Data-driven forecasting of naturally fractured reservoirs basedon nonlinear autoregressive neural networks with exogenous input

L. Sheremetov a,n, A. Cosultchi a, J. Martínez-Muñoz a, A. Gonzalez-Sánchez a,M.A. Jiménez-Aquino b

a Mexican Petroleum Institute, Eje Central Lázaro Cárdenas 152, San Bartolo Atepehuacan, Distrito Federal, Mexicob PEMEX Exploración y Producción, Mexico

a r t i c l e i n f o

Article history:Received 31 January 2014Accepted 12 July 2014Available online 30 July 2014

Keywords:time series forecastingoil production predictionNARX neural networksnaturally fractured reservoirs

a b s t r a c t

In this paper we discuss the results of the modeling of naturally fractured reservoir based on theapplication of the nonlinear autoregressive neural network with exogenous inputs (NARX). We showthat the NARX network can be efficiently applied to multivariate multi-step ahead prediction of reservoirdynamics. Predictability of the time series is studied using the Hurst exponent. We show thatpreliminary clustering of the time series can increase the precision of the forecasting. We evaluate theproposed approach using a real-world data set describing the dynamic behavior of a naturally fracturedoilfield asset located in the coastal swamps of the Gulf of Mexico. This paper is not only intended forproposing a new model but to study carefully and thoroughly several aspects of the application of ANNmodels to reservoir modeling and to discuss conclusions that could be of the interest for petroleumengineers.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

The reservoir is described by a set of time series (TS) of fluidsfrom petroleum wells, which are characterized by different start-ing points and mutual influence. Production performance is bothcontrolled by the reservoir properties and is also affected byoperational constraints and surrounding wells performance. Therock and fluid properties of the reservoirs are highly nonlinear andheterogeneous in nature. The situation is even worse for naturallyfractured reservoirs (NFR), where natural fractures and faults(created over geologic time) are the primary channels both forhydrocarbon migration and for water breakthrough and gasconing. Thus, production TS comprise high-frequency multipoly-nomial components, represent a long memory process and areoften discontinuous (or piecewise continuous) which make diffi-cult to get the best model for such data.

Several important tasks of petroleum reservoir engineering areconcerned with the forecasting of oil production. Usually, produc-tion prediction problem is considered within several differentsettings (He et al., 2001). The first is the prediction of existingwells which is based on that well's previous production data. Theother one is the spatial prediction of a new infill drilling well

which is based on nearby wells' history production data. Finally,the problem of backward prediction, known as “backcasting”, canalso arise for some brown fields with no record of the measuredwells' production. In this paper, due to the space limits we focuson the former case.

Traditional methods of forecasting in petroleum engineeringinclude DCA, black oil model history matching, exploration analo-gies and exploration trend extrapolations (Weiss et al., 2002).These tools are based on subjective data interpretation: to pick theproper slope, to tune the parameters of the numerical simulationmodel in such a way that they keep the reasonable values tointerpret reservoir geology.

TS forecasting, along with clustering and classification, is one ofthe traditional time series data mining tasks (Batyrshin andSheremetov, 2008). Traditional prediction techniques based onTS analysis usually establish some requirements that should befulfilled. For instance, the use of the ARMA method is limited tostationary TS (the ARIMA model assumes that the data becomestationary after differencing), that implies that the mean, varianceand autocorrelation structure do not change over time (Peña et al.,2001). Such assumptions do not fulfill for the TS describing thebehavior of the reservoir.

For the past few decades, artificial neural networks (ANNs), amongother artificial intelligence techniques, have been extensively appliedin petroleum engineering due to their potential to handle nonlinea-rities and time-varying situations along with their ability to learn andadapt to new dynamic environments (Sheremetov et al., 2005;

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/petrol

Journal of Petroleum Science and Engineering

http://dx.doi.org/10.1016/j.petrol.2014.07.0130920-4105/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail addresses: sher@imp.mx (L. Sheremetov), acosul@imp.mx (A. Cosultchi),

jmmunoz@imp.mx (J. Martínez-Muñoz), agonzal@imp.mx (A. Gonzalez-Sánchez),marco.antonio.jimenez@pemex.com (M.A. Jiménez-Aquino).

Journal of Petroleum Science and Engineering 123 (2014) 106–119

Mohaghegh, 2005; Bravo et al., 2013). Several ANN topologies havebeen studied in their application to short-term (1–2 years) univariateand multivariate prediction of oil production. Though, nonlinearautoregressive neural network with exogenous inputs (NARX) havebeen studied for univariate forecasting of TS (Menezes and Barreto,2008), their application in multivariable settings in multi-step-aheadforecasting schemes has not been fully explored yet (Diaconescu,2008).

In this paper we analyze the problem of flow rate forecasting ofnaturally flowing wells under a limited availability of operationaldata (irregularities in operational conditions, lack of productionwell tests, etc.) by using ANN models. Though we try to summarizethe aspects that should be studied in order to thoroughly validatethe application of the ANN for modeling of the oilfield behavior, inthis paper we take as modest goals (i) analysis of the predictabilityof the production TS; (ii) applicability of the univariate andmultivariate forecasting, (iii) analysis of different topologies ofthe NARX networks, and (iv) the application of clustering techni-ques to improve forecasting results. We will not attempt to studythe feature selection process, the prediction capabilities of theNARX networks to forecast infill wells or thoroughly study long-term forecasting (for the periods up to 10 years). Obviously, toproperly estimate the utility of the obtained conclusions we wouldrequire many tests.

The rest of the paper is organized as follows. In the next sectionweexplain the motivation for research emphasizing both application-related and model-related aspects of production prediction. The basicsof TS forecasting with NARX networks is considered in Section 3.Section 4 resumes the results of the provided experiments with realdata from Jujo-Tecominoacán oilfield located in the coastal swamps ofthe Gulf of Mexico, used to validate the proposed approach. Section 5provides a review of the related work followed by conclusions.

2. The NARX network in time series forecasting

We start with a short introduction to NARX networks just tomake the motivation for this research more clear.

Recurrent neural network (RNN) is a class of neural networkwhere connections between units form a directed cycle. Thiscreates an internal state of the network which allows it to exhibitthe dynamic temporal behavior. Unlike feedforward neural net-works, RNNs can use their internal memory to process arbitrarysequences of inputs. RNNs cannot be easily trained for largenumbers of neuron units nor for large numbers of input units.Successful training has been mostly in time series problems withseveral inputs. Such kind of architectures is usually trained bymeans of temporal gradient-based variants of the backpropagationalgorithm. However, learning to perform tasks, in which thetemporal dependencies present in the input/output signals spanlong time intervals, can be quite difficult using gradient-basedlearning algorithms (for networks like Time Delay Neural Network– TDNN). Learning of such long-term temporal dependencies ismore effective with Nonlinear Autoregressive with eXogenousinput (NARX) architectures because their input vector is builtthrough two delay lines: sliding over the input and output signals(Menezes and Barreto, 2008).

The NARX is a recurrent neural network which has beendemonstrated being well suited for modeling nonlinear systemsand specially time series. Compared to classical prediction modelsof time series such as linear parametric autoregressive (AR),moving-average (MA) and autoregressive moving-average (ARMA)models (Box and Jenkins, 1970) recurrent NN (RNN) with asufficiently large number of neurons is a realization of the non-linear ARMA process (Haykin, 1999). Compared to feedforwardANN, they have the following advantages: (i) learning is more

effective in NARX networks because the gradient descent is betterand (ii) because of a feedback, these networks converge muchfaster and generalize better than other networks (Lin et al., 1996;Gao and Er, 2005). Embedded memory in recurrent NARX alsohelps reducing the effect of vanishing gradient (when the outputof a system at time instant k depends on network inputs presentedat times rook). In our previous paper, we show that NARX ANNoutperformed considerably the traditional TDNN network for theproblem at hand (Sheremetov et al., 2013). That is why NARXnetworks are used in this study.

In long-term prediction, the model's output is fed back to theinput for a fixed number of time steps. This way, input compo-nents, previously composed of actual sample points, are graduallyreplaced by predicted values1. The output of the network is

yðtþkÞ ¼ f ðuiðt�1Þ;…;uiðt�nÞ; yðt�1Þ; …;

yðt�nÞ; yðt�1Þ; …; yðt�nÞÞwhere i¼1,…,m.

The transfer function of the network f is the same as that of aone-output feedforward neural network; for more than one out-put f should have a subindex i:

f ðUÞ ¼ g ∑wh;jhqðU Þ� �

where g is the activation function of the output node, hyperbolictangent, hyperbolic logarithm or linear, depending upon the sumof the activation functions of the nodes in the hidden layer. wh,j isthe weight corresponding to the hidden node, h is the number ofthe hidden node. hqðU Þ is the activation function of the hiddennode. This activation function h( � )is defined as

hðU Þ ¼ net ∑wi;hui� �

;

where net is one of the hyperbolic tangent or hyperbolic logarithmfunctions:

tanhðxÞ ¼ ex�e� x

exþe� x; loghðxÞ ¼ 11þe�x

wi,h is the weight of input i that goes to the hidden node h. ui is thei-th input to the network, whether any of the input variables or afeedback.

Linear activation functions are not used in the hidden layer,since they would not contribute to the nonlinearity of the transferfunction of the network. However, the linear function can be usedat the output node in those cases where the characteristic functiondoes not contain a very high degree of nonlinearity. In this case theoutput layer absorbs the function linearly accumulating the con-tributions of the nodes of the hidden layers.

TS forecasting is usually performed in two different settings:(a) in a univariate setting, when the ANN is trained only with thetime series data and (b) in a multivariate setting, when othervariables (static and dynamic nature) are added as additionalinputs. Those variables that change monthly enter the networkjust as TS. Variables regarded as static are converted to TS copyingthe same values month by month and changing them if anyvariation occurs. The process of the selection of the most appro-priate set of input variables called “feature selection” is out of thescope of this paper.

In Fig. 1 a network topology is illustrated for the latter case. Foreach variable vi there is a number of n delays and the outputforecast k months ahead from the current month. First, based informer studies on forecasting oil production, it was decided to usea two-layer feedforward network (Schrader et al., 2005; Menezes

1 If the prediction horizon tends to infinity, from some time in the future theinput regressor is composed only of estimated values and the multi-step-aheadprediction task becomes a dynamic modeling task while the ANN model emulatesthe dynamic behavior of the system.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 107

and Barreto, 2008). However, after a series of tests on the numberof nodes in the hidden layer and activation functions, the networktopology with the smallest MSE was one layer with 20 hiddennodes. For the experiments, different combinations of threeactivation functions were used for hidden layer and the output:hyperbolic tangent (tan), hyperbolic logarithmic (log), and linear(lin). For each experiment we selected the four combinations.

The training horizon is set manually separated by k time units,where k is determined by the separation between the last knownsample before the test segment, and each sample in the testsegment. In the same way, internally the amount of delays is setthrough commands in order to change the window for monthsbefore and months after the current time.

3. Motivation for research

Though ANN models have been reported in the last decade asviable forecasting there are still a number of open questions/challenges for their wider acceptance by petroleum engineers andpetroleum companies. These questions can be divided into twosections as application-related and model-related.

3.1. Application related issues of production prediction

In brown fields, almost all wells produce with an increasingamount of gas and water especially when the pressure dropsbelow the fluid bubble pressure, so that gas and oil are flowingseparately within the well or even within the reservoir. As aconsequence, oil and gas production curve patterns lose theirsynchrony and behave erratically. On the other hand, waterproduction curves show an increasing pattern which, in somecases, can be modeled with a logistic curve (Cosultchi et al., 2012).However, as a consequence of gas injection we observed thatwater production can also behave erratically suggesting a discon-tinuous water plug flow within the reservoir. This natural behaviorof the fluids raises several questions to TS forecasting. How theprediction models behave with different types of fluids (oil, gas,

water)? Should there be different forecasting models for each fluidof the multiphase flow?

The field exploitation usually follows through primary produc-tion, with both natural and artificial lift techniques, secondaryproduction, and enhanced recovery stages. From the forecastingpoint of view, it is worth to know how to train the model of amature field to respond at any stages of production history? If theproduction history is split into different stages, what time horizonis essential for training a model? Can a unique model be useful topredict for each of the production stages?

Finally, there are several questions on the prediction of TSrelated to the production intervals corresponding to differentformations. In mature fields usually there are several wellsproducing simultaneously from different formations and layers.For sandy reservoirs, where each production layer is separated byimpermeable barriers, it can be reasonable to account separatelythe production of each layer. For NFR, instead, fractures may crossthe production zones making such assumption difficult to beaccepted. The TS behavior coming from different formations,layers or geological sections could be both independent (due todifferent geological composition and petrophysical characteristics)or highly influenced by each other. So, what should be thegranularity of oil production prediction for NFR: at well, formationor production interval level? Are there any forms of TS´s groupingfavorable for forecasting?

3.2. ANN-based model related issues

Building a neural network is a balancing act between the data, thenumber and topology of nodes, and the training algorithm employed.All these aspects should be evaluated before embracing a certainmodel. We divide such aspects into several groups: data-relatedissues, time series related issues and ANN-related issues as follows.

3.2.1. Data-related issuesData issues are related to the use of pre-processing techniques

usually recommended before the application of forecasting. Such

Fig. 1. Structure of the NARX network for oil production forecasting.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119108

pre-processing can be related both to the uncertainties in data andto the assumptions that a particular model requires. The opera-tional practices of production measurement (at the output of theseparation battery instead of the well head), along with theintrinsic difficulty of direct measurement of flow rates of a multi-phase flow (gas–oil–water), are the main source of uncertainty inhistorical flow rate data. So, the identification of data discrepanciesin the well behavior is critical for data cleansing. Well testing datahistory can be used for this purpose. Also the application ofexponential smoothing techniques, originated more than 30 yearsago as an extrapolation method for univariate time series, can beapplied to substitute the outliers (De Gooijer and Hyndman, 2006).When the input variables are measured on different scales, thenormalization or standardization of the inputs and target valuesby mapping them into some interval (usually [0,1] or [�1, 1],respectively) are required to bring all of the variables into propor-tion with one another.

In the case of multivariate prediction, another critical point isthe feature selection. The purpose of feature selection is theoptimization of the number of inputs of the model by selectingthe most important ones. For production prediction there are anumber of candidate variables describing well operating variables(wellhead pressure, flowline pressure, temperature, choke size,completed intervals, etc.) and reservoir conditions (static andflowing bottomhole pressure, temperature, porosity, permeability,initial water saturation, etc.). For instance, the Pearson correlationcoefficient and multivariate PCA analysis are useful techniques toshorten the list of variables to those that contribute to maximizethe variance between the cases (wells, in our case).

3.2.2. Predictability analysis of TSAn oilfield contains many wells and the behavior of their

production TSs can be quite different depending on natural orartificial factors. To reduce the risk of wrong forecasting, one needsto identify the wells with low and high levels of predictability.An accurate metric of time series predictability provides a measureof confidence in the accuracy of a prediction. Predictabilityanalysis indicates to what extent the past can be used to deter-mine the future and helps to distinguish random from non-random systems and identify the persistence of trends. The mostwidely used methodology is known as Rescaled Range analysis orR/S analysis. In general, the real TS lay between the following twoextremes: a time series generated by a deterministic linear processhas high predictability, while a time series generated by anuncorrelated process has low predictability, and its past valuesprovide only a statistical characterization of the future values.There are several metrics and tests which can be applied, so ourpurpose will be to check their performance for nonlinear oilproduction forecasting problem.

In this paper, two metrics are used for the analysis of predict-ability. The first one, a Hurst exponent will be used as a measure ofthe long-term memory of a time series, which helps to infer thelevel of difficulty in predicting and choosing an appropriate model.The rescaled range and chunk size follows a power law, and theHurst exponent is given by the exponent of this power law. AHurst exponent close to 0.5 is indicative of a Brownian time series.Production TS belong to persistent time series (0.5oHo1). Asshown in Kantz and Schreiber (2006), chaotic time series exhibitlong-range dependence due to their self-similar nature whereNARX networks could show their potential. Given the number ofobservations in a TS n, the Hurst exponent H is estimated by fittingthe power law E½RðnÞ=SðnÞ� ¼ CnH to the data, where E is theexpected value, C is a constant, R is the range of the first n valuesand S is standard deviation.

Another metric called η-metric was introduced by Kaboudan(1999) to measure the probability that a time series is predictable,approaching 1 for strongly deterministic signal and 0 for acomplex signal that is badly distorted by noise. The metriccompares the best fit models before and after shuffling. Theshuffling process is done by randomly re-sequencing an observeddata set using Efron's bootstrap method. This metric unfortunatelyhas a serious drawback: for nonstationary TS (our case) the valueof the metric tends to increase with the length of the time series.So, a variation of η-metric, a local predictability metric is used.Thus, the variation of the predictability over time can be observed,and the overall predictability of a specific time series can beestimated by calculating the average η over all windows.

η¼ 1�ffiffiffiffiffiffiffiffiffiffiSSEYSSES

s;

where SSEY ¼ ∑N

i ¼ 1ðyt� ytÞ2 and

SSES ¼ ∑N

i ¼ 1ðSt� StÞ2; S�shuf f led Y

The sum of squared errors of prediction (Sum of Squares Error –

SSE) is a measure of the discrepancy between the data and anestimation model. The original long-term TS is divided into Qsamples and the sample series is shifted by τ, and the η-metric iscalculated again on the new sample.

3.2.3. ANN settingsOne of the difficulties that arises when using ANNs is the

selection of the best ANN type, architecture, the training algorithmand its parameters. Although many different approaches exist inorder to find the optimal architecture of an ANN, none of thesemethods can guarantee the optimal solution for real forecastingproblems (Khashei and Bijari, 2010). The usual procedure is to testnumerous networks and select the network with the lowestgeneralization error.

There are many parameters of the ANN predictor which caninfluence the prediction capacity of the network and thus shouldbe studied. These parameters are network type, architecture (thenumber of hidden layers, the number of neurons in each layer,transfer function), learning algorithm, memory length, etc. Thereare several rules to fix the initial architecture, which should befurther tuned through the experiments. In the case of multivariateforecasting, the number of neurons comprising the input layer iscompletely and uniquely determined by the number of features inthe data. It is well known from practice of ANN that the inputvariables of ANN should not to be correlated, because highlycorrelated input variables may degrade the forecasting by inter-acting with each other as well as with other elements andproducing a biased effect (Farawey and Chatfield, 1998; Zhang,2003). The number of neurons of the input layer is also defined bythe memory size and by the number of lags of the dynamic inputvariables.

For prediction ANN (the so called Regression Mode), the outputlayer has a number of nodes corresponding to the output para-meters. Once again, in the case of TS forecasting the number ofnodes depends upon the length of the output pattern. In thetypical case of one-step ahead prediction, only one value ispredicted. Multi-step ahead prediction, though possible, tends todiverge rapidly from the true pattern due to the accumulation oferrors. The question of the separation value and the number ofpredicting points deserves special attention. In the case of one-stepahead prediction, obviously the predicted values are used for theupcoming predictions. Instead,multi-step ahead prediction permitsto use the real values but with larger separation.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 109

Another question is memory length. Long memory process is aprocess with a random component, where a past event has adecreasing/decaying effect on future events. The process has somememory of past events, which is “forgotten” as the time movesforward. The mathematical definition of long memory process isgiven in terms of autocorrelation: a value yt at time ti is correlatedwith a value ytþd at time tiþd, where d is some time increment inthe future. In a long memory process the decrease of theautocorrelation function (ACF) for a time series is a power law.

The question of hidden layers topology is the balance betweenthe computational and resolution performance. ANNs with twoand more hidden layers are extremely hard to train (“deeplearning”) and thus, problems that require more than a hiddenlayer are considered “non-solvable”. Nevertheless, one hiddenlayer is sufficient for the large majority of problems. Concerningthe size of the hidden layer, in practice it is defined by some rule-of-thumb. Another parameter is the transfer function (both forhidden and output layers).

The only valid architecture optimization technique is testing.The network topology can be tuned, i.e. removing nodes from thenetwork during training by identifying those nodes which, ifremoved from the network, would not noticeably affect networkperformance (i.e., resolution of the data). Though formal pruningtechniques exist, a rough idea of which nodes are not importantcan be obtained by looking at the weight matrix after training (thenodes on either end of the weights very close to zero can beremoved).

Another crucial issue related to TS forecasting is the trainingperiod duration, which impacts the nature of patterns obtainedand their predictability over time (Mehta and Bhattacharyya,2004). Longer training horizons are generally sought, in order todiscern sustained patterns with robust training data performanceextending into the predictive period. However, in dynamic envir-onments patterns that persist over time may be unavailable, andshorter-term patterns may hold higher predictive ability, albeit forshorter predictive periods. Such potentially useful short-termpatterns may be lost when the training duration covers muchlonger periods. A too short training duration can, of course, besusceptible to over-fitting to noise.

4. Experimental settings and results

In the following section we report the main results of theempirical evaluation of the NARX network for production fore-casting of the oilfield. The results of the predictive capabilities ofthe developed NARX-based models, tested on the available bench-marks (CATS ANN Competition and Mackey-Glass equation), havebeen reported before by the authors (Sheremetov et al., 2013).Therefore, in this paper we only provide the results of theexperiments over the oilfield production TS.

4.1. The reservoir under study

The Jujo-Tecominoacán oilfield is located 63 km SE of Villaher-mosa. It comprises a NWSE oriented anticline with double closure,with Jujo in the SE and Tecominoacán in the NW. The Jujo-Tecominoacán field is affected by a series of normal and inversefaults form individual blocks apparently in hydraulic communica-tion. The rock composition corresponding to the three productionformations, Upper Jurassic Kimmeridgian (UJKim), Upper JurassicTithonian (UJTith) and Lower Cretaceous (LC) are presented inTable 1. Fromwell log´s, 10 lithostratigraphic reservoir layers (fromLC at the top to UJK9 at the bottom) had been defined across thereservoir.

Production started on October 1980. The main productionmechanisms are fluid expansion and moderate water drive. Oilproduction is irregularly distributed over the field both vertically(it mostly comes from dolomites of UJKim age, specifically fromthe uppermost Kimmeridgian beds UJK5 and UJK6 with minorcontribution from dolomites of UJTith and LC ages) and horizon-tally forming zones of diverse productivity (Fig. 2). The reservoirrock is characterized as a highly heterogeneous system with dualporosity and permeability, which resulted in different types ofproduction behavior.

The initial pressure was 707 kg/cm2 and the crude oil bubblepoint pressure of 262 kg/cm2 was reached by 2002 in most of thewells. For this study, production history of 42 wells, active during2003, was selected. The selection, till 2003, covers the primary andartificial production type from the reservoir, before pressuremaintenance project started a year later. Statistical characteristicsfor these wells can be found in Table A1 of the Appendix.

4.2. NARX architecture

The main characteristics that correlate to production and can beconsidered as potential model inputs, include geophysical data,formation thickness, source rock data, and log data. Production dataare completed with geological (formation, completion thickness of theproduction intervals defined by the top and the bottom depths, maxvolume of calcite VLIS and dolomite VDOL – characterizing a typicalcarbonate reservoir rock) and geophysical data (matrix and fracturepermeability) coming from well log interpretation and numericalsimulation. Though production data represent small time series, therest of the variables are numeric, while formation is categorical.

A total of 31 variables describing (a) oil, gas and waterproductions, (b) flowing bottom-pressure, (c) chokes sizes,(d) completion interval depth, and (e) cleaning and stimulationtreatments were used in the experiments. After correlation ana-lysis and the set of the simulation runs with the network the list ofthe input variables was reduced to 10 variables. Table 2 lists thevariables chosen as inputs to the model.

The training horizon was set manually to 12 time units. Twoseparations for one-step ahead and multi-step ahead predictions wereset to 1 and 6, given the former better results during the experiments.MATLAB Neural Network Toolbox™ 7 was used to run the experi-ments. We limited our experiments to the Levenberg–Marquardtlearning algorithm since it was reported to outperform other compe-titors (Qian and Rasheed, 2004).

Table 1Production formations of Jujo-Tecominoacán oilfield and their lithology.

Formation Statisticparameters

Depth Lithology

(m) (vol. fraction)

Top Bottom Dolomia Limestone Claystone

LC Min 4558.52 4618.15 0.00536 0.017786 0.051127Max 5906.25 6012.68 0.87894 0.915867 0.251616Mean 5143.49 5244.9 0.33467 0.524485 0.103123Std Dev. 300.812 309.677 0.29885 0.303871 0.036394

UJTith Min 4618.15 5023 0.15328 0.003323 0.034717Max 6012.68 6674.65 0.90029 0.763751 0.161923Mean 5241.02 5611.89 0.77362 0.110607 0.07793Std Dev. 305.021 329.094 0.16916 0.163439 0.025037

UJKim Min 5023 5231 0.77606 0.004457 0.02034Max 6674.65 7399.9 0.93207 0.080057 0.095406Mean 5594.47 6081.71 0.87707 0.030084 0.053726Std Dev. 324.688 417.144 0.03284 0.013388 0.018266

Note: UJKim¼Upper Jurassic Kimmeridgian; UJTith¼Upper Jurassic Tithonian;LC¼Lower Cretaceous.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119110

Though our main interest is focused on oil production, we willalso model the behavior of the other 3 dynamic variables: gas,water and bottom-hole pressure (BHP), which are settled asoutput variables. Firstly, we tried to generate a model with multi-ple outputs but did not manage to get good results. So in thefollowing we generated individual models for each fluid andpressure (which, as shown below, happens to have differentarchitectures). In order to forecast each variable the four modelswere integrated to generate a concurrent model of the reservoir.

4.3. Experimental results

The idea of the experiments was to forecast two time period: Jan. –Dec. 2003 and May 2010 – Apr. 2011. The selection is related to thebeginning of the pressure maintenance program by gas (HC andnitrogen) injection in 2004. For the basic set of experiments weselected all the wells of the oilfield producing in 2003 and having atleast 24 months of production, in total 42 wells (see Table A1 of theAppendix). We evaluated different configurations of the model:

– Univariate and multivariate forecasting,– Four different architectures of the network transfer functions

for hidden and output layers respectively: tan–tan, log–tan,tan–lin, log–lin,

– Different combinations of the input variables,

– Different data groupings (by formation, by interval, by well orby frequency distribution patterns),

– Two types of data preprocessing: normalization and stand-ardization.

Table A2 in the Appendix summarizes experimental results. Wealso used the oilfield model created with IMAGINE softwareimplementing Top-down modeling approach (Mohaghegh, 2011)to compare the predictions with those obtained with NARXmodels. The inputs of the IMAGINE model were selected after aset of experiments and are the following: X/longitude, Y/latitude,oil, FBH pressure, and footage.

Many forecasting techniques do not easily provide a measure ofthe prediction error, and there is no single best approach todetermining this error, yet for many uses of time series forecastingknowing the uncertainty is as important as knowing the predic-tion. To measure the precision of the forecasting, two metrics wereused. The mean square error is computed on the missing values:

EMSE ¼1n

∑n

i ¼ 1ðyi� yiÞ2;

where yi is the actual and yi is the predicted value.Adjusted MAPE introduced by Armstrong, also known as

symmetric MAPE (though it is not symmetric since over- andunder-forecasts are biased) is calculated as follows:

ESMAPE ¼1n

∑n

i ¼ 1

yi� yiyiþ yi

��������

MSE can only be used to compare different models, meanwhileSMAPE can give us more general information about the forecastingperformance. In general a SMAPE of 10% is considered to be verygood, and below 20% is acceptable.

As the first step we have developed a univariate model of theoil production of the field to use the results as reference for themultivariate model (see Table A2 of the Appendix). While compar-ing between original, normalized (over the [0, 1] interval), stan-dardized and exponentially smoothed data, the use of original datashowed slightly better results for all the fluids and pressure, whiledata smoothing produced almost no effect.

4.3.1. Model decompositionIn order to build the best model of a reservoir, we studied

different options for organizing the model by considering the

Fig. 2. Areal view of Jujo-Tecominoacán oilfield with the distribution of cumulative oil production Np (MMBBL) over the field (rbf interpolation).

Table 2Input variables of multivariate ANN.

Exhaustive list of input variables Optimized list of input variables

Longitude LongitudeLatitude LatitudeQ-oil/gas/water (t) Q-oil/gas/water (t)Bottom-hole flowing pressure (t) Bottom-hole flowing pressure (t)Bottom-hole static pressure (t) Fracture permeabilityCompleted interval Max depth of completed intervalsDepth VLIS and VDOLChoke size (production and casing tubing)Treatment (stimulation, cleaning, workovers)NGNet payPermeability (fracture, matrix)Porosity (fracture, matrix)Water saturationLithology

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 111

following options: (a) monthly oil production at the well level,(b) monthly oil production split by each production interval,(c) monthly oil production by formation (one model for eachformation), and (d) producing wells slit by the similar productionpatterns (one model for each group). In the case of univariateforecasting instead of developing one model for each well, wedeveloped the model for the entire field.

4.3.2. Selection of the best fitted models for each fluidTo provide this type of experiments, we used a multivariate

model. As we can see from Table 3, the log–tan model showed thebest performance for pressure prediction, while log–lin modelsbehaved better for water, gas and oil forecasts. As expected, theprediction of water entrance was the most difficult task for themodel. One of the decisive factors was the absence of the reliabledata on water production.

4.3.3. Comparison of the forecasting results of univariate andmultivariate models

As we can see from Table A2 all the models reveal similar results.The most interesting thing is that the predictions for each well arealso very similar. However, 8 wells (19% of the data set) have higherror values –let us call them “outliers” – these wells are the same forall the models. Thus, if we exclude these outliers from the set, wereduce the SMAPE almost by half. These outliers are characterized by“an atypical behavior” compared to the rest of the wells. Suchatypical behavior can be either a rapid production declination or adrastic increase or very insignificant production rate which arereflected in their statistical characteristics as follows: quadratic orbimodal frequency histograms of oil production (W23, W28, W32and W37 wells), oil and gas rates with high kurtosis (displacementtowards very low production rate for W25 well), high correlationbetween oil and gas production rates (W28, W37 and W32 wells). Inorder to improve the capabilities of the model we propose to identifysuch atypical wells at the preliminary stages by grouping the data setby following multivariate analysis. The procedure and obtainedresults are discussed in the following section.

The IMAGINE model shows different results on each of the errormetrics. According to the SMAPE, it shows the worst result, thoughvery similar to the competitors. It is mostly due to its poor perfor-mance on four wells (W18, W32, W36, and W40). On the other hand,using MSE metrics it shows the best performance (MSE¼8.22Eþ07).

As we can also see from Table A2, the univariate model showsbetter results for 1-year period than the multivariate one. Though thedifference is relatively large (2%), it is mostly explained by the poorforecasting of the TS W12 (SMAPE of 84.5%) under multivariatesettings due to the well shut down during several months. If thesemonths are not considered (it is feasible, since the model tries topredict the natural behavior and is not fitted in by operationalconstraints), the SMAPE for this well is as low as 19% and the averageSMAPE is reduced to 10.67%, just 0.17% below the univariate model.The multivariate model suffers from the following drawbacks:

� since it has more parameters and every additional parameter has tobe estimated, it brings in an additional source of error;

� being more complex, the model can miss some nonlinearities thatare handled properly by the univariate models; and� an additional complication is conditional forecasting, when a flowrate is predicted, while values for static variables are assumed over theprediction interval.

Nevertheless the univariate model also has several importantlimitations:

� it can only be applied to the existing wells;� for longer periods, this model does not reflect properly thechanges in the well's behavior.

Another interesting point from Table A2 is that, apparently, TSdata grouping based on either producing formations (used togenerate the IMAGINE model of the oilfield), producing intervals(multivariate model), or wells production (univariate model) hasmixed effect on the forecasting accuracy. The univariate modelshows better forecasting (lower SMAPE) than the bivariate one for26 wells.

4.3.4. Predictability of the time seriesWe analyzed the dependencies between the statistical char-

acterization of the TS under study (see Table A1 from theAppendix) and the aspects of their predictability. Extreme valuesof the parameters are marked by a shadow. In Fig. 3 the results ofthe predictability analysis are shown both for univariate andmultivariate settings. For the TS of the wells under study, themean value of Hurst exponent is 0.658, with standard deviation of0.032. It is worth to mention that, only after 2004 when thepressure maintenance program started, several oil production TSexhibit fractal properties (with HE0.5). However, since this studycovers only the natural production period (up to 2003), therefore,none of those wells are included.

It can be clearly seen that TS with higher values of Hurstexponent (H40.69 in our case) indicating a smoother trend, lessvolatility, and less roughness, show better forecasting results, withthe SMAPE less than 5%. At lower values of H (o0.69), however, noconclusion on the relationship between variables can be made.

The η-metric was calculated only for the univariate best-fittedmodel (log–tan). Default parameters were used: the sample sizeQ¼20, and the shift step τ¼5. As we can see from Table A2, thereare four outliers, W21, W25, W28, and W32. The rest of the resultsshowed positive correlation of η-metric predictability and fore-casting error with coefficient¼�0.59. It is negative since lowerSMAPE correlates to higher predictability.

Usually it is stated that short TS are less suitable for forecasting.For the wells from case study, no dependency between theprecision of the forecasting and the TS length was found. Forillustrative purposes, the relationship between the length of the TSand the error metrics is shown in Fig. 4.

4.3.5. Forecasting results on longer time horizonsIn the next set of experiments we extended the forecasting

period until the last day of production of each of the 42 wells.

Table 3The best fitting models (shadowed cells) for each fluid and the pressure (standardized).

tan–tan log–tan tan–lin log–lin

MSE SMAPE MSE SMAPE MSE SMAPE MSE SMAPE

Pressure 3.23Eþ02 3.47% 1.41Eþ02 1.81% 4.20Eþ02 4.22% 4.30Eþ02 3.71%Gas N/A N/A 2.35Eþ02 15.11% 3.46Eþ02 22.89% 2.39Eþ02 14.93%Water N/A N/A 3.09Eþ09 43.42% 2.11Eþ08 37.11% 1.21Eþ08 24.30%Oil N/A N/A 1.42Eþ08 16.80% 1.72Eþ08 15.64% 9.87Eþ07 12.56%

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119112

Fig. 5 shows the oil and gas forecasting results for the field and, forillustrative purposes, for two representative wells (W20 and W26,see Table A3 of the Appendix). The vertical line separates theproduction curves used for training and forecasting. As it can beseen, the model for the whole oilfield has very competitiveforecasting. Obviously, some other wells forecasting does notmatch the real patterns, but it is also worth to mention thatproduction TS beyond 2004 are affected by gas injection andunexpected gas channeling. In order to see how the precision ofthe forecasting model degrades with time, Fig. 6 shows the SMAPEbehavior over the 8 years period.

Finally, we have also conducted the experiments on forecastingof the second period, May 2010 – Apr 2011, corresponding to thepressure maintenance. It should be mentioned that these resultsare obtained on a different data set, since in this case the wellsactive in 2010 were considered, and thus they cannot be comparedto the previous results considering only 2003 years wells. Thewells from the second dataset have more erratic behavior due tothe water and gas channeling because of injection, so the averageerrors are higher. Two models were compared: the first one,trained over the complete period of the wells history, and thesecond one – only trained on the data belonging to the pressuremaintenance period (after 2004). The second model also consid-ered the nitrogen and hydrocarbon gas injection from the nearestwell. As we can see from Table 4, the model trained on the shorterperiod of pressure maintenance gives better results. It means thatthe training and forecasting periods should belong to the sametype of oil production.

4.3.6. Forecasting improvement with clusteringIn the final set of experiments we divided the data set into

different groups based on different behavior patterns of the wells.71 Wells producing before 2004 were studied. Since oil productionpatterns show such diverse frequency distribution, a method toclassify wells of similar production behavior is required. Multi-variable analysis as principal component analysis (PCA) wasapplied to select the main variables followed by k-mean analysis.

In petroleum engineering, the production performance of thewells is represented by a productivity index, PI, or performanceindexes such as inflow performance relationship (IPR) or verticalflow performance (VFP) which qualify the production potential of awell. We choose instead as well performance indicators: (a) the oilproduction at 3, 6, 9, 12, 36 months independently of the startingdate; (b) the total accumulated production Np; (c) the number ofproducing months and, (d) the static initial pressure Pws. PCA showshigh correlation between the production variables and, conse-quently, the number of variables was reduced to four: accumulatedproduction of the first year, the total accumulated production at2003, number of production months and the initial static pressure,Pws. This selection allowed separating the wells applying Clusteranalysis (CA). This technique is an explorative multivariate dataanalysis used to classify entities with similar properties based ontheir nearness or similarity (Romesburg 1984; Lim, 2003). Thismethod is used in this paper to split the wells based on theirbehavior. Hierarchical agglomerative clustering is performed on thenormalized data by means of Ward's methods (Hervada-Sala andJarauta-Bragulat, 2004) and using Manhattan distances. The resultsare represented as a dendrogram where the objects are linkedtogether based on the distances between them (Dlink) and themaximum calculated distance (Dlink/Dmax)�100. Statistically sig-nificant clusters selected in this case have a cut-off value of 20% (seeresults in Fig. 7).

The members of each group and the mean characteristics wereobtained by applying the K-means clustering algorithm, one of thesimplest unsupervised learning methods with the k number ofgroups fixed a priori. The assignment of members of each group isbased on the minimum distance between the value of the selectedvariables and the means of the group. The results are presented inTable 5.

The wells classified in the first two groups, A and B, havesimilar mean values of production indices but a quite differentinitial Pws. The wells of the last two groups, C and D, have similarinitial Pws but the mean values of the production indices are threetimes higher for wells in D-group that those of C-group. Formodeling, we trained three models (as 16 wells from group Bhave already been shut-down by 2003). The results of oil fore-casting are shown in Table 6. In order to compare the obtainedresults, the first raw contains the errors for the reservoir model(JUJO-TECO).

As we can see, clustering of the TS based on production indicesprovides a small increment on the precision of the forecasts(1.13%). Only in the case of Group D the SMAPE is above theaverage. To a great extent it is due to the contribution of wells W6and W12. We have already mentioned that the W12 TS was founddifficult to predict. The interesting observation is that these twowells have relatively high Hurst exponent (0.684 and 0.765).Another direction for a future work could be the TS groupingbased just on their properties.

5. Related work and discussion

We based the following analysis on the literature available tothe authors taking OnePetro as a basic reference source, with aclear understanding that some papers could be found out of thescope of this paper.

Neural networks have been applied to TS analysis and predic-tion for many years. In petroleum engineering, ANN has beenapplied both to estimate and predict particular reservoir proper-ties (PVT, viscosity, permeability, etc.) and to predict flow ratesbased on the reservoir model. ANN's capability to estimate the PVTproperties and to predict petrophysical parameters based on welllogging data and available core plug analyses converted them in

Fig. 3. Distribution of the SMAPE errors as a function of the Hurst exponent.

Fig. 4. Distribution of the SMAPE errors as a function of the TS length.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 113

one of the best estimating method with high performance (Gharbiet al., 1999; Saemi et al., 2007; Lashin and El-Din, 2012).

Compared to numerical simulation, flow rates prediction withANN, both training and forecasting, can take just minutes whilenumerical reservoir simulators, though being very flexible, usuallyrequire several months to build and history match a flow model ofa mature field and update it (once a year) with recently collected

data. Another argument is the availability of ANN modeling toolsfor any engineer equipped by a standard desktop computer with-out any need in expensive specialized software. Finally, as theresults reported in the literature show, ANN models can exhibitvery competitive precision compared to the classic methods,outperforming the traditional DCA method (da Silva et al., 2007).

As we see from the analysis, different ANN architectures,univariate and multivariate, one- and multi-step-ahead TS fore-casting models are used in several applications related to produc-tion prediction. Such applications include oil and gas production

Fig. 5. Forecasting of oil and gas production for a period of 7 years (2004–2010): (a) for the whole field, and (b) for selected wells (W20 and W26).

Fig. 6. SMAPE behavior on the 8 years horizon for three best-fitting models.

Table 4Forecasting accuracy on oil production period from May 2010 till Apr 2011 fordifferent training periods.

Training SMAPE MSE

Complete history 21% 2.62Eþ07May 2004 – April 2010 15% 1.74Eþ07

Fig. 7. Classification of production wells using Cluster analysis.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119114

prediction (Garcia and Mohaghegh, 2004; Chakra et al., 2013), theallocation of production with greater reliability (Olivares-Velazquez et al., 2012), recovery prediction (Weiss et al., 2002),history matching (Ramgulam et al., 2007), missing reservoirreserves calculation (Mohaghegh, 2011), infill oil well prediction(He et al., 2001), and candidate wells selection for interventions(González-Sánchez et al., 2012), to mention a few. In theseapplications, both cumulative and monthly productions are usedas outputs, being the latter case a TS forecasting discussed in thepaper. Nevertheless, these models are difficult to compare sincethey use both different configurations and different data sets.

Feed-forward networks (FFN) with a multi-layer back-propagationlearning algorithm often called a multi-layer perceptron (MLP) can beapplied directly to problems of production TS forecasting provided thedata is suitably pre-processed (pre-processing of input data includesfeature extraction, dimensional reduction, outlier rejection etc.). Thistype of ANN is used in the commercially available petroleumengineering tools like DECIDE! from Schlumberger. Specialized ANNpackages focused on data mining and forecasting, both commercialand free, like NeuroShell, STATISTICA Neural Network, etc. offer awider range of ANN models and even enabling custom networks but,though having some preprocessing capabilities, require a time-consuming data-preparation stage.

He et al. (2001) describe a backpropagation ANN forecastingmodel to predict existing and infill oil well performance using onlyproduction data (without any reservoir data). The hidden layerwith 10 neurons and sigmoid transfer function define the ANN'sarchitecture. The experiments included two data sets (for existingand infill wells prediction) of 9 wells each from a low permeablecarbonate reservoir. The primary production history data wereused to train and test (1.5 year period) the neural networks. Forthe cumulative production, for testing data points correlationcoefficient of 0.99 and mean percentage error of 4.57% wereobtained. For the 3 month interval production output, a correla-tion coefficient of 0.93 was obtained for testing data points, whilethe average percentage error was 22.82% for training data set and16.06% for testing data points. The authors conclude good pre-dictive capacities of their model for short term (1–2 years)prediction. Only cumulative production was studied in the caseof infill well with only qualitative estimation of the results.

Recursive networks of NARX, Elman and Jordan type, capable ofsequence-prediction that is beyond the power of a standard MLP,are another good candidates for multi-step-ahead predictions ofTS. To our knowledge, this type of networks has not been appliedto production prediction so far but Elman RNN was successfullyused for seismic data inversion (Baddari et al., 2010). The authorsstudied the effects of network architectures on the rate ofconvergence and prediction accuracy. They tested the architec-tures for their ability to learn and predict, concluding thatincreasing of the number of the nodes in the hidden layer (from10 to 90) permitted to considerably improve the generalizationability of the Elman RNN.

As for the model inputs, though feature selection is out of thescope of this paper, here we just mention that the reportedforecasting models range between univariate (only TS of thepredicted variable are considered), through dynamic multivariatemodels (only dynamic variables are considered) to different multi-variate configurations integrating both dynamic and static vari-ables. For instance, He et al. (2001) describe a backpropagationANN forecasting model to predict existing and infill oil wellperformance using only production data (without any reservoirdata). On the other hand, in Weiss et al. (2002), the authors study atotal of 30 variables which were analyzed and ordered through aprocess of fuzzy ranking, but for the model inputs some variableswith low ranking but suggested by the experts were also selected.

The use of another type of ANN, higher order neural networks(HONN), to forecast production of water, oil and gas is reported inChakra et al. (2013). That type of neural networks has a goodcapacity to capture non-linear relationships between inputs andweights of the network. Two case studies illustrate the behavior ofthe proposed model with (i) only one dynamic parameter data, oilproduction data, and (ii) three dynamic parameter data, oil, gasand water production data used for forecasting. The oldest well ofthe field was used for the case study, for which 9 years productionhistory was used with 77 data points for training and months 78–94 (the last 16 months) to validate HONN models. The authorsreport the use of data preprocessing techniques: a moving averagefilter with a time span of five-points found to be optimal forreducing the random noise associated with production data; whilethe most significant input variables were selected by employingauto-correlation function (ACF) for single parameter data andcross-correlation function (CCF) for multiple parameter data. Thenetwork topology included one hidden layer with varied numberof neurons (1–5). Hyperbolic tangent function was used for hiddenlayers and a linear function for the output layer (tan–lin model).The HONN was used as a one-step-ahead predictor.

In da Silva et al. (2007) the authors study both short-term (fortwo mature onshore fields: Bonsucesso and Carmópolis) and long-term (for a synthetic reservoir) forecasting. The authors useddifferent ANN available from DECIDE!, NeuroShell, and STATISTICANeural Networks but do not specify their type and topology.Both fields are under waterflooding. For Bonsucesso, cumulative

Table 5Classification of the producing wells of two fields (Jujo and Teco) by cluster analysis.

Variables Group A Group B Group C Group D

Acum. Oil first year, Bbs 777,800.67 748,576.36 829,641.99 2,576,017.93Np at 2003, Bbs 3,400,455.89 3,750,363.84 12,418,651.55 39,126,918.05Months of production 65.54 82.18 202.11 236.77Initial Pws, kg/cm2 303.70 614.22 598.94 660.60Total producing wells 13 17 28 13

JUJO field 6 8 15 11TECO field 7 9 13 2

Wells shut-in by 2003 4 16 7 2

Table 6Best-fitting NARX models for each group obtained by cluster analysis.

Unit Model SMAPE MSE

JUJO-TECO log–lin 12.56% 9.87Eþ07GROUP A tan–tan 8.67% 1.63Eþ08GROUP C log–lin 11.94% 3.61Eþ07GROUP D log–tan 13.71% 8.76Eþ08GROUP AVG 11.43% 3.59Eþ08

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 115

production curves were used. Model inputs were the following:produced cumulative oil and water, cumulative injected water andnumber of days of production. For Carmópolis, one well wasstudied taking the information about its neighbors injecting wellsinto account. The results show that manual tuning of neuralnetworks can affect considerably the precision.

To address this problem, Saemi et al. (2007) proposed to use aGenetic Algorithm (GA) to determine the number of neurons in thehidden layers, the momentum and the learning rates for minimiz-ing the time and effort required to find the optimal architecture.Such hybrid approaches could further enhance the performance ofANN by choosing appropriate ANN parameters, e.g. number oflayers, nodes in the hidden layer, training algorithm, etc.

TS predictability deserves a further discussion, since we did notobserve a good correlation between Horst exponent and forecastingresults. Other methods include Lyapunov exponent (λ) which showsthe long term behavior of the time series and the metric (Kolmo-gorov–Sinai) entropy that characterizes chaotic motion of a system inan arbitrary-dimensional phase space. Originating from Shannon'sinformation theory, metric entropy is a measure of the rate of lost ofpredictability, which indicates that how far into the future can bepredicted with given initial information. In random systems, themetric entropy is equal to infinity (1/0, only zero step ahead can bepredicted), and in periodic systems, the metric entropy is equal to zero(1/1, all the information can be predicted). Metric entropy cannottake negative values and can be large for a chaotic system.

6. Conclusions

In spite of several studies with promising results on oil produc-tion prediction using methods of time series forecasting (revisedwith more details in the forthcoming related work section), there isno solid understanding in petroleum engineering community thatthis approach can have real applicability in practice. In this paperwe tried to make a step ahead to such understanding.

The main problem of the wide-spread application of ANN inengineering practice is that ANN architecture considerably affectsthe performance of prediction. The behavior of the NARX model is

dependent upon the dimension of embedded memory of inputand output, the number of neurons in the input and hidden layers,the activation function, etc. Though several aspects of this problemhave been studied in the paper and several solutions have beenanalyzed, the determination of these architectural elements, in anoptimal way, is a critical and difficult task for the NARX model, andremains an objective for future works.

Initiating each experiment with ANN, weight matrices arereshuffled causing different downhill paths, which can affect thetraining results. That is why, probably, we obtained differentwinning activation functions' combinations for different types ofexperiments. There are no clear guidelines on how to establish theinitial conditions, which requires more research work.

The advantage of ANN forecasting models is that predictionallows local variation instead of smooth curve projection as whenDCA pattern is used. The use of both static and dynamic dataextends the predictive capabilities of the ANN model from simpleTS forecasting to spatial prediction. Preliminary clustering could bea good means to increase the precision of the forecasting.

The case studies come from the same oilfield, nevertheless, theobtained conclusions may be applied to other NFR with primaryproduction and under gas injection for pressure maintenance.Further study is necessary to investigate the robustness of theproposed methods and to establish data-driven ANN-based reser-voir models as a practical, cost-effective and robust tool for oilfieldproduction management.

Acknowledgments

Partial support for this work was provided by CONACYT-SENER-Hidrocarburos project 146515. The authors would also liketo thank PEMEX for the permission to publish the results.

Appendix

See appendix Tables A1–A3.

Table A1Statistics of the wells from the case study.

Well number Months of production Skewness Kurtosis Best fit frequency distribution Completion intervals Hurst exp 2003

W1 326 �0.22 �0.22 Gaussian mixture UJK 0.643W2 300 0.55 �0.11 Extreme value UJK/UJT 0.604W3 222 1.51 2.79 Gaussian mixture UJK/UJT 0.658W4 293 2.29 6.06 Gaussian mixture LC 0.633W5 199 2.13 3.21 Gaussian mixture UJK/LC 0.684W6 292 0.37 �0.50 Gaussian mixture UJK/LC 0.643W7 301 �0.80 1.12 Gaussian mixture UJK 0.620W8 316 �0.01 �0.32 Normal UJT 0.643W9 297 �0.11 �1.10 Gaussian mixture UJK/UJT 0.645W10 282 1.47 1.06 Gaussian mixture UJK 0.644W11 97 �0.31 �1.01 Extreme value UJK/UJT 0.765W12 254 �0.78 2.72 Gaussian mixture UJK 0.643W13 321 0.77 0.69 Gaussian mixture UJK 0.664W14 339 1.34 2.37 Gaussian mixture UJK 0.620W15 296 0.87 �0.55 Gaussian mixture UJK 0.649W16 295 0.92 1.62 Gaussian mixture UJK/UJT 0.633W17 340 3.33 11.24 Gaussian mixture UJK 0.643W18 319 1.38 2.67 Gaussian mixture UJT 0.650W19 292 0.31 �0.76 Johnson UJK 0.633W20 108 �0.89 �0.61 Gaussian mixture UJK 0.723W21 74 0.37 �0.57 Gaussian mixture UJT 0.683W22 109 �0.16 �1.02 Gaussian mixture UJK 0.743W23 269 0.33 �1.05 Gaussian mixture UJK/UJT 0.640W24 252 0.48 �0.70 Pareto UJT 0.658W25 217 2.77 7.21 Gaussian mixture UJK/UJT 0.660W26 336 0.51 �1.13 Gaussian mixture UJK 0.633

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119116

Table A1 (continued )

Well number Months of production Skewness Kurtosis Best fit frequency distribution Completion intervals Hurst exp 2003

W27 307 2.32 8.33 Gaussian mixture UJK 0.649W28 250 �0.05 �1.60 Extreme value UJK/UJT 0.653W29 127 �1.80 3.38 Gaussian mixture UJK 0.691W30 95 �1.24 0.96 Gaussian mixture UJK 0.687W31 199 2.26 5.02 Extreme value UJK 0.641W32 304 0.69 �0.96 Pareto UJK/UJT 0.649W33 197 1.40 0.78 Extreme value UJK 0.658W34 186 �0.30 �0.20 Gaussian mixture UJK 0.703W35 163 1.67 3.92 Gaussian mixture UJK 0.691W36 288 0.25 �0.57 Gaussian mixture UJK 0.649W37 224 �0.41 �0.46 Extreme value UJK 0.645W38 236 �0.35 0.26 Normal UJK 0.640W39 298 0.75 �0.01 Gaussian mixture UJK 0.647W40 177 1.06 1.24 Gaussian mixture UJK/UJT 0.664W41 266 0.56 �1.23 Gaussian mixture UJK 0.647W42 262 0.78 �0.71 Gaussian mixture UJK 0.645

Extreme values of the parameters are marked by a shadow.

Table A2Comparative forecasting results per well. In case of univariate and multivariate forecasting only the best-fitting model is considered.

Well IMAGINE Univariate, log–tan Multivariate, log–lin η

RMSE SMAPE MSE SMAPE MSE SMAPE log–tan

W1 8.76Eþ07 2.43% 6.20Eþ07 2.05% 7.04Eþ07 2.71% 0.90W2 6.96Eþ07 16.47% 3.44Eþ08 13.50% 2.17Eþ08 11.08% 0.82W3 3.93Eþ06 5.50% 1.72Eþ06 2.95% 2.28Eþ06 3.30% 0.90W4 2.25Eþ07 9.13% 4.35Eþ07 8.14% 4.09Eþ07 6.93% 0.95W5 6.54Eþ06 5.52% 1.37Eþ07 10.45% 3.26Eþ07 17.85% 0.97W6 4.20Eþ07 8.21% 8.36Eþ07 4.55% 1.91Eþ08 7.97% 0.85W7 7.40Eþ08 7.39% 1.35Eþ08 2.38% 9.47Eþ07 1.81% 0.91W8 2.08Eþ08 4.71% 2.77Eþ08 5.23% 2.27Eþ08 5.16% 0.88W9 5.25Eþ06 6.50% 2.78Eþ06 2.74% 3.67Eþ07 9.22% 0.98W10 8.70Eþ07 2.97% 6.09Eþ07 2.66% 6.60Eþ07 2.64% 0.95W11 5.65Eþ07 2.71% 2.31Eþ09 4.92% 6.04Eþ07 2.84% 0.73W12 4.77Eþ08 24.94% 7.62Eþ06 29.90% 1.56Eþ09 84.45% 0.58W13 4.03Eþ07 5.23% 1.47Eþ08 1.97% 1.14Eþ07 2.26% 0.95W14 2.04Eþ08 17.75% 2.77Eþ06 12.34% 1.30Eþ08 12.25% 0.83W15 1.74Eþ06 4.11% 1.17Eþ07 4.11% 3.88Eþ07 15.10% 0.97W16 2.12Eþ07 2.20% 1.98Eþ07 1.71% 1.21Eþ07 1.77% 0.95W17 5.57Eþ07 6.71% 7.36Eþ07 3.88% 2.11Eþ07 3.57% 0.73W18 5.87Eþ07 29.91% 5.32Eþ07 9.98% 6.00Eþ07 8.42% 0.75W19 9.73Eþ06 14.74% 4.25Eþ08 11.97% 6.04Eþ07 12.65% 0.90W20 2.41Eþ07 2.34% 5.72Eþ07 3.45% 5.60Eþ06 1.47% 0.85W21 4.68Eþ08 23.96% 5.85Eþ06 20.04% 5.34Eþ08 22.27% �0.01W22 2.72Eþ07 3.11% 8.21Eþ06 6.00% 5.51Eþ07 3.95% 0.73W23 4.22Eþ07 20.22% 6.02Eþ06 22.34% 1.01Eþ08 31.48% 0.68W24 9.51Eþ06 13.99% 5.72Eþ06 9.69% 5.92Eþ07 23.16% 0.83W25 1.87Eþ07 26.13% 2.07Eþ07 18.81% 6.07Eþ07 38.84% 0.90W26 1.17Eþ07 2.35% 9.87Eþ06 1.90% 6.99Eþ06 1.80% 0.98W27 9.36Eþ06 2.36% 1.83Eþ07 1.66% 1.11Eþ07 2.75% 0.92W28 1.86Eþ07 49.43% 2.12Eþ06 48.92% 4.13Eþ06 25.24% 0.94W29 9.78Eþ06 2.37% 1.79Eþ07 2.55% 1.76Eþ07 3.56% 0.87W30 9.82Eþ06 3.29% 2.77Eþ07 4.34% 5.55Eþ07 7.21% 0.84W31 7.25Eþ06 7.78% 1.18Eþ07 4.63% 2.17Eþ07 10.65% 0.96W32 4.35Eþ07 47.09% 7.99Eþ06 30.21% 1.27Eþ07 23.42% 0.91W33 5.89Eþ07 59.38% 1.53Eþ08 58.86% 1.06Eþ07 41.11% 0.67W34 3.29Eþ07 2.03% 2.52Eþ07 1.97% 2.10Eþ07 1.52% 0.77W35 2.22Eþ07 3.63% 1.17Eþ07 2.14% 1.65Eþ07 3.01% 0.82W36 1.64Eþ08 10.52% 3.02Eþ06 1.77% 8.40Eþ06 1.58% 0.93W37 9.90Eþ07 45.30% 1.21Eþ07 44.16% 8.18Eþ07 41.01% 0.87W38 1.71Eþ07 3.26% 2.41Eþ07 3.22% 2.29Eþ07 3.93% 0.86W39 2.84Eþ07 4.91% 1.45Eþ08 2.96% 2.10Eþ07 3.55% 0.92W40 1.08Eþ08 25.67% 2.51Eþ07 2.47% 1.10Eþ07 5.00% 0.88W41 3.98Eþ06 4.56% 6.97Eþ07 7.22% 3.34Eþ07 10.23% 0.95W42 2.10Eþ07 8.44% 6.82Eþ07 6.38% 3.71Eþ07 8.79% 0.95

Average 8.22Eþ07 13.08% 1.14Eþ08 10.50% 9.87Eþ07 12.56%

The minimum errors for each well are shadowed. For the η-metric, the outliers are shadowed.

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 117

Table A3Original monthly oil and gas production and forecasting results (best fitting model) for the wells W20 and W26 (82 months).

Month W20 W26

Qo log–lin Qg log–lin

1 66,404 61,274 65 662 66,315 64,809 58 633 63,775 63,428 66 604 66,087 62,400 65 655 62,614 63,406 65 636 63,738 60,601 57 687 66,303 61,536 57 598 60,057 63,256 66 589 66,385 58,449 62 66

10 63,212 63,443 68 6311 66,526 61,875 66 6812 64,508 64,604 63 6413 66,124 62,012 73 6314 66,492 64,648 53 7315 62,453 63,203 53 5316 61,719 61,218 54 5217 56,833 60,087 45 5218 60,775 56,192 54 4719 60,666 59,161 54 5720 55,219 57,365 59 5321 61,141 51,235 55 5522 58,234 52,495 54 5223 60,057 49,794 52 6024 58,120 51,572 57 6225 59,074 51,021 56 6126 59,773 57,664 53 4527 57,960 56,268 57 5228 60,021 58,913 56 5529 57,366 64,736 55 5330 58,276 60,049 53 5531 59,924 53,577 44 5132 54,083 55,511 48 4433 59,540 52,768 49 4834 58,290 59,138 49 4535 58,178 56,762 48 5036 54,205 56,787 50 4737 55,629 50,250 50 4838 56,089 53,151 48 4839 52,603 52,636 50 4840 53,742 53,486 48 4941 51,752 54,397 50 4742 44,576 52,969 53 5043 38,450 44,394 51 5244 34,658 38,263 56 5045 36,983 33,589 54 5546 35,801 33,515 56 5247 37,048 32,389 54 5548 35,350 34,813 54 5249 29,986 34,194 57 5350 25,473 30,777 52 5551 30,333 27,406 51 5252 33,513 31,179 49 5153 32,409 32,645 43 4854 23,419 33,142 49 4455 21,589 26,859 45 5056 11,565 25,108 45 4657 16,599 18,915 39 5058 18,579 20,782 36 4259 17,499 21,205 42 4260 4227 18,856 42 4561 0 11,890 53 4362 3124 10,425 45 5563 3022 11,042 39 4664 3124 11,203 45 4265 1850 10,875 47 4866 0 10,357 48 4867 0 8333 44 5268 0 7065 39 4869 0 6890 32 4370 0 7589 33 3771 0 9957 32 3972 2642 9708 38 3773 2695 7307 37 43

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119118

References

Baddari, K., Ferahtia, J., Aifa, T., Djarfour, N., 2010. Acoustic impedance inversion byfeedback artificial neural network. J. Pet. Sci. Eng. 71, 106–111.

Batyrshin, I., Sheremetov, L., 2008. Perception based approach to time series datamining. J. Appl. Soft Comput. Elsevier Sci. 8 (3), 1211–1221.

Box, G., Jenkins, G., 1970. Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco.

Bravo, C.E., Saputelli, L., Rivas, F., Pérez, A.G., de Guzmán, N., Nickolau, M., Zangl, G.,Mohaghegh, S.D., Nunez, G., 2013. State of the Art of Artificial Intelligence andPredictive Analytics in the E&P Industry: A Technology Survey. SPE Paper150314.

Chakra, C., Song, K.-Y., Saraf, D.N., Gupta, M.M., 2013. Production forecasting ofpetroleum reservoir applying higher-order neural networks (honn) withlimited reservoir data. Int. J. Comput. Appl. 72 (2), 23–35.

Cosultchi, A., Sheremetov, L., Velasco-Hernandez, J., Batyrshin, I., 2012. A methodfor aquifer identification in petroleum reservoirs: a case study of Puerto Ceibaoilfield. J. Pet. Sci. Eng., Elsevier Sci. vol. 94–95, 55–65.

De Gooijer, J.G., Hyndman, R.J., 2006. 25 Years of time series forecasting. Int. J.Forecast. 22, 443–473.

Diaconescu, E., 2008. The use of NARX neural networks to predict chaotic timeseries. WSEAS Trans. Comput. Res. 3 (3), 182–191.

Farawey, J., Chatfield, C., 1998. Time series forecasting with neural networks: acomparative study using the airline data. Appl. Stat. 47 (2), 231–250.

Gao, Y., Er, M.J., 2005. NARMAX time series model prediction: feedforward andrecurrent fuzzy neural network approaches. Fuzzy Sets Syst. 150 (2), 331–350.

Gharbi, R.B., Elsharkawy, A.M., Karkoub, M., 1999. Universal neural-network-basedmodel for estimating the PVT properties of crude oil systems. Energy Fuels 13 (2),454–458.

Garcia, A., Mohaghegh, S.D., 2004. Forecasting of Natural Gas Production into Year2020: a Comparative Study. SPE Paper 91413.

González-Sánchez, A., Cosultchi-Marcu, A., Vargas-Castro, F., 2012. Effect of treat-ments on oil well production by ANN. In: Sheremetov, L., Haasis, H.-D., Villa-Vargas, L. (Eds.) Operations Research and Data Mining, ORADM´2012, Work-shop Proceedings, IPN, pp. 128–139.

Haykin, S., 1999. Neural Networks: A Comprehensive Foundation, 2nd ed. PearsonEducation, NJ, USA.

He, Z., Yang, L., Yen, J., Wu, C., 2001. Neural-Network Approach to Predict WellPerformance Using Available Field Data. SPE Western Regional Meeting, SPE68801. Bakersfield, California, 26–30 March.

Hervada-Sala, C., Jarauta-Bragulat, E., 2004. A program to perform Ward's clusteringmethod on several regionalized variables. Comput. Geosci. 30 (8), 881–886.

Kaboudan, M., 1999. A measure of time-series' predictability using geneticprogramming applied to stock returns. J. Forecast. 18, 345–357.

Kantz, H., Schreiber, T., 2006. Nonlinear Time Series Analysis, 2nd ed. CambridgeUniversity Press, Cambridge.

Khashei, M., Bijari, M., 2010. An artificial neural network (p, d, q) model fortimeseries forecasting. Expert Syst. Appl. 37, 479–489.

Lashin, A., El Din, S.S., 2013. Reservoir parameters determination using artificialneural networks: Ras Fanar field, Gulf of Suez, Egypt. Arab. J. Geosci. 6 (8),2789–2806.

Lim, J.S., 2003. Multivariate statistical techniques including PCA and rule basedsystems for well log correlation. In: Nikravesh, M., Aminzadeh, E., Zadeh, L.A.(Eds.), Soft Computing and Intelligent Data Analysis in Oil Exploration (bookseries Developments in Petroleum Science, 51. Elsevier, Amsterdam,pp. 673–688.

Lin, T., Horne, B.G., Tino, P., Giles, C.L., 1996. Learning long-term dependencies inNARX recurrent neural networks. IEEE Trans. Neural Netw. 7 (6), 1329–1351.

Mehta, K., Bhattacharyya, S., 2004. Adequacy of training data for evolutionarymining of trading rules. Decis. Support Syst. 37, 461–474.

Menezes, J.M.P., Barreto, G.A., 2008. Long-term time series prediction with theNARX network: an empirical evaluation. Neurocomputing 71 (16–18),3335–3343.

Mohaghegh, S.D., 2005. Recent developments in application of artificial intelligencein petroleum engineering. J. Pet. Technol. 57 (4), 86–91.

Mohaghegh, S.D., 2011. Reservoir simulation and modeling based on artificialintelligence and data mining (AI&DM). Special issue – Artificial Intelligenceand Data Mining, Mohaghegh, S.D., Camacho-Velazquez, R.G., Behbahani, K.(Eds.) Journal of Natural Gas Science and Engineering, vol. 3, No. 6, pp. 697–705.

Olivares-Velazquez, G., Escalona-Quintero, C., Gimenez, E., 2012. Production Mon-itoring using Artificial Intelligence. SPE Paper 149594.

Peña, D., Tiao, G.C., Tsay, R.S. (Eds.), 2001. A Course in Time Series Analysis. JohnWiley and Sons, New York.

Qian, B., Rasheed, K., 2004. Hurst exponent and financial market predictability. In:Proceedings of the IASTED Conference on Financial Engineering and Applica-tions (FEA 2004). pp. 203–209.

Ramgulam, A., Ertekin, T., Flemings, P.B., 2007. Utilization of Artificial NeuralNetworks in the Optimization of History Matching. Latin American and theCaribbean Petroleum Engineering Conference, SPE Paper 107468.

Romesburg, H.C., 1984. Cluster analysis for researchers, Belmont, Calif.: LifetimeLearning Publications.

da Silva, L.C.F., Portella, R.C.M., Emerick, A.A., Ebecken, N.F.F., 2007. Predictive data-mining technologies for oil-production prediction in petroleum reservoir. In:Proceedings of the Latin American and the Caribbean Petroleum EngineeringConference, SPE 107371.

Saemi, M., Ahmadi, M., Varjani, A.Y., 2007. Design of neural networks using geneticalgorithm for the permeability estimation of the reservoir. J. Pet. Sci. Eng. 59,97–105.

Schrader, S.M., Balch, R.S., Ruan, T., 2005. Using Neural Networks to EstimateMonthly Production: a Case Study for the Devonian Carbonates, Southeast NewMexico. SPE Production Operations Symposium, SPE 94089. Oklahoma City,Oklahoma, 16–19 April.

Sheremetov, L., Alvarado, M., Bañares-Alcántara, R., Aminzadeh, F., 2005. Intelligentcomputing in petroleum engineering (editorial). J. Pet. Sci. Eng., Elsevier Sci. 47(1–2), 1–3.

Sheremetov, L.B., González-Sánchez, A., López-Yáñez, I., 2013. Time series forecast-ing: applications to the upstream oil and gas supply chain. In: Proceedings ofthe IFAC Conference on Manufacturing Modelling, Management, and Control,Elsevier, vol. 7, Part 1, pp. 957–962.

Weiss, W.W., Balch, R.S., Stubbs, B.A., 2002. How artificial intelligence methods canforecast oil production. In: SPE/DOE Improved Oil Recovery Symposium. Tulsa,Oklahoma, 13–17 April 2002, pp. 1–16.

Zhang, G.P. (Ed.), 2004. Neural Networks in Business Forecasting. Idea Group Inc.,Hershey, PA.

Table A3 (continued )

Month W20 W26

Qo log–lin Qg log–lin

74 0 7750 36 3875 164 5940 34 4576 4906 5254 32 5477 5070 6615 35 3678 5065 8866 33 3779 5430 8848 47 6080 7149 10,097 63 5681 7013 11,133 61 9882 0 7664 0 102

L. Sheremetov et al. / Journal of Petroleum Science and Engineering 123 (2014) 106–119 119

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页 文献云下载 图书馆入口 外文数据库大全 疑难文献辅助工具