Characterization and estimation of reservoir properties in a carbonate reservoir in Southern Iran by fractal methods
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Abstract
Reservoir heterogeneity has a major effect on the characterization of reservoir properties and consequently reservoir forecast. In reality, heterogeneity is observed in a wide range of scales from microns to kilometers. A reasonable approach to study this multiscale variations is through fractals. Fractal statistics provide a simple way of relating variations on larger scales to those on smaller scales and vice versa. Simple statistical fractal models (fBm and fGn) can be useful to understand the model construction and help the reservoir structure characterization. In this paper, the fractal methods (fGn and fBm) have been applied to characterize and to estimate of reservoir properties. Three methods, namely boxcounting, variogram, and R/S analysis, were carried out on log and core data for porosity and permeability data to estimate fractal dimension; a high fractal dimension estimated in this study reveals a high heterogeneity in the reservoir. Moreover, sampling from simulated fractal data at nonexisting data depths enables us to generate appropriate realizations of reservoir permeability with suitable accuracy at a proper computational time.
Keywords
Porosity Permeability Fractal geometry fGn fBmIntroduction
Reservoir characterization is one of the important tasks in petroleum engineering studies. An accurate characterization of reservoir dynamic behavior can be dependent on the exact characterization of the spatial distribution of reservoir properties throughout the reservoir (Hurtado et al. 2009). The porosity, permeability, and water saturation are key reservoir properties which affect the reservoir flow behavior and fluid recovery (Xiao et al. 2012). Among these, permeability is of prime importance, especially in carbonate reservoirs; the reason is that carbonate reservoirs have a more complex internal structure and more heterogeneous (Gholinezad and Masihi 2012; AlKhansi and Blunt 2007).
Methods for accessing the reservoir properties can be divided into direct and indirect methods. Direct methods include coring, well logging, and well tests; In practice, coring is the most reliable method, but it is expensive and time consuming; Moreover, it may be impossible to take samples in some intervals of a well. Indirect methods are based on estimation and forecast. Geostatistics as a branch of statistical science is one of the tools for indirect methods, which studies spatial structures that vary in pace and/or in time. Geostatistics can also be used for quantity description and local relationship between data within the reservoirs (Deutsch 2002). Methods based on geostatistics, despite their advantages, have drawbacks such as smoothing data; thus, they cannot show details and local heterogeneity (Rasouli and Tokhmechi 2010).
The term “fractal” was coined by Mandelbrot in the 1970s. Fractal geometry can be used to describe irregular and fragmented patterns in nature (Mandelbrot 1983; Aasum et al. 1991). In fact, fractal geometry presents a mathematical model for most of the complex phenomena such as coastline, mountain, cloud, and fracture patterns (Yin 1996; Li et al. 2009; Kim and Schechter 2009).
Recent studies have shown that statistical fractals are useful for modeling natural phenomena (Arizabalo et al. 2006; Vadapalli et al. 2014). Because of the nature of variability in reservoir data, statistical fractals such as fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) are candidate approaches to characterize porosity and permeability and, in general, the hydraulic property variations on the subsurface data(Zeybek and Onur 2003; LozadaZumaeta et al. 2012).
The porosity and horizontal component of petrophysical properties such as permeability are usually modeled using krigingbased methods (i.e., Geostatistics). The property values along the well can be used as conditions through these approaches. Although these properties are inferred from well log data, there are some practical cases, where the data are missing in some well intervals. Due to significant variability of petrophysical properties in vertical direction as a result of reservoir layering in reservoirs, fractal models can be used to characterize them and to help model them in these intervals.
The purpose of this paper is to employ the statistical fractals (fBm and fGn) for the characterization and their estimation at nonexisting data depths. In this study, three wells in a carbonate reservoir in the southern part of Iran were studied. Once the R/S analysis and spectral density were carried out on logderived porosity data and corebased porosity, and permeability data, fractal properties were generated based on FFT(fast Fourier transform); next, the fractal structures were investigated. Finally, appropriate realizations of the reservoir permeability were generated.
The fractal theory
Methods of determination of fractal dimensions
Method  Summary method of fractal dimension (D) 

Boxcounting  \(D = \mathop {\lim }\limits_{r \to 0} \frac{\log N\left( r \right)}{{\log \left( {\frac{1}{r}} \right)}}\) N(r): number of boxes required to completely cover original shape r: reverse of mesh size for grid that is covering of shape 
Variogram  \(m = 4  2D\) m: slope of Logarithmic graph of variogram versus distances of separation 
Fractal modeling
Fractal modeling is an important class of simulations. One of the advantages of fractals is that the fractal dimension can be kept constant during the simulation and this resolves the smoothing shortcoming associated with kriging, as in fact the fractal dimension carries the data variation property. However, in fractal simulation, the minimized estimation variance is not guaranteed and more importantly, fractalbased approaches are actually simulators and not estimators; thus, with rerunning the fractal simulation, different realizations can be obtained in general (Rasouli and Tokhmechi 2010; Yu and Cheng 2002). The other advantage of fractals is that fractals are independent of scale, which allows one to utilize any data regardless of the measurement scale (Kim and Schechter 2009).
In petroleum engineering, especially where risks and cost are high, fractal models will take their place along with existing models as a powerful tool for the reservoir engineering because of old customary models and assuming that homogenous reservoirs cannot be exact. As an alternative method, a heterogeneous reservoir with a fractal structure can have a better accommodation for reservoir data.
The discovery that well logs often have a fractal character was made by Hewett (Hardy and Beier 1994). It is surprising that the logs analyzed by many authors in many different formations show similar to fGn; for example, Carne and Tubman analyzed three horizontal wells and four vertical wells in a carbonate formation. They found that all the logs behave as fGn (Crane and Tubman 1990). Moreover, Hardy found similar results by analyzing some wells in sandstone formations (Hardy 1993). The analysis of porosity logs of several Iranian reservoirs, which are also mostly carbonate formations, revealed that most of the logs can reasonably well be described by either fGn or fBm models (Sahimi 2000).
One should consider that not all logs are adapted to the fGn models, but one may apply fractal modeling like the fGn and fBm to evaluations of reservoirs properties (Hardy and Beier 1994). It should be noted the details of the application of fractal distributions can vary from one reservoir to another.
Fractional Brownian motion(fBm)
Brown motion is a random motion of a particle in an interval or distance in a random direction with a step length determined by a Gaussian (normal) probability distribution function (Yin 1996). Mandelbrot and Van Ness applied the term fractional Brownian motion (fBm) in 1968. fBm comprises a family of random functions described by an index H (0 < H<1) (Mandelbrot and Van Ness 1968; Luo et al. 2005; Dieker 2004; Sheng et al. 2012).
Consider a Gaussian random process B _{H}(r) and \(r = \left( {x,y,z} \right)\;{\text{and}}\;r_{0} = \left( {x_{0} ,y_{0} ,z_{0} } \right)\) are two arbitrary points in space and H is called the Hurst exponent.
 1.
The mean value is zero:
$$\left\langle {B_{\text{H}} \left( r \right)  B_{\text{H}} \left( {r_{0} } \right)} \right\rangle = 0$$(1)  2.
The variance value is:
$$\text{var} \left( {r  r_{0} } \right) = \left\langle {\left( {B_{\text{H}} \left( r \right)  B_{\text{H}} \left( {r_{0} } \right)} \right)^{2} } \right\rangle \cong \left( {r  r_{0} } \right)^{{2{\text{H}}}}$$(2)  3.
The correlation function will be:
$$C\left( r \right) = \frac{{\left\langle {  B_{\text{H}} \left( {  r} \right) \times B_{\text{H}} \left( r \right)} \right\rangle }}{{\left\langle {B_{\text{H}} \left( r \right)^{2} } \right\rangle }} \cong 2^{{2{\text{H}}  1}}$$(3)

H = 0.5 corresponding to an uncorrected trend and model in this condition is completely random (Dashtian et al. 2011; Ostvand 2014; Jafari et al. 2011).

H > 0.5 is characterized by persistence, i.e., the process shows a clear trend; in other words, a trend at x is likely to be followed by a similar trend at \(x + \Delta x\) (FonseaPinto et al. 2009; Gaci and Zaourar 2011).

H < 0.5 means that it shows an antipersistent behavior (FonseaPinto et al.2009; Delignieres 2015)
Fractional Gaussian noise(fGn)

A fGn model with H close to 1 is similar to a fBm model with H < 0.4.

The histogram of the fGn model is Gaussian; however, the histogram of the fBm is boxshaped.
Methods of generation of fractal structure
There are different methods for the generation of fractal data based on both fBm and fGn models (Babadagli 2005). In this work, fast Fourier transform (FFT) method was used to generate fractal data. The algorithm based on the FFT has lower computational efforts, less complexity and more suitable performance than other methods.
Based on the Fourier method, one should first generate random data between 0 and 1. The number of random data will be equal to number of the expected simulated data. Once the Fourier transform of random data must be calculated, this Fourier transform is multiplied by root of S(ω); finally, the data are returned to the original space by using a reverse Fourier transform, and thereby the resulting array containing data with fractal structures having an ideal size.
Methods of determination of fractal dimension
As summarizes in Table 1, the fractal dimension can be calculated by the following two methods.
Boxcounting
Variogram
Analysis methods of fractal data
There are different methods for the analysis of fractal data (Jafari et al. 2011; Kaya 2005), In this paper, both rescaled range (R/S) analysis and spectral density analysis were employed.
R/S analysis
The rescaled range (R/S) analysis is a measure of how a sequence varies as lag increases. In fact, it is a measure of the cumulative fluctuation of the sequence. The range is rescaled by dividing it by the standard deviation. In the R/S analysis, the rescaled range is plotted versus the lag on a log–log scale and the slope of the line fitted gives the Hurst exponent (H).The R/S algorithm is presented in Appendix (Hardy and Beier 1994; Kirichenko et al. 2011; Li et al. 2011; Arizabalo et al. 2004).
Spectral density
The spectral density is a measure based on the Fourier transform of the original data. The analysis of spectral density can provide insights into the ordering of the sequence of data (Harcarik et al. 2012).
Field data description
Range of log data in each of wells (A, B, and C)
Well  Depth 

A  3927–4173.8 
B  3943.9–4477.6 
C  4051.9–4489.9 
Statistical analysis
Statistical analysis of log data in each of wells (A, B, and C)
Statistical parameter  Well A  Well B  Well C  

Porosity \(\left( {\frac{\text{V}}{\text{V}}} \right)\)  Mean  0.0302  0.0521  0.0333 
Standard deviation  0.0425  0.0427  0.0354  
Irreducible water saturation \(\left( {\frac{\text{V}}{\text{V}}} \right)\)  Mean  0.787  0.538  0.818 
Standard deviation  0.289  0.310  0.227  
Rock density \(\left( {\frac{\text{gr}}{{{\text{cm}}^{3} }}} \right)\)  Mean  –  2.73  2.730 
Standard deviation  –  0.0257  0.032 
Statistical analysis of core data in well C
Statistical parameter  Porosity \(\left( {\frac{{\mathbf{V}}}{{\mathbf{V}}}} \right)\)  Permeability (mD)  Rock density \(\left( {\frac{{{\mathbf{gr}}}}{{{\mathbf{cm}}^{3} }}} \right)\) 

Mean  0.0446  1.764  2.720 
Standard deviation  0.0028  0.4893  0.0007 
Estimation of fractal dimension
Hurst exponents in each of wells (A, B, and C)
Well  Hurst exponent(H) 

A  0.4661 
B  0.4671 
C  0.4987 
Determination of fractal dimension in each of wells (A, B, and C) by log data
Well  Fahliyan thickness in the well (m)  \(\varvec{D}_{\varvec{F}}\) (by boxcounting method)  \(\varvec{D}_{\varvec{F}}\) (by variogram method)  \(\varvec{D}_{\varvec{F}}\) (by R/S analysis) 

A  246.8  1.69  1.829  1.9322 
B  530.6  1.753  1.956  1.9342 
C  438  1.918  1.904  1.997 
Estimation of β for fBm, fGn models by log data
Well  β (For fBm model)  β (For fGn model) 

A  −1.932  0.0678 
B  −1.934  0.0658 
C  −1.997  0.0028 
Estimation of H, β for core data
Distribution  H  β (For fBm model)  β (For fGn model) 

Core porosity  0.434  −1.868  0.133 
Core permeability  0.349  −1.698  0.302 
Determination of fractal dimension \((D_{F}\)) in well C for core data
Distribution  \(\varvec{D}_{\varvec{F}}\) (by variogram method)  \(\varvec{D}_{\varvec{F}}\) (by R/S analysis) 

Core porosity  1.844  1.868 
Core permeability  1.851  1.698 
Generation of fractal models for distribution of reservoir properties
Pseudopermeability log of Fahliyan reservoir
Statistical analysis of simulated permeability distribution in well C
Method  Mean  Standard deviation  Error of mean 

Fractal(fBm)  1.7645  0.4893  0.0025 
Fractal (fGn)  1.7506  0.4894  0.0075 
Conclusions

Fractal theory, a main branch of mathematics that can consider most of geometrics, can be used as a powerful method in petroleum engineering. The application of fractal modeling, because of the heterogeneous nature of reservoirs, has an appropriate adaption to real reservoirs. In this work, fractal modeling proved efficiently suitable for the characterization of reservoir properties;

In this study, Hurst exponents were calculated for wells A, B, and C and were about 0.5; thus, the porosity in these wells was approximately randomly distributed;

The fractal dimension was calculated by using three methods, namely boxcounting, variogram, and R/S analysis, and the results confirmed that the determination of the fractal dimension by variogram method, because of doing it by the GS + software package, could have higher accuracy and lower computational time than the other manual methods;

In the current work, an appropriate model for the distribution of permeability in Fahliyan reservoir was presented by using fBm and fGn data in nonsampled intervals. The fractal data used showed a suitable coordination with the core data in well C.
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