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WATER RESOURCES RESEARCH, VOL. 41, W10421, doi:10.1029/2004WR003838, 2005

Tracer model identification using artificial neural networks Serhat Akin Petroleum and Natural Gas Engineering Department, Middle East Technical University, Ankara, Turkey Received 23 November 2004; revised 8 June 2005; accepted 19 July 2005; published 26 October 2005.

[1] The derivation of transport parameters from tracer tests conducted in geothermal

systems will depend strongly on the conceptual and mathematical model that is fitted to the data. Depending on the model employed the estimation of transport parameters (porosity and dispersivity of the fracture network, porosity of the matrix) may result in a significant variation in dispersivity. If the results from such tracer tests are to be used in parameter selection for larger-scale models, it is crucial that the tracer test is itself interpreted with an appropriate model. In order to tackle this problem, artificial neural network (ANN) technology is proposed. A dual-layer neural network model was trained using synthetic tracer test data generated using analytical one-dimensional homogeneous, double-porosity pseudosteady state, multifracture, and fracture matrix models. The developed model was then used to identify several actual tracer tests conducted in various geothermal reservoirs reported in the literature. In most cases it was observed that the model successfully identified a wide variety of reservoir models. In some cases the model decreased the number possible models to two. It was also observed that ANN results were in accord with least squares analysis. Citation: Akin, S. (2005), Tracer model identification using artificial neural networks, Water Resour. Res., 41, W10421, doi:10.1029/2004WR003838.

1. Introduction [2] Tracer study is an important technique for reservoir characterization, particularly in geothermal engineering [Akin and Okandan, 1995], petroleum reservoir engineering [Brigham and Abbaszadeh, 1987], and hydrology [Nishikawa et al., 1999]. Several processes generally act simultaneously on a chemical constituent while it is transported through a porous medium. Among these, the two primary processes are the physical phenomena of convection and hydrodynamic dispersion. While convection deals with the bulk movement of fluids, hydrodynamic dispersion describes the actions of molecular dispersion and shear or mechanical mixing. These transport processes normally are represented adequately by the well-known convectiondispersion diffusion equation with or without chemical reactions. Most of the time, these diffusion equations are based on linear or one-dimensional geometry largely because of the relative ease with which such equations can be solved analytically. [3] The analysis of a tracer test is a modeling exercise consisting of two steps: diagnostics and parameter estimation. The first step is to choose a model to represent flow and transport in the rock mass. The choice of model is based on knowledge of the rock mass and the behavior of the test response. After a model has been chosen, the second step is to determine values of model parameters such that model computed responses match field responses. In such analyses, the first step consists of conventional type curve analysis [Akin and Okandan, 1995]. Initial parameter estimates for several reservoir parameters are obtained at this stage. Then these estimates are fine tuned using an Copyright 2005 by the American Geophysical Union. 0043-1397/05/2004WR003838

automated history matching technique. Genetic algorithms, simulated annealing or a Gauss-Newton optimization technique with or without Marquardt’s modification can be used as an automated history matching procedure [Akin, 2001]. The analyses are usually conducted using a computer code where a high-level computer language (i.e., FORTRAN, C, Pascal, Basic or Java) is used. The conventional fitting procedure can be very cumbersome and can involve prohibitive computer costs. If on the other hand, the model generated response cannot be made to match the field response, the model must be revised. In other words, model selection and parameter estimation are an iterative process. In some cases there may be insufficient information to identify a unique model or a unique set of parameters for a given model [Brusseau, 1998]. When faced with such nonuniqueness adopting only one model may lead to statistical bias and underestimation of uncertainty. Thus several possible models and/or parameter sets may need to be considered and ranked until additional information is collected to better define the flow system. [4] In order to tackle ranking of competing models several optimum decision rules were proposed: Akaike information criterion (AICk) [Akaike, 1974], Bayesian information criterion (BICk) [Akaike, 1977; Rissanen, 1978; Schwarz, 1978], Kashyap information criterion (KICk) [Kashyap, 1982; Carrera and Neuman, 1986; Samper and Neuman, 1989] and Hannan information criterion (HICk) [Meyer et al., 2004; Hannan, 1980]. The AICk statistic is calculated as the sum of the maximum likelihood objective function and 2 times the number of parameters (equation (1)). Similarly, the BICk statistic is based on the maximum likelihood objective function (equation (7)). When AICk and BICk statistics are compared among models, the model with lower values may be more suitable to the given scenario

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with the available observations. Most of the time other factors such as specific observations to which the model adequately fits, or differences in observation weighting schemes need to be considered together to make a choice. AICk ¼ NLLk þ 2Nk HICk ¼ NLLk þ 2Nk lnð ln N Þ

ð1Þ ð2Þ

where NLLk is the negative log likelihood of Mk given by equation (6), Nk is the dimension of qk (Number of parameters associated with model Mk) and N is the dimension of D (number of discrete data points). More recently, Neuman [2002, 2003] proposed an alternative strategy based on the posterior model probability, p(MkjD), based on a result due to KICk and referred to resulting method as maximum likelihood Bayesian model averaging (MLBMA).   1 exp 2DKICk pðMk Þ pðMk j DÞ ¼ K   P 1 exp 2DKICl pðMl Þ

ð3Þ

DKICk ¼ KICk  KICmin

ð4Þ

l¼1

 KICk ¼ NLLk þ Nk ln

     N    þ lnFk D^qk ; Mk  2p

      NLLk ¼ 2 ln p D^qk ; Mk  2 ln p ^qk jMk

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[6] As can be seen from the aforementioned discussion, there is a need for robust model selection procedure. In this study, a new approach based on the power of artificial neural networks (ANN) is proposed. ANNs have proved to be valuable pattern recognition tools. They are capable of finding highly complex patterns within large amounts of data. A relevant example is well test analysis. ANN techniques have been successfully used to identify reservoir and well bore flow models using pressure transient and pressure derivative data [Allain and Horne, 1990; Anraku and Horne, 1993]. Similarly, model selection problem in tracer test data analysis could be tackled using ANN modeling. On the basis of shape of tracer test curve, ANN model can identify type of describing mathematical transport model. [7] The paper is structured in the following manner. First the feed forward artificial neural network that was trained using one dimensional tracer test models including homogeneous, fracture-matrix, multifracture and double-porosity pseudosteady state models is described. Then, the uses of the ANN model selection technique is demonstrated using published tracer test data covering a wide range of tracer tests conducted in several geothermal reservoirs. The tracer test model fits using a generalized reduced gradient (GRG2) nonlinear optimization code [Lasdon et al., 1978] are presented and compared to ANN results and classical model ranking schemes including AICk and BICk.

ð5Þ

2. Artificial Neural Network Models ð6Þ

where given a data set D, M = (M1, . . ., Mk) is the set of all models considered, at least one of which must be correct. In these equations, Fk is Fisher information matrix, and NLLk is the negative log likelihood of Mk, evaluated at ^qk. The highest-ranking model is that corresponding to minimum KICk. The Fisher information matrix term tends to a constant as N becomes large, so that KICk becomes asymptotically equivalent to the BICk [Meyer et al., 2004]. BICk ¼ NLLk þ Nk ln N

ð7Þ

  1 exp 2DBICk pðMk Þ pðMk j DÞ ¼ K   P 1 exp 2DBICl pðMl Þ l¼1

ð8Þ

DBICk ¼ BICk  BICmin

ð9Þ

where BICmin is the smallest value of BICk over all candidate models [Burnham and Anderson, 2002, p. 297]. [5] Carrera and Neuman [1986] found KICk to provide more reliable rankings of alternative groundwater flow and geostatistical models than do BICk or AICk and HICk. Most of the aforementioned model selection procedures use a fixed penalty penalizing an increase in the size of a model. Most of the time nonadaptive selection procedures perform well only in one type of situation. For instance, BICk with a large penalty performs well for ‘‘small’’ models and poorly for ‘‘large’’ models, and AICk does just the opposite [Shen and Ye, 2002].

[8] Neural computing is based on a mathematical model inspired by biological models [Hecht-Neilsen, 1990]. This computing system is made up of a number of artificial neurons and a large number of interconnections between them. On the basis of the structure of the connections, two different classes of network architecture can be identified: layered feed-forward neural networks and nonlayered recurrent neural networks (Figure 1). In layered feed-forward neural networks, the neurons are organized in the form of layers. The topmost layer is called the input layer and it is made up of special input neurons, transmitting only the applied external input to their outputs. The neurons in a layer get input from the previous layer and feed their output to the next layer, but connections to the neurons in the same or previous layers are not permitted. The last layer of neurons is called the output layer and the layers between the input and output layers are called the hidden layers. The output of every hidden node passes through a sigmoid function. Standard sigmoid function (equation (10)) is logistic and its range changes between 0 and 1. In single layer networks there is only the layer of input nodes and a single layer of neurons constituting the output layer. If there are one or more hidden layers, such networks are called multilayer networks. Structures, in which connections to the neurons of the same layer or to the previous layers are allowed, are called recurrent networks [Hornik et al., 1989]. f ðI Þ ¼

1 1 þ expðI Þ

ð10Þ

The lines represent weighted connections (i.e., a scaling factor) between processing elements. The performance of a network (Figure 1) is measured in terms of a desired signal and an error criterion. The output of the network is

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a mechanism of providing the network with the desired output either by manually ‘‘grading’’ the network’s performance or by providing the desired outputs with the inputs. Unsupervised training is where the network has to make sense of the inputs without outside help. During training the ANN there are two major pitfalls: memorization and over saturation. In both cases global optimum cannot be reached. For a particular ANN if the training errors are significantly smaller than the validation errors then the ANN is over trained. In other words, the ANN memorizes the training data but cannot interpret any other data. The customary practice is to set all the free parameters of the network to random numbers that are uniformly distributed inside a small range of values. This is so because if the weights are too large, the sigmoids will saturate from the very beginning of training and the system will become stuck in a kind of saddle point near the starting point [Hecht-Neilsen, 1990]. In other words, use of large weights at the early stages of the training, results in high activation levels (i.e., high values of the sigmoid function) which causes the components of weight update to attain values of the same order of magnitude. In this case, the updated weights may suddenly produce a large in the net input. This change can be such that it causes the activation levels of a particular unit approach to 0 or 1 overshooting their target values. This phenomenon is known as premature saturation. Premature saturation is avoided by choosing the initial values of the weights and threshold levels of the network to be uniformly distributed inside a small range of values. This is so because when the weights are small the units operate in their linear regions and consequently it is impossible for the activation function to saturate.

3. Mathematical Models

Figure 1. (a) Layered feed-forward neural network and (b) nonlayered recurrent neural network. compared with a desired response to produce an error. In a typical neural network application two types of error measures are used: mean squared error (MSE), which is the squared difference between the actual output and the predicted output and absolute relative error (ARE), which is the absolute value of ((actual output - predicted output)/ actual output). ARE expressed in terms of percent represents the average size of prediction error, relative to the actual output. An algorithm called back propagation [Haykin, 1994; Rumelhart et al., 1986] is then used to adjust the weights a small amount at a time in a way that reduces the ARE. The network is trained by repeating this process many times. Each time a record goes through the net, it is one trial, one sweep of all records is called an epoch. So the total number of trials is equal to number of records multiplied by the number of epochs. The goal of the training is to reach an optimal solution based on the performance measurement. There are two approaches to training: supervised and unsupervised. Supervised training involves

[9] In this study, four different models were considered: the multifracture model [Fossum and Horne 1982], the fracture matrix model [Bullivant and O’Sullivan, 1989], the uniform porous model [Sauty, 1980], and the double-porosity pseudosteady state model [Bullivant and O’Sullivan, 1989]. In each model it is assumed that there is a good connection between the injection and production wells along a streamline which is surrounded by a stream tube of constant cross section. The tracer is injected as a slug from the injection well and the response is recorded in the observation well. The description of each model is given below. 3.1. Multifracture Model [10] This model, as reported by Fossum and Horne [1982], assumes a single-fracture/multifracture system, joining the injection and observation wells. Dispersion is due to the high-velocity profile across the fracture and molecular diffusion, which moves tracer particles between streamlines (Taylor dispersion). The transfer function Ct is given by the following expression; Ct ¼

n X

ei Cr ðLi =ui ; Pei Þ

ð11Þ

i¼1

where n is number of flow channels in the fracture system, ei is the flow contribution coefficient, Li is the apparent fracture length, ui is the velocity, and Pei is the Peclet number of the ith flow channel. Therefore, if n is taken as

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one then only a single fracture is present. It should be noted that for all practical purposes, a multifracture system must be represented with at least two fractures since it was reported by Akin [2001] that the value of the transfer function, Ct does not change much as n increases. The form of relative concentration, Cr for each of the paths for a mass of tracer concentrated at point x = 0 at time = 0 is



2b2 u2 105Dm

ð13Þ

3.2. Fracture Matrix Model [11] In this model, as reported by Bullivant and O’Sullivan [1989], there is a large fracture with microfracturing in the rock matrix on either side. Tracer particles leave the main fracture and enter the microfracture network (there is a small amount of fluid exchange), stay for a while, and then return to the main fracture. Longitudinal dispersion due to the velocity profile across the fracture is ignored in order to give a clear distinction from the single fracture model. A fracture with fluid velocity constant across the thickness and with diffusion perpendicular to the fracture into an infinite porous medium is used in this model. The solution is in the following form: tb wðt  tb Þ

ð14Þ

Here U is the Heaviside step distribution, w is a ratio of transport along the fracture to transport out of the fracture, tb is the response start time, and J is a model parameter. 3.3. Uniform Porous Model [12] In the uniform, homogeneous porous model, it is assumed that a slug of tracer is instantaneously injected into a system with constant thickness. It is also assumed that, the flow is rapid allowing the kinematic dispersion components to be predominant. For purely hydrodispersive transfer the solution for one dimensional flow as reported by Sauty [1980] is   K Pe 2 Cr ¼ Exp  ð1  tr Þ 4tr tr

ð15Þ

where K¼

pffiffiffiffiffiffi trm Exp

trm



Pe ð1  trm Þ2 4trm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ Pe2  Pe1

Cr ¼

  K Pe Exp  ð1  tr Þ2 4tr tr

ð12Þ

where Pe is the dimensionless Peclet number and tr is the mean arrival time. Using the above model, by knowing the coefficient of molecular diffusion, Dm, it is possible to obtain the average velocity, length, mean arrival time, and inferred fracture aperture, b for each flow channel by using the following definition for dispersivity, h [Horne and Rodriguez, 1983].

Cr ¼ JU ðt  tb Þ1=2 Exp

[1980] also reported an analytical expression for the slug injection of a tracer solution into a two dimensional field on the flow axis as shown below. ð18Þ

where

1  ð1  t r Þ2 Cr ¼ rffiffiffiffiffi Exp 4 tr tr 2 P Pe e



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 ð16Þ ð17Þ

In the above equations Pe is the dimensionless Peclet number and tr is the mean arrival time. Similarly, Sauty

pffiffiffiffiffiffi trm Exp

trm ¼



Pe ð1  trm Þ2 4trm



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4Pe2  2Pe1

ð19Þ

ð20Þ

3.4. Double-Porosity Pseudosteady State Model [13] For this model the reservoir contains uniformly distributed high-permeability microfractures which divide the reservoir into low-permeability blocks that consist of unswept pores by the fluid flow. Similar to the mechanism defined for the fracture-matrix model, the tracer leaves the microfractures and then returns again. However the effect is different, such that the blocks may be filled with tracer. Longitudinal dispersion due to the movement of fluid into the microfracture network is neglected. The solution for this case [Bullivant and O’Sullivan, 1989] is  1=2  Cr ¼ J Expðam t ÞU ðt  tb Þ1=2 I1 2 tb af am ðt  tb Þ

ð21Þ

In equation (21), af is the rate of tracer interchange per unit fracture volume and am is the rate of tracer interchange per unit matrix volume.

4. Methodology 4.1. Input Data and Training [14] The training data covered synthetic tracer test data generated using the aforementioned analytical models. The concentration and time data were normalized by dividing the test data to maximum data obtained for a wide range of Peclet numbers and porosities (see Table 1 for mean, standard deviation, minimum, and maximum of variances of training data). The purpose of normalization stage was to prevent saturating the network. Over saturation is observed when a network stops learning as the derivatives of the sigmoid function (equation (10)) become very small. Two different training strategies were used: batch training and individual training. In batch training a total of 80 data sets (20 from each model) were used for training the ANN. Twelve additional data sets (Figures 2a and 2b) were used for verifying the results. The ANN model had one input layer, 2 hidden layers and one output layer. The ANN properties are given in Table 2. Each data set consisted of 29 time-concentration pairs. Less number of pairs resulted in increased uncertainty as reported by Allain and Horne [1990] and Anraku and Horne [1993] for pressure transient model identification exercises. The output of the ANN changed from 1 to 4 representing multifracture model, fracture-matrix model, uniform porous model, and doubleporosity pseudosteady state model respectively. Individual

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Table 1. Variances in Training Data Model\Variance

Mean

Standard Deviation

Minimum

Maximum

Multifracture Fracture matrix Homogeneous DPPSSa

0.0660 0.0814 0.0946 0.1150

0.0197 0.0415 0.0288 0.0187

0.0356 0.0350 0.0345 0.0649

0.1180 0.1767 0.1237 0.1307

a

Double-porosity pseudosteady state.

training strategy [Sultan and Al-Kaabi, 2002] on the other hand considered each of the models separately (i.e., 20 data sets for each model) and a separate network is formed for each model. The output of the ANNs was either 0 or 1 (i.e., true or false). Batch training was as successful as the individual training option; however, the training time was somewhat longer. Since, batch training strategy was simpler and it produced similar results when compared to individual training, batch training was used in the study. The absolute relative error (ARE) per input for training and for verification at the end of 500 epochs was less than 5 and 8%, respectively. [15] The developed ANN model for tracer model identification was first verified using tracer test data that was not used for training. Twelve synthetic tracer test data (Figures 2a and 2b) were generated using the aforementioned analytical models for verification and validation purposes. The results of the ANN model for these verification runs were 1.195, 1.043 and 1.017 for the multifracture, Figure 2b. Synthetic tracer test data used in verification runs. (top) Homogeneous model. (bottom) Double-porosity pseudosteady state model. 3.085, 3.01 and 2.86 for homogeneous 1-D, 1.98, 1.90 and 2.16 for the fracture-matrix and 3.75, 3.76 and 3.96 for the double-porosity pseudosteady state model. The largest error was observed for the multifracture and double-porosity pseudosteady state models. These results verify that the developed ANN model is capable of finding the correct mathematical model using its pattern recognition abilities. ANN performance could have been further improved by providing more training and validation data. Yet another way to improve the results may be the use of more than 29 data points in training. However, use of more than 29 data points will result in significant increase in CPU time. 4.2. Application to Real-World Data Sets [16] The developed ANN model was used to identify several geothermal tracer test reported in the literature. The first set of tracer test data (Figure 3) was from Laugaland (Iceland) geothermal field [Axelsson et al., 2001]. Fluorescein was injected at a constant rate and the peaks observed Table 2. ANN Model Properties Property

Figure 2a. Synthetic tracer test data used in verification runs. (top) Multifracture model. (bottom) Fracture-matrix model.

Input nodes Output nodes Hidden layer 1 Hidden layer 2 Momentum factor Learning parameter

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channel volume and dispersion. They used three separate flow channels that are assumed to connect the different feed zones of the injection and production wells in the simulation. The ANN model developed in this study was used to identify the physical model. First, 29 samples from the tracer test data were acquired. Random sampling was not preferred for keeping the observed peaks and irregularities due to keeping multichannel response and production changes. For tracer test 1 and 3, the ANN result was 2.05 (in other words fracture-matrix model) and 2.35 respectively. The ambiguity for the third tracer test was somewhat larger compared to the first one but the result is in accord with the reservoir description reported by Axelsson et al. Note that channelized transport may arise from the nonuniform velocity of fluid and solute transport in a variable aperture fracture or from the narrow pathways of least

Figure 3. Laugaland (Iceland) LN-12 tracer test data. (top) First tracer test. (bottom) Third tracer test [Axelsson et al., 2001]. in the tracer concentration-time plot were believed to be due to shallow feed zones. In their analysis Axelsson et al. reported that the injected water travels through the area bedrock by two modes: first, along direct, small volume flow paths, such as fractures or interbeds; second, by dispersion and mixing throughout a large volume of the reservoir. Axelsson et al. analyzed these tracer test data using a one-dimensional flow channel model, where the tracer return is controlled by the distance between injection and production zones in the corresponding wells, the flow Table 3. Sum of Squares Residuals and AICk and BICk Statistics of Laugaland (Iceland) LN-12 Tracer Test Data Fits Normalized Sum of Sum of Squares Residual Squares Residual Model\test 1 Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS Model\test 3 Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS

AICk

BICk

7506614.76 35671739.5 6399837.35 45634809.2 54518189.5

0.14 0.65 0.12 0.84 1.00

123.38 105.90 105.36 106.08 108.25

147.25 111.87 111.33 112.05 116.20

26319266.4 61010066.4 8726680.69 91105290.6 51228398

0.29 0.67 0.10 1.00 0.56

103.32 85.70 85.12 86.03 87.59

124.45 90.98 90.41 91.31 94.64

Figure 4. Dixie Valley tracer test data. (top) Well 41-18, (middle) well 65-18, and (bottom) well 45-5 [Rose et al., 2004].

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of squares residual values for both tracer test 1 and test 3 (Table 3). Furthermore, low AICk and BICk statistics of the fracture-matrix model confirmed ANN results. Thus fracture-matrix model was identified as the best model representing the Laugaland tracer test data. [18] The second set of data was from Dixie Valley geothermal field. Tracer test data included in the analysis consisted of wells 41-18, 65-18, and 45-5 (Figure 4). In these tests Rose et al. [2004] reported the use of 1,5-naphtalene disulfonate, 1,3,6-naphtalene trisulfonate and 1-naphtalene sulfonate tracers for the prediction of swept reservoir volume. They analyzed the tracer returns using the TOUGH2 simulator using an injector/producer doublet. The grid was designed to simulate the damage zone around a subvertical fault. The permeability, porosity and the injector/producer flow rates were kept constant. In a separate study, Adams et al. [1989] used a numerical model that assumed that fluid flow at the Dixie Valley geothermal field was primarily through high-permeability channels associated with the SWNE trending range front fault that separates Dixie Valley from the adjacent Stillwater Range to model the aforementioned tracer tests. Similar to the previous example 29 points from these tracer tests were sampled while keeping the irregularities present in the data. As can be seen from Figure 4 the data were somewhat noisy and showed several flow paths. ANN analysis resulted in 1.924 (fracture-matrix model), 1.05 (multifracture model) and 1.106 (multifracture model) for wells 41-18, 65-18, and 45-5. As can be seen, as the concentration-time plot gets noisier or in other words displays multiple peaks, the ANN model shows a tendency to select the multifracture model. This is expected as the ANN model interprets these peaks as separate flow channels and responds by selecting the multifracture model. It is however possible that some of these peaks are due to measurement errors or noise in the data. An exponential smoothing algorithm given in Microsoft Excel’s data analysis package with a dampening factor of 0.5 was used to smooth Dixie Valley tracer test data (Figure 5). ANN model response when the noise was removed was equal to 2.268, 1.019, and 1.143 for wells 41-18, 65-18, and 45-5 respecFigure 5. Sampled and smoothed Dixie Valley tracer test data. (top) Well 41-18, (middle) well 65-18, and (bottom) well 45-5. resistance. Fracture channels are fixed in orientation and position, whereas pathways of least resistance vary according to the flow direction. Since Laugaland geothermal reservoir is believed to be fractured, in this regard, the ANN model results are in agreement with the model reported by Axelsson et al. [17] The Laugaland tracer test data were analyzed using conventional least squares analysis coupled with generalized reduced gradient algorithm (GRG2) developed by Lasdon et al. [1978]. The least squares objective function (F = (Cmodel  Cfield)2) was used without weighting factors. In order to validate the method, synthetic tracer test data used in ANN validation were fit (Figures 2a and 2b) to corresponding analytical models. An excellent match was obtained for all cases (see inset bar charts in Figures 2a and 2b). After verification of the least squares tracer test analysis method, Laugaland data sets were analyzed (Figure 3). Fracture-matrix model resulted in smallest sum

Table 4. Sum of Squares Residuals and AICk and BICk Statistics of Dixie Valley Tracer Test Data Normalized Sum of Sum of Squares Residual Squares Residual Model\well 41-18 Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS Model\well 65-18 Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS Model\well 45-5 Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS

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AICk

BICk

60.29 62.34 41.40 220.77 313.65

0.19 0.20 0.13 0.70 1.00

163.31 147.37 126.43 305.80 400.68

184.45 152.65 131.71 311.09 407.72

8.40 33.71 32.85 16.79 71.85

0.12 0.47 0.46 0.23 1.00

118.78 126.09 125.23 109.17 166.23

140.98 131.64 130.78 114.72 173.63

10.94 39.93 18.36 74.11 114.93

0.10 0.35 0.16 0.64 1.00

101.10 112.09 90.52 146.27 189.09

120.11 116.84 95.27 151.03 195.42

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Figure 6. Cove Fort tracer test data. (top) Well Olga, (middle) well Linda, and (bottom) well Clara [Bloomfield and Moore, 2003]. 8 of 11

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Table 5. Sum of Squares Residuals and AICk and BICk Statistics of Cove Fort Tracer Test Data

Model\well Olga Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS Model\well Linda Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS Model\well Clara Multifracture Single-fracture Fracture-matrix Uniform 1-D DPPSS

Sum of Squares Residual

Normalized Sum of Squares Residual

AICk

BICk

0.17 0.20 0.23 0.28 0.36

0.46 0.54 0.63 0.76 1.00

70.11 52.14 52.17 52.22 54.31

84.74 55.80 55.83 55.88 59.19

0.07 0.11 0.14 0.22 0.16

0.33 0.50 0.64 1.00 0.76

66.34 48.38 48.41 48.49 50.43

79.97 51.79 51.81 51.89 54.98

2.37 2.91 2.88 2.79 3.45

0.69 0.84 0.84 0.81 1.00

79.67 62.21 62.18 62.08 64.75

96.08 66.31 66.29 66.19 70.22

tively. As can be seen the ANN model has not changed its decision. Thus this example shows the success of ANN model’s identification abilities for situations where background noise may dominate the tracer test data. Least squares analysis of the tracer test data (Figure 4) indicated that fracture-matrix model is the best model for well 41-18 and the second best model for wells 65-18 and 45-5 (Table 4). For these tracer tests when least squares residuals were considered the multifracture model ranked the 2nd best. However, high AICk and BICk statistics associated with the multifracture model indicated that the increased number of parameters did not justify the improved fit over the other models. Low AICk and BICk statistics of tracer test data fits of wells 41-18 and 45-5 implied that the fracture-matrix model is better than the other models. Interestingly, homogeneous 1-D model resulted in the lowest AICk and BICk statistics for well 65-18. It must be noted that the BICk statistics for the fracture-matrix model ranked the second for this test. It was concluded that ANN results are in accord with the least squares results. [19] The last data set used was from the Cove FortSulphurdale geothermal resource, Utah (USA). Bloomfield and Moore [2003] reported that the reservoir is producing from a shallow steam cap and a deeper liquid resource. Vapor phase tracer, R-134a was injected in dry steam wells named Olga, Linda, and Clara (Figure 6). Tracers test results, production data and aquifer configurations were matched using the 3-D, multicomponent, thermal reservoir simulator TETRAD. The simulations incorporated the phase behavior of the vapor phase tracer, R-134a but properties like permeability and porosity were kept as constant. No attempt was made in the numerical simulations to modify the simplifying assumption of homogeneous permeability, despite geologic and geophysical evidence for the presence of faults. The results showed that the predicted tracer return curves were sensitive to the positions of the discharge and recharge aquifers. The best match of the measured and predicted tracer concentrations was obtained when the recharge aquifer was placed to the east of the field and discharge occurred to the west. The ANN model results were 1.098, 1.04, and

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1.036 for Olga, Linda, and Clara wells, respectively. Because of multiple peaks present in the data the ANN model decided that multifracture model represented the tracer tests better than the other models. Numerical smoothing did not change the results. In a separate study, Moore et al. [2000] concluded that early tracer returns had come from a location where the reservoir tapped a small fraction of the slowly moving liquid injection flow close to the injection well, and that the rest of the tracer may arrive later, and by a different route. Least squares fits (Table 5) to tracer test data supported the ANN results such that for all tracer tests multifracture model resulted in smallest sum of squares residuals. On the other hand, similar to Dixie Valley tracer test analyses, the multifracture model fits resulted in high AICk and BICk statistics showing that the increased number of parameters did not justify the improved fit over the other models. Singlefracture model fits ranked the first among the other models for Olga and Linda but Clara tracer test. [20] ANN analyses reported above shows a tendency to select more parameter rich models when compared to AICk and BICk statistics. When used as a pattern recognition tool, the ANN tries to find out the best model that fits the analyzed pattern. In doing so, the ANN does not pay attention to the number of parameters used in the model simply because the ANN is trained while minimizing a MSE criterion only. It is possible to train the ANN to find the best model with minimum number of parameters by including a penalty in MSE function as the number of parameters increases. This topic which was not considered in this study needs further attention. [21] Finally, it should be mentioned that the results of the tracer test analysis and the model rankings based on the ANN analysis presented above should not be looked upon as unique solutions, even though they are considered to be the most likely ones. In addition to distance between wells and volume of flow paths, mechanical dispersion is the only factor assumed to control the tracer return curves in the interpretations presented above. Retardation of the tracers by diffusion into the rock matrix is neglected. Through this effect, the chemicals used as tracers diffuse into the rock matrix when the tracer concentration in the flow path is high. As the concentration in the flow path decreases, the concentration gradient eventually reverses, causing diffusion from the rock matrix back into the fracture. This will, of course, affect the shapes of the tracer return curves obtained. In particular, it may cause the flow, through the channels discussed above, to be underestimated. Robinson and Tester [1984] postulate that matrix diffusion should be negligible in fractured rock, whereas Grisak and Pickens [1980] point out that it may be significant when fracture apertures are small, flow velocities are low and rock porosity is high. This clearly indicates that the tracer test data require further analysis and interpretation, which is beyond the scope of the present work.

5. Conclusions [22] An ANN model was proposed to identify tracer test models. The developed model can successfully identify several models including multifracture, homogeneous, double-porosity steady state and fracture-matrix flow. It was observed that the developed ANN model is not sensitive to noise present in the data; however, as the concentration-

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time plot displays multiple peaks, the ANN model shows a tendency to select the multifracture model. For some cases the ANN model identified two different models for the tracer tests conducted in the same field decreasing the number of possible models to two. ANN model results agreed well with all least squares analysis but for some of the tracer tests AICk and BICk statistics did not agree with ANN results. It was concluded that the developed ANN could be used as a model identifier and a diagnostic tool to decrease the model possibilities, thus eliminating the iteration process commonly used during the interpretation of tracer test conducted in geothermal reservoirs. Although the cases presented here are for geothermal reservoirs the developed model can be readily used for tracer tests conducted elsewhere.

Notation af rate of tracer interchange per unit fracture volume. am rate of tracer interchange per unit matrix volume. ^qk maximum likelihood estimate of qk based on p(Djqk, Mk). p(MkjD) posterior model probability for a given model M with D discrete data points. H dispersivity. u mean velocity. D difference. qk number of parameters associated with model Mk. BICmin smallest value of BICk over all candidate models. C concentration. Cmodel model concentration. Cfield observed concentration. D number of discrete data points. Dm coefficient of molecular diffusion. ei flow contribution coefficient. F least squares objective function. Fk Fisher information matrix. I input to the neurons. J model parameter. KICmin smallest value of KICk over all candidate models. Li apparent fracture length. M model. N dimension of D. n number of flow channels in the fracture system. Nk dimension of qk. Pe Peclet number. Pei Peclet number of the ith flow channel. tb response start time. tr mean arrival time. U Heaviside step distribution. ui velocity. w ratio of transport along the fracture to transport out of the fracture.

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Sultan, M. A., and A. U. Al-Kaabi (2002), Application of neural network to the determination of well-test interpretation model for horizontal wells, SPE paper 77878 presented at SPE Asia Pacific Oil and Gas Conference, Soc. of Pet. Eng., Melbourne, Victoria, Australia, 8 – 10 Oct.

 

S. Akin, Petroleum and Natural Gas Engineering Department, Middle East Technical University, Inonu Bulvari, 06531Ankara, Turkey. (serhat@ metu.edu.tr)

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