Arab J Geosci (2012) 5:579–586 DOI 10.1007/s12517-010-0214-2

ORIGINAL PAPER

Fracture analysis in Tabnak hydrocarbon field of Iran by using fractal geometry and multi-fractal analysis Hamid Sarkheil & Hossain Hassani & Firuz Alinia & Aliasghar Enayati & Abbasali Nikandish

Received: 25 July 2010 / Accepted: 4 October 2010 / Published online: 19 October 2010 # Saudi Society for Geosciences 2010

Abstract High gas production from the Dashtak formation of Tabnak hydrocarbon field in Fars province, Iran, indicates the presence of natural fractured reservoir whose production potential is dominated by the structural fracture. The connectivity of fractured media depends upon the power–law exponent and the fracture density. Fracture pattern traces obtained from the outcrops of producing formations of six different stations in Tabnak hydrocarbon field. 2D fracture network maps of Tabnak hydrocarbon field have been analyzed from their scaling properties. The fractal analysis of fracture intensity showed heterogeneous multi-fractal structure characterized by generalized dimensions. Distribution of fracture lengths exhibits power–law behavior with specific exponent. Scaling laws serve to make extrapolations and to study the fracture connectivity related to scale. Fracture distribution model and reservoir productivity can be estimated, which are of great interest in decision making to optimize gas production. Keyword Fractals . Fracture network . Multi-fractals . Power–law exponent . Tabnak hydrocarbon field

Introduction Fractal behavior has been observed in natural fracture patterns (Barton and Larson 1985 and Barton and Hsieh 1989) and fractal geometry provides a quantification of size scaling or H. Sarkheil : H. Hassani (*) : F. Alinia Mining and Metallurgical Engineering Faculty, Amirkabir University of Technology, Tehran, Iran e-mail: [email protected] A. Enayati : A. Nikandish Iranian Central Oil Fields Company, Tehran, Iran

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. So that, 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). The main objective is to analyze the fractal (Mandelbrot 1982; Turcotte 1997) and multi-fractal properties of fracture network in the Tabnak hydrocarbon reservoir (South of Iran). 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 Tabnak anticline that located in the eastern part of Asaloyeh hydrocarbon field. High gas production from the some geology formations of 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, b). The quantification and modeling of fracturing is thus of great interest for studying connectivity, transport properties, and production rate.

Geological setting The area of province of culminations in situation

study is located in South of Zagros in Fars Iran (Fig. 1). Tabnak structure has two that separated from each other with a saddle of wells No. 25. Carbonate evaporates of

580

Arab J Geosci (2012) 5:579–586

Fig. 1 A map of southern Iran, the location of the studied area and the situation of well No. 25 in UGC map (Tabnak hydrocarbon reservoir location)

Dashtak formation are important seal interval in the Zagros basin. The Dashtak formation consists of dolomite, anhydrite, shale, and limestone. It comprises of Aghar shale member, A member, B member, C member, Sepidar dolomite, and D member. Dashtak formation comprises eight facies deposited in inner middle homocline ramp (Sarkheil et al. 2009a, b). Four sequences were distinguished in the Dashtak formation and two surfaces sequence boundaries type I and three sequence boundaries type II. 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. 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 brecciaed 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 strongly related by lithology. Thus, structural assessment is very important to achieved 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 analyses 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

Arab J Geosci (2012) 5:579–586

581

Fig. 2 Total trend of fractures in station number 1 and 2

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 out in Eocene marls, fold at Cretaceous level, which do not coincide with fold in deeper horizons, and finally fold in Early Cretaceous and older rocks.

Surface fracture and methodology The present investigation measures the fractures specification (e.g., dip, strike, and density) of measurement in six different stations along various location in Tabnak hydrocarbon field (Figs. 2, 3, 4, 5, and 6) using

Fig. 3 Total trend of fractures in station number 3 and 4

the result for fractal analysis; related from data are shown in Table 1. Data were collected from six stations along the 20-km transect in an area of approximately 75 km2. A total of 932 fracture plane measurements from six stations were recorded for evaluation to facilitate analyzing data on the scale of Tabnak hydrocarbon field, the stations were divided into six sectors based on regional structures and topographic expression (Figs. 5 and 6). Data within each sector was summarized for interpretation. Therefore, Rose diagrams may be used to conduction with 1% counter plots for interpretation at a larger scale. When the fracture data from each of these regions is compiled into a plot which covers the study area, two orientations of strike dominate: a set which strikes northeast at NW-SE, sets striking NE-SW and E-W also appear throughout the Tabnak hydrocarbon field. The only substantial set which trends to the central division in the Tabnak hydrocarbon field fracture system is oriented at NE-SW.

582

Fig. 4 Total trend of fractures in station number 5 and 6

Fig. 5 Total trend and strike of open fractures on surface in six surface stations, Tabnak field, Fars, Iran

Arab J Geosci (2012) 5:579–586

Arab J Geosci (2012) 5:579–586

583

Fig. 6 Trace map of fracture network in Tabnak hydrocarbon field, Iran (S.1 to S.6 are the numbers of station locations)

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 repeat over various scales. This scaling behavior is described by a non-integer (or fractal) dimension varying 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

ð1Þ

The fractal dimension is: D¼

LogN Logð1="Þ

ð2Þ

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 Table 1 Data were collected from six stations in Tabnak hydrocarbon field

the slope of straight line segments fitted to the N, (1/ε) plotted on logarithmic axes (Barton 1995). D is the so called box-counting dimension or capacity dimension, generally noted D0. The application analysis to a fracture traces from Tabnak region map (Fig. 6) gives an estimated fractal dimension D0=1.82 with R2=0.9762, where R2 is a curve fitting error (Fig. 7).

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 (Halsey et al. 1989). 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 spectrum (Halsey et al. 1989; Mandelbrot 1982; Schertzer and Lovejoy 1989; Evertsz and Mandelbrot 1992). Application from the map, we obtained the multi-fractal exponents D(q) by using the boxcounting algorithm (Walsh and Watterson 1993). The fracture map is covered by Nn boxes of size εn, where the subscript n

Station No.

Total trend of fractures

Dip (degree)

Trend

Filling material

1 2 3 4 5 6

NW-SE NW-SE NE-SW E-W NE-SW NW-SE

85 60 70 and 75–80 75–80 60 70

N NE N N N N

Anhydrite No fill Anhydrite Anhydrite No fill Anhydrite

584

Arab J Geosci (2012) 5:579–586

6

D0 =1.82

Log N(E)

5 4 3

y = 1.8786x + 0.2384 R2 = 0.9769

2 1 0 0

0.5

1

1.5

2

2.5

3

Log 1vE

Fig. 8 Multifractal spectrum (D(q) is the dimension of order q)

Fig. 7 Box-counting fractal dimension estimation

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: li L

ð3Þ

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 "ntðqÞ

ð4Þ

It can be calculated using the following expression. #ðqÞ ¼

Nn X

Piq

N ð> l Þ1l c ð5Þ

i¼1

where, Pi is from the probability distribution of mass for all boxes, i, at this ε. Finally we define the fractal dimension by (Chhabra and Jensen 1989; Posadas et al. 2001): DðqÞ ¼

tðqÞ ð q  1Þ

ð6Þ

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.

Result of fractal and multi-fractal analysis

ð7Þ

N is the number of fractures having a length equal or greater than l, and c an exponent varying generally between 2 and 3. The exponent c is linked to the amount of short and large fractures, and it has important consequence on connectivity properties (Bour and Davy 1997). The cumulative frequency distribution of fracture length (n=932) shows a power–law behavior with an exponent c~2.92 (Fig. 9) which agree well with literature (Cladouhos and Marrett 1996). 3 2.5

Log (N>1)

Fi ¼

The results suggest that the spatial distribution of fracture intensity has not a homogeneous fractal structure but a heterogeneous one, with generalized positive fractal dimensions (D(0)=1.74>D(1)=1.68>D(2)=1.63>..........>D(12)= 1.40). D(0), D(1), and D(2) are respectively the capacity, information, and correlation dimension (Hentshel and Procaccia 1983). The value D(12)=D(∞) is the fractal dimension of the most intensive clustering in the heterogeneous set. It obvious that one fractal dimension is not enough to describe the fracture properties related to scale. A full spectrum of generalized dimensions is then required to take into account the fracture clustering which affect connectivity. Several field studies have demonstrated that fracture populations have a power–law length distribution (Scholz and Cowie 1990; Davy 1993). Natural fracture length distributions thus show to obey to this fractal law.

y = -2.9223x + 2.5944 R2 = 0.9934

2 1.5 1 0.5

Figure 8 displays the multi-fractal spectrum of fracture traces of Fig. 6. D(q) exhibits a characteristic behavior, decreasing with increasing q (Hentshel and Procaccia 1983).

0 -0.1

0.1

0.3

0.5

Log l (Km)

Fig. 9 Fracture length distribution

0.7

0.9

Arab J Geosci (2012) 5:579–586

Over more than two decades, a best fit was obtained with a good linearity between 1 and 10 km length. The maximum extrapolated fracture length is about 32 km. This power–law can also be used to estimate the number of small fractures. The same law and the same fractal exponent will be valid for the Tabnak hydrocarbon reservoir, Fars, Iran. According to numerical modeling (Bour and Davy 1997) and on the basis of the percolation theory (Stauffer and Aharony 1992), the connectivity of fractured media depends upon the power–law exponent and the fracture density. The small and large fractures will contribute to connectivity with an amount or ratio depending on exponent c. In our case, c~3 may be a critical value where 50% of fractures belong to the infinite cluster or correlation length. Berkowitz et al. (2000) analyzed the fracture connectivity on the light of relation of exponent c to capacity fractal dimension D(0), for c>D(0), the connectivity does not depend of scale, inversely for c
Conclusions Fracture networks in the Tabnak hydrocarbon reservoir have multi-fractal structure characterized by the full singularities spectrum and exhibiting its scaling behavior. The power–law fracture length distribution shows scale invariance. The fractal exponent could be attributed to the entire hydrocarbon reservoir. The fracture connectivity seems to be independent of investigation scale. The extrapolation to 3D analysis based on fractal mathematics and percolation theory will constitute an interesting field of research for a better understanding of fracture geometry and reservoir conductivity properties. The management of data on a GIS could help to a better optimization of fault and big fracture prospecting of this region, focusing in fractured connected media to reduce low hydrocarbon producer wells.

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

References Barton CC (1995) Fractal analysis of scaling and spatial clustering of fractures. In: Barton CC, LaPointe PR (eds) Fractals in the earth sciences. Plenum Press, New York, London, pp 141– 178

585 Barton CC, Larson E (1985) Fractal geometry of two-dimensional fracture networks at Yucca Mountain, southwestern Nevada. In: Stephannson O (ed) Proceeding, International Symposium on Fundamentals of Rock Joints. Bjorklinden, Sweden, pp 77– 84 Barton CC, Hsieh PA (1989) Physical and hydrological-flow properties of fractures, In International Geological Congress, 28th, field trip guidebook T385. American Geophysical Union, Washington D.C, 36 p Berkowitz B, Bour O, Davy P, Odling G (2000) Scaling of fracture connectivity in geological formations. Geophys Res Lett 27 (14):2061–2064 Bour O, Davy P (1997) Connectivity of random fault networks following a power–law fault length distribution. Water Resour Res 33(7):1567–1583 Chhabra A, Jensen RV (1989) Direct determination of the ƒ(α) singularity spectrum. Phys Rev Lett 62:1327 Cladouhos TT, Marrett R (1996) Are fault growth and linkage models consistent with power–law distribution of fault length. J Struct Geol 17:863–873 Davy P (1993) On the frequency–length distribution of the San Andreas fault system. J Geophys Res 98:12414–12451 Evertsz CGC, Mandelbrot BB (1992) Multi-fractal measures. In: Peitgen HO, Jurgens H, Saupe D (eds) Chaos and Fractals. Springer, Berlin, pp 849–881 Giorgilli A, Casati D, Galgani L, Sironi L (1986) An efficient procedure to compute fractal dimesions by box counting. Phys Lett A 115(5):202–206 Halsey TC, Jensen MH, Kadanoff LP, Procaccia I, Shraiman BI (1989) Fractal measures and their singularities: the characterization of strange sets. Phys Rev Lett A 33:1141–1151 Lyakhovsky V (2001) Scaling of fracture length and distributed damage. Geophys J Int 144:114–122 Liebovitch LS, Toth T (1989) A fast algorithm to determine fractal dimensions by box counting. Phys Lett A 141(8/9):386–390 Lisle RJ (1994) Detection of zones of abnormal strains in structures using Gaussian curvature analysis. AAPG Bull 78:1811–1819 Lovejoy S, Schertzer D, Tsonis AA (1987) Functional box-counting and multiple elliptical dimensions in rain. Science 235:1036–1038 Posadas AND, Gimenez D, Bittelli M, Vaz CMP, Flury M (2001) Multi-fractal characterization of soil particle-size distributions. Soil Sci Soc Am J 65:1361–1367 Mandelbrot BB (1982) The fractal geometry of nature. Freeman, NewYork Matsumoto N, Yomogida K, Honda S (1992) Fractal analysis of fault systems in Japan and the Philippines. Geophys Res Lett 19 (4):357–360 Scholz CH, Cowie PA (1990) Determination of total strain from faulting using slip measurements. Nature 346:873–879 Price NJ, Cosgrove JW (1990) Analysis of geological structures. Cambridge University Press, Cambridge Sanders CAE, Fullarton L, Calvert S (2002) Modelling fracture systems in extensional crystalline basement. Geological society of London Publications. Hydrocarbons Cryst Basement 147:145–162 Sarkheil H, Hassani H, Alinia F (2009a) The fracture network mmodeling in naturally fractured reservoirs using artificial neural network based on image loges and core measurements. Aust J Basic Appl Sci 3(4):3297–3306 Sarkheil H, Hassani H, Alinia F, Enayati AA and Motamedi H (2009b) A forecasting system of reservoir fractures based on artificial neural network and borehole images information_exemplified by reservoir fractures in Tabnak field, Fars, Iran, 9th International Multidisciplinary Scientific Geo-Conference & EXPO SGEM 2009, Vol.1, pp.563–570 Schertzer D, Lovejoy S (1989) Nonlinear variability in geophysics: multi-fractal analysis and simulation. In: Pietronero L (ed)

586 Fractals physicals origin and properties. Plenum Press, New York, pp 49–79 Stauffer D, Aharony A (1992) Introduction to percolation theory. Taylor and Francis, Bristol Tsuchiya N, Nakatsuka K (1995) Fractal analysis and modeling of a two-dimensional fracture network in a geothermal reservoir. Geoth Resour Counc Trans 19:547–552

Arab J Geosci (2012) 5:579–586 Turcotte DL (1997) Fractals and chaos in geology and geophysics. 2d Ed., Cambridge Walsh JJ, Watterson J (1993) Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies. J Struct Geol 15(12):1509–1512 Wang Sh and Wu Ch (2001) Self-affine multi-fractal spectrum and Levy stability index from NA27 data, Chin Phys Lett 18 (1)