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IDENTIFICATION AND CHARACTERIZATION OF NATURALLY FRACTURED RESERVOIRS USING CONVENTIONAL WELL LOGS

Liliana P. Martinez, Richard G. Hughes and Michael L. Wiggins The University of Oklahoma

ABSTRACT

In petroleum exploration and production, fractures are one of the most common and important geological structures, for they have a significant effect on reservoir fluid flow. Despite their importance, detection and characterization of natural fractures remains a difficult problem for engineers, geologists and geophysicists.

This paper presents a technique for the identification and characterization of naturally fractured reservoirs using conventional well logs. Logs are the most readily available source of information, however they are seldomly used in a systematic manner for quantitative analysis of naturally fractured reservoirs.

Since all well logs are affected in one way or another by the presence of fractures, a Fuzzy Inference System is implemented in this study to obtain a fracture index using only data from conventional well logs. Additionally, a self-consistent model from O’Connell and Budiansky for the prediction of elastic properties of fractured porous rocks is inverted using genetic algorithms to obtain crack density and crack aspect ratio.

The proposed algorithms are tested using data available from the Mills McGee # 1, an Austin Chalk formation well in Milam County, Texas. The results obtained are compared with core information available.

INTRODUCTION

The word “fracture” is used as a collective term representing any of a series of discontinuous features in rocks such as joints, faults, fissures and/or bedding planes. Fractures have a significant effect on both the mechanical and hydraulic properties of rock masses. They can have either a positive or negative impact on flow rates and recoveries from fractured reservoirs depending on the mechanical and flow characteristics of the fractures and the operating conditions of the field. Natural fracture systems not only control the performance and the state of depletion in reservoirs under primary, secondary or tertiary recovery, but also influence flow patterns for production, cementing and completion techniques and the trajectory and quality of the wellbore during drilling operations.

Natural fracture systems can be identified and evaluated by several techniques, with the most common being core analysis, well log analysis and pressure transient analysis. Core analysis can provide quantification of fracture geometry, fracture frequency and the nature of any filling material. The disadvantages of core analysis for naturally fractured reservoir evaluation are that it is difficult to assess how representative the core plugs are of the entire reservoir, cores that contain fractures of practical significance are often lost in the process of recovery, mechanical fractures are often induced due to the release of stress as the core is brought to the surface and core analysis is costly, labor intensive and subject to the availability of drilled rocks.

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These factors have directed the industry to employ well logs that are cost-effective and readily available. Among the advantages of well logs over core analysis are that, while cores are typically obtained over a portion of a single well, well logs are run over a significantly larger portion of nearly every well. Well logs also provide in-situ measurements of the formation at reservoir conditions and provide a consistent, one-dimensional profile of rock properties expressed in terms of a consistent length scale. Because they are highly affected by the borehole condition, well logs may not be the most suitable method for reservoir evaluation.

Pressure transient testing includes generating and measuring pressure variations in wells. Subsequent analysis of these pressure variations provides estimates of rock, fluid, and reservoir properties. These properties are averaged properties over the megascopic scale of the interwell spacing or the formation in hydraulic communication with the well. Thus, this technique may be somewhat unreliable for identifying fractures at the well scale (Elkewidy and Tiab, 1998).

In this paper, methods that use conventional well logs to develop a consistent tool for fracture identification and characterization are presented. A fuzzy inference system will be used to obtain a fracture intensity index and a model presented by O’Connell and Budiansky (O’Connell, 1984) will be used to obtain crack density and aspect ratio. To obtain these parameters requires an inversion of the model where gradient methods cannot be used. Thus, a genetic algorithm approach is utilized. These two models will be evaluated with Austin Chalk formation data from a well in Milam County, TX belonging to the Anadarko Petroleum Corporation.

WELL LOG RESPONSE TO FRACTURES

A well log is a record of formation characteristics made by a tool as it rises in a wellbore. They provide a means to evaluate certain formation parameters and the hydrocarbon production potential of a reservoir. Logging tools can be grouped into two categories: conventional and unconventional. Conventional well logs are those that are routinely collected at almost all industry boreholes. Unconventional well logs are either too specialized, expensive or too recently developed to be run in every well. For the purposes of this paper, the caliper, gamma ray (GR), spontaneous potential (SP), sonic transit time, density and neutron porosity logs are classified as conventional well logs, while the litho-density, spectral gamma ray, borehole televiewer (BHTV), and formation microscaner (FMS) logs (among others) are classified as unconventional well logs.

Caliper tools measure hole size and shape. Fractured zones may exhibit one of two basic patterns on a caliper log. The caliper log may indicate a slightly reduced borehole size due to the presence of a thick mud cake, particularly when using lost circulation material or heavily weighted mud. Alternatively, borehole elongation may be observed which occurs preferentially in the dip direction of fractures due to crumbling of the fractured zone during drilling (Fertl, 1980). Neither of these log response patterns can be taken as a conclusive evidence of the presence of fractures. Highly permeable and underpressured formations can also cause mud cake build-up, which would result in the caliper log recording a hole size smaller than the bit size. Unconsolidated formations can also show borehole elongation effects.

The SP log is a measurement of the natural potential differences or self-potentials between an electrode in the borehole and a reference electrode at the

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surface. The response of the SP curve in front of fractured zones may exhibit either erratic behavior or a more systematic negative deflection due to a streaming potential (the flow of mud filtrate ions into the formation). However streaming potentials can also occur near silt beds (Crary et al., 1987).

The GR log is a record of a formation’s radioactivity. The GR log is principally used quantitatively to calculate shale volume. In fractured reservoirs an increase in the gamma ray without concurrently higher formation shaliness, is frequently observed. This increase has been explained by the deposition of uranium salts along the discontinuity surfaces of a fracture or within the crack itself (Fertl, 1980).

Natural Gamma Ray Spectroscopy records the individual mass concentrations of uranium, thorium and potassium. A high uranium content may reflect the effect of organic shales or the depositation of uranium salts in fractures (Serra et al., 1980). The solubility of uranium compounds accounts for their transport and their frequent occurrence in fractures.

The Density Log is a continuous record of the formation’s bulk density. The dual detector density tool reports two values: a value of uncompensated density using a long-spaced detector response, and a value of density correction ∆ρ. The correction is added to the uncompensated values to obtain the compensated bulk density, ρb (Bassiouni, 1994):

..........................................................................................................(1)

where ρls is the long-spaced detector, uncompensated density. The ∆ρ term is a measure of the correction made to the bulk density to compensate for mudcake and for the density tool not seating perfectly against the borehole wall. It will also respond to a fluid filled fracture. An active, erratic ∆ρ curve may therefore indicate fractures when the hole is in gauge. Since the density logs are a measure of total reservoir porosity, fractures filled with fluid will decrease the recorded bulk density, creating a sharp negative peak on the density curve, and a corresponding peak on the ∆ρ correction.

The measurement principle of the neutron log is based on the fact that hydrogen is very efficient in slowing down fast neutrons. Similar to the density log, any neutron-type log is a measure of the total reservoir porosity in fluid saturated formations. Therefore, in the presence of fractures, the neutron log is expected to have a behavior similar to that of the density log.

The litho-denstiy tool reports the measurement of the effective photoelectric absorption cross-section index for the formation, Pe. The Pe index will report anomalously high values near mud invaded open fractures. Therefore, a high reading of Pe, with good tool-borehole contact established by the caliper curve, may be a good indicator of fractures (Ellis, 1987).

Acoustic logging for formation evaluation can be defined as the recording of one or more parameters of acoustic wave trains for use in estimating fundamental rock properties. Acoustic logging includes the measurement of both interval transit time and amplitude/attenuation logging mainly for compressional and shear waves.

Sonic interval transit time logging records the time required for an acoustic wave to transverse a given length of formation. Most acoustic logging tools are designed to detect the first arriving compressional wave only when the energy level

ρρρ ∆+= lsb

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reaches a certain threshold. (Jorden et al., 1986). If the energy level does not reach the threshold value, cycle skipping occurs. Cycle skipping may occur when the attenuation in a formation is abnormally high (due to under-compaction, light hydrocarbons, or fractures) or when the mud is gas-cut. In hard rock (i.e. fast formations), cycle skipping may be a good indicator of fractures (Fertl, 1980).

Acoustic amplitude logging records the energy level of an acoustic wave while acoustic attenuation logging records the decrease in amplitude across a specified distance in the medium. The acoustic amplitude log delineates fractures by measuring the energy loss caused by the mode conversion that occurs when an acoustic wave reaches a fluid filled fracture. The signal amplitude is affected by the dip angle of the fracture, the number of fractures, the shape of the fracture faces and the nature of the material within the fracture (Guyod, et al, 1969). However, considerable care is necessary in the interpretation of the amplitude log because changes in lithology or porosity can produce effects that are similar to the response from fractures (Aguilera, 1976).

Resistivity logs are measurements of the ability of the fluids in a formation to conduct electricity. The dual laterlog generally provides three resistivity measures: the deep laterlog (which investigates about 10 ft. into the formation), the shallow laterlog (which investigates 3 to 6 ft. into the formation), and the microspherically focused log (MicroSFL) which measures resistivity in the invaded zone. The effect of fractures on resistivity logs will depend primarily on the fracture direction, size (aperture size and height), length, and the fluid inside the fracture. A fractured zone should appear as a very conductive anomaly to the microresistivity tools because they see the fractures as entirely filled with mud filtrate.

Wellbore imaging involves recording downhole information of the borehole surface and converting depth, orientation, and caliper data into two-dimensional images. The most widely used imaging devices are the Formation MicroScanner (FMS) and the Borehole Televiewer (BHTV). The FMS is a resistivity tool with arrays of electrodes where each array is located on pads orthogonal to one another. Porosity, formation fluid, rock textural characteristics, and borehole rugosity affect the quality of the FMS images. The BHTV uses an ultrasonic transducer to send a short acoustic pulse out to the borehole-casing wall (Zemanek, et al, 1969). The transducer rotates rapidly in the borehole. The amplitude and transit time signals can be made with varying frequency transducers.

Both the FMS and the BHTV logs provide high-resolution images of the wellbore, therefore they are considered the most direct and effective methods for detecting fractures in boreholes. However, image distortion and the presence of induced fractures can produce significant bias in the characterization of fracture populations. In addition, the sampling area for these logs is restricted to the immediate vicinity of the borehole. Finally imaging logs are not as widely available as conventional well logs.

Since conventional well logs are readily available, it is of great importance to understand the effects of fractures on them to try to use the information they provide to attempt a better characterization of naturally fractured reservoirs. A model to integrate conventional well logs into a Fuzzy Inference System to identify the presence of fractures is proposed.

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FUZZY LOGIC AND FUZZY INFERENCE SYSTEMS

Real world problems are characterized by the need to be able to process incomplete, imprecise, vague or uncertain information. One approach to account for this uncertainty, developed by Zadeh (1974), is based on a concept known as fuzzy sets. A fuzzy set allows for an object to be a member of a set to some degree or membership grade. Thus, fuzzy sets differ from classical sets in that they allow for an object to be a partial member of a set.

A Fuzzy Inference System (FIS) is a system that uses fuzzy sets to make decisions or draw conclusions. A FIS can be defined as the base fuzzy sets that are to be used (as defined by their membership functions), the rules that combine the fuzzy sets, the fuzzy composition of the rules and the defuzzification of the solution fuzzy set. Each of these components require that appropriate choices be made for the rules and mathematical manipulations to allow the system to provide insight into the particular problem being solved.

A fuzzy set is fully defined by its membership function. For some applications the sets that will have to be defined are easily identifiable. For other applications they will have to be determined by knowledge acquisition from an expert or group of experts. For identification of fractures from well logs, the fuzzy sets are the data at each depth from each of the logs available. Normalized log response values are used in this work. The normalization uses a two-step process. First, a 6-point moving average filter (3 ft. total length) is applied to obtain a background value to which the current data value is compared. The difference between the log data value and the background value is then scaled to be a value between 0 and 1. Data scaling is necessary for two reasons. First, it is desired to account for essential variability in the filtered log data, and, without some type of scaling process, those logs with the largest original variance would dominate the subsequent analysis. Second, it is desired to have all logs measured in similar units because they will be easier to compare in the FIS and the analysis will not be biased towards those with higher absolute values. In this study a linear scaling method that maps the maximum log value to one and the minimum log value to zero will be used. The linear scaling has the following form:

ab

axz i

i −−= .................................................................................................................(2)

where zi is the scaled value, xi is the original value, a and b are scaling constants. In this study, a is the minimum log value and b is the maximum log value.

Once the names of the fuzzy sets have been established, their associated membership functions must be considered. In a fuzzy rule-based system the rules can be represented in the following way (Nauck, et al, 1997):

If (a is A) AND (b is B)……AND…..THEN (z is Z)

where a, b, and z represent variables (e.g. distance, size) and A, B and Z are linguistic variables such as ‘cold’, ‘warm’, ‘hot’. Then the phrase “a is A” is an abbreviation of the compete statement “a belongs to the fuzzy subset A with a corresponding membership value µA(a) (or to a degree µA(a))”. Two membership functions have been designed for each well log. Each membership function maps the scaled log data

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to a fuzzy subset, which expresses whether the probability of fractures is ‘high’ or ‘low’. Membership functions are designed for each log according to the response of the log to the presence of fractures.

When high or low log response values may be related to fractures (sonic, caliper, gamma ray, density, neutron porosity, and resistivity logs) a sigmoidal membership function is used. This membership function is defined as (Roger et al, 1997):

))(exp(1

1),;(

cxacaxsig

−−+= ................................................................................(3)

where a is the slope at the crossover point, x = c. Depending on the sign of the parameter a, a sigmoidal MF is inherently open right or left. An open right sigmoidal MF indicates high likelihood of fractures, while an open left sigmoidal MF indicates low likelihood of fractures. The sigmoidal MFs are shown in Fig. 1.

When erratic log response values or small variations about the background value may be related to a high probability of fractures (SP and MSFL logs), the generalized bell MF seems to be appropriate. A generalized bell MF is specified by three parameters (Roger et al, 1997)

b

a

cxcbaxbell

2

1

1),,;(

−+= ...........................................................................................(4)

where c represents the MFs center, a determines the MF width and b is an additional parameter related to the slope at the point c+a. The bell MFs are shown in Fig. 2.

The membership function for the output variable-fracture index will indicate the probability of fractures according to the logs analyzed. In this case, sigmoidal membership functions are used to indicate high and low fracture index.

Note that these membership functions are choices that were made based on the previous discussion on typical log responses to fractures. Other MF could be chosen. For instance, if it is known that one of the logs respond in an atypical fashion in a particular field, another MF that captures the response should be chosen.

Once the membership functions have been defined for each one of the input and the output variables, the next step is to set the rules required by the FIS to identify fracture intensity from conventional well logs. In this study only those rules that reflect the response of conventional well logs to fractures will be taken into account. Again, the set of rules that is appropriate for fracture characterization is likely to be different for different fields and/or formations. Examples of the rules that can be used to obtain a fracture intensity index are:

- If (caliper is high) and (sonic transit time is high) then (fracture index is high).

- If (resistivity is high) and (MSFL is high) then (fracture index is high). - If (resistivity is low) and (MSFL is low) then (fracture index is low).

Aggregation is the process by which the fuzzy sets that represent the outputs of each rule are combined into a single output fuzzy set. The input of the aggregation process is the list of truncated output functions returned by the implication process for each rule. The output of the aggregation process is one fuzzy set for each output variable.

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Once the rules have been composed, the solution, as has been seen, is a fuzzy set. However, for most applications there is a need for a single action or ‘crisp’ solution to emanate from the inference process. This will involve the ‘defuzzification’ of the solution set. Lee (1990) describes the three main approaches as the max criterion, mean of maximum and the center of area. The max criterion method finds the point at which the membership function is a maximum. The mean of maximum takes the mean of those points where the membership function is at a maximum. The center of area method finds the center of gravity of the solution fuzzy sets. For this work, the center of gravity method is used to obtain the output fracture index.

Figure 3 shows three of the FIS rules that have been put together to show how the output of each rule is combined into a single fuzzy set, and how the output fuzzy set was defuzzified using the centroid calculation method. Note that for this example, the value for the fracture index at this depth is 0.3. This procedure is repeated at each depth in the input log data and a log of the fracture index is the resulting output.

Though the Fuzzy Inference System is able to provide some quantification as to the degree of the presence of fractures in a given well, the FIS cannot give quantitative information about the fracture characteristics that may be indicated. It may be possible to obtain certain characteristics through the analysis of the acoustic well logs. A method to obtain crack density and crack aspect ratio using a model by O’Connell and Budiansky was presented elsewhere (Martinez et al., 2001). In this work, the same approach is used, but the solution to the problem is obtained using genetic algorithms.

O’CONNELL AND BUDIANSKY SELF CONSISTENT MODEL.

This model assumes that the effective moduli of a composite of porous elastic materials depend on the properties of the individual components, the volume fractions of the components and the geometric and spatial characteristics of the components.

O’Connell (1984) presents the mathematical formulation for the model which is based on basic energy considerations for the effects of inclusions of simple geometric shapes in a matrix. The following presentation of the model follows that of O’Connell (1984) and also that found in Martinez et al. (2001). The main advantage of this method over previously presented models is its ability to account for interaction between cracks and pores.

The model considers a solid, permeated with two classes of porosity: crack-like, characterized by a crack density with fluid pressure equal to the applied normal stress on the crack face, and pore-like (i.e. tubes or spheres) characterized by a volume porosity, with fluid pressure substantially less than the applied hydrostatic stress. Fluid is allowed to flow between cracks at different orientations and between cracks and pores in response to pressure differences.

The parameters for this model are the crack density, ε, the porosity of the spherical pores, φ, the fluid bulk modulus, Kf, the bulk and shear moduli of the uncracked non-porous matrix material, Ko and Go, the frequency, ω and the characteristic frequency for fluid flow between cracks, ωs. The crack density is defined by

⟩⟨=P

AN

22*

πε ......................................................................................................... (5)

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where N is the number of cracks per unit volume; A is the area in plain-form of the crack, and P is the length of the perimeter of the crack. The characteristic frequency for fluid flow between cracks can be estimated from:

34 αµ

ϖ

≈ K

s .............................................................................................................(6)

where µ is the viscosity of the fluid, α is the aspect ratio of the crack which is given by: α = c/a, with c and a the minor and major axis lengths of the crack (O’Connell, 1984). With the previous definitions, the crack porosity is (Mavko et al., 1998):

πεαφ3

4=c ...................................................................................................................(7)

The moduli are considered to be a function of the frequency, ω, and are complex quantities (i.e. K = Kr + iKc, and G = Gr + iGc). The real parts of these moduli represent effective elastic moduli, and the imaginary parts represent anelastic energy dissipation. The complex bulk modulus, K, is given by (O’Connell, 1984):

φεφ

εφ

Ω+

−−+

−−+

Ω+

−−+

−−

−

−=

iK

K

v

v

v

v

K

K

iv

v

v

v

K

K

K

K

ff

o

f

o

121

1

9

16

21

1

21

121

1

9

16

21

1

2

31

12

2

......................................(8)

With:

s

o

K

K

v

v

ϖϖ

−−=Ω

21

1

9

16 2

.................................................................................................(9)

The shear modulus G is determined using (O’Connell, 1984):

εφ

−+

Ω+−−

−−−=

’2

3

’1

1

45

)’1(32

’57

)’1(151

vi

v

v

v

G

G

o

.............................................. (10)

Ω+

−−−

−−−=

’1’21

’1

9

16

’21

’1

2

31

’

’ 2

iv

v

v

v

K

K

o

εφ ..........................................................(11)

where ν’ is a fictitious Poisson ratio that satisfies:

GK

GKv

2’6

2’3’

+−= ............................................................................................................ (12)

The same type of expression relates the moduli and Poisson’s ratio of the porous solid:

GK

GKv

26

23

+−= .............................................................................................................(13)

A direct solution of this model is not possible and an iterative procedure is required. In Martinez, et al. (2001), a solution was obtained by a trial-and-error method. In this paper, the O’Connell and Budiansky model was inverted using a genetic algorithm to obtain fracture porosity and fracture aspect ratio using conventional well logs.

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INVERSION OF THE O’CONNELL AND BUDIANSKY MODEL

The inversion of the O’Connell and Budiansky self consistent model to obtain crack density, ε and aspect ratio (c/a) can be seen as an optimization problem, where the goal is to minimize an objective function. If the real parts of the moduli (Kr and Gr) are known, the inversion of the model will consist of obtaining the parameters ε and (c/a) that will minimize an objective function given by:

rcalrrcalr GGKKF −+−= ................................................................................... (14)

where Krcal and Grcal are the real portions of the bulk and shear moduli obtained from the O’Connell and Budiansky model.

This particular problem is not suitable to be solved through the conventional gradient optimization methods. Therefore a genetic algorithm approach is implemented and programmed using Fortran.

A 6×8 bit binary string formed the chromosomes, which represent the 6 unknown variables, K’r, K’c, Kc, Gc, α, ε, each with an eight-bit resolution. An initial population of 30 chromosomes was generated. Each chromosome was first randomly generated and then evaluated in order to guarantee that it was within the solution space. The randomly generated chromosome was decoded to obtain the generated values for K’r, K’c, Kc, Gc, α, and ε. Equations (12) and (13) were then used to verify that the Poisson’s ratios were in the range between 0 and 1. If the chromosome satisfied these constraints, it was allowed into the population. Otherwise the chromosome was rejected and a new chromosome was randomly generated and evaluated until a population size of 30 chromosomes was obtained. Every chromosome within the population was evaluated using Eqs. (8), (10) and (11) with the real portions of the bulk and shear moduli as “output” parameters (Krcal and Grcal). The Kr and Gr terms are either obtained experimentally or can be computed when the compressional and shear wave velocities and the bulk density are known. The F value computed from Eq. (14) was taken as a fitness value for the generation; smaller values of F are “better” or “more fit” than chromosomes that have higher values for F. A new generation of chromosomes was then created based on the original population according to the following procedure: - A set of two parents was selected from the population according to their fitness

value. - The two parents were combined randomly to generate two new offspring. These

two new chromosomes were evaluated for fitness. A new set of parents was selected, and the process was repeated until a new population of 30 chromosomes was obtained.

- Once the new population was obtained, mutation was applied randomly to some of the chromosomes in the new population at a mutation rate of 0.01.

The process described was repeated for 100 generations. At the end of the 100th generation, if the fitness value of the best chromosome was greater than 0.001, the process was repeated. The optimum solution to the problem was the chromosome with the lowest fitness value amongst all of the 100 generations.

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CASE STUDY

Boreholes that are suitable for case study are rare. In order to apply the Fuzzy Inference Algorithms, several conventional well logs that are useful for fracture detection must be available, and in order to apply O’Connell and Budiansky inverse algorithm, the suite of logs available must be complete enough to allow for the calculation of the lithology and saturations. Data quality must be carefully examined since reliable log information cannot be obtained under adverse borehole conditions. Additionally, some calibration of the log results to actual data is recommended. In order to do so, the borehole must either be continuously cored, or contain a detailed imaging log that can be used for comparison.

Data from a well drilled by Union Pacific Resources (now Anadarko Petroleum Corporation) was obtained and used in this study. Additionally, John Lorenz, geologist at Sandia National Laboratories, provided special core analysis characterizing the natural fractures for this well (Lorenz, 1997).

The Mills-McGee #1 is on a lease located in Milam County, TX. This well produces from the Cretaceous Austin Chalk formation in the Giddings (Austin Chalk-3) Field (Figures 14 and 15). In the area of the well, the formation produces from fine-grained limestone and chalk. Productivity and fluid flow directions are predominantly controlled by the presence of fractures.

The well was logged using a comprehensive suite of state-of-the-art tools. In addition, 220 ft of core was obtained and analyzed by the UPR rock lab and by Sandia National Laboratory. Anadarko Petroleum Corp. provided the well log digital data, and Sandia National Laboratory provided the description and interpretation of fractures from the core.

Two categories of fractures were recognized. The first was “hairline” fractures, which were healed, fully mineralized fractures that had widths of 0.1 mm or less. The second were “Semi-open” fractures, which were somewhat wider than the hairline fractures. These two fracture types occurred in the same zones and were not mutually exclusive. They were inferred to be components of a single fracture population since they had similar characteristics. Of the 775 fractures that were observed, 36 were of the semi-open variety. The remainder were hairline fractures. Mechanically induced fractures were identified from the cores and excluded from the analysis (Lorenz, 1997).

Well logs available and used in this study for the Mills McGee well are listed in Table 1. Only those logs that are known to be affected by the presence of fractures were chosen for this analysis. The original well log data is displayed in Fig. 4.

The caliper log displayed in Fig. 4a shows that although this is a well in a fractured formation, the borehole is in gauge with no enlargement or reduction in the zone between 5860 and 6040 ft. The good borehole condition means that the log measurements should be very reliable. The conventional SP log is presented in Fig. 4b. It is possible to observe several zones with an erratic SP behavior presumably due to the presence of fractures. The behavior of the gamma ray log as shown in Fig. 4c is not a conclusive fracture indicator. The peaks that this log displays in the interval between 5870 ft and 6010 ft might be due to thin shale beds or to the presence of fractures. Figure 4d shows the sonic log. In this case cycle skipping is not clearly observed, and the compresional travel time values are those corresponding to

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chalk/shale lithology. Neutron porosity and density porosity are presented in Fig. 4e. In this case the density porosity value does not report large changes. On the other hand, the neutron porosity is not as smooth as the density porosity, possibly due to the presence of fractures. Figures 4e and 4f present the density and density correction logs. The density log reports an almost constant value of bulk density in the interval between 5870 and 6070 ft.; however, the density correction log reports large correction values in the interval between 5850 and 5940 ft. Since the caliper in this interval reports a gauge hole, the abnormalities in the density correction values may be due to the presence of fractures in this interval. Induction logs are shown in Fig. 4g. In fractured formations these logs indicate the presence of fractures if the spherically focused log (SFL) reads less than the deep induction log (ILD). According to this criterion, it is possible to distinguish several fractured zones in the interval shown. Figure 4i presents the shallow (AT10), medium (AT60) and deep (AT90) resistivity logs. It is possible to observe some separation between the shallow and deep resistivity logs, especially between 5860 and 5920 ft. In general though, there is no significant separation between the two resistivity measurements. The litho-density (Pe index) log, Fig. 4j, reports a fairly constant value in the range between 4.5 and 5.5, which is in agreement with the lithology expected in this well.

No individual log shows fracturing directly, but through the use of FIS, inference of the presence of fractures in the well may be possible. The FIS was applied under seven different scenarios (set of rules). Figure 5 reports the results obtained for the different cases. The fractures reported by the core description are displayed in the figures as dots. Only open and semi open fractures are shown. Hairline fractures are not displayed because they are very numerous, most of them are sealed and they are not analyzed in the core description. The rules for each of the cases are presented in Table 2. Each set of rules is defined somewhat arbitrarily based on the discussion about the effects of fractures on the different logs.

For the six cases analyzed using the FIS, there is good correlation between the fracture detection algorithm and the core analysis, specifically in the interval between 5910 and 5930 ft. All the cases presented in Fig. 5, are able to recognize that interval as one with high fracture presence. The correlation between the core analysis and the log analysis for the set of fractures between 5965 and 5990 ft is not as clear. The only case that is detecting this fractured zone is Case 3. This case has a high noise level, however, which makes this combination of rules poorly suited for a FIS for fracture detection. Case 5 does not seem to be the most appropriate suite of rules for the fracture detection algorithm because it is not identifying the main fractured intervals.

The fractured interval reported by the core analysis between 5965 and 5975 ft is not recognized by any of the cases analyzed. This may indicate a depth shift between the logs and the core analysis that has not been taken into account. The high fracture frequency observed in zones where core analysis does not report fractures might indicate the presence of mechanically induced fractures that are excluded from the core description. However, the FIS algorithm may be responding to hairline fractures that are not accounted for in the core description provided by Sandia National Labs. Case 6 seems to be the most appropriate for this specific example. Case 6, besides having the same rules as Case 5, has two additional rules involving the resistivity and the SFL log. It is then possible to conclude that induction and

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resistivity logs play a very important role in detecting fractures for this well. Case 7 is a combination of all the logs available for this well. From Fig. 5g it is possible to conclude that even though this combination of logs seems to be identifying most of the fractured intervals in the well, there is also a relatively high noise level in the fracture index. However this noise could also be due to the mechanical or hairline fractures ignored in the core analysis.

Figure 6 presents the results obtained using the O’Connell and Budiansky model. The fracture porosity reported in Fig. 6c clearly identifies several of the high fracture frequency zones reported by the core analysis, specifically the intervals at 5920-5925 ft and 6000-6007 ft. However, there are zones where it is difficult to correlate the fracture porosity obtained with what was reported by the core analysis. The main reason that can explain this difficulty is the presence of more than 700 hairline fractures that are not classified as open or semi open fractures in the interval analyzed and may be affecting the fracture porosity results. The whole interval reports a relatively high crack density and fracture porosity that may be reflecting the presence of these hairline fractures.

Figure 7 combines the results obtained for case 6 of the FIS (the scale has been reduced from 0 to 1 to 0 to 0.01 using a linear scaling as in Eq. (2)), and the fracture porosity. From Fig. 7, it is possible to observe a fairly good correlation between the results obtained with the FIS and the inversion of the O’Connell and Budiansky model in most of the intervals where the FIS reports a high fracture intensity index. This may lead us to the conclusion that the FIS system proposed might be used as fracture porosity indicator.

Discrepancies between core description of fractures and indicators of fractures using well logs are expected. The fracture indicators in this work are based on well log data that reflect bulk rock properties up to several feet away from the borehole while the core analysis is only reflecting the characteristics of the borehole itself. In addition, in highly fractured zones it is often difficult to obtain reliable core samples. That does not appear to be the case in this example, but there may be some effects since there was rubble recovered in the core barrel. Depth shifts between recorded core depths and well log depths (due to unfilled space in the core barrel, core expansion, stretch in the wireline logging cable, operator error, etc.) can also cause discrepancies between the core description and the log results.

CONCLUSIONS AND RECOMMENDATIONS

No single conventional well log provides reliable characterization of the distribution and geometric characteristic of the fractures in the wellbore, however all logs are affected in one or another way by the presence of fractures.

This work has shown that it is possible to obtain a good estimate of important parameters such as fracture index, crack porosity and crack aspect ratio, required to approach a characterization of naturally fractured reservoirs, using only information that can be obtained from conventional well logs.

Fuzzy logic provides an alternative method to handle uncertainty. This is especially useful in reservoir characterization where the data available is not always very precise and does not usually provide direct indication of the reservoir properties that govern the flow of fluids in the porous media. Detection and description of naturally fractured reservoirs remains a difficult task. However in this work it has

13

been shown that Fuzzy Inference Systems can be successfully used to integrate the different well logs that may be available into a single tool to identify the presence of fractures. Furthermore, the fracture index obtained through the FIS may give a direct indication of the fracture porosity.

The FIS is very sensitive to the set of logs selected for the analysis. In the case analyzed in this study, the best results were obtained with the combination of caliper, gamma ray, sonic, spontaneous potential and resistivity logs. However this combination may not be the optimum one in another location. Each field is unique and requires individual analysis.

There is a good correlation between the fracture index obtained with the Fuzzy Inference algorithm and the ones observed in the provided core analysis, indicating that the algorithm is able to detect open and semi-open fractures when the appropriate suite of conventional logs are provided.

A methodology not only to identify the presence of fractures, but also to quantify the fracture porosity and fracture aspect ratio has also been presented. The O’Connell and Budiansky model, a rock physics model that has been implemented for the interpretation of seismic data, was successfully used at the wellbore to obtain important information about the fractures in the well, namely the crack density, aspect ratio and crack porosity.

The FIS can be tuned in several ways. In this study the only tuning was done through the different combination of rules. For future work it is recommended that the FIS also be tuned through the membership functions and through the implication method used for the evaluation of the rules. In order to generalize the FIS used to obtain the fracture index, and the method proposed to invert O’Connell and Budiansky model, as reliable methods for the characterization of naturally fractured reservoirs, extensive field data along with experimental data is needed for analysis.

ACKNOWLEDGMENTS

We are grateful to the U.S. Department of Energy for financial support of this project through U.S. DOE Grant No. DE-AC26-99BC15212. We would also like to thank Anuj Gupta for initiating this project and Ray Brown and Carl Sondergeld for helpful discussions on this topic. Thanks also go to John Lorenz with Sandia National Laboratory and Jeff DeJarnett with Anadarko Petroleum Corporation for providing the core analysis and log data files respectively.

NOMENCLATURE

Symbols A = Area in plain-form of the crack. a = Crack radius. FI = Fracture index. G = Shear modulus. Go = Shear modulus of mineral material making up rock. Grcal = Calculated real portion of the shear modulus. h = Formation thickness K = Effective bulk modulus of the rock with pore fluid. Kf = Effective bulk modulus of pore fluid.

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Ko = Bulk modulus of mineral material making up rock. Krcal = Calculated real portion of the bulk modulus. N = Number of crack per unit volume. P = Perimeter of the crack. Vp = P-wave velocity. Vs = Shear wave velocity. v = Poisson ratio. v’ = Fictitious Poisson ratio.

Well log symbols AT10, AT60, AT90 = Shallow, medium and deep resistivity logs. CAL = Caliper log. DPHI = Density porosity. DRHO = Density correction. DT = Sonic transient time. GR = Gamma Ray. ILD = Deep induction log. ILM = Medium induction log. NPOR = Neutron porosity. PEF = Lithodensity log. RHOB = Bulk density. SFL = Spherically focused log. SP = Spontaneous potential.

Greek symbols α = Crack aspect ratio. ∆ρ = Density correction. ε = Crack density. φf = Fracture porosity. φm = Matrix porosity. ρ = Bulk density. µA(x) = Degree of membership of element x in the fuzzy set A. µ = Viscosity of the fluid. ω = Frequency. ωs = Characteristic frequency for fluid flow between cracks.

REFERENCES

Aguilera, R.: “Analysis of Naturally Fractured Reservoirs From Conventional Well Logs”, Journal of Petroleum Technology, p. 764-772, July 1976.

Bassiouni, Z., Theory, Measurement, and Interpretation of Well Logs. Society of Petroleum Engineers, SPE Textbook Series Vol. 4, 1994.

Crary, S. et al.: “Fracture Detection With Logs”, The Technical Review. V. 35, no. 1, p. 23-34, 1987.

Elkewidy, T. I. and Tiab, D.: “An Application of Conventional Well Logs to Characterize Naturally Fractured Reservoirs with their Hydraulic (Flow) Units; A

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Novel Approach”, SPE paper 40038 presented at the SPE Gas Technology Symposium held in Calgary, Canada, 15-18 March, 1998.

Ellis, D. V. Well Logging For Earth Scientists. Elsevier Science Publishing Co., New York, 1987.

Fertl, W. H.: “Evaluation of Fractured Reservoirs Using Geophysical Well Logs”, SPE paper 8938 presented at the 1980 SPE/DOE Symposium on Unconventional Gas Recovery held in Pittsburgh, Pennsylvania, 18-21 May, 1980.

Guyod, H. and Shane, L. E., Geophysical Well Logging, Houston, TX., 1969.

Lee, C. C.: “Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part II”, IEEE Transactions on Systems, Man and Cybernetics, 20(2):419--435, 1990.

Lorenz, J.: “Description and Preliminary Interpretations of Fractures in the Mills McGee #1 Core”, Personal Communication to Tom Zadick. August 18, 1997.

Martinez, L., Gupta, A. and Brown, R., “Interpretation of Important Fracture Characteristics from Conventional Well Logs”, SPE paper 67280 Presented at the Production and Operations Symposium, 25- 28 March, 2001, Oklahoma City, OK

Mavko, G., Mukerji, T. and Dvorkin, J., The Rock Physics Handbook: Tools For Seismic Analysis In Porous Media, Cambridge University Press, 1998.

Nauck, D., Klawonn, F. and Kruse, R., Foundations of Neuro-Fuzzy Sistems, John Wiley & Sons, U.K., 1997.

O’Connell, R.J.: “A Viscoelastic Model of Anelasticity of Fluid Saturated Porous Rocks”, Physics and Chemistry of Porous Media, AIP Conf. Proceedings, p.166-175. 1984.

Roger, J.S., Sun, C.T. and Mizutani, E., Neuro-Fuzzy and Soft Computing. Prentice Hall, Upper Saddle River, NJ., 1997.

Schlumberger, Log Interpretation Principles/Applications. Schlumberger Educational

Serra, O., Balwin, J. and Quirein, J.: “Theory, Interpretation and Practical Application of Natural Gamma Ray Spectroscopy”, Paper F presented at the 21st Society of Professional Well Log Analysts Logging Symposium. 1980.

Zadeh, L. A.: “Fuzzy Logic And Its Application To Approximate Reasoning”, Information Processing, 74:591-594, 1974.

Zemanek, J. and Caldwell, R. L.: “The Borehole Televiewer –A New Logging Concept For Fracture Location and Other Types of Borehole Inspection”, Journal of Petroleum Technology, p. 762-774, June 1969

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Fig. 1: Sigmoidal Membership Function

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Scaled Deviation From Background

High likelihood

Low likelihood

Fig. 2: Generalized Bell Membership Function

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Scaled Deviation From Background

High likelihood

Low likelihood

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fracture Index = 0.3

Fracture Index

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fracture Index = High

Fracture Index

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

GR = High

GR

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

SP=High

SP

If GR is high And SP is high Then Fracture Index is high

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Resistivity=Low

Resistivity

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fracture Index=Low

Fracture Index

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

MSF=Low

MSF

If Resistivity is low And MSF is low Then Fracture Index is low

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Caliper=Low

Caliper

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

∆T = High

∆T

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Fracture Index=High

Fracture Index

If ∆T is high And Caliper is low Then Fracture Index is high

Fig. 3: Fuzzy Inference Process

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5850

5950

6050

-20 0 20

SP

5850

5950

6050

8 10 12 14

CALI

5850

5950

6050

0 40 80

GR

5850

5950

6050

40 60 80 100

DT

5850

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6050

0 0.1 0.2 0.3 0.4

NPOR DPHI

5850

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6050

0 2 4 6 8 10

ILM ILD

SFL

5850

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6050

0 0.1

DRHO

5850

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6050

2 2.3 2.6 2.9

RHOB

5850

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6050

0 2 4 6 8 10

AT90 AT60

AT10

5850

5950

6050

3 4 5 6

PEF

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

Fig. 4: Mills-McGee #1 raw well log data

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5850

5950

6050

0 0.5 1

Case1

Open fractures

5850

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Case2

Open fractures

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Case3

Open fractures

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Case4

Open fractures

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Case5

Open fractures

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Case6

Open fractures

5850

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Case7

Open fractures

(a) (b) (c) (d) (e) (f) (g)

Fig. 5: Fuzzy Inference System Cases

20

5850

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6050

0 0.05 0.1 0.15

Fracture density

Open fractures

5850

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6050

0 0.05 0.1

Aspect Ratio

Open fractures

5850

5950

6050

0 0.005 0.01

Fracture porosity

Open fractures

(a) (b) (c)

Fig. 6: O’Connell and Budiansky model inverted parameters

21

5850

5950

6050

0 0.005 0.01

case6

Open fractures

Fracture porosity

Fig. 7: Fracture Porosity and Fracture Index

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Table 1. Well Logs available for the Mills McGee Well #1

Well log Fracture detection significance

Caliper Registers borehole enlargement that may be caused by the presence of fractures.

Spontaneous Potential May indicate the presence of fractures, but, is not considered a reliable fracture detection tool. (Schlumberger Ltd., 1989)

Gamma Ray Without the spectral gamma ray data, the gamma ray log by itself is not conclusive in fracture detection.

Bulk Density Open fractures filled with drilling fluid may cause a reduction in bulk density (Fertl, 1980).

Density correction It is considered one of the best fracture detection tools among the conventional well logs (Serra, 1986)

Photoelectic factor Has been recognized as a useful fracture detection tool (Schlumberger, 1989)

Sonic transit time Fractures are know to cause cycle skipping (Bassiouni, 1994)

Shallow/Deep Induction combination The resistivity of the deep induction tool will exceed the one for the shallow induction tool

Spherically Focused Log Often exhibits erratic values in the presence of fractures, but is sensitive to poor borehole conditions. (Schlumberger, 1989)

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Table 2. Set of Rules used in the different cases for the FIS

CASE 1 IF SFL is high and AT10/AT90 is high THEN FI is high IF DRHO is high and PEF is high THEN FI is high

CASE 2 IF DT is high and DRHO is high and CAL is high THEN FI is high IF GR is high and SP is high and SFL is high THEN FI is high IF DRHO is high and CAL is medium THEN FI is high

CASE 3 IF DT is high and SFL is high THEN FI is high IF PEF is high and DRHO is high THEN FI is high IF AT10/AT90 is high and ILD/ILM is high THEN FI is high

CASE 4 IF GR is high and PEF THEN FI is high IF DT is high and SFL high THEN FI is high IF PORDIF is high and DRHO is high THEN FI is high

CASE 5 IF CAL is high and GR is high and SP is high THEN FI is high IF DT is high and GR is high THEN FI is high IF DT is high and SP is high THEN FI is high

CASE 6

IF CAL is high and GR is high and SP is high THEN FI is high IF DT is high and SFL is high and PEF is high THEN FI is high IF DT is high and SP is high THEN FI is high IF SFL is high and GR is high and SP is high THEN FI is high IF SFL is high and AT10/AT90 is high THEN FI is high

CASE 7

IF CAL is high and GR is high and SP is high THEN FI is high IF DT is high and GR is high THEN FI is high IF NPOR-DPHI is high and DRHO is high THEN FI is high IF SFL is high and AT10/AT90 is high and ILD/ILM is high THEN FI is high