Arab J Geosci (2013) 6:945–956 DOI 10.1007/s12517-011-0400-x

ORIGINAL PAPER

Fractures distribution modeling using fractal and multi-fractal–neural network analysis in Tabnak hydrocarbon field, Fars, Iran Hamid Sarkheil & Hossain Hassani & Firuz Alinia

Received: 20 January 2011 / Accepted: 2 September 2011 / Published online: 14 September 2011 # Saudi Society for Geosciences 2011

Abstract Modeling of fractures distribution in naturally fractured hydrocarbon reservoirs is a complex process that contains large amount of uncertainty. High gas production from the Dehram group of Tabnak hydrocarbon reservoir in Fars province indicates the presence of natural fractures. This study presents a novel methodology that integrates various features of geological, statistical and artificial intelligence techniques in a nested loop to characterize field fractures and then to model them. Their characterization is, to some extent, technique dependent. Secondary properties such as fracture density, fractal dimension and fractal spectrum are there defined for better description of fractures’ spatial distribution. Irregular geometry laws showed that the connectivity of fractured media depends on power-law exponent and some fracture characteristics. A neural network is incorporated in the proposed methodology to determine these relationships, by processing field data available from the six outcrops with similar lithology, image logs and core analyses. The value of 1/82 was calculated for fractal dimension of surface fractures. The fracture density in the range of 0.2 to 1.4, fractal dimension and fractal spectrum (D(q=0)) of fractures in the range of 1.0 to 1.6 and 1.8 to 3.0 respectively were calculated for the 10-m intervals within the well and extended with fractal– neural network algorithm. Determination of fractal dimension and fractal spectrum from image logs and dual application of neural network and fractal geometry in fracture modeling are innovative in this study. Finally fracture distribution models estimated are of great interest H. Sarkheil : H. Hassani (*) : F. Alinia Mining and Metallurgy Engineering Faculty, Amirkabir University of Technology, Tehran, Iran e-mail: [email protected]

with gas production rates of the Tabnak hydrocarbon reservoir and have more reality results compared with DFN algorithm models using Open Flow software. Keywords Fractures distribution . Tabnak . Fractal dimension . Neural network . DFN algorithm

Introduction Natural fractures are mechanical breaks in rocks. They occur at different scales and are highly heterogeneous. Rock fracturing is a complicated process, which is sensitive to changes in geological conditions under lithostatic, fluid pressures, tectonic, thermal and other geological stresses such as uplifting, volcanoes and salt intrusion. Generally fractures initiate and propagate when stresses become equal or greater than the rock strength. Naturally fractured reservoirs (NFRs) play an important role in hydrocarbon exploration and production, because a large number of oil and gas reservoir are naturally fractured (Aguilera 1995; Nelson 2001). Production from those reservoirs is usually affected by presence of a system of connected fractures. This fracture network can contribute significant porosity and permeability to a reservoir. Understanding and quantifying the fracture distribution, such as identifying the location of highly fractured zones with the reservoir and determining the degree of reservoir anisotropy, are very important for NFR exploration and development. For a particular geological setting, the spatial distribution of fractures can be a very complex function of different geological factors. The question of how fracture properties change with an increase of folding, as a function of bed thickness and lithology, is of critical importance not only

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for the prediction of the fracture distribution within the reservoir but also for understanding the development of the fracture system within the geological setting. Conventional statistical analysis with linear relationships between variables is usually used to examine the relationships between these variables. During the last decade, the application of neural networks for identification of nonlinear and non-stationary systems has increased. Artificial neural networks have been used widely in finance, engineering, medicine and management (Garson 1998). Earlier mathematical models for fracture distribution networks and flow through them include equivalent continuum models (Heieh and Neuman 1985), discrete network models (Long et al. 1992) and hybrid techniques (Cacas et al. 2001). Based on fractal and multi-fractal concept, equivalent discontinuum fractal models are also available (Watanabe and Takahashi 1995). Recently, attempts have been made to characterize natural fractures applying fuzzy logic and neural networks, where an integrated approach is adopted to encompass various information available from the field (Ouenes 2000; Quintero et al. 2001). The data include seismic, porosity, permeability, lithology, bed thickness, state of stress, fault patterns and production data. The outcome is a network map of fracture index/intensity for each discrete block. The approach uses fuzzy logic to quantify and rank to importance of each geological parameter on fractures and neural networks to account for complex, nonlinear relationships between these geological parameters and the fracture index. Thus, it took into account the overall effect of fracture network upon fluid flow in the reservoir with a primary emphasis on predicting well performance. Their purpose, however, did not require generation and treatment of discrete fractures, as the production performance was assessed for the reservoir at its in situ condition (i.e., without any hydraulic stimulation). Neural networks, which can extract relationships among multiple variables underlying observed data, are seen to be an excellent tool for investigating the fracture density as a function of multiple geologic parameters. The neural network is attractive for analyzing the complex nonlinear system because it does not require a priori knowledge about the functional structure among variables. Ouenes et al. (1998) used neural networks to analyze the ultimate recovery distribution within the naturally fractured reservoir as a function of the bed thickness, formation resistivity and formation curvature. It is commonly observed that fractures naturally develop in sets, each having different orientation, density and geometric parameters. The interaction of the fractures is effective in exploration and development of NFRs. The flow characteristic of the fracture system can be assessed by using discrete fracture modeling (Chiles and de Marsily

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1993; Karpov 2001). In this model a fracture system with required statistical characteristics is statistically generated. The hydraulic isotropy of the system can be estimated by analyzing the flow conductance between specified source and sink within the system. Fractal behavior has been observed in natural fracture patterns (Barton and Larson 1985; Barton and Hsieh 1989; Sarkheil et al. 2010), and fractal geometry provides a quantification of size scaling or scale dependency of the complex fracture systems. Fractal analysis relies on the estimation of a non-integer number, i.e., fractal dimension (D). Typically box-counting technique is applied to measure the fractal dimension of the fracture network (Barton and Larson 1985). The point of a multi-fractal analysis is to detect the multi in multi-fractal scaling. In multi-fractal analysis, the box-counting grid technique has been used to gather information about the distribution of pixel values (called the “mass distribution”), which becomes the basis for a series of calculations that reveal and explore the multiple scaling rules of multi-fractals (Chhabra and Jensen 1989). Structural style and its relationship to morphology of Zagros fold–thrust belt is dominated by magnificent exposures of NWSE trending folds. These folds differ in their size and geometry in this area. One of the regions of this fold–thrust belt is the Tabnak anticline that is located in the eastern part of the Asaloyeh hydrocarbon field (Fig. 1). High gas production from some geology formations of the Tabnak hydrocarbon field in Fars province, Iran indicates the presence of natural fractured reservoir whose production potential is dominated by the structural fracture where the gas flow occurs essentially along open fractures (Sarkheil et al. 2009a, 2009b, 2010). The quantification and modeling of fracturing is thus of great interest for studying connectivity, transport properties and production rate. The present study investigates the use of neural network and fractal (Mandelbrot 1989; Turcotte 1997) and multifractal (Sarkheil et al. 2010) mathematics in fracture modeling. The results are used to investigate the effects of porosity, permeability, fracture density, volume of shale factor, fractal dimension and multi-fractal spectrum on the part of backlimb of Tabnak fold that is used to determine fracture distribution pattern.

Geological setting The area of study is located in south of Zagros in Fars province of Iran (Fig. 1). Tabnak structure has two culminations that separated from each other with a saddle in situation of well No. 25. Carbonate evaporates of Dehram group are important seal intervals in the Zagros basin (Fig. 2). The Dehram group consists of dolomite, anhydrite, shale and limestone. It comprises Aghar shale member, A member, B

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Fig. 1 Location map of studied area, main structural subdivisions of the Zagros fold and thrust belt (Berberian and King 1981)

Fig. 2 The map of southern Iran, the location of the studied area and the position of well No. 25 in UGC map (Tabnak hydrocarbon reservoir)

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member, C member, Sepidar dolomite and D member. Dashtak formation comprises eight facies deposited in the inner middle homocline ramp (Sarkheil et al. 2009a, 2009b, 2010). Four sequences were distinguished in the Dashtak formation and Dehram group and two surfaces sequence boundaries type I and three sequence boundaries type II. The first sequence begins at the lowermost disconformity at the base of Aghar shale up to massive anhydrite A. During relative sea-level highstand, the lagoon was then gradually filled to the intertidal level. The lagoon and mid-ramp facies are known for maximum flooding surfaces (MFS) at this sequence and apparently correlateable to Tr50 of Ladnian stage. The second sequence begins from the base of B member up to the top of B massive anhydrite. MFS of the upper boundary is a sequence boundary type II with brecciated dolomite. In order to predict regional fracture patterns, it is necessary to identify the main processes responsible for fracturing. No straightforward answer exists as different processes may be dominant in various geological settings and is highly dependent on the various phases and characteristics of the local geological history (Sanders et al. 2002). Fracturing also takes place in essentially undeformed regions (Price and Cosgrove 1990) and is strongly related to lithology. Thus, structural assessment is very important to achieve natural fractured reservoir (Lyakhovsky 2001). Though traditional curvature analysis has often been used for fracture assessment (Price and Cosgrove 1990), curvature analysis on the present-day geological model gives limited insight for fracture purposes as it only analyzes the geometry on the present-day model statically (Lisle 1994). The Tabnak anticline and these geological layers are characterized by simple fold within Cretaceous levels. These folds are asymmetric, verging southwest with a typical wavelength of about 10 km. Oligocene– Lower Miocene Asmari carbonates in this domain are shortened by folds with wavelengths of 1–2 km on the flank of the main structures, which are locally breached by thrust faults. Our structural interpretation for the deeper horizons is based on information provided by seismic data, which indicate larger structures at depth. The Albian Kazhdumi and Eocene Pabdeh formations are intermediate decollement horizons that separate three distinct sets of structures in the area simple fold with short wavelength in Oligocene carbonates that tend to be out in Eocene marls, fold at Cretaceous levels, which do not coincide with fold in deeper horizons, and finally fold in Early Cretaceous and older rocks.

Data sources In this study, different field data sources cover different scales (microscopic to regional) and are of different

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resolution. Moreover, no single tool can provide all information needed to fully characterize a fracture network distribution. Thus, in order to portray the multifaceted characteristics of natural fractures, an integrated approach is required, where all available field data from a wide range of geological, petrophysical, drilling, production and other sources are utilized. There are two main types of data sources for fracture characteristics in fracture network distribution. The first group includes outcrop, facies and other geological sources, which are used in studying reservoir geological features. This group of data is at a large scale of reservoir, with resolution ranges from a few inches to several feet. They reveal reservoir structure, bed thickness, lithology and curvature of various formations. Also outcrops data describe various fracture characteristics, such as orientation, size and spacing. In this study some features of fractures are measured (e.g. dip, strike and density) at 6 different stations along various locations in the Tabnak hydrocarbon field. Data is collected from 6 stations along the 20-km transect in a region with an area of approximately 75 km2. A total of 932 fracture plane measurements from 6 stations were recorded for evaluation to facilitate analyzing data on the scale of the Tabnak hydrocarbon field. The stations were divided into six sectors based on regional structures and topographic expression. The other group contains the data available at the well site, such as image logs, core derived data, petrophysical data and well testing. Among the most efficient logging tools for fracture characterization are borehole televiewer, formation micro-scanner and core. It is possible to determine properties such as fracture density, volume of shale, geology facies, mud loss, porosity and permeability.

The proposed modeling technique This study presents a systematic procedure to fractures distribution modeling in reservoir. Output fractures are conditional to the available traces for finding the optimized production well station. In order to achieve this, we first review the field data sources that are typically available in a petroleum exploration and that could reveal information about key fracture properties of location, orientation and size. Statistical distributions of those properties are then determined by integrating data from the different sources at different scales. Finally, they are simulated to model the fracture distribution based on fractal and multi-fractal. The procedure includes three major computational modules: backpropagation neural network (BPNN), fractal dimension and fractal spectrum simulation (Fig. 3).

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Fig. 3 Flowchart of the fractal and multi fractal–neural network modeling procedures

Artificial neural network The artificial neural network (ANN) is based on the modern biological research on the hemopoietic tissue of the human

brain. It attempts to imitate the way in which a human brain works in processes such as studying, memorizing, reasoning and inducing with a complex network, which is performed by extensively connecting various processing

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units (Nikravesh 1998; Jiao 1990). ANN is a dynamic system with mighty functions in the nonlinear and superscale-consecutive-temporal processing; at the core of it are the concepts of nonlinear of consecutive temporal, the global effect of the Internet, the parallel processing ability of super-scale and the imaginary ability of study. ANN has the advantage of the common nature of the nonlinear dynamic system as well, such as unpredictability, attractability, dissipatedness, imbalance, irreversibility, multidimension, broad-interlinkage and self-adaptability. Although there are many kinds of ANN, back propagation (BP) is one of the most effective and extensively used methods. BP is a mapping network with a multi-layered structure. Compared to the conventional model, it has the characteristics of a longer duration and better timing prediction. Therefore, a BP neural network is developed in this study to cope with the processing of complex nonlinearity in the research of reservoir fracture modeling (Neural Ware 1995; Wei 1990).

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Fig. 4 Architecture of BP network (Neural Ware 1995)

The relationship of input and hidden layers: Hidden ½ j ¼ f

n X

! Wij ai

ð1Þ

i¼1

Foundation of the back-propagation (BP) algorithm The relationship of hidden and output layers: Perhaps the most widely used training algorithm for feed forward neural network is the back propagation method (Rumelhart and McClelland 1986; Ripley 1996; Bishop 1995). This method propagates the error produced by the network for a given input backward through the layers to adjust the network weights. The training of a network by the back propagation algorithm involves three stages: the feed forward of the input training pattern, the calculation and back propagation of the associated error, and the adjustment of the connection weights. During feed forward, each input node receives an input signal (actual value of the explanatory variable) and feeds this value to each of the hidden nodes. Each hidden node then computes its activation function and sends the result to each output node. Each output node computes its activation to form the response of the net for the given input pattern. The output of the net is compared to the observed value to determine the associated error. This error is used to calculate the update of the connection weight. The BP neural network model is selected in the analysis of reservoir fracture modeling in the Tabnak anticline, where nodes are organized in input, hidden and output layers. The input and hidden layers have four nodes each, and the output layer has one node. Unlike two-layer networks, the three-layer network has a middle hidden layer. Every neural unit can divide input-space in a different way. Some hiding characters can be obtained from the input-layer, and more complex districts can be formed to improve the classifying ability. The structure of the BP neural network is shown in Fig. 4.

Out ½l  ¼ f

q X

! Vji bj

ð2Þ

i¼1

Here, ai bj Wij Vji

is is is is

the the the the

input node i of the input layer output node j of the hidden layer weight between input and hidden layer, and weight between hidden and output layer.

The outlines of the algorithm are as follows. Firstly, a starting weight value is selected and the output value obtained according to the above two equations. The error of counting model output and ideal model output can be obtained. If it is not suitable, it can be adjusted to improve the prediction. The error function (Ek) is constructed to make the error decrease with the weight adjustment (Neural Ware 1995). If the Sigmoid function is used as the responding function f(x)=1/(1+e−x), then the absolute deviation is: Ek ¼

q X  k 2 d i =2

! ð3Þ

i¼1

Then the adjusted weight from the input layer to the hidden layer is: $V ¼ bðgradvEk Þ ¼ bdik bj

ð4Þ

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where β denotes the studying rate; the adjusted weight from the hidden layer to the output layer is: " # q X   k $Wij ¼ aj Vji di f 0 s0 j aki ð5Þ i¼1

(a denotes the studying rate). Validation of the BP model The multi-layer network is trained by repeating this process many times. The goal of the training is to reach an optimal solution based on the performance measurement. Firstly, groups of training samples must be provided. In addition, each sample group is made up of an input sample and an ideal output sample. The ideal output value of this index is 1 among the 39 points in the Tabnak hydrocarbon field. According to these sample data, the learning-ability and self-applicability of ANN can be used. Through iteration of the simple function, it can be a prediction model. With 39 points the training processor can be made. The training is finished when the output value is equal to the ideal output value. Otherwise, the output value can be made equal to the ideal output value. At the start of the training, the maximum of the studying rate is 0.9. After 4,000, the learning rate has slowed down to 0.7. Then after 2,784, the error Ek = 7.370E04. When it is acceptable, the prediction coefficient could be obtained. Fractal analysis Fractures exist over a wide range of scales, from microfractures to largest faults. Fracture patterns in nature display a self-similar geometry, at least at statistical sense, that repeats over various scales. This scaling behavior is described by a non-integer (or fractal) dimension varying Fig. 5 Spatial fractures distribution of Tabnak reservoir

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in 2D from 1 to 2. The box-counting method (Giorgilli et al. 1986; Liebovitch and Toth 1989; Lovejoy et al. 1987; Matsumoto et al. 1992; Tsuchiya and Nakatsuka 1995; Walsh and Watterson 1993) is generally used to measure the fractal dimension of the spatial distribution and scaling of fractures. A sequence of grids of different cell size ε is placed over the fracture map; the number of cells intersected or containing a fracture is counted. The fractal relation is: N "D ¼ 1

ð6Þ

The fractal dimension is: D¼

LogN Logð1="Þ

ð7Þ

where N is the number of cells containing fractures, ε is the length of the side of the cell and the fractal dimension D is the slope of straight line segments fitted to the N, 1/ε plotted on logarithmic axes (Barton 1995). D is the socalled box-counting dimension or capacity dimension, generally noted D0. The application analysis to fracture traces from Tabnak region map gives an estimated fractal dimension D0=1.82 with curve fitting error R2 =0.9762 (Figs. 5 and 6). Multi-fractal analysis The major characteristic of fractal sets is their scaling properties related to self-similarity. For mono-fractals, scaling can be described by only one exponent (fractal dimension). This is not the case for natural fractal sets like fracture networks, which are multi-fractals. These objects can then be entirely described by a spectrum D(q) of fractal exponents, the generalized dimensions, where the fractal dimension is D0 and the function D(q) is the multi-fractal

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D0=1.82

5

Log N(E)

4 3 y = 1.8786x + 0.2384 R2 = 0.9769

2 1 0 0

0.5

1

1.5 Log 1/E

2

2.5

3

Fig. 7 Multi-fractal spectrum (D(q) is the dimension of order q

Fig. 6 Box-counting fractal dimension (D) estimation

spectrum (Halsey et al. 1989; Mandelbrot 1989; Schertzer and Lovejoy 1989 ; Evertsz and Mandelbrot 1992). With application from the map, we obtained the multi-fractal exponents D(q) by using the box-counting algorithm. The fracture map is covered by Nn boxes of size εn, where the subscript n indicates the nth generation scale. We associate the content of each box with measures Fi, which in our case a fracture intensity index defined as: Fi ¼

li L

ð8Þ

where li is the total fracture length in the ith box (i=1,…, Nn) and L is the total length of fractures on the map. Thus, the fracture intensity is the measure, and its support is the natural area represented by the 2D map itself. The generating function χ(q,ε) defines the multi-fractal spectrum in terms of the mass exponents τ(q), where τ(q) is a real parameter given by

# ðqÞ ¼ "tðqÞ n

ð9Þ

It can be calculated using the following expression:

# ðqÞ ¼

Nn X

Piq

consists of interconnected fractures. In this study we used two approaches to simulate the fracture distribution. The first model was prepared by the result of fracture density, and the other models contain the fractal dimension and fractal spectrum at the well site. From available field data, fracture density (fracture area in unit volume, square meter/cubic meter) can be calculated directly from image logs. Fractal dimension can be estimated using the box-counting method, and fractal spectrum can be determined using the mathematical relations. The ANN technique is applied in computing the distribution of the three key fracture properties: fracture density, fractal dimension and fractal spectrum. Using these wellbore values for training and validation, ANN estimates fracture density, fractal dimension and fractal spectrum values over the sample block in the Tabnak hydrocarbon reservoir. This procedure is expected to make fractures distribution modeling significantly efficient. To demonstrate capability of the proposed models, firstly, fracture density, fractal dimension and fractal spectrum for simulating in the block sample of reservoir (Fig. 8) are estimated by a BPNN (Table 1). A set of input data of lithology index (facies), volume of

ð10Þ

i¼1

Finally we define the fractal dimension by: DðqÞ ¼

tðqÞ ð q  1Þ

ð11Þ

Multi-fractals have multiple dimensions in the D(q) vs q spectra. The procedure of calculating the multi-fractal spectrum is carried out with q in the range −12 to 12 with an increment of 1 (Fig. 7). Fracture modeling Fracture network distribution studies require threedimensional modeling of the fracture system, which

Fig. 8 Location of wells on Tabnak hydrocarbon reservoir

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Table 1 The data set for Dashtak and Kangan formations in well No. 3 (for example), in Tabnak hydrocarbon reservoir Well x

y

Depth Fr ф

k

Mud Loss (bbl) Fractal dim. Fractal spectrum Fracture density Volume of shale

3 3 3 3 3 3 3 3

621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82

2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03

2468 2478 2488 2498 2508 2518 2528 2538

1 1 1 1 1 1 1 1

0.0350 0.0350 0.0340 0.0340 0.0340 0.0340 0.0356 0.0356

1.4020 1.4020 1.5070 1.5070 1.5070 1.5070 1.6400 1.6400

1147 1147 1200 1150 1120 1450 1530 1450

1.476 1.457 1.42 1.418 1.416 1.418 1.41 1.409

1.812 1.967 1.881 2.419 2.517 2.619 2.611 2.61

0.56 0.57 0.54 0.56 0.58 0.56 0.64 0.65

−9999.99 0.040703 0.084354 0.030340 0.038870 0.047435 0.065578 0.036205

3 3 3 3 3 3 3 3 3 3 3 3 3

621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82 621087.82

2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03 2297619.03

2548 2558 2568 2578 2588 2598 2608 2618 2628 2638 2648 2658 2668

1 1 1 1 1 1 2 2 2 2 2 2 2

0.0253 0.0240 0.0244 0.0244 0.0260 0.0350 0.0364 0.0378 0.0380 0.0400 0.0400 0.0460 0.0460

0.8640 0.8640 0.7050 0.7050 0.7103 1.0819 0.9846 0.9846 0.9850 0.9850 0.9850 1.4450 1.4450

850 850 850 850 756 560 582 582 1039 1039 1039 1300 1230

1.433 1.438 1.442 1.444 1.442 1.4273 1.426 1.4273 1.426 1.4287 1.4187 1.419 1.414

2.634 2.639 2.643 2.645 2.643 2.6283 2.627 2.6283 2.627 2.6297 2.6197 2.02 2.015

0.41 0.36 0.32 0.3 0.32 0.467 0.48 0.467 0.48 0.453 0.553 0.55 0.6

0.035757 0.074495 0.060106 0.578270 0.921092 0.671134 0.330453 0.207526 0.400543 0.654190 0.863016 0.838824 0.826956

Fr Formation, dim dimension

shale, is chosen among the relevant data sources mentioned earlier, as they are widely available and reliable and with high resolution. Through training, the ANN establishes the complex relationships between input data, fracture density, fractal dimension and fractal spectrum. It then gener-

Fig. 9 Characterized 3D fracture density (square meter/ cubic meter)

ates spatial distribution of fracture density, fractal dimension and fractal spectrum in the reservoir (Figs. 9, 10 and 11). Comparing the computed outputs with the input data, we can test the accuracy of BPNN. Both graphs give close correlations, with R2 being very close to 1.

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Fig. 10 Characterized 3D fractal dimension (D0)

Characterization of fracture networks using a DFN approach Discrete fracture network (DFN) approach is an ideal numerical tool with which to synthesize realistic fracture network models from digitally and conventionally mapped data. Discrete fracture network-related research carried out over the past 20 years has focused on identifying those individual discrete features which provide discrete connections which carry the most important portion of flow. The task of geology then becomes to characterize these individual pathways rather than the average properties of the medium. To achieve this, it is necessary to understand the geometry and properties of the discrete fractures which form these pathways. In this study, also, by modeling the fracture distribution stochastically with FRACA (OPEN FLOW V. 2.0.3) software in the Tabnak hydrocarbon reservoir based on Fig. 11 Characterized 3D multi-fractal spectrum (D(q=0))

DFN method (Figs. 12, 13, 14 and 15), this software produce a set of micro-fractures in randomized procedures. The discrete feature conceptual model is the essence of the DFN approach. The DFN approach provides the three-dimensional framework of discrete features which concentrate on grid generation system: conventional rectangular grid system, hexagonal grid system, radial grid system and also the flow barriers such as faults and argillaceous layers which provide partial or complete seals (Winberg et al. 2003).

Conclusions As previously discussed, a case study was performed over a limited area in backlimb of the Tabnak hydrocarbon reservoir, in order to demonstrate the ability of the proposed research in comparison with DFN model (using the Open Flow software) that is based on the random

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Fig. 12 DFN gridding approach with randomly distributed points

distribution of points. Firstly, the fracture density and the fractal dimension and multi-fractal spectrum were estimated by BPNN. A set of input data of lithology index, volume of shale, mud loss, porosity and permeability were chosen among the relevant data sources that were previously described, as they were widely available and reliable and had high resolution. Through training, the NN establishes the complex relationships between input data set and output data set including fracture density, fractal dimension and multi-fractal spectrum. It then generates spatial distribution of fracture density, fractal dimension and multi-fractal spectrum.

Fig. 13 Fractures distribution model with DFN algorithm (section A, depth Z1=2350 m)

Fig. 14 Fractures distribution model with DFN algorithm (section B, depth Z2=2550 m)

Comparing the computed output data with the input data can show the accuracy of BPNN. Both graphs give close correlation, with R2 being very close to 1. Finally, the result of fracture density, fractal dimension and multi-fractal spectrum spatial distribution characterized in 3D formats in comparison with DFN model shows that 3D fractal– neural network fractures distribution models are in agreement of well test analysis and production data results. The fractal and multi-fractal–neural network fractures distribution modeling process can be repeated and improved throughout a Tabnak or any hydrocarbon field.

Fig. 15 Fractures distribution model with DFN algorithm (section C, depth Z3=2750 m)

956 Acknowledgments The authors would like to acknowledge the financial support provided by the Iranian Central Oil Fields Company.

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