Mathematical Geology, Vol. 38, No. 1, January 2006 (
C 2006) DOI: 10.1007/s11004-005-9002-z

Effective Permeability Estimation for 2-D Fractal Permeability Fields1 Tayfun Babadagli2 Hurst exponents (H) of the distribution of permeability at micro (pore) scale were measured as close to 0.1 for sandstone and limestone samples. Based on these observations and previously reported H values for field scale permeability distribution ranging between 0.6 and 0.9, square permeability fields at different scales varying between 1 and 100 ft were generated for the H values of 0.1, 0.5, and 0.9. The study also considered different permeability fields and number of grids ranging from 10 to 500 md and from 8 × 8 to 64 × 64, respectively. The effective permeability of fractally distributed 2-D fields was calculated using different averaging techniques and compared to the actual (equivalent) permeability obtained through numerical simulation. The geometric mean and power averaging techniques as well as the perturbation theory yielded the most reasonable agreement between the actual and calculated effective permeabilities. The accuracy of these techniques increases with increasing average model permeability. It was also observed that as the H decreases, the permeability values obtained were higher than the actual values. Two extreme values of the number of grids (8 × 8 and 64 × 64) yielded the highest error percentages. Thus, the optimum number of grids was found to be 16 × 16 and 32 × 32 depending on the average permeability of the model. The exponent of the power law model was correlated to the fractal dimension of the permeability field for 8 × 8 and 64 × 64 grids. While a good correlation exists for 8 × 8 number of grids, no correlation was obtained for 64 × 64. Hence, an alternate model was proposed for 8 × 8 grids but for grid numbers higher than 32 × 32, no technique was found suitable for averaging of the fractal permeability fields. KEY WORDS: effective permeability; upscaling; averaging techniques; fractal distribution of permeability; grid size and dimension; Hurst exponent; fractal dimension; micro scale.

INTRODUCTION Permeability mapping is one of the most critical aspects of modeling subsurface reservoirs. In preparation of the permeability maps, core level permeability data need to be up-scaled to a grid scale to reduce the number of grids to an optimal number. This entails accurate and reliable averaging of permeability. 1 Received

7 November 2003; accepted 5 March 2005; Published online: 18 April 2006. of Civil and Environmental Engineering, School of Mining and Petroleum, University of Alberta, 3-112 Markin CNRL/Natural Resources Engineering Facility, Edmonton, Alberta, T6G 2W2, Canada; e-mail: [email protected].

2 Department

33 C 2006 International Association for Mathematical Geology 0882-8121/06/0100-0033/1


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To date, many different averaging techniques were proposed for permeability up-scaling. Permeability, unlike porosity and saturation, which are additives, cannot easily be averaged using these techniques (Richardson and others, 1987). Classical averaging techniques (static techniques), i.e., arithmetic, geometric, and harmonic were found suitable for flow parallel (arithmetic) and perpendicular (harmonic) to layering or purely random distribution of permeability. The arithmetic and harmonic averaging techniques are applicable for perfectly layered systems. More accurate ways of averaging permeability are to solve the flow equation with no-flow boundaries. This technique, known as dynamic averaging, requires extensive computing time. In terms of accuracy, it is more reliable than static techniques but the computing time and effort are still problems even though millions of cells can be handled in the dynamic technique (Li and others, 1999). Other static techniques such as power law averaging (McGill and others, 1993), modified version of power law averaging for log-normal medium (Abramovich and Indelman, 1995; Neuman, 1994), renormalization (Aharony and others, 1991; King, 1989), and perturbation theory (King, 1989) were proposed for permeability averaging. Effective medium theory can also be used to calculate the average permeability (McGill and others, 1993). Renard and de Marsily (1997) reviewed these techniques critically. As a relatively faster and reliable technique, power law averaging is preferable over the other techniques but it needs an empirical determination of the power averaging exponent. Permeability distributions in natural rocks are often observed to be correlated in space. It has been shown that the permeability distribution exhibits fractal characteristics at field scale and the quantification of heterogeneity through fractal dimension has been useful in modeling studies (Aasum and others, 1991; Babadagli, 1999; Beier and Hardy, 1993; Berta and other, 1994; Chang and Mohanty, 1994; Crane and Tubman, 1990; Emanual and others, 1989; Hewett, 1986; Hewett and Behrens, 1990; Hird and Dubrule, 1995; Kelkar and Shibli, 1994; Perez and Chopra, 1991 ). Studies at smaller scales are limited (Makse and others, 1996; Tidwell and Wilson, 1996). Therefore, experimental observations at micro scale were used to quantify the permeability distribution by measuring the Hurst exponent (H) of the distribution in this study. It was observed that the permeability at micro/core scale (mm cm) shows a fractal behavior with H values varying between 0.1 and 0.25 for sandstone and limestones. Pressure solving technique was applied to compute the average permeability of synthetically generated fractal permeability distributions (actual values) and tested against the existing averaging techniques (calculated values). A sensitivity study was performed for different model sizes, number of grids, average permeabilities and the Hurst exponent of the permeability field. For the grid sizes and average permeability cases that yield more than 5% difference between the actual and calculated values, a correlation between the H of the distribution and power law exponent was sought.

Effective Permeability Estimation for 2-D Fractal Permeability Fields

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FRACTAL DISTRIBUTION OF PERMEABILITY The Hurst exponent (H) of permeability distribution was computed after mapping the permeability field at micro/core scale. For mapping, permeability was measured using a probe permeameter 1 cm apart. Then, different techniques were applied to measure the H of the distribution. Experimental Observation (Micro-Pore Scale) Permeability field maps were constructed for two different rock samples, namely a sandstone and a limestone. Permeability was measured at every centimeter using a probe permeameter to obtain 1024 data points in a square shaped area. 1-D measurements on sandstones were also performed for 32 data points that are 1 cm apart. After measuring the permeability to air, it was corrected to convert into liquid permeability. Average liquid permeabilities of the sandstone and limestone are 500 mD and 7 mD, respectively. Figures 1(a) and (b) show the distribution of permeability. A similar measurement technique was applied in several studies (Hurst and Rosvoll, 1991; Jacquin and others, 1991; Jensen and Corbett, 1993; Makse and others, 1996; Manrique and others, 1996; Noetinger and Jacquin, 1991; Tidwell and Wilson, 1997). These studies carried out the measurements from micro to core scales and provided statistical evaluations of the permeability field. Only Makse and others (1996) tested the long range correlations in permeability fluctuations for sandstone samples with much higher permeabilities than used in this study. The Hurst exponent of the permeability distribution was calculated using different techniques as explained in the next section. Measurement of Hurst Exponent The techniques previously applied and evaluated for the measurement of the Hurst exponent (H) of the joint roughness were applied for the data shown in Figures 1(a) and (b) (Babadagli and Develi, 2001). The first technique applied to measure the H of the permeability distribution is the variogram analysis. The variogram is defined as the mean squared increment of points: 1
[V (xi ) − V (xi+h )]2 γ (h) = 2n(h) i=1 n(h)

(1)

where γ (h) is the semi-variogram and n is the number of pairs at a lag distance, h and V(xi ) and V(xi+h ) are the sample values at location xi and xi+h . If the variogram γ (h) is plotted against lag distance, h on a log–log paper, the slope yields the Hurst exponent, H according to the following relationship: γ (h) = γo h2H

(2)

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Figure 1. (a) Representation of one of the permeability distribution realizations with different Hurst exponent, H (light areas represent the higher permeabilities) and an experimental observation for a sandstone sample. (b) Representation of one of the permeability distribution realizations with different Hurst exponent, H (light areas represent the higher permeabilities) and an experimental observation on a limestone sample.

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The H values were obtained for different lag distances, namely 4, 5, and 6, then averaged. The H of 1-D permeability distribution of two different sandstones measured throughout a core at every cm for 32 points was found to be 0.14 and 0.25. The rescaled range (R/S) technique (Feder, 1988 ) was also applied and the H values were found to be 0.3 for both data sets. The variogram analysis was also applied repeating the same procedure to measure the H values of each of 32 profiles. The average of 32 H values was assumed to be the H in this direction (vertical). For the sandstone sample the H values were found to be 0.274 and 0.166 using the data points in vertical and horizontal directions, respectively. Since the R/S technique is not applicable to 2-D sets, the power spectral density analysis was applied (see Babadagli and Develi (2001) for details of computation) to measure the H of the permeability distribution for comparison. The H was found to be 0.205, which is very close to the ones obtained from the variogram analysis. The same technique was applied for the limestone data (Fig. 1(b)). The vertical and horizontal H values were found to be 0.124 and 0.108, respectively for the limestone sample. Having H < 0.5 implies that the correlation at this scale is antipersistent for these particular samples.

Field Scale (Mega to Giga Scale) At larger scales, the Hurst exponent (H) of permeability distribution has a greater value. In his pioneering study, Hewett (1986) showed that 1-D permeability distribution has an H value of 0.87. Later, Hewett and Behrens (1990) reported that the H varies between 0.7 and 0.9 for topographic features of the Earth’s surfaces. Emanuel and others (1989) measured the H of well log values, that could be representative of permeability distribution, and found that the H varies between 0.9 and 0.6. For a layered sandstone, they noted that this value is around 0.85. In the succeeding studies, Berta and others (1994), Perez and Chopra (1991), Crane and Tubman (1990), Aasum and others (1991), Beier and Hardy (1993), Kelkar and Shibli (1994), Tubman and Crane (1995), and Babadagli (1999) measured the H of vertical permeability distributions varying between 0.9 and 0.6 for different formations. These values are significantly lower than the values reported in the previous section at smaller scales implying that (H > 0.5) a correlation exists at this scale. Hence, one can conclude that the permeability distribution possesses greater H values at macro-giga scales than at micro scale. Typically, rock properties at pore scale posses H values lower than 0.5, (Katz and Thompson, 1995; Krohn, 1988; Muller and McCauley, 1992), which is consistent with our observations. This could be true for permeability distribution but not for the distribution of other properties, i.e., pore sizes or fracture properties. For example, Babadagli (2001) reported the opposite behavior for the H values of fracture networks measured at different scales from micro to giga, i.e., the higher the scale, the lower the H.

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Based on these observations, it can be generalized that the micro/macro scale Hurst exponents of different rock properties take values lower than 0.5 while at mega/giga-scales it is greater than 0.5. This is a critical issue and the averaging techniques should be tested not only for different H values but also the scale that is represented by the model size. In other words, in averaging permeability field, the same average permeability and type of the permeability field represented by the same H are not the only critical parameters and the effect of the total length of the model and the number of grid points should be tested. GENERATION OF FRACTAL PERMEABILITY FIELDS It was previously suggested that the permeability distribution in the x-y direction (horizontal) represents a fractional Brownian motion (fBm) (Berta and others, 1994; Hewett, 1986; Hewett and Behrens, 1990; Perez and Chopra, 1991; Tubman and Crane, 1995). This might be fractional Gaussian noise (Berta and others, 1994; Hewett, 1986; Hewett and Behrens, 1990; Kelkar and Shibli, 1994; Perez and Chopra, 1991; Tubman and Crane, 1995) or fBm (Perez and Chopra, 1991) in the x-z direction (vertical). Fractal permeability fields were generated for different Hurst exponents (H) to be used in permeability averaging. In this generation process fBm concept was used. Several methods have been proposed to generate fBm traces (Feder, 1988). Here, midpoint displacement and successive random addition algorithms introduced by Voss (1985) were selected to obtain 2-D self-affine distributions. An fBm is a generalization of the random function. Consider a stationary stochastic process BH (x) with the following mean: BH (x) − BH (xo ) = 0

(3)

and a variance of increments is given by [BH (x) − BH (0)]2  = |x|2H σo2

(4)

where x and xo are the arbitrary points in space and H is called the Hurst exponent or intermittency coefficient. An independent Gaussian variable with zero mean and unit variance is assigned at the central point of the lattice in the first generation. In the next generation the elevations at the four corner points are interpolated and at each generation the variance, σ , is reduced as σn2

 2 Hn 1 2 = σn−1 2

(5)

This process generates 2-D self-affine distributions. The details of the algorithm can be found in Saupe (1988). Using this method, four different sets of 2-D

Effective Permeability Estimation for 2-D Fractal Permeability Fields

39

synthetic surfaces with known H values using four different random number seeds were generated. After generating the fBm distributions, they were converted to permeability fields. To be able to compare the cases generated using different random number seeds, the same mean value of permeability and uniform variance were applied in each generation process as suggested by Hewett (1986) and applied by Hewett and Behrens (1990). This procedure was repeated for four different average permeability values: 10, 50, 200, and 500 mD. For each realization of the permeabilities generated using different random number seeds, the same arithmetic average was used and the permeability was normalized to be distributed between the same minimum and maximum. For example, the minimum permeability was set to 1 and the maximum to 25 md for the average permeability of the 10 md case. The same normalization process was followed for different random permeability fields generated using different random number seeds. Permeability distributions were generated by Hewett (1986), Chang and Mohanty (1994), and Babadagli (1999) after a similar normalization procedure. Figures 1(a) and (b) show different realizations of synthetically generated permeability distributions with different H values. As the H decreases, the distribution becomes rougher and it approaches random noise-type distribution. In order to indicate this, 3-D views of the distribution were also added for the sandstone case (Fig. 1(a)). When compared to the synthetic permeability fields, one can observe that the roughness of the experimentally measured distribution resembles to the distribution with the corresponding H (close to 0.1 for both sandstone and limestone). EFFECTIVE PERMEABILITY CALCULATION Lab scale observations in this study and field scale measurements published in the literature showed that the permeability distributions in micro/macro scale and mega/giga scale exhibit fractal behavior. However, as the scale increases, the Hurst exponent (H) increases. Therefore, a wide range of H values (between 0.1 and 0.9) was considered as well as the size (scale) of the model (between 1 and 100 ft). The average permeabilities were also selected as between 10 and 500 mD representing the limestone and sandstone samples used in the measurements, respectively. To fill the gap between these two values of average permeabilities, models with average permeabilities of 50 and 200 mD were also used and numerically calculated average permeabilities were compared to the ones obtained from the existing techniques. Description of the Methodology and the Numerical Model For the numerical estimation of the effective permeability, a commercial black-oil simulator (ECLIPSE, 1996) was utilized. The model parameters are

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Figure 2. Fluid and reservoir properties used in the estimation of average permeability by numerical simulator (pressure averaging).

given in Figure 2. The simulations were carried out using a constant rate for a homogeneous system assigning the same permeability values to each grid block for four permeabilities (20, 50, 100, and 500 mD) to test and verify the model. At constant injection rate, the pressure drop across the two ends of the model was used and the permeability was calculated using the following form of Darcy’s equation: kaverage = c

qµh (P2 − P1 )LW

(6)

where k is the grid block permeability; µ is the fluid viscosity; P1 and P2 are the upstream and downstream pressures, respectively; q is the constant injection rate; L, W, and h are the length (x-direction), width (y-direction), and thickness (z-direction), respectively; and c is the proportionality constant for unit conversion. After testing the model for the homogeneous cases (the same permeability for each grid), typical injection rates yielding a stabilized pressure drop were determined for each permeability level. The permeability obtained from the numerical run was compared to the original permeability to verify the model.

Effective Permeability Estimation for 2-D Fractal Permeability Fields

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The fractally distributed permeability values for three different H values (0.1, 0.5, and 0.9) were entered as data to the simulator. The runs were performed for different number of grids, average permeabilities, distributions generated using different random number seeds, and model sizes.

Application of Existing Permeability Averaging Techniques The average permeabilities obtained from the numerical runs from Eq. (6) were compared to the ones obtained through the averaging techniques listed in Table 1. In addition to the arithmetic, geometric, and harmonic averaging techniques, perturbation, power average, and normalization techniques were applied. As shown in Table 1, modified versions of the perturbation theory, i.e., PERT.2B (Dagan, 1993; Matheron, 1968; ) and PERT.1B (King, 1989) were also tested. In all these equations, D denotes the Euclidean dimension. When D = 2 in PERT.2B and POW.AV.2B, the averaging technique turns out to be the geometric mean. In lieu of the Euclidean dimension, (D = 3−H) value was used as well (PERT.1A, PERT.2A, PERT.3A, POW.AV.1A, 1B, 2A, and 2B). This value can be considered as a local fractal dimension of the permeability field. As per the power averaging technique, the classical definition (POW.AV.1A) was tested first. In this equation the fractal dimension of the medium was used for p. Noetinger (1994) suggested p = 1–2/D, D being the Euclidean dimension of the medium. This was tested as well using the fractal dimension for the D (3−H) in the equation (POW.AV.2A). Another approximation of power averaging was suggested by Ababou and others (1989) (POW.AV.1B). For p in this equation both the fractal dimension (POW.AV.1B) and p = 1−2/D, as D being the fractal dimension (3−H) of the medium (POW.AV.2B), were used. Comparisons were made for different number of grids, model sizes, H of the distribution, and average model permeabilities (see Fig. 2 for the different values of these). The results are given in Figures 3 and 4 for different random number seeds that yield different realizations of the distribution for the same H value. It was observed that the geometric mean, perturbation, and power average techniques, except POW.AV.1A, showed a reasonable agreement between the calculated and actual average permeabilities for all number of grids (Fig. 3). Note that a deviation less than 5% was accepted as a reasonable agreement. As a result, one can observe that, as the average permeability increases, a better agreement between the calculated and actual average permeability values is obtained for the 8×8 grid case. The same can be said for increasing grid size. Hence, the best agreement between the actual and calculated permeability values was obtained when L = 100 ft (Figs. 3(a)) and k = 500 md (Fig. 3(c)). If L = 1 and k = 10 or 50 md (Fig. 3(b)) the deviation from the actual effective permeability is greater than 5% for the H values smaller than 0.5.

PERT.3B

POW. AV. 1A

Perturbation-3B

Power Average-1A

KING NORM

PERT.3A

Perturbation-3A

King’s Normalization

PERT.2B

Perturbation-2B

POW. AV. 2B

PERT.2A

Perturbation-2A

Power Average-2B

PERT.1B

Perturbation-1B

POW. AV. 1B

PERT.1A

Perturbation-1A

POW. AV. 2A

HARM.

Harmonic Average

Power Average-2A

GEOM.

Geometric Average

Power Average-1B

ARIT.

Abbreviation

Arithmetic Average

Method

−

1 2 4 δln k ] D

2 + δln k 1 2 2

1 −

1 2 4 δln k ] D

4(k1 + k3 )(k2 + k4 )[k2 k4 (k1 + k3 ) + k1 k3 (k2 + k4 )] k¯ = [k2 k4 (k1 + k3 ) + k1 k3 (k2 + k4 )][k1 + k2 + k3 + k4 ] +3(k1 + k2 )(k3 + k4 )(k1 + k3 )(k2 + k4 )

k¯ = k p 1/p  pδ 2 ln k k¯ = kgeom exp 2 ¯k = k p 1/p  pδ 2 ln k k¯ = kgeom exp 2



2

1 D

1

  k¯ = kgeom 1 + 12 −

1 2

2 + δln k



1 D

  k¯ = kgeom 1 + 12 −



1 D

k1 k3

k2 k4

p = 1–(2/D); D = Fractal Dimension (D = 3–H)

p = 1–(2/D); D = Fractal Dimension (D = 3–H)

p = D, D = Fractal Dimension (D = 3–H)

2 = Variance in logarithm of perm. D = Fractal δln k Dimension (D = 3–H) 2 = Variance in logarithm of perm. D = 2 δln k (Euclidean Dimension) 2 = Variance in logarithm of perm. D = Fractal δln k Dimension (D = 3–H) 2 = Variance in logarithm of perm. D = 2 δln k (Euclidean Dimension) p = D, D = Fractal Dimension (D = 3–H)

 2 1 k¯ = kgeom exp δln k 2 −



1 D

p = −1

lim k¯

p→0

p=1

δk2 = Perm. variance, D = 2 (Euclidean Dim.)

p

ki

 p1

p

ki

 p1

 p1

2 ) k¯ = karit (1 − δk2 /D k¯ arit  2 1 k¯ = kgeom exp δln k 2 −

i=1

N 

i=1

N 

p

ki

δk2 = Perm. variance, D = Fractal Dim. (D = 3–H)

1 N

1 N

i=1

N 

Definitions

2 ) k¯ = karit (1 − δk2 /D k¯ arit

k¯ harm =



1 N





k¯ geom =

k¯ arit =

Equation

Table 1. Permeability Averaging Techniques Tested on Fractal Permeability Fields

42 Babadagli

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Figure 3. Comparison of the numerically measured (actual) effective permeabilities to the calculated ones using different methods for permeability fields with different fractal dimension (random number seed A).

In general, the low H value cases did not show a good agreement with the actual values for the low permeability-small model sizes (8 × 8 grids). In other words, the biggest deviation from the actual values was obtained for the lowest H value, which was 0.1. Also, as the H value of the distribution decreases, the

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Figure 4. Comparison of the numerically measured (actual) effective permeabilities to the calculated ones using different methods for permeability fields with different fractal dimension for different random number seeds (seeds B and C).

calculated values of the average permeability increase. This is an important issue to be taken into account in upscaling the fractal permeability fields because the H values of permeability distribution for this scale (model size, L = 1 ft) were measured around 0.1 (see Fig. 1). Also, note that, the highest average permeability case presented a different trend, i.e., the low H case showed a lower ratio of calculated to actual permeability (Fig. 3(c)) for 8 × 8 grids and L = 1. This is the only case breaking the general rule; the lower the H, the higher the calculated effective permeabilities. However, the deviation is very small being within the acceptable limits (less than 5%). As the number of grid points increases, the deviation from the actual values becomes less. For 16 × 16 and 32 × 32 grids, a good agreement was observed for the geometric mean, perturbation, and power average techniques, except POW.AV.1A (Figs. 3(d), (e), and (f)). For the 64 × 64 grids, as the size of the model and the average permeability increase, the deviation from the actual values increases (Figs. 3(g) and (h)). Replacement of the Euclidean dimension, D, by the fractal dimension (D = 3−H) of the system yielded a better result for 8 × 8 and 16 × 16 number of grids and high permeability (500 md) range for all H values, especially for PERT.1A and POW.AV.1B. In conclusion, as the average permeability of the model decreases, it is recommended to use 16 × 16 and 32 × 32 grids for smaller models sizes (L = 1

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and 10 ft). 16 × 16 grids is recommendable for low permeability media (10 md). A higher number of grid points is more desirable for the higher permeability cases. For the higher average permeability cases (k > 10 md), 32 × 32 grid points are recommendable for small model sizes (L = 1 ft). One can select the optimal values of the number of grid points and the method depending on the size of the model and permeability range of the medium to start the up-scaling. For higher number of grid points (64 × 64), high permeability, and larger models sizes, overestimated values of permeability were calculated. It is obvious that for this number of grid models, one should pay attention to the value of the average permeability as the higher permeabilities yield a deviation from the actual effective permeability. Similar observations were made for the same cases when different random number seeds were used in the generation of the permeability fields. Two examples are given in Figs. 4(a) and (b) for comparison of the two identical cases possessing the same characteristics but different random numbers seed. At this point, one has to refer to two previous publications, which reported a similar analysis. McGill and others (1993) tested different methods and observed that the most consistent technique was the geometric mean and effective medium theory for log-normally distributed permeability fields. They observed similar deviations for renormalization (King’s normalization in this study). In the present study, we obtained the best agreement for the highest H value (0.9) for the renormalization technique. Harmonic average and POW.AV.2B methods yielded the worst agreement. This is also in accordance with the McGill and others’ (1993) observations. Mukhopadhyay and Sahimi (2000) reported another case where they used the fractal permeability distribution as similar to this study. They observed that the renormalization and effective medium approximation yielded a good agreement with the actual average permeability values. When the field contains very low permeability zones that can be interpreted as low H value fields, the predictions are not very accurate. This is consistent with the observations of this study as well.

Use of Fractal Dimension in Effective Permeability Estimation The power law technique is not easily applicable, as it requires a correct determination of the power exponent. Technique abbreviated as POW.AV.2B in Table 1 yielded a good agreement between the actual and calculated values when the-local-fractal dimension (D = 3−H) is used for D in p = 1−(2/D). An alternative technique could be to seek a correlation between the fractal dimension of 2-D permeability distribution and the power exponent (p) in k¯ = k p 1/p yielding the best agreement with the actual permeability value. We observed through the above analysis that the lowest and highest number of grid points (8 × 8 and 64 × 64), specifically for smaller models of 8 × 8 grid points (L = 1) and low permeability

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Figure 5. Relationship between the exponent (p) in power averaging and the inverse of the fractal dimension of permeability distribution (D = 3 – H) for different average model permeabilities and three different random number seeds used to generate the permeability fields.

(k = 10 md) along with the larger model sizes and higher permeabilities (200 md and above) of 64 × 64 grid points (L = 10 and 100), yielded the worst agreement. Therefore, a correlation between the fractal dimension of the permeability field and the exponent of the power averaging equation (k¯ = k p 1/p ) was sought for these cases. Figure 5 shows four cases of 8 × 8 grids. Three cases in each figure represent the three different random number seeds. It was observed that there is a correlation between (1/D) and the exponent (p). The slope decreases with increasing model size for the low permeability case. For higher permeability case no significant change on the slope was obtained. The relationship is almost a straight line with zero slope. For more detailed analysis, the same plots were done for 8 × 8 and 64 × 64 grid points for only one random number seed but for three different average model permeabilities (Fig. 6). One can observe that the slope decreases with increasing average model permeability for 8 × 8 grid points. For the L = 10 ft case, the change in the slope with increasing permeability is higher than that in the case of L = 1 ft and would eventually approach a zero slope straight line as suggested by Figure 6. This indicates that the average permeability, model size, and the number of grids are critical in up-scaling the fractal permeability fields. For the 64 × 64 case, no correlation was observed and the slope values for especially higher permeability case (k = 200 md) did not follow a trend, unlike the previous 8 × 8 case.

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Figure 6. Relationship between the exponent (p) in power averaging and the inverse of the fractal dimension of permeability distribution (D = 3 – H) for different average model permeabilities but the same random number seed used to generate the permeability fields.

CONCLUSIONS The effective permeability of the fractal permeability fields was calculated and compared to the actual (equivalent) values obtained from the numerical simulations. Overall, the perturbation theory (PERT. 2A, 2B, 3A, and 3B in Table 1) yielded the best results for all Hurst exponent (H) values along with the geometric mean and power averaging (POW. 2B in Table 1). Lower number of grid points (8 × 8) with low average permeability (10 md) at small scales (L = 1 ft) yielded a remarkable deviation form the actual values when the H is smaller than 0.5. The H of the permeability field was measured for both a sandstone and limestone as being around 0.1 at this scale. Therefore, one has to pay attention to the scaling techniques at this scale. Deviations were observed for a higher number of grids 64 × 64, especially for high average model permeabilities (>200 md). 16 × 16 and 32 × 32 grid points were found to be optimal model size for any average permeability for especially lower H values. An alternative approach correlating the exponent of the power averaging and the fractal dimension of the permeability field was presented. A correlation was observed for 8 × 8 grids, which was one of the problematic cases in the comparison analysis, whereas no correlation was found for 64 × 64 grids. This showed that replacing the power exponent in power averaging techniques with a correlation to the H value could be an alternative for 8 × 8 grids, especially at smaller scales.

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