construction of fracture network model using static and...

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Abstract This study reports a new process of constructing a discrete fracture network (DFN) model, which reflects various statistical properties of fractures extracted from static data such as borehole images and dynamic data such as pressure derivative curves.

A DFN model is constructed by generating a number of disc-shaped fractures based on their statistical properties. The process of constructing DFN model proposed consists of two streams. One stream is for the fractures with wide apertures, which work as main flow path of fluid. Another stream is for the fractures with narrow apertures, which work as main storage of fluid. In the each stream, a DFN model is constructed by applying fractal theory and geostatistics that define the size and spatial distribution of fractures, respectively. The overall DFN model is obtained by merging the two types of DFN models. Then, the DFN model is converted to a continuum model with equivalent permeable blocks based on cubic law and is evaluated by the fluid- flow behavior.

Japex has been operating the Yufutsu fractured basement gas reservoir since 1996. A total of approximately 8,000meters of electrical borehole images from eleven wells is measured to understand the fracture system. According to the proposed process, the DFN models are constructed. To evaluate the DFN models by comparing the simulated pressure derivative curves with the observed curve, flow simulations are carried out as a simulation of a single porosity type. The observed pressure derivative curve shows a concave shaped curve like a dual porosity behavior, but the curve never reaches the second flat state. Through the parameter studies of the statistical properties, which relate to the size distribution and the spatial distribution of fractures, the observed curve was successfully

reproduced and the quantitative relationship between the fractures with wide apertures and the fractures with narrow apertures was found. Introduction JAPEX has been operating the Yufutsu fractured basement gas reservoir since 1996, which is located in central Hokkaido, northern Japan. The reservoir extends over a depth range of 4,000 to 5,000 meters. Totally 14 wells have been drilled in this reservoir up to now. The measured pore pressure at each well follows the same pressure trend so that each well is inferred to be drilled in an identical pressure system. However, some of wells are almost none productive in spite of the existence of gas showing and fractures. To study the behavior of the reservoir and to optimize the operation and development of the Yufutsu field, a reliable fluid flow model is anticipated.

Heterogeneity of fractured rock masses makes it difficult to construct a mathematical model to represent fluid flow. The recent studies focus on discrete fracture network (DFN) models constructed by generating a number of disc-shaped fractures based on their statistical properties.2, 3, 6, 8, 11 The advantage of the DFN model is to accommodate more realistic fracture network geometries and characteristics than conventional dual porosity models. It is available to put various kind of information of fractures observed in fields into the DFN model with respect to their statistical properties.

One of the important issues in constructing a reliable DFN model is what kinds of data should be used for the construction of DFN models. In the case of the Yufutsu field, seismic data is not suitable for the quantitative analysis of the fractures due to its low quality influenced by the overlaid thick volcanic rocks. The static data such as borehole images are used for detecting each fracture in the reservoir to extract statistical properties of fractures. The dynamic data such as transient single well tests are used for evaluating the fluid flow behavior of DFN models constructed by the extracted statistical properties. Another issue is what process for constructing DFN models should be adapted to reflect the features of their statistical properties.

The purposes of this study are to propose a process for constructing DFN models using electrical borehole images of multiple wells, to incorporate the variation of productivity of each well into the DFN models, and to study parameters of statistical properties of DFN models by comparing with the observed fluid flow behavior.

SPE 77741

Construction of Fracture Network Model Using Static and Dynamic Data T. Tamagawa, SPE, T. Matsuura, SPE, T. Anraku, SPE, K. Tezuka and T, Namikawa, JAPEX

2 T. TAMAGAWA, T. MATSUURA, T. ANRAKU, K. TEZUKA, AND T. NAMIKAWA SPE 77741

In this paper, we first describe the result of interpretation of electrical borehole images focusing on the relationship between well productivity and apertures of fractures. Secondly, a process of constructing DFN models incorporating the result of the interpretation is proposed. Finally, we discuss parameters of the extracted statistical properties by comparing with the observed fluid-flow behavior. Interpretation of electrical borehole images Fig. 1 shows the geological structure of the Yufutsu fractured basement gas reservoir. This reservoir extends over a depth range of 4,000 to 5,000 meters and is comprised of Cretaceous granite overlain by an alluvial / fluvial conglomerate of Eocene age. The general structure of the reservoir is that of a north-south trending horst complex with several north-south or northeast-southwest trending normal faults. Gas is accumulated to the fractures formed in the horst blocks.

A total of approximately 8,000 meters of electrical borehole images from 11 wells was investigated to understand a relationship between the productivity of well and the characteristics of fractures. Each fracture is picked on electrical borehole images and its dip, dip azimuth and depth are recorded in the software. Apertures of fractures are also estimated by using a technique proposed by Luthi and Souhaite5 based on the resistivity contrast between borehole mud and the formation under the assumption that a shape of a fracture is planar and parallel plates. A total number of picked fractures is about ten thousand. The results of the interpretation are as follows:

1) The productivities of the wells are correlated with the existence of the fractures with wide apertures. Fig. 2 shows typical electrical borehole images of a productive well and none productive well. The productive well has a fracture with wide aperture, which looks to have the width of several pixels on dynamically normalized image. On the other hand, none productive well has a lot of fractures but their apertures are very narrow. Fig. 3 shows the aperture distribution calculated from the resistivity contrast with respect to depth. Square symbols denote the fractures with apparent wide apertures identified on dynamically normalized images. The fracture density of a productive well (Well-A) is the same as that of none productive well (Well-B), but the averaged apertures of fractures in Well-A are obviously wider than those in Well-B. Well-C meets fractures with rather wider apertures than those in Well B and penetrates fractures which have around 0.1 millimeter apertures. However, Well-C is also none productive well. Therefore, in a lot of fractures observed in Well-A, a small number of fractures with wide apertures is inferred to control the productivity of well. We interpreted that the value of an aperture to contribute to a productivity of a well is 0.2 millimeter. 2) The orientation of the fractures with wide apertures is concentrated on high dip angle. Fig. 4 shows the poles of fractures plotted on an upper hemisphere. Square symbols denote fractures with wide apertures and dot symbols

denote fractures with narrow apertures. Cross symbols show resistive fractures filled with minerals. We can see that the orientations of fractures with wide apertures are concentrated on high dip and north-south or northeast-southwest dip azimuth. 3) The fractures with wide apertures tend to make clusters. Fig. 5 shows the observed depth of fractures along wellbore. The distributions of the fractures with wide apertures are concentrated on certain zones, while the fractures with narrow apertures are uniformly distributed.

From the results of the interpretation mentioned above, the

fracture network system of the Yufutsu reservoir is inferred that a small number of fractures with wide apertures contribute mainly to flow and a huge number of fractures with narrow apertures contribute mainly to storage. Moreover, the fractures with wide apertures tend to make clusters. This means that the density of main flow path should vary in space, namely heterogeneity. Construction of DFN models using static data Process of constructing DFN model. The process of constructing DFN model proposed by Watanabe and Takahashi11and Tezuka and Watanabe8 is modified to reflect the result of interpretation of electrical borehole images. The basic assumptions are as follows:

1) A shape of a fracture is a disc.8 The disc is defined by its thickness (aperture), radius and orientation. 2) A radius of fracture is proportional to its aperture.8,9 3) An aperture distribution is defined by a single fractal dimension in the scale of the whole reservoir. To incorporate the variation of productivity at each well

into a DFN model, we introduce two processes as follows: 1) The final DFN model is constructed by merging the two types of DFN models. One is a DFN model for the fractures with wide apertures and the other model is a DFN model for the fractures with narrow apertures. 2) The spatial distribution of fractures is determined by geostatistics. A similar approach was tried by Long and Billaux.4 They assume that small subregions of the area of interest have homogeneous statistical properties and that a simulation of the spatial variation of the statistical properties defined by each subregion can be made with geostatistics. In the case of the Yufutsu reservoir, owing to only data from the sparse and irregular locations of wells, we try to divide into a lot of quite small subregions and determine the existences of the centers of discs in each subregion by using Sequential Indicator Simulation (SISIM).1

Fig. 6 shows the process of constructing DFN model. First

step is to calculate apertures of fractures from resistivity contrast. Then the fracture size distribution is defined by using a proportional relationship. Next, the flow separates into two

SPE 77741 CONSTRUCTION OF FRACTURE NETWORK MODEL USING STATIC AND DYNAMIC DATA 3

streams. The left stream is for the fractures with wide apertures (Family-1). The right stream is for the fractures with narrow apertures (Family-2). The DFN model of each family is constructed independently. Then, the two streams come together and the two DFN models are merged to produce the final DFN model. Lastly, The final DFN model is converted to a continuum model with fine grids, which have equivalent permeability define by cubic law.11 Then, the fine gird model is upscaled to the coarse grid model for flow simulation. Fracture size. We define a radius of fracture from its aperture using the following equation:

where a is an aperture, r is a radius of disc, α is a conversion parameter. The conversion parameter depends on the field. In this study, our initial estimation of α is 2.0E-6. This value will be investigated by using dynamic data hereafter. Fracture Density. We first derive the cumulative frequency of the fracture apertures. To correct the crossing probability caused by an inclination of a fracture, the following equation is used according to Terzaghi style7:

where NBa is a frequency of fracture crossing the unit length of well whose aperture is wider than a, n is a number of observed fractures, L is total length of borehole, wi is a weighting factor for a crossing probability caused by an inclination of a fracture, Bhdev is a borehole deviation, Fdip is a fracture dip, BHaz is a borehole azimuth, Faz is a fracture azimuth.

According to the crossplot between the fracture aperture and the corrected cumulative frequency, the relationship is expressed by the following equation (Fig. 7):

where C is a constant and m is an exponential constant (fractal dimension). From Fig. 7, C and m are estimated as follows:

C：0.067 m:1.1506

By substituting Eq. (1) into Eq. (3), the frequency of fractures crossing the unit length of well, whose radius are larger than r, is given by:

Then, considering the crossing probability of the fracture size, the frequency relationship of fractures along the borehole can be expanded to 3D space as follows:

The number of generated fractures follows Equation (5). The boundary of the aperture between the narrow fractures and the wide fractures is set to 0.2 millimeter. This value corresponds to a radius of 100 meter (Fig. 7). Fracture orientation. Fig. 8 shows the orientation distribution of fracture in each family after correcting the crossing probability. Dark tone denotes a concentration of an orientation. We can see that the orientations of the fractures in Family-1 are more concentrated on the area of high dip than those in Family-2. Fracture spatial function. It is necessary to estimate a variogram, which is used for SISIM to predict the locations of the fractures. We first divide into windows of 5m intervals along borehole to determine the existence of fracture. Zero is assigned if there is no fracture and one is assigned if there is more than one fracture. This process is applied to eleven wells for each fracture family (wide/narrow) independently. Then, the experimental variogram of each family is calculated from the data and is approximated by a spherical variogram as an analytical function. Fig. 9 shows the results of the experimental variogram and the approximated spatial functions. We can see that the variogram of Family-1 shows the lower value of nugget and shorter range of influence than those of Family-2. It means that Family 1 fractures are more clustered than Family-2 fractures.

In this estimation, the variograms have large uncertainties because these locations of fractures are evaluated mainly from one dimensional borehole data. Horizontal data from sparsely located wells are more limited than vertical data. Especially, the range of Family-1 variogram decides the size of fracture network for main flow and strongly affects the fluid flow behavior in the reservoir. This parameter will be investigated by using dynamic data hereafter. DFN model. We adopt SISIM conditioned by well data so as to reflect the productivities of wells on a DFN model. Fig. 10 shows the centers of fractures predicted by SISIM. Dots denote the centers of fractures. We can see that the centers of fractures tend to make clusters in Family-1, while the centers of fractures in Family-2 are uniformly distributed. According to these spatial distributions, we generate discs following the statistics of the size and orientation distribution shown in Fig. 7 and Fig. 8, respectively. Fig. 11 shows the result of DFN model of each family. The generated number and the generated radius of discs in the families are as follows: Generated Number, Generated Radius Family-1 : 32891, 500 - 100 meter Family-2 : 1436443, 100 - 30 meter

The final fracture network model is constructed by merging the two DFN models. And the DFN model is converted to a continuum model with equivalent permeable fine grids. The grid size is 12.5m by 25.0m by 5.0m and the total number of fine grids is about one hundred and fifty million. Fig. 12 shows the cross section of the permeability distribution along a line including both the productive and none productive wells. Red and blue parts denote higher and

ra ⋅= α

)180cos()sin()sin()cos()cos(

−−=⋅=⋅=×+=

= ∑=

BHazFazCFdipBHdevBFdipBHdevA

maCNBa −⋅=

mm rCNB −− ⋅⋅= α

2−−

⋅+⋅⋅⋅

lower permeable zones, respectively. Two important features are clearly seen in the figure. First, the fractures with wide apertures, indicated by high permeability, are making clusters. Second, this DFN model reflects the variation of productivities of wells. In fact, Well-A, which is a highly productive well, is penetrating high permeable zone and Well-B, which is no productive well, dose not meet high permeable fractures. Parameters studies using dynamic data Parameters and flow simulation. The final step is to study the parameters of the extracted statistical properties by comparing the simulated pressure derivative curve of the build-up test with the observed data. We focus on two parameters to evaluate the effect of the size distribution and the spatial distribution of fractures. One parameter is a conversion parameter α shown in Eq. 1. Another parameter is a range of Family-1 variogram RV. Both parameters have large uncertainties because the statistics are derived from one dimensional borehole data.α relates a fracture aperture to its size. The three different values of α are selected as follows: Conversion Parameter α：2.5E-6, 2.0E-6, 1.5E-6

Theαvalue of 2.0E-6 is our initial estimation. The small value of α leads a larger size of a fracture from a given aperture. RV defines an extent of a cluster formed by Family-1 fractures. The three different values of RV are selected as follows: Range of Family-1 Variogram RV : 100m, 200m, 500m

The value of 200m is the extracted value from the electrical borehole images. Table 1 shows details of nine DFN models, which is used for the studies using dynamic data. In every DFN model, the other statistical parameters are exactly the same.

The simulated region is 3.0 km in N-S by 3.0 km in E-W by 0.9 km in depth. A permeability distribution in this region is extracted from a fine gird model after the conversion of a DFN model. The fine model is up-scaled to the coarse model. The size of a coarse gird is 50m by 50m by 10 m. The total number of grids is about three hundred thousand. The flow simulations are carried out with a concept of a single porosity model by assigning a constant porosity to all of the grids.

Fig. 13 shows high permeable zones formed by Family-1 fractures in each coarse model. Upper, middle and bottom models are constructed by using α of 2.5E-6, 2.0E-6, 1.5E-6, respectively. Left, center and right models are constructed by using RV of 100m, 200m and 500m, respectively. In this figure, we can see that the distributions and the extents of clusters formed by Family-1 fractures change with the parameters of α and RV. DFN models constructed by the Rv of 100m have many relatively small clusters (model1, model4 and model7), while DFN models constructed by the RV of 500m have a few relatively large clusters (model3, model6 and model9). The extent and the number of the clusters in the DFN

models vary based on the same crossing probability of fractures in the field. If we select a short range of Family-1 variogram, a narrow extent of each cluster is constructed. This means a reduction of crossing probability of a cluster penetrated by a well. To maintain a crossing probability of fractures in the field to an observed value, a lot of narrow clusters should be generated. Derivative curves. Fig. 14 shows the comparison of the observed and simulated derivative curves. Square symbols denote the field observations. Cross, circle and triangle symbols denote the simulated curves from the models constructed by the different RV of 100m, 200m and 500m, respectively. Upper, middle and Bottom plots show the results of the different α of 2.5E-6, 2.0E-6 and 1.5E-6.

The observed derivative curve shows a concave shaped curve like a dual porosity behavior, but never reaches the second flat state. From the comparison of the observed and the simulated derivative curves, the following results come out:

1) The derivative curves simulated from all models also show concave shaped curves like dual porosity behaviors. 2) All models are not able to reproduce the observed shape at early time within one hour. This is owing to the effect of the wellbore storage and the large grid size near wellbore to simulate derivatives at early time. 3) The observed derivative was successfully reproduced by the DFN model (model4), which has the parameters of the α of 2.0E-6 and the RV of 100m. The best-matched RV is found to be shorter than the extracted value from electrical borehole images. In addition to the above results, we can see from the

variation of the simulated derivative curves with the differentαand RV as follows:

1) The concavities of the derivatives appear late as the estimated sizes of fractures become large and the ranges of Family-1 variograms become long. 2) Under the same range of Family-1 variogram, the upward slopes at late time after the appearance of the concavities becomes gentle with increasing the estimated size of fractures. 3) Under the same estimated sizes of fractures, the steepest slopes at late time are given by the model2, model5 and model8, which have the RV of 200 m. Warren and Root10 studied a dual porosity model

comprised of a fracture and a matrix. In their study, the contrast of the properties between a fracture and a matrix produces a concave shape in a pressure derivative curve. In our DFN models, we suppose that a cluster formed by Family-1 fractures works as a fracture part in Warren and Root’s model and uniformly distributed Family-2 fractures work as a matrix part in their model. However, a contrast of permeability at the boundaries between clusters formed by Family-1 fractures and uniformly distributed Family-2 fractures does not produce the second flat state implied by the Warren and

Root model. According to the understanding mentioned above, we interpret the variation of the simulated curves as follows:

1) Both large estimated size of fractures and long range of Family-1 variogram create a wide extent of a cluster formed by Family-1 fractures. It means that a distance from a shut-in well to the outer boundaries of clusters become long. Consequently, the concavities of the derivatives appear late. 2) An upward slope of a derivative curve at late time reflects an intensity of a permeability contrast at the outer boundaries of clusters. A gentle slope means a little permeability contrast. The larger estimated size of fractures causes higher connectivity between clusters (Fig. 13). This is because that a permeability contrast at the outer boundaries of clusters reduces. Under the same range of Family-1 variogram, the upward slopes at late time after the appearance of the concavities becomes gentle with increasing the estimated size of fractures. 3) As mentioned above, the Rv of 100m causes a lot of uniformly distributed narrow clusters (model1, model4 and model7), while the RV of 500m causes a few relatively large cluster (model3, model6 and model9). Both uniformly distributed narrow clusters and a few relatively large clusters are inferred to result in reducing a heterogeneous character. Consequently, DFN models constructed by the Rf1of 200m show the steepest slopes under the same estimated size of fractures (model2, model5 and model8).

Conclusions We proposed the process of constructing a DFN model using electrical borehole images of multi wells. The constructed DFN model was found to be able to simulate the observed pressure response in the field. This means that the constructed DFN model represents not only the static characteristics of the fracture system, which is found in electrical borehole images, but also the dynamic characteristics of the reservoir, which shows a concave shaped pressure derivative curve. As the result of the parameters studies, which relate to the geometries of fracture networks such as the size and spatial distribution of fractures, the concave shaped pressure derivative curves are found to be affected mainly by a permeability contrast between clusters formed by the fractures with wide apertures and the uniformly distributed fractures with narrow apertures. In the future work, the uncertainty of the statistical parameter of fractures will be reduced by using the dynamic measurements such as the interference test. Nomenclature a = aperture of fracture, m = conversion parameter which relates a fracture

aperture to its size C = constant of exponential equation to express a

relationship between an aperture of fracture and a cumulative frequency

L = total length of borehole, m

m = exponential constant of exponential equation to express a relationship between an aperture of fracture and a cumulative frequency

N = cumulative frequency of fracture in a unit volume, m-3

NB = cumulative frequency of fracture crossing a unit length of well, whose radius are lager than R, m-1

NBa = cumulative frequency of fracture crossing a unit length of well, whose aperture is wider than a, m-1

r = radius of fracture (disc), m Rv = range of variogram of fractures with wide

apertures wi = weighting factor for correcting a crossing

probability with an inclination of a fracture Acknowledgments The authors would like to thank Thohoku University for developing the basic idea of a fracture network model and Japan Petroleum Exploration Co., Ltd. for permission to publish the data from the Yufutsu field within this paper. References 1. Deutsch, C. V. and Journel, A. G.: GSLIB geostatistical

software library and user’s guide, Oxford University Press, (1998).

2. Dershowitz, W. and LaPointe, P.: “Discrete fracture approaches for oil and gas applications,” NARMS ’94, Northern American Rock Mechanics Symposium, Austin, Texas (1994).

3. Long, J. C. S., Gilmour, P. and Witherspoon, P. A.: “A model for steady state flow in random, three dimensional networks of disk-shaped fractures,” Water Resource Research, 21(8), (1985) 1150-1115.

4. Long, J. C. S. and Billaux, D. M.: “From field data to fracture network modeling: an example incorporating spatial structure,” Water Resources Research, 23(7), (1987) 1201-1216.

5. Luthi, S. M. and Souhaite, P.: “Fracture apertures from electrical borehole scans,” Geophysics, 55(7), (1990) 821-833.

6. Parney, R, Cladouhos, T., La Pointe, P., Dershowitz, W.: “Fracture and Production Data Integration Using Discrete Fracture Network Models for Carbonate Reservoir Management, South Oregon Basin, Wyoming”, paper SPE60306 presented at the 2000 Rocky Mountain Section SPE Meeting.

7. Terzaghi, R.: “Sources of errors in joint surveys,” Geotechnique, (1965) 15287-303.

8. Tezuka, K. and Watanabe, K.: “Fracture Network Modeling of Hijiori hot dry rock reservoir by Deterministic and Stochastic Crack network simulator (D/SC),” Proc. WGC 2000, (2000) 3933-3938.

9. Vermilye, J. M. and Sholtz, C. H.: “Relation between vein length and aperture,” Journal of Structural Geology, 17(3), (1995) 423-434.

10. Warren, J. E. and Root, P. J.: “The behavior of naturally fractured reservoir,” Society of Petroleum Engineers Journal, (1963) 245-255.

11. Watanabe, K. and Takahashi, H.: “Fractal geometry characterization of geothermal reservoir fracture networks,” Journal of Geophysical Research, 100(B1), (1995) 521-528.

Table 1- Details of DFN models for the parameter studies using dynamic data

Family-1 Family-2 Family-1 Family-2

model1 2.5E-06 400-80 80-24 51391 2220819 100

model2 2.5E-06 400-80 80-24 51391 2220819 200

model3 2.5E-06 400-80 80-24 51391 2220819 500

model4 2.0E-06 500-100 100-30 32891 1436443 100

model5 2.0E-06 500-100 100-30 32891 1436443 200

model6 2.0E-06 500-100 100-30 32891 1436443 500

model7 1.5E-06 677-133 133-40 18502 807999 100

model8 1.5E-06 677-133 133-40 18502 807999 200

model9 1.5E-06 677-133 133-40 18502 807999 500

Range of Variogram of Family-1 Fracture (m)Model ID

Conversion Parameter: α

Generated Radius of Fractures (m) Generated Number of Fractures

10km16km

Fig. 1- The geological structure of the Yufutsufractured basement gas reservoir

None Productive Productive

Fig. 2- Typical electrical borehole images.Productive wells have fractures with wideapertures.

0.0001

0.0010

0.0100

0.1000

1.0000

4300 4350 4400 4450

0.0001

0.0010

0.0100

0.1000

1.0000

4300 4350 4400 4450

0.0001

0.0010

0.0100

0.1000

1.0000

4700 4750 4800 4850

Fig. 3- Apertures of Fractures calculated fromResistivity Contrast. Square symbols denote wideapertures of fractures identified at electrical boreholeimages shown in Fig.2. Broken lines denote lowerlimits of aperture, which is interpreted to contributeto productivities of wells. This value is 0.2 millimeter.

Measured Depth [m]

Well A: Productive

Well B: None Productive

Well C: None Productive

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IIIIII

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XXXXXXXXXXXXXXXXXXXXX

Fig. 4- Pole plots of fractures on UpperHemisphere. Square, dot and crosssymbols denote fractures with wide, narrowapertures and apertures filled withminerals respectively.

Well A: Productive

Well B: None Productive

Well A Well B Well C Well D Well A Well B Well C Well D

Fig. 5- Observed location of fractures along wellbore. Black bars denote the locations of fractures with wide apertures andGrey bars denote those with narrow apertures. The fractures with wide apertures tend to make clusters.

Fractal

Final Fracture Network Model (Family1+Family2)

Geostatistics Sequential Indicator Simulation

Size distribution

Calculate apertures from Electrical Borehole

Orientation Distribution

Spatial Distribution

(Fractures with Wide Apertures)

Fracture Network of Family1

Family 1 Fracture

Orientation Distribution

Spatial Distribution

(Fractures with Narrow Apertures)

Fracture Network of Family2

Family 2 Fracture

a=α ・ｒ (Velmily and Sholtz,1995)

a:Aperture, r:radius of fracture , α :Conversion

Continuum fine girds with equivalent permeability

Continuum coarse girds for fluid flow simulator

Up-Scale

Fig. 6- Process of constructing discrete fracture network models. The process flow consists of two streams. One stream isfor the fractures with wide apertures. Another stream is for the fractures with narrow ones. Each DFN model is constructedby applying fractal rule and geostatistics, which define the size and spatial distribution of fractures respectively. The finalDFN model is constructed by merging the two types of DFN model that are outputs of the two streams.

Fig. 7- Estimated size distribution on the assumption that a fracture size is proportional to its aperture. The boundaryof the aperture between the narrow fractures and the wide fractures is set to 0.2 millimeter, which gives a radius of100 meter using the value of conversion parameter of 2.0E-6.

y = 0.0067x-1.1506

0.001 0.01 0.1 11.0E-111.0E-091.0E-071.0E-051.0E-031.0E-01

1 10 100 1000

h Family1 (Wide Apertures)

Family2 (Narrow Apertures) A critical value for productivity

Aperture [mm] Fracture Size [m]

Family-1: Fractures with wide apertures Family-2: Fractures with narrow

Fig.8-Extracted orientation distributions on upper hemisphere. Dark tone denotes more concentration of theorientation of fractures.

Distance [m] Distance [m]

Nuget:0.051Range:200

Nuget:0.145Range:350

Fig. 9- Variograms estimated along wellbore. The experimental variogram of each family is calculated from Datashown in Figure 5 and a spherical variogram as an analytical function is fitted to these experimental data. Family-1variogram shows lower value of nugget and shorter range of influence than these of Family-2 variogram. It meansthat Family-1 fractures are more clustered than Family-2 fractures.

Family-1: Fractures with wide apertures Family-2: Fractures with narrow apertures

0 500 1000 1500 0.0

0 500 1000 1500

6.4 km 6.

Fig. 10- Result of estimation for the centers of fractures using SISIM conditioned by well data. Grey dots denote theestimated centers of fractures

Family-1: Fractures with wide apertures Family-2: Fractures with narrow apertures

Family-1: Fractures with wide apertures The generated number of discs: 32891 The generated radius of discs: 500-100

Family-2: Fractures with narrow apertures The generated number of discs: 1436443 The generated radius of disc: 100-30 meter

6.4 km

Fig. 11- Result of generation of fractures following the size and orientation distribution shown in Figure 7and 8. The finalDFN model is constructed by merging Family-1 DFN model and Family-2 DFN model.

Fig. 12- Cross section of permeability distribution of north-south component across the productive well and noneproductive well. Red parts and blue parts denote higher and lower permeability zone, respectively. The equivalentpermeability of the fine grids is calculated by using cubic law. The fine grid size is 12.5 by 25 by 5 meter. The total numberof the fine grid in the whole reservoir is about one hundred and fifty million. In this DFN model, Well-A, which is highlyproductive well, is penetrating high permeable zone and Well-B, which is none productive well, does not meet highpermeable fractures. It means that this DFN model reflects the variation of productivity at each well.

Well-A High Productivity

Well-B None productivity

model1 model2 model3

Fig. 13- High permeable zone formed by Family-1 fractures in the coarse models

WWeellll

α =2.5E-6

α =2.0E-6

α =1.5E-6

Rv=100m Rv=200m Rv=500m

Range of variogram of Family-1 Long Short

0.01 0.1 1 10 100 1000

Square: observed Cross: model1 Circle: model2 Triangle: model3

Square: Observed Cross: model4 Circle: model5 Triangle: model6

Square: Observed Cross: model7 Circle: model8 Triangle: model9

α =2.5E-6

α =2.0E-6

α =1.5E-6

Fig. 14- The observed and simulated derivative curves.

Time [hour]

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