DEVELOPMENT OF INFILL DRILLING RECOVERY MODELS FOR CARBONATE … · 2016-07-06 · his work introduces a novel methodology to improve reservoir characterization models. In this methodology

DEVELOPMENT OF INFILL DRILLING RECOVERY MODELS FOR CARBONATE RESERVOIRS

CT&F - Ciencia, Tecnología y Futuro - Vol. 1 Núm. 5 Dic. 1999

R. SOTO*, CH. H., WU�, and A. M. BUBELA �

� Texas A&M University, College Station, Texas 77843-3116, USAEcopetrol - Instituto Colombiano del Petróleo, A.A. 4185 Bucaramanga, Santander, Colombia

E-mail: [email protected]

* A quien debe ser enviada la correspondencia

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DEVELOPMENT OF INFILLDRILLING RECOVERY

MODELS FOR CARBONATERESERVOIRS USING NEURAL

NETWORKS AND MULTIVARIATESTATISTICAL AS A NOVEL

METHOD

his work introduces a novel methodology to improve reservoir characterization models. In thismethodology we integrated multivariate statistical analyses, and neural network models for forecastingthe infill drilling ultimate oil recovery from reservoirs in San Andres and Clearfork carbonate formations

in West Texas. Development of the oil recovery forecast models help us to understand the relative importance ofdominant reservoir characteristics and operational variables, reproduce recoveries for units included in thedatabase, forecast recoveries for possible new units in similar geological setting, and make operational (infilldrilling) decisions. The variety of applications demands the creation of multiple recovery forecast models. Wehave developed intelligent software (Soto, 1998), Oilfield Intelligence (OI), as an engineering tool to improvethe characterization of oil and gas reservoirs. OI integrates neural networks and multivariate statistical analysis.It is composed of five main subsystems: data input, preprocessing, architecture design, graphic design, andinference engine modules. One of the challenges in this research was to identify the dominant and the optimumnumber of independent variables. The variables include porosity, permeability, water saturation, depth, area, netthickness, gross thickness, formation volume factor, pressure, viscosity, API gravity, number of wells in initialwaterflooding, number of wells for primary recovery, number of infill wells over the initial waterflooding, PRUR,IWUR, and IDUR. Multivariate principal component analysis is used to identify the dominant and the optimumnumber of independent variables. We compared the results from neural network models with the non-parametricapproach. The advantage of the non-parametric regression is that it is easy to use. The disadvantage is that itretains a large variance of forecast results for a particular data set. We also used neural network concepts todevelop recovery models. The neural network infill drilling recovery model is capable of forecasting the oilrecovery with less error variance compared with non-parametric, fuzzy logic and regression models.

Keywords: neural networks, carbonate reservoirs, recovery models, infill drilling

T


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ste trabajo introduce una metodología novedosa para mejorar los modelos de caracterización deyacimientos. En esta investigación se usaron técnicas de estadística multivariada y redes neuronalespara desarrollar modelos de predicción de los recobros primarios de aceite (PRUR), recobros de aceite

al inicio de la inyección de agua (IWUR) y recobros de aceite debido a la perforación de pozos de relleno (IDUR)en yacimientos de carbonatos localizados en el este de Texas. Los modelos desarrollados fueron comparadoscon los modelos de regresión no-lineal y con los de regresión no-paramétrica. Uno de los desafíos en estainvestigación fue identificar las variables independientes dominantes y el número óptimo de estas. Para ello sedesarrolló un sistema inteligente (Soto, 1998), Oilfield Intelligence (OI), que integra conceptos de componen-tes principales, análisis de factores y redes neuronales. OI está compuesto por cinco subsistemas: carga ypreprocesamiento de los datos, diseño de la arquitectura de la red neuronal, diseño gráfico y una máquina deinferencia. El análisis multivariado de componentes principales permite resolver el problema de dimensionalidad.Cuántas y cuáles variables deberían usarse en la obtención de cada modelo. Después se utilizaron las redesneuronales para desarrollar modelos capaces de predecir los recobros primarios, de inyección de agua ydebido a la perforación de pozos de relleno en las formaciones de carbonato de San Andrés y Clearfork en eleste de Texas. Los coeficientes de correlación son del orden del 99% con errores absolutos no mayores del 3%comparados con coeficientes de correlación del orden de 0.91 y errores absolutos alrededor del 27% de otrosmodelos publicados internacionalmente en los últimos 15 años. Las variables consideradas en esta investiga-ción fueron porosidad, permeabilidad, saturación de agua, profundidad, área, espesor total, espesor neto,factor volumétrico de formación, presión, viscosidad, gravedad API, número de pozos al inicio de la inyecciónde agua, número de pozos para la recuperación primaria, número de pozos de relleno al inicio de la inyecciónde agua, PRUR, IWUR, e IDUR. Obviamente el desarrollo de un modelo en redes neuronales que represente conalta precisión los datos requiere experiencia del ingeniero para realizar un control de calidad de los datos,determinar las variables dominantes y optimizar la estructura o topología de la red neuronal.

E



INTRODUCTION

The amount of oil that can be recovered from an oilreservoir is dependent on the reservoir characteristics,recovery method, number of wells, and operationsefficiency. Waterflooding is often used as a secondaryrecovery process after the reservoir has been producedduring primary recovery. Since most reservoirs are heter-ogeneous, infill drilling after an initial water flood permitsproduction of oil from parts of the reservoir that mightotherwise have been bypassed (Wu et al. 1992; Wu etal., 1988; Shao et al., 1994a; French et al., 1991).Researchers and engineers have shown that the infilldrilling in the West Texas carbonate reservoirs hasindeed accelerated and increased oil recovery (Lu et al.,1993, 1994). However, the dominant mechanisms andparameters (independent variables) that affect the oilrecovery are not fully identified or may be subjected tovarying degrees of uncertainty. To forecast or predictinfill drilling recovery efficiency of an individual unit ora reservoir is a difficult task for most reservoir-engi-neering professionals.

Many approaches are used to evaluate infill drillingrecovery efficiency. One is based on unit recovery effi-ciency and reservoir analysis, plus the reservoir engi-neer�s experience and intuition. A more elaborateapproach is the use of a �reservoir simulator�, whichrequires detailed and yet uncertain input reservoir and

NOMENCLATURE

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ACE alternating conditional expectationsAPI API gravity of crude oilAREA productive area (acres)BASIN basin index: 1 for San Andres, 2 for ClearforkE performance indexf transformed functionFVF oil formation volume factor (RB/STB)GROSS gross pay (m)IDUR infill drilling ultimate oil recovery (MSTB)IWUR initial waterflood ultimate oil Recovery (MSTB)k node index of output layern index of independent variablesNET net pay (m)NOIW number of wells after infill drillingNOPW number of wells for primary RecoveryNOWW number of wells for initial WaterfloodingPRUR primary ultimate oil recovery (MSTB)o output value from hidden nodes

OOIP original oil in place (MSTB)p index of training setPERM permeability (mD)POR porosity (fraction)PRESS initial reservoir pressure (Pa)PRUR primary ultimate recovery (MSTB)RB reservoir barrelSTB stock tank barrel, oil production unitSW initial water saturation (%)VIS oil viscosity (Pas)W weightsx independent variabley dependent variablez value of individual transformd error signalh learning rateDIWUR IWUR - PRUR (MSTB)DIDUR IDUR - IWUR (MSTB)

production data. The approach we take here is statis-tical. It is based on an oil recovery efficiency databasedeveloped for specific producing formations in a parti-cular geological basin.

The field data from waterflood units in the PermianBasin were gathered and used to develop the oil recov-ery forecast models. The database includes reservoirand production data of 21 units in the San Andres forma-tion, and 23 units in the Clearfork formation. Estima-tion of ultimate and incremental infill drilling recoveryhas been a difficult task because of the limited data andthe uncertainties in the independent variables (Maliket al., 1993; Shao et al., 1994b; Wu et al., 1993, Wu et al.,1997). Many attempts have been made to develop theinfill drilling oil recovery forecast models with referenceto reservoir rock and fluid properties. Non-linear re-gression, statistical analysis, and fuzzy logic were usedto analyze the oil recovery data. While these reportedmodels have improved over the years, they are not en-tirely satisfactory due to the inexact nature of the dataset and the inherent limitations in the models themselves.

With the development of linear regression (Wu et al.,1998; Shao et al., 1994a) or statistical analysis (Frenchet al., 1991) models of infill drilling ultimate recovery (IDUR),a substantial effort was put into determining which inde-pendent variables are the most important and what isthe optimum number of the independent variables. Some


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of these independent variables go beyond the basicreservoir and field properties. Additional parameterssuch as the primary ultimate recovery (PRUR) andthe initial waterflood ultimate recovery (IWUR) wereneeded to succesfully forecast IDUR (Lu, 1993).

In this work non-parametric regression analysis andneural network modeling are used to develop forecastmodels that improve the degree of consistency and accu-racy. The non-parametric approach is based on the workof Breiman and Friedman (1985). Xue and Datta-Gupta(1996) applied this approach to integrate seismic datain reservoir characterization. While the non-parametricregression analysis provided better identification of thedominant independent variables and more consistentforecast, it did not improve the variance of forecastresults.

The application of neural networks for modeling non-linear systems has been improved substantially in recentyears (Nikravesh et al., 1996; Bomberger et al., 1996;Rogers and Dowla, 1994; Al-Kaabi et al., 1990; Azimi-Sajadi and Liou, 1989; Johnson and Wichern, 1998). How-ever, one of the problems that still had to be addressedwas the determination of the dominant variables, andthe optimum number of independent variables. Someintuitive insight and functional knowledge of the physicalbehavior of the system turned out to be helpful for iden-tifying the dominant independent variables. A non-parametric regression analysis and the principal compo-nents and factor analysis of multivariate statistical anal-ysis were used to identify the dominant independentvariables so we could develop more efficient and realis-tic neural-networks models.

DATA SET

The input data are shown in Tables 1, 2, 3, and 4.Tables 1 and 2 list the reservoir and operational pro-perties of the San Andres units. The dependent varia-bles are primary ultimate oil recovery (PRUR), initialwaterflood ultimate recovery (IWUR) and infill drillingultimate recovery (IDUR). The labels and units of eachvariable are referred to the Nomenclature. During thedata analysis and model development, it is found thatIWUR is strongly dependent on PRUR; and IDUR onPRUR and IWUR. The sequential dependency com-plicates the development of dependable IDUR models.Table 3 and 4 list the properties of Clearfork units.

A REVIEW OF THE INFILL DRILLINGRECOVERY MODELS

The updated non-linear standard regression forecastmodels for primary ultimate oil recovery (PRUR), initialwaterflooding ultimate oil recovery (IWUR), and infilldrilling ultimate oil recovery (IDUR) of San Andresand Clearfork units are summarized in Table 5. Foreach model, we included the independent variables, thecoefficient of determination and the average absoluteerror. Figure 1 shows a comparison of actual and pre-dicted PRUR. Apparently, the correlation betweenpredicted values and actual values is good but the aver-age absolute error is about 23.46%. Figures 2, 3 and 4show residual plots for the statistical model to predictPRUR, IWUR, and IDUR. The residual plots do nothave a constant variance and the upward trend suggeststhat the models may need additional terms. From theseplots and the average absolute errors, we could concludethat it was necessary to search other modeling techni-ques.

NON-PARAMETRIC REGRESSION APPROACHFOR ESTIMATING OPTIMAL TRANSFOR-MATIONS FOR MULTIPLE REGRESSIONS

Non-parametric regression is one of the novelapproaches to constructing a suitable model descriptionfrom available information. It is developed to alleviatethe problem of parametric regression that often leadsto erroneous results caused by the mismatch betweenassumed model structure and the physical relationshipsof the actual data. In non-parametric regression we donot fix a priori the form of the dependency of the depen-dent variable on the independent variables. In fact, oneof the main results of non-parametric regression is theform of the relationship.

Non-parametric regression is intended to build amodel in the form,

(1)

where, the inverse transformation, f0-1 ,and thetransform sum of the independent variables, z0 ,areselected to maximize the correlation between the right-hand and left-hand sides of the relation:

(2)

y = f -1 (z0)0

z0 = z1 + z2 + ... + zn


CT&F - Ciencia, Tecnología y Futuro - Vol. 1 Núm. 5 Dic. 1999 9

subject to some constraints. The data transforms iscalculated:

(3)

In this case the symbol fn(xn) does not necessarilymean a certain algebraic expression. It is rather a rela-tionship defined point-wise. The method of alternatingconditional expectations (ACE) (Breiman and Friedman1985), constructs and modifies the individual transfor-mations in order to maximize the correlation in the trans-formed space. Certain trivial constraints (zero meanand unit variance for the individual transformations)assure that the solution is almost unique. To make the

ACE algorithm really work, however, one has to implysome kind of restriction on the smoothness of the indivi-dual transformations, and this is done somewhat hidden,the way a certain �smoother� is used to construct andimprove the transformations.

One of the great advantages of non-parametricregression is that it provides an insight into the influenceof the individual variables. The shape of the point-to-point transformation is very informative, and the rangeof the transformed variable zi tells a lot about the relativesignificance of the independent variables.

We used a dummy variable called BASIN withvalues 1 for San Andres and 2 for Clearfork. The coef-ficients of determination and the average absolute errors

z1 = f1(x1), z2 = f2(x2), ..., zn = fn(xn) and z0 = f0(y)


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for PRUR, IWUR, and IDUR are 0.9786, 0.933 and0.9436, and 21.2%, 28.1% and 29.4 %, respectively.Figure 5 shows a comparison of actual and predictedPRUR for non-parametric model using GRACE. Fig-ure 6 shows the residual plot for IDUR to check foradequacy of each model. We can see from this plotthat the errors do not have zero mean, neither constantvariances, and the model underestimates the IDUR asthe actual IDUR increases.

THE OILFIELD INTELLIGENCE

The neural network simulator developed in thisresearch is called Oilfield Intelligence (OI) (Soto, 1998).The neural network simulator was built in MATLAB, a

high-performance language for technical computing thatintegrates computation, visualization, and programmingin an easy-to-use environment.

OI is composed of five main subsystems: Loadingdata, preprocessing, architecture design, graphic design,and inference engine modules. We wrote more than1,200 lines of programming as M-files using MATLABas a platform. Figure 7 illustrates the architecture OIand Figure 8 shows the graphical user interface (GUI)with each of these modules.

THE DOMINANT INDEPENDENT VARIABLES

A multivariate statistical analysis was also performedto determine the dominant independent variables with

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reference to that investigated by the non-parametricregression analysis. Table 6 shows the results of a prin-cipal component analysis for PRUR. As can be seen,seven principal components could explain about 90%of the total variance of the data. To describe the possiblerelationships among the variables and determine if thereis any possibility for grouping variables, we used theconcept of factor analysis (Johnson and Wichern 1998).Table 7 shows an output of the rotated factor loading.The first factor grouped the primary ultimate oil recovery(PRUR) with the productive area and the number ofprimary recovery wells (NOPW) as indicated by the

loading values. The first factor shows the dependenceof the primary ultimate oil recovery on the productivearea and the number of wells for primary recovery.According to this factor analysis, we decided to usearea, because of the highest coefficient in factor1, asindependent variable to predict PRUR. The secondfactor grouped BASIN, DEPTH and initial reservoirpressure (PRESS). For this group, we selected BASINas independent variable because it was easier for theneural network to recognize that pattern. The third fac-tor grouped the API gravity and FVF. The fourth fac-tor shows that gross thickness (GROSS) has the highest

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loading factor. Factors five and six are explained bywater saturation (SW) and porosity (POR) variablesrespectively. Similar principal component and factoranalysis was performed for IWUR and IDUR.

Results of the principal component and factor anal-ysis indicate that the dominant independent variablesfor PRUR are: productive area (AREA), BASIN, APIgravity, FVF, gross thickness (GROSS), and initial watersaturation (SW). The dominant independent variablesfor IWUR are: productive area, the number of initialwaterflood wells (NOWW), DEPTH, the number ofprimary wells (NOPW), API, FVF, net pay (NET),

porosity (POR), permeability (PERM), and viscosity(VIS). The dominant independent variables for IDURare: BASIN, productive area (AREA), and the numberof infill wells (NOIW), PRESS, API, FVF, NET, VIS,SW, and POR. Those dominant independent variableswere used to develop neural network infill drillingrecovery models.

NEURAL NETWORKS

A neural network is a series of layers with nodesand weights that represent complex relationships among

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input and output variables. The first layer has inputnodes representing the input variables (independent var-iables) specified by the problem. The node of a hiddenlayer uses the sum of the weighted outputs of previouslayer and sigmoid function to provide output for thenodes in the subsequent hidden layer. The number ofnodes in each layer and the weights are determined bytrials and by optimization. The objective of the neuralnetwork is to obtain optimal weights to give a best valuefor the nodes (the dependent variable) of the output layer.

The advantages of the neural network approach areseveral. It does not require an explicit functional rela-tionship between the input and output variables. It canbe trained from past available data to learn and approxi-mate the nonlinear relationships to any degree ofaccuracy. It is applicable to multivariate systems. Oneof the significant drawbacks of the neural networkapproach is that the input nodes must be specified apriori. The type of input variables and the optimalnumber of input variables can not easily be determined

from neural network analysis.Performance learning is one of the most important

steps where network parameters are adjusted in aneffort to optimize the performance of the network. Twosteps are involved in this optimization process. The firststep is to define a quantitative measure of networkperformance, performance index. It represents a glo-bal error (E) of the neural network defined as,

(4)

Where the inner summation is over all nodes in theoutput layer, and the outer sum is over the number ofthe training set. During training, the size of error gener-ally decreases until it reaches a threshold level.

The second step of the optimization process is tosearch the parameter space to reduce the performanceindex. A steepest descent algorithm is often used forthe optimization. Mathematically, the weights (W) areadjusted as follows:

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(y real - y real)E = å åp k pk pk


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(5)

Where h is learning rate and o is the output valuefrom hidden nodes and d is the error signal termproduced by the nodes in the hidden or output layer,

(6)

Before training the network, the learning rate (h)for the network must be specified. The new weightsare then used to calculate the new output. The procedureis repeated until a tolerance is satisfied.

We used a backpropagation algorithm with theLevenberg-Marquardt procedure as an optimizationmethod for convergence was used. The 44-sample dataset was divided into three subsets for training set (72%),validation (14%) and testing (14%). A post-trainingevaluation of the performance of the trained neuralnetworks is carried out by calculation of the errors forthe training, validation, and testing data sets.

The final topology of the neural network for pre-diction of PRUR has 7 neurons (independent varia-bles) in the input layer and two hidden layers with 12and 10 neurons, respectively. For IWUR, the topologyhas 9 neurons in the input layer and two hidden layerswith 10 and 6 neurons respectively. The topology ofthe neural network for IDUR has 9 neurons in the inputlayer, and two hidden layers with 10 and 8 neurons,respectively. After the neural networks were �trained,�the weight and bias vectors were incorporated intoFortran-90 and Visual Basic interfaces so that the resultscould be used in a practical manner.

The neural network model showed very good per-formance for prediction of PRUR, IWUR, and IDUR.

¶ Ed = -¶ (åWold ·o)

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Wnew = Wold + h ·d ·o


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The comparison of calculated and measured PRUR,IWUR and IDUR are presented in Figures 9 to 11.

The coefficients of determination and the averageabsolute errors for PRUR, IWUR, and IDUR are 0.998,

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0.992 and 0.9995, and 1.0%, 2.1% and 3.3%, respecti-vely. Using the same scales of the residual plots fornon-parametric regression models, we plotted residualplots for the neural network models of PRUR, IWUR,and IDUR (Figure 10 shows an example). Checking

the adequacy of each model we can see from this plotthat the errors do have zero mean and constant variance.

The dominant independent variables identified foreach model are used to develop the neural network oilrecovery models. A series of sensitivity analysis with

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different neural network topologies is performed todevelop the best neural network models. The approachhelped eliminate over-fitting and meaningless depen-dency of certain independent variables. For the sensitiv-ity analysis of variable dependency, we made a seriesof runs for each basin (San Andres and Clearfork) byvarying the value of each independent variable whilekeeping other independent variables at individual mean.As an example, Figure 12 shows the dependency ofcalculated IDUR on the productive area. The monotone-ly increasing relationship indicates physically mea-ningful dependency of the variables. Figure 13 showsthe dependency of the calculated IDUR on the numberof infill wells. Similar sensitivity analysis was made forother independent variables. Table 8 shows a summaryof the oil recovery forecast model performance.

CONCLUSIONS

· The correlation coefficients of the non-linear re-gression models for predicting the infill ultimate oilrecovery for both San Andres and Clearfork carbon-ate formations in West Texas apparently are goodbut the average absolute error is about 23.46%

· One of the significant constraints for the model devel-opment is the limited number of field data that areinexact and often exhibit uncertain relationships.Principal components and factor analysis help un-derstand the relative importance of dominant reser-voir characteristics and operational variables toimprove the modeling.· The advantage of the non-parametric regression is

that it is easy to use and can quickly provide resultsthat reveal the dominant independent variables andrelative characteristics of the relationships. Thedisadvantage is retaining a large variance of forecastresults for a particular data set. The average absoluteerrors for PRUR, IWUR, and IDUR are 21.2%,28.1% and 29.4%, respectively. The residual plotsshowed that the errors do not have zero mean, norconstant variances.· Multivariate principal component and factor anal-

yses were employed to develop an effective neuralnetwork. The neural network infill drilling recoverymodel is capable of forecasting the oil recovery withless error variance. The average absolute errors forPRUR, IWUR, and IDUR are 1.0%, 2.1% and3.3% respectively. The residual plots showed that




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the errors do have zero mean and constant variances.· The novel methodology applied in this research can be

used to get better models for a reservoir characterization.

ACKNOWLEDGMENT

This material is based in part upon work supportedby the Texas Advanced Technology Program underGrant No. ATP-036327-066 1998. The Instituto Co-lombiano del Petroleo support for part of the works isalso appreciated. We would like to acknowledge PeterValko�s contribution to the development and applicationof multiple forecast models.

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DEVELOPMENT OF INFILL DRILLING RECOVERY MODELS FOR CARBONATE … · 2016-07-06 · his work introduces a novel methodology to improve reservoir characterization models. In this methodology