180

160

140


Figure 13 Bulk modulus


mathematical representations of backpropagation methodusing the input dataset

119895= input vector for unit 119895

119895=Weight vector for unit 119895

119911119895= 119895sdot 119895 the weighted sum of inputs for unit 119895

119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)

where 119905119895is the actual value or the target value that we

wish to achieve using the backpropagation method Sincewe update after each training example we can simplify thenotation somewhat by imagining that the training set consistsof exactly one example and so the error can simply be denotedby 119864 Downstream (119895) = set of units whose immediate inputsinclude the output of 119895 Outputs = set of output units in thefinal layer

We want to calculate 120597119864120597119908119895119894for each input weight 119908

119895119894

for each output unit 119895 Note first that since 119911119895is a function of

119908119895119894regardless of where in the network unit 119895 is located

120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)

Furthermore 120597119864120597119911119895is the same regardless of which input

weight of unit 119895 we are trying to update So we denote thisquantity by 120575

119895 Consider the casewhen 119895 isin OutputsWe know

119864 =

1

2

sum

119896isinOutputs(119905119896minus 120590 (119911

119896))2

(A3)

Since the outputs of all units 119896 = 119895 are independent of119908119895119894 we

can drop the summation and consider just the contributionto 119864 by 119895

120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)

= minus (119905119895minus 119900119895) (1 minus 120590 (119911

119895)) 120590 (119911

119895)

= minus (119905119895minus 119900119895) (1 minus 119900

119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)

Now consider the case when 119895 is a hidden unit Like beforewe make the following two important observations

For each unit 119896Downstream from 119895 119911119896is a function of 119911

119895

The contribution of error by all units 119897 = 119895 in the samelayer as 119895 is independent of 119908

119895119894


119895119894

for each hidden unit 119895 Note that 119908119895119894influences just 119911

119895which

influences 119900119895which influences 119911

119896for all 119896 isin Downstream

each of which influence 119864 So we can write120597119864

120597119908119895119894

= sum

119896isinDownstream(119895)

120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)

Again note that all the terms except 119909119895119894in the above product

are the same regardless of which input weight of unit 119895 weare trying to update Like before we denote this commonquantity by 120575

119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900

119895) Substituting

120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum

119896isinDownstream(119895)120575119896119908119896119895119900119895(1 minus 119900

119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum

119896isinDownstream(119895)120575119896119908119896119895 (A8)

Nomenclature

119883119895 Input vector for unit 119895

119882119895 Weight vector for unit 119895

119885119895 Weighted sum of inputs for unit 119895

119874119895 Output of unit 119895

119879119895 Laplace transform parameter119864 Calculated error for each unitMAPE Mean absolute percentage error119860119905 Actual value

119865119905 Predicted value by data-driven model

Highlights

Advanced artificial intelligence and data mining techniqueis used to develop data-driven models in order to generatesynthetic geomechanical well logs

Highly accurate results from data-driven models areachieved that are validated against blindwells that have actualfile data in the Marcellus shale asset

The geomechanical distributions created with field-widedata demonstrate much better consistency and improvementcompared to using just partial field data

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] C L Cipolla E P Lolon J C Erdle and B Rubin ldquoReservoirmodeling in shale-gas reservoirsrdquo SPE Reservoir Evaluation ampEngineering vol 13 no 4 pp 638ndash653 2010


[2] W Yu and K Sepehrnoori ldquoSimulation of gas desorption andgeomechanics effects for unconventional gas reservoirsrdquo inProceedings of the SPE Western RegionalPacific Section AAPGJoint Technical Conference Energy and the EnvironmentWorkingTogether for the Future pp 718ndash732Monterey Calif USA April2013

[3] S Esmaili A Kalantari-Dahaghi and S D Mohaghegh ldquoFore-casting sensitivity and economic analysis of hydrocarbon pro-duction from shale plays using artificial intelligenceampdatamin-ingrdquo in Proceedings of the Canadian Unconventional ResourcesConference Calgary Canada October-November 2012 paperSPE 162700

[4] T W Patzek F Male and M Marder ldquoGas production in theBarnett Shale obeys a simple scaling theoryrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 110 no 49 pp 19731ndash19736 2013

[5] U Aybar M O Eshkalak K Sepehrnoori and T W PatzekldquoThe effect of natural fracturersquos closure on long-term gasproduction from unconventional resourcesrdquo Journal of NaturalGas Science and Engineering 2014

[6] MO Eshkalak ldquoSimulation study on theCO2-driven enhanced

gas recovery with sequestration versus the re-fracturing treat-ment of horizontal wells in the US unconventional shalereservoirsrdquo Journal of Natural Gas Science and Engineering2014

[7] M O Eshkalak U Aybar and K Sepehrnoori ldquoAn integratedreservoir model for unconventional resources coupling pres-sure dependent phenomenardquo in Proceedings of the the SPEEastern Regional Meeting pp 21ndash23 Charleston WV USAOctober 2014 paper SPE 171008

[8] M O Eshkalak U Aybar and K Sepehrnoori ldquoAn economicevaluation on the re-fracturing treatment of the US shale gasresourcesrdquo in Proceedings of the SPE Eastern Regional MeetingCharleston WV USA October 2014 paper SPE 171009

[9] M O Eshkalak S D Mohaghegh and S Esmaili ldquoSyntheticgeomechanical logs for marcellus shalerdquo in Proceedings of theSPE Digital Energy Conference and Exhibition The WoodlandsTexas USA March 2013 paper SPE 163690

[10] M Omidvar Eshkalak Synthetic geomechanical logs and dis-tributions for marcellus shale [MS thesis] West Virginia Uni-versity Libraries West Virginia University Morgantown WVUSA 2013

[11] L Rolon S D Mohaghegh S Ameri R Gaskari and BMcDaniel ldquoUsing artificial neural networks to generate syn-thetic well logsrdquo Journal of Natural Gas Science and Engineeringvol 1 no 4-5 pp 118ndash133 2009

[12] S Mohaghegh R Arefi S Ameri K Aminiand and R NutterldquoReservoir characterization with the aid of artificial neuralnetworkrdquo Journal of Petroleum Science and Engineering vol 16pp 263ndash274 1996

[13] S MohagheghM Richardson and S Ameri ldquoVirtual magneticimaging logs generation of synthetic MRI logs from conven-tional well logsrdquo in Proceedings of the SPE Eastern RegionalMeeting pp 223ndash232 Pittsburgh Pa USA November 1998

[14] I A Basheer and Y M Najjar ldquoA neural network for soilcompactionrdquo in Proceedings of the 5th International Symposiumon Numerical Models in Geomechanics G N Pande and SPietrusczczak Eds pp 435ndash440 Balkema Roterdam TheNetherlands 1995

[15] A J Maren C T Harston and R M Pap Handbook of NeuralComputation Applications Academic Press San Diego CalifUSA 1990

[16] Y Khazani and S D Mohaghegh ldquoIntelligent productionmodeling using full-field pattern recognitionrdquo SPE ReservoirEvaluation amp Engineering vol 14 no 6 pp 735ndash749 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Artificial intelligence and data mining have been usedwithin last 20 years in reservoir modeling and characteri-zation to perform analysis on formation of interest [11 12]Also some studies indicated that artificial neural network(ANN) is a powerful tool for pattern recognition and systemidentification such asmethodology developed byMohagheghet al in 1998 [13] to generate synthetic magnetic resonanceimaging (MRI) logs using conventional logs such as SP GRand resistivity Their methodology incorporated an artificialneural network as itsmain tool to generate the target variableThe synthetic magnetic resonance imaging logs were gener-ated with a high degree of accuracy even when the modeldeveloped used data not employed during model devel-opment Moreover Basheer and Najjar demonstrated thatANN is suitable to predict and classify soil compaction androcks characteristics as well as determining somemechanicalparameters such as Youngrsquos modulus Poissonrsquos ratio [14]They mainly investigated the neural network capability insolving geotechnical engineering problems and they provideda general view of some neural network application in theirfield of research

In this study it is demonstrated that AIampDM technologyis able to develop data-driven models for generating rockgeomechanical properties The overall work-flow includesdevelopment of synthetic geomechanical well logs from con-ventional logs such as gamma ray and bulk density that arecommonly available These data-driven models used around30 percent of data (coming from geomechanical logs) of theentire asset which were available to expand them for the restof the field with conventional logs but no geomechanical logsData-driven models have been validated with blind wellsBlind wells are wells with actual data which are selecteddue to different locations in the asset of 100 horizontal gaswells Moreover the logs generated from data-driven modelsare used to build an integrated field-wide geomechanicaldistribution (maps and volumes) for rock geomechanicalproperties In this work the ultimate purpose is meant topropose a technique in order to omit the necessity of runninggeomechanical logs for the entire asset once such logs areobtained for some portion of the asset to be used in thedata-driven models The number of well logs required to runthe data-driven models that leads to an accurate result isdetermined to be 30 percent of the wells in an unconventionalasset In this study just 30 out of 100 horizontal wells in theMarcellus shale asset in southern Pennsylvania own actualgeomechanical well logs that are provided by a major servicecompany

2 Methodology

In order to accomplish the objectives of this study a method-ology and thorough procedure are required to be definedThemethodology used to accomplish the objectives of this studyincludes four steps as indicated below A detailed descriptionof each step is explained afterwards

(a) data preparation

(b) data-driven model development using AIampDM

(c) validation of data-driven models(d) geomechanical property distribution

21 Step (a) Data Preparation Data preparation is the mostimportant step in developing data-driven models due to thefact that all the other steps are using the data prepared inthis step Data preparation involves checking the data foraccuracy entering data in a right format in a computer fileand developing and documenting a database structure thatintegrated the various properties used in the next stepsIn this study a dataset form the available well logs arerequired to be prepared that portrays specific property ofthe rock versus depth First production pay zone must beidentified from the conventional well logs along with thehorizontal wells trajectory After identifying the depth ofthe producing zones for Marcellus shale from conventionalwell logs such as gamma ray available information of eachindividual well is extracted from well logs in every oneor half a foot according to its log characteristics and toolsused Also in order for the models to understand thedifferences between production pay zones and while usingthese datasets it is essential to specify the contrast betweenpay zones (upper Marcellus Purcell lower Marcellus andOnondaga) and the adjacent rocks nonshale To account forthis contrast 50 feet depth of log data located above andbelow the pay zones of interest is also added to the maindataset

The prepared dataset contains rows and columns con-sisting of following data that are recorded versus depth thewells name the well coordinates the values for gamma ray(GR) bulk density (BD) sonic porosity bulk modulus (BM)shear modulus (SM) Youngrsquos modulus (YM) Poissonrsquos ratio(PR) and total minimum horizontal stress (TMHS) for eachhorizontal shale well It must be emphasized that not all thewells include geomechanical well logs thus geomechanicalvalues are only recorded for the wells that have such real data30 wells

22 Step (b) Data-Driven Models Development In this stepthe prepared dataset was processed using backpropagationalgorithm of neural network into two main parts

Part 1 (Conventional Models) In order to have consistentconventional logs data for all wells for the shale productionpay zone part 1 is defined and the conventional logs aregenerated for those wells that missed some conventional logsAs it is shown in Figure 1 the bulk density and sonic porosityfor 30 wells were produced by using two different data-driven models First model (neural network model 1) usedgamma ray depth location and well coordinates as an inputto develop training calibration and verification segmentsfor generating the bulk density of around 30 wells in theasset where bulk density and sonic data weremissing Secondmodel (neural network model 2) used also bulk density asinput beside inputs of first step to generate sonic porosity forthe part of asset without this property (around 30 wells alsoused in this part)



depth for 70 wells





depth for 30 wells




Generate



for 30 wells



wells



wells


for 30 wells





Longitude

Latit

ude






MAPE = 100119899

119899

sum

119905=1

1003816100381610038161003816100381610038161003816

119860119905 minus 119865119905

119860119905

1003816100381610038161003816100381610038161003816

(1)




(2)








6220

6230

6240

6250

6260

6270

6280

Bulk modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Shear modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280


05

07

09

11

13

15

6220

6230

6240

6250

6260

6270

6280

Young modulus

1 2 3 46220

6230

6240

6250

6260

6270

6280

Poissonrsquos ratio

0 01

02

03

04

05












6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio



6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05








6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











depth for 70 wells





depth for 30 wells




Generate



for 30 wells



wells



wells


for 30 wells





Longitude

Latit

ude






MAPE = 100119899

119899

sum

119905=1

1003816100381610038161003816100381610038161003816

119860119905 minus 119865119905

119860119905

1003816100381610038161003816100381610038161003816

(1)




(2)








6220

6230

6240

6250

6260

6270

6280

Bulk modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Shear modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280


05

07

09

11

13

15

6220

6230

6240

6250

6260

6270

6280

Young modulus

1 2 3 46220

6230

6240

6250

6260

6270

6280

Poissonrsquos ratio

0 01

02

03

04

05












6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio



6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05








6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











MAPE = 100119899

119899

sum

119905=1

1003816100381610038161003816100381610038161003816

119860119905 minus 119865119905

119860119905

1003816100381610038161003816100381610038161003816

(1)




(2)








6220

6230

6240

6250

6260

6270

6280

Bulk modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Shear modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280


05

07

09

11

13

15

6220

6230

6240

6250

6260

6270

6280

Young modulus

1 2 3 46220

6230

6240

6250

6260

6270

6280

Poissonrsquos ratio

0 01

02

03

04

05












6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio



6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05








6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6220

6230

6240

6250

6260

6270

6280

Bulk modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280

Shear modulus

0 1 2 3 46220

6230

6240

6250

6260

6270

6280


05

07

09

11

13

15

6220

6230

6240

6250

6260

6270

6280

Young modulus

1 2 3 46220

6230

6240

6250

6260

6270

6280

Poissonrsquos ratio

0 01

02

03

04

05












6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio



6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05








6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6203

6223

6243

6263

6283

6303

6323

6343

6363

Bulk modulus

0 2 46203

6223

6243

6263

6283

6303

6323

6343

6363

Shear modulus

0 1 2 3 46203

6223

6243

6263

6283

6303

6323

6343

6363


0 05 16203

6223

6243

6263

6283

6303

6323

6343

6363

Young modulus

1 2 3 546203

6223

6243

6263

6283

6303

6323

6343

6363

Poissonrsquos ratio



6274

6294

6314

6334

6354

6374

6394

6414

Bulk modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414

Shear modulus

0 1 2 3 46274

6294

6314

6334

6354

6374

6394

6414


0 02

04

06

08

1

6274

6294

6314

6334

6354

6374

6394

6414

Young modulus

1 2 3 546274

6294

6314

6334

6354

6374

6394

6414

Poissonrsquos ratio

0 01

02

03

04

05








6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6343

6353

6363

6373

6383

6393

6403

Shear modulus

0 1 2 36343

6353

6363

6373

6383

6393

6403


05 07 096343

6353

6363

6373

6383

6393

6403

Young modulus

1 2 3 546343

6353

6363

6373

6383

6393

6403

Bulk modulus

0 1 2 3 46343

6353

6363

6373

6383

6393

6403

Poissonrsquos ratio

0 01

02

03

04



61445

61645

61845

62045

62245

62445

Shear modulus

1 15

25

2 3

61445

61645

61845

62045

62245

62445


05 07 0961445

61645

61845

62045

62245

62445

Young modulus

2 3 5461445

61645

61845

62045

62245

62445

Bulk modulus

0 2 461445

61645

61845

62045

62245

62445

Poissonrsquos ratio




4 Conclusions





425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










425400375350325300275250225200



425400375350325300275250225200



220210200190180170160150140130


220210200190180170160150140130




Appendix



026

024

022

020

018

016

014

012

010


026

024

022

020

018

016

014

012

010



090

085

080

075

070

065

060

055


090

085

080

075

070

065

060

055



300

280

260

240

220

200

180

160

140


300

280

260

240

220

200

180

160

140








119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119900119895= Output of unit 119895 (119900

119895= 120590 (119911

119895))

119905119895= target for unit 119895

(A1)




119895119894



120597119864

120597119908119895119894

=

120597119864

120597119911119895

119909

120597119911119895

120597119908119895119894

=

120597119864

120597119911119895

(A2)




119864 =

1

2

sum


119896))2

(A3)



120575119895=

120597119864

120597119911119895

=

120597

120597119911119895

1

2

(119905119895minus 119900119895)

2

= minus (119905119895minus 119900119895)

120597119900119895

120597119911119895

= minus (119905119895minus 119900119895)

120597

120597119911119895

120590 (119911119895)


119895)) 120590 (119911

119895)


119895) 119900119895

(A4)

Δ119908119895119894= minus120578

120597119864

120597119908119894119895

= 120578120575119895119909119895119894 (A5)



119895


119895119894


119895119894


119895which




120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot

120597119911119895

120597119908119895119894

= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

sdot 119909119895119894

(A6)



119895 Also note that 120597119864120597119911

119896= 120575119896 120597119911119896120597119900119895= 119908119896119895

and 120597119900119895120597119911119895= 119900119895(1 minus 119900


120575119895= sum


120597119864

120597119911119896

sdot

120597119911119896

120597119900119895

sdot

120597119900119895

120597119911119895

= sum


119895)

(A7)

thus120575119895= 119900119895(1 minus 119900

119895) sum


Nomenclature




119874119895 Output of unit 119895



Highlights






References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article geomechanical properties of ...well logs such as gamma ray, available information...

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