IDENTIFICATION AND CHARACTERIZATION OF NATURALLY FRACTURED RESERVOIRS USING CONVENTIONAL WELL LOGS

Transcription

1 IDENTIFICATION AND CHARACTERIZATION OF NATURALLY FRACTURED RESERVOIRS USING CONVENTIONAL WELL LOGS Liliana P. Martinez, Richard G. Hughes and Michael L. Wiggins The Uniersity of Oklahoma ABSTRACT In petroleum exploration and production, fractures are one of the most common and important geological structures, for they hae a significant effect on reseroir 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 reseroirs using conentional well logs. Logs are the most readily aailable source of information, howeer they are seldomly used in a systematic manner for quantitatie analysis of naturally fractured reseroirs. 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 conentional well logs. Additionally, a self-consistent model from OConnell and Budiansky for the prediction of elastic properties of fractured porous rocks is inerted using genetic algorithms to obtain crack density and crack aspect ratio. The proposed algorithms are tested using data aailable from the Mills McGee #, an Austin Chalk formation well in Milam County, Texas. The results obtained are compared with core information aailable. INTRODUCTION The word fracture is used as a collectie term representing any of a series of discontinuous features in rocks such as joints, faults, fissures and/or bedding planes. Fractures hae a significant effect on both the mechanical and hydraulic properties of rock masses. They can hae either a positie or negatie impact on flow rates and recoeries from fractured reseroirs 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 reseroirs under primary, secondary or tertiary recoery, 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 ealuated by seeral techniques, with the most common being core analysis, well log analysis and pressure transient analysis. Core analysis can proide quantification of fracture geometry, fracture frequency and the nature of any filling material. The disadantages of core analysis for naturally fractured reseroir ealuation are that it is difficult to assess how representatie the core plugs are of the entire reseroir, cores that contain fractures of practical significance are often lost in the process of recoery, 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 intensie and subject to the aailability of drilled rocks.

2 These factors hae directed the industry to employ well logs that are costeffectie and readily aailable. Among the adantages of well logs oer core analysis are that, while cores are typically obtained oer a portion of a single well, well logs are run oer a significantly larger portion of nearly eery well. Well logs also proide in-situ measurements of the formation at reseroir conditions and proide 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 reseroir ealuation. Pressure transient testing includes generating and measuring pressure ariations in wells. Subsequent analysis of these pressure ariations proides estimates of rock, fluid, and reseroir properties. These properties are aeraged properties oer 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, 998). In this paper, methods that use conentional well logs to deelop 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 OConnell and Budiansky (OConnell, 984) will be used to obtain crack density and aspect ratio. To obtain these parameters requires an inersion of the model where gradient methods cannot be used. Thus, a genetic algorithm approach is utilized. These two models will be ealuated 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 proide a means to ealuate certain formation parameters and the hydrocarbon production potential of a reseroir. Logging tools can be grouped into two categories: conentional and unconentional. Conentional well logs are those that are routinely collected at almost all industry boreholes. Unconentional well logs are either too specialized, expensie or too recently deeloped to be run in eery 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 conentional well logs, while the litho-density, spectral gamma ray, borehole teleiewer (BHTV), and formation microscaner (FMS) logs (among others) are classified as unconentional 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 heaily weighted mud. Alternatiely, borehole elongation may be obsered which occurs preferentially in the dip direction of fractures due to crumbling of the fractured zone during drilling (Fertl, 98). Neither of these log response patterns can be taken as a conclusie eidence 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 selfpotentials between an electrode in the borehole and a reference electrode at the

3 surface. The response of the SP cure in front of fractured zones may exhibit either erratic behaior or a more systematic negatie deflection due to a streaming potential (the flow of mud filtrate ions into the formation). Howeer streaming potentials can also occur near silt beds (Crary et al., 987). The GR log is a record of a formations radioactiity. The GR log is principally used quantitatiely to calculate shale olume. In fractured reseroirs an increase in the gamma ray without concurrently higher formation shaliness, is frequently obsered. This increase has been explained by the deposition of uranium salts along the discontinuity surfaces of a fracture or within the crack itself (Fertl, 98). Natural Gamma Ray Spectroscopy records the indiidual 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., 98). The solubility of uranium compounds accounts for their transport and their frequent occurrence in fractures. The Density Log is a continuous record of the formations bulk density. The dual detector density tool reports two alues: a alue of uncompensated density using a long-spaced detector response, and a alue of density correction ρ. The correction is added to the uncompensated alues to obtain the compensated bulk density, ρ b (Bassiouni, 994): ρ b = ρ ls + ρ...() where ρ ls 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 actie, erratic ρ cure may therefore indicate fractures when the hole is in gauge. Since the density logs are a measure of total reseroir porosity, fractures filled with fluid will decrease the recorded bulk density, creating a sharp negatie peak on the density cure, and a corresponding peak on the ρ correction. The measurement principle of the neutron log is based on the fact that hydrogen is ery efficient in slowing down fast neutrons. Similar to the density log, any neutron-type log is a measure of the total reseroir porosity in fluid saturated formations. Therefore, in the presence of fractures, the neutron log is expected to hae a behaior similar to that of the density log. The litho-denstiy tool reports the measurement of the effectie photoelectric absorption cross-section index for the formation, Pe. The Pe index will report anomalously high alues near mud inaded open fractures. Therefore, a high reading of Pe, with good tool-borehole contact established by the caliper cure, may be a good indicator of fractures (Ellis, 987). Acoustic logging for formation ealuation can be defined as the recording of one or more parameters of acoustic wae trains for use in estimating fundamental rock properties. Acoustic logging includes the measurement of both interal transit time and amplitude/attenuation logging mainly for compressional and shear waes. Sonic interal transit time logging records the time required for an acoustic wae to transerse a gien length of formation. Most acoustic logging tools are designed to detect the first arriing compressional wae only when the energy leel 3

4 reaches a certain threshold. (Jorden et al., 986). If the energy leel does not reach the threshold alue, 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, 98). Acoustic amplitude logging records the energy leel of an acoustic wae 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 conersion that occurs when an acoustic wae 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, 969). Howeer, 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, 976). Resistiity logs are measurements of the ability of the fluids in a formation to conduct electricity. The dual laterlog generally proides three resistiity measures: the deep laterlog (which inestigates about ft. into the formation), the shallow laterlog (which inestigates 3 to 6 ft. into the formation), and the microspherically focused log (MicroSFL) which measures resistiity in the inaded zone. The effect of fractures on resistiity 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 ery conductie anomaly to the microresistiity tools because they see the fractures as entirely filled with mud filtrate. Wellbore imaging inoles recording downhole information of the borehole surface and conerting depth, orientation, and caliper data into two-dimensional images. The most widely used imaging deices are the Formation MicroScanner (FMS) and the Borehole Teleiewer (BHTV). The FMS is a resistiity 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, 969). The transducer rotates rapidly in the borehole. The amplitude and transit time signals can be made with arying frequency transducers. Both the FMS and the BHTV logs proide high-resolution images of the wellbore, therefore they are considered the most direct and effectie methods for detecting fractures in boreholes. Howeer, 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 icinity of the borehole. Finally imaging logs are not as widely aailable as conentional well logs. Since conentional well logs are readily aailable, it is of great importance to understand the effects of fractures on them to try to use the information they proide to attempt a better characterization of naturally fractured reseroirs. A model to integrate conentional well logs into a Fuzzy Inference System to identify the presence of fractures is proposed. 4

5 FUZZY LOGIC AND FUZZY INFERENCE SYSTEMS Real world problems are characterized by the need to be able to process incomplete, imprecise, ague or uncertain information. One approach to account for this uncertainty, deeloped by Zadeh (974), 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 proide insight into the particular problem being soled. A fuzzy set is fully defined by its membership function. For some applications the sets that will hae to be defined are easily identifiable. For other applications they will hae 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 aailable. Normalized log response alues are used in this work. The normalization uses a two-step process. First, a 6-point moing aerage filter (3 ft. total length) is applied to obtain a background alue to which the current data alue is compared. The difference between the log data alue and the background alue is then scaled to be a alue between and. Data scaling is necessary for two reasons. First, it is desired to account for essential ariability in the filtered log data, and, without some type of scaling process, those logs with the largest original ariance would dominate the subsequent analysis. Second, it is desired to hae 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 alues. In this study a linear scaling method that maps the maximum log alue to one and the minimum log alue to zero will be used. The linear scaling has the following form: xi a zi =...() b a where z i is the scaled alue, x i is the original alue, a and b are scaling constants. In this study, a is the minimum log alue and b is the maximum log alue. Once the names of the fuzzy sets hae 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, 997): If (a is A) AND (b is B) AND..THEN (z is Z) where a, b, and z represent ariables (e.g. distance, size) and A, B and Z are linguistic ariables such as cold, warm, hot. Then the phrase a is A is an abbreiation of the compete statement a belongs to the fuzzy subset A with a corresponding membership alue µ A (a) (or to a degree µ A (a)). Two membership functions hae been designed for each well log. Each membership function maps the scaled log data 5

6 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 alues may be related to fractures (sonic, caliper, gamma ray, density, neutron porosity, and resistiity logs) a sigmoidal membership function is used. This membership function is defined as (Roger et al, 997): sig( x; a, c) = + exp( a( x c))...(3) where a is the slope at the crossoer 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.. When erratic log response alues or small ariations about the background alue 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, 997) bell( x; a, b, c) =...(4) b x c + a 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.. The membership function for the output ariable-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 preious 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 hae been defined for each one of the input and the output ariables, the next step is to set the rules required by the FIS to identify fracture intensity from conentional well logs. In this study only those rules that reflect the response of conentional 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 (resistiity is high) and (MSFL is high) then (fracture index is high). - If (resistiity 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 ariable. 6

7 Once the rules hae been composed, the solution, as has been seen, is a fuzzy set. Howeer, for most applications there is a need for a single action or crisp solution to emanate from the inference process. This will inole the defuzzification of the solution set. Lee (99) 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 graity of the solution fuzzy sets. For this work, the center of graity method is used to obtain the output fracture index. Figure 3 shows three of the FIS rules that hae 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 alue for the fracture index at this depth is.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 proide some quantification as to the degree of the presence of fractures in a gien well, the FIS cannot gie quantitatie 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 OConnell and Budiansky was presented elsewhere (Martinez et al., ). In this work, the same approach is used, but the solution to the problem is obtained using genetic algorithms. OCONNELL AND BUDIANSY SELF CONSISTENT MODEL. This model assumes that the effectie moduli of a composite of porous elastic materials depend on the properties of the indiidual components, the olume fractions of the components and the geometric and spatial characteristics of the components. OConnell (984) 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 OConnell (984) and also that found in Martinez et al. (). The main adantage of this method oer preiously presented models is its ability to account for interaction between cracks and pores. The model considers a solid, permeated with two classes of porosity: cracklike, 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 olume 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, f, the bulk and shear moduli of the uncracked non-porous matrix material, o and G o, the frequency, ω and the characteristic frequency for fluid flow between cracks, ω s. The crack density is defined by A ε = N *... (5) π P 7

8 8 where N is the number of cracks per unit olume; 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: 3 4 α µ ϖ s...(6) where µ is the iscosity of the fluid, α is the aspect ratio of the crack which is gien by: α = c/a, with c and a the minor and major axis lengths of the crack (OConnell, 984). With the preious definitions, the crack porosity is (Mako et al., 998): πεα φ 3 4 = c...(7) The moduli are considered to be a function of the frequency, ω, and are complex quantities (i.e. = r + i c, and G = G r + ig c ). The real parts of these moduli represent effectie elastic moduli, and the imaginary parts represent anelastic energy dissipation. The complex bulk modulus,, is gien by (OConnell, 984): φ ε φ ε φ Ω Ω + + = i i f f o f o (8) With: s o ϖ ϖ Ω = (9) The shear modulus G is determined using (OConnell, 984): φ ε + Ω + = 3 45 ) 3( 5 7 ) 5( i G G o... () Ω + = i o ε φ...() where ν is a fictitious Poisson ratio that satisfies: G G =... () The same type of expression relates the moduli and Poissons ratio of the porous solid: G G =...(3) A direct solution of this model is not possible and an iteratie procedure is required. In Martinez, et al. (), a solution was obtained by a trial-and-error method. In this paper, the OConnell and Budiansky model was inerted using a genetic algorithm to obtain fracture porosity and fracture aspect ratio using conentional well logs.

9 INVERSION OF THE OCONNELL AND BUDIANSY MODEL The inersion of the OConnell 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 objectie function. If the real parts of the moduli ( r and G r ) are known, the inersion of the model will consist of obtaining the parameters ε and (c/a) that will minimize an objectie function gien by: F = + G G... (4) r rcal r rcal where rcal and G rcal are the real portions of the bulk and shear moduli obtained from the OConnell and Budiansky model. This particular problem is not suitable to be soled through the conentional 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 ariables, r, c, c, G c, α, ε, each with an eight-bit resolution. An initial population of 3 chromosomes was generated. Each chromosome was first randomly generated and then ealuated in order to guarantee that it was within the solution space. The randomly generated chromosome was decoded to obtain the generated alues for r, c, c, G c, α, and ε. Equations () and (3) were then used to erify that the Poissons ratios were in the range between and. 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 ealuated until a population size of 3 chromosomes was obtained. Eery chromosome within the population was ealuated using Eqs. (8), () and () with the real portions of the bulk and shear moduli as output parameters ( rcal and G rcal ). The r and G r terms are either obtained experimentally or can be computed when the compressional and shear wae elocities and the bulk density are known. The F alue computed from Eq. (4) was taken as a fitness alue for the generation; smaller alues of F are better or more fit than chromosomes that hae higher alues 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 alue. - The two parents were combined randomly to generate two new offspring. These two new chromosomes were ealuated for fitness. A new set of parents was selected, and the process was repeated until a new population of 3 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.. The process described was repeated for generations. At the end of the th generation, if the fitness alue of the best chromosome was greater than., the process was repeated. The optimum solution to the problem was the chromosome with the lowest fitness alue amongst all of the generations. 9

10 CASE STUDY Boreholes that are suitable for case study are rare. In order to apply the Fuzzy Inference Algorithms, seeral conentional well logs that are useful for fracture detection must be aailable, and in order to apply OConnell and Budiansky inerse algorithm, the suite of logs aailable 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 aderse 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, proided special core analysis characterizing the natural fractures for this well (Lorenz, 997). The Mills-McGee # 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 4 and 5). In the area of the well, the formation produces from fine-grained limestone and chalk. Productiity and fluid flow directions are predominantly controlled by the presence of fractures. The well was logged using a comprehensie suite of state-of-the-art tools. In addition, ft of core was obtained and analyzed by the UPR rock lab and by Sandia National Laboratory. Anadarko Petroleum Corp. proided the well log digital data, and Sandia National Laboratory proided 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. 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 exclusie. They were inferred to be components of a single fracture population since they had similar characteristics. Of the 775 fractures that were obsered, 36 were of the semi-open ariety. The remainder were hairline fractures. Mechanically induced fractures were identified from the cores and excluded from the analysis (Lorenz, 997). Well logs aailable and used in this study for the Mills McGee well are listed in Table. 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 586 and 64 ft. The good borehole condition means that the log measurements should be ery reliable. The conentional SP log is presented in Fig. 4b. It is possible to obsere seeral zones with an erratic SP behaior presumably due to the presence of fractures. The behaior of the gamma ray log as shown in Fig. 4c is not a conclusie fracture indicator. The peaks that this log displays in the interal between 587 ft and 6 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 obsered, and the compresional trael time alues are those corresponding to

11 chalk/shale lithology. Neutron porosity and density porosity are presented in Fig. 4e. In this case the density porosity alue 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 alue of bulk density in the interal between 587 and 67 ft.; howeer, the density correction log reports large correction alues in the interal between and 594 ft. Since the caliper in this interal reports a gauge hole, the abnormalities in the density correction alues may be due to the presence of fractures in this interal. 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 seeral fractured zones in the interal shown. Figure 4i presents the shallow (AT), medium (AT6) and deep (AT9) resistiity logs. It is possible to obsere some separation between the shallow and deep resistiity logs, especially between 586 and 59 ft. In general though, there is no significant separation between the two resistiity measurements. The litho-density (Pe index) log, Fig. 4j, reports a fairly constant alue in the range between 4.5 and 5.5, which is in agreement with the lithology expected in this well. No indiidual 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 seen 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 ery 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. 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 interal between 59 and 593 ft. All the cases presented in Fig. 5, are able to recognize that interal as one with high fracture presence. The correlation between the core analysis and the log analysis for the set of fractures between 5965 and 599 ft is not as clear. The only case that is detecting this fractured zone is Case 3. This case has a high noise leel, howeer, 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 interals. The fractured interal 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 obsered in zones where core analysis does not report fractures might indicate the presence of mechanically induced fractures that are excluded from the core description. Howeer, the FIS algorithm may be responding to hairline fractures that are not accounted for in the core description proided by Sandia National Labs. Case 6 seems to be the most appropriate for this specific example. Case 6, besides haing the same rules as Case 5, has two additional rules inoling the resistiity and the SFL log. It is then possible to conclude that induction and

12 resistiity logs play a ery important role in detecting fractures for this well. Case 7 is a combination of all the logs aailable for this well. From Fig. 5g it is possible to conclude that een though this combination of logs seems to be identifying most of the fractured interals in the well, there is also a relatiely high noise leel in the fracture index. Howeer 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 OConnell and Budiansky model. The fracture porosity reported in Fig. 6c clearly identifies seeral of the high fracture frequency zones reported by the core analysis, specifically the interals at ft and 6-67 ft. Howeer, 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 7 hairline fractures that are not classified as open or semi open fractures in the interal analyzed and may be affecting the fracture porosity results. The whole interal reports a relatiely 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 to to to. using a linear scaling as in Eq. ()), and the fracture porosity. From Fig. 7, it is possible to obsere a fairly good correlation between the results obtained with the FIS and the inersion of the OConnell and Budiansky model in most of the interals 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 seeral 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 recoered 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 conentional well log proides reliable characterization of the distribution and geometric characteristic of the fractures in the wellbore, howeer 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 reseroirs, using only information that can be obtained from conentional well logs. Fuzzy logic proides an alternatie method to handle uncertainty. This is especially useful in reseroir characterization where the data aailable is not always ery precise and does not usually proide direct indication of the reseroir properties that goern the flow of fluids in the porous media. Detection and description of naturally fractured reseroirs remains a difficult task. Howeer 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 aailable into a single tool to identify the presence of fractures. Furthermore, the fracture index obtained through the FIS may gie a direct indication of the fracture porosity. The FIS is ery sensitie 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 resistiity logs. Howeer this combination may not be the optimum one in another location. Each field is unique and requires indiidual analysis. There is a good correlation between the fracture index obtained with the Fuzzy Inference algorithm and the ones obsered in the proided core analysis, indicating that the algorithm is able to detect open and semi-open fractures when the appropriate suite of conentional logs are proided. 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 OConnell 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 seeral 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 ealuation of the rules. In order to generalize the FIS used to obtain the fracture index, and the method proposed to inert OConnell and Budiansky model, as reliable methods for the characterization of naturally fractured reseroirs, extensie field data along with experimental data is needed for analysis. ACNOWLEDGMENTS We are grateful to the U.S. Department of Energy for financial support of this project through U.S. DOE Grant No. DE-AC6-99BC5. 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 proiding the core analysis and log data files respectiely. NOMENCLATURE Symbols A = Area in plain-form of the crack. a = Crack radius. FI = Fracture index. G = Shear modulus. G o = Shear modulus of mineral material making up rock. G rcal = Calculated real portion of the shear modulus. h = Formation thickness = Effectie bulk modulus of the rock with pore fluid. = Effectie bulk modulus of pore fluid. f 3

14 o rcal = Bulk modulus of mineral material making up rock. = Calculated real portion of the bulk modulus. N = Number of crack per unit olume. P = Perimeter of the crack. V p = P-wae elocity. V s = Shear wae elocity. = Poisson ratio. = Fictitious Poisson ratio. Well log symbols AT, AT6, AT9 = Shallow, medium and deep resistiity 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. φ f = Fracture porosity. φ m = Matrix porosity. ρ = Bulk density. µ A (x) = Degree of membership of element x in the fuzzy set A. µ = Viscosity of the fluid. ω = Frequency. = Characteristic frequency for fluid flow between cracks. ω s REFERENCES Aguilera, R.: Analysis of Naturally Fractured Reseroirs From Conentional Well Logs, Journal of Petroleum Technology, p , July 976. Bassiouni, Z., Theory, Measurement, and Interpretation of Well Logs. Society of Petroleum Engineers, SPE Textbook Series Vol. 4, 994. Crary, S. et al.: Fracture Detection With Logs, The Technical Reiew. V. 35, no., p. 3-34, 987. Elkewidy, T. I. and Tiab, D.: An Application of Conentional Well Logs to Characterize Naturally Fractured Reseroirs with their Hydraulic (Flow) Units; A 4

15 Noel Approach, SPE paper 438 presented at the SPE Gas Technology Symposium held in Calgary, Canada, 5-8 March, 998. Ellis, D. V. Well Logging For Earth Scientists. Elseier Science Publishing Co., New York, 987. Fertl, W. H.: Ealuation of Fractured Reseroirs Using Geophysical Well Logs, SPE paper 8938 presented at the 98 SPE/DOE Symposium on Unconentional Gas Recoery held in Pittsburgh, Pennsylania, 8- May, 98. Guyod, H. and Shane, L. E., Geophysical Well Logging, Houston, TX., 969. Lee, C. C.: Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part II, IEEE Transactions on Systems, Man and Cybernetics, (): , 99. Lorenz, J.: Description and Preliminary Interpretations of Fractures in the Mills McGee # Core, Personal Communication to Tom Zadick. August 8, 997. Martinez, L., Gupta, A. and Brown, R., Interpretation of Important Fracture Characteristics from Conentional Well Logs, SPE paper 678 Presented at the Production and Operations Symposium, 5-8 March,, Oklahoma City, O Mako, G., Mukerji, T. and Dorkin, J., The Rock Physics Handbook: Tools For Seismic Analysis In Porous Media, Cambridge Uniersity Press, 998. Nauck, D., lawonn, F. and ruse, R., Foundations of Neuro-Fuzzy Sistems, John Wiley & Sons, U.., 997. OConnell, R.J.: A Viscoelastic Model of Anelasticity of Fluid Saturated Porous Rocks, Physics and Chemistry of Porous Media, AIP Conf. Proceedings, p Roger, J.S., Sun, C.T. and Mizutani, E., Neuro-Fuzzy and Soft Computing. Prentice Hall, Upper Saddle Rier, NJ., 997. 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 st Society of Professional Well Log Analysts Logging Symposium. 98. Zadeh, L. A.: Fuzzy Logic And Its Application To Approximate Reasoning, Information Processing, 74:59-594, 974. Zemanek, J. and Caldwell, R. L.: The Borehole Teleiewer A New Logging Concept For Fracture Location and Other Types of Borehole Inspection, Journal of Petroleum Technology, p , June 969 5

16 High likelihood Low likelihood.. Scaled Deiation From Background Fig. : Sigmoidal Membership Function High likelihood Low likelihood.. Scaled Deiation From Background Fig. : Generalized Bell Membership Function 6

17 GR = High SP=High Fracture Index = High.. GR.. SP.. Fracture Index If GR is high And SP is high Then Fracture Index is high Resistiity=Low.. Resistiity MSF=Low.. MSF Fracture Index=Low.. Fracture Index If Resistiity is low And MSF is low Then Fracture Index is low T = High Caliper=Low Fracture Index=High T Caliper Fracture Index If T is high And Caliper is low Then Fracture Index is high Fracture Index =.3.. Fracture Index Fig. 3: Fuzzy Inference Process 7

18 CALI 8 4 SP - GR 4 8 DT NPOR DPHI...3 RHOB DRHO. ILM ILD SFL AT9 AT6 AT PEF (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Fig. 4: Mills-McGee # raw well log data 8

19 Case Open fractures Case Open fractures Case3 Open fractures Case4 Open fractures Case5 Open fractures Case6 Open fractures Case7 Open fractures (a) (b) (c) (d) (e) (f) (g) Fig. 5: Fuzzy Inference System Cases 9

20 Fracture density Open fractures Aspect Ratio Open fractures Fracture porosity Open fractures (a) (b) (c) Fig. 6: OConnell and Budiansky model inerted parameters

21 case6 Open fractures Fracture porosity.5. Fig. 7: Fracture Porosity and Fracture Index

22 Well log Caliper Spontaneous Potential Gamma Ray Bulk Density Density correction Photoelectic factor Sonic transit time Table. Well Logs aailable for the Mills McGee Well # Shallow/Deep Induction combination Spherically Focused Log Fracture detection significance Registers borehole enlargement that may be caused by the presence of fractures. May indicate the presence of fractures, but, is not considered a reliable fracture detection tool. (Schlumberger Ltd., 989) Without the spectral gamma ray data, the gamma ray log by itself is not conclusie in fracture detection. Open fractures filled with drilling fluid may cause a reduction in bulk density (Fertl, 98). It is considered one of the best fracture detection tools among the conentional well logs (Serra, 986) Has been recognized as a useful fracture detection tool (Schlumberger, 989) Fractures are know to cause cycle skipping (Bassiouni, 994) The resistiity of the deep induction tool will exceed the one for the shallow induction tool Often exhibits erratic alues in the presence of fractures, but is sensitie to poor borehole conditions. (Schlumberger, 989)

23 CASE CASE CASE 3 CASE 4 CASE 5 CASE 6 CASE 7 Table. Set of Rules used in the different cases for the FIS IF SFL is high and AT/AT9 is high THEN FI is high IF DRHO is high and PEF is high THEN FI is high 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 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 AT/AT9 is high and ILD/ILM is high THEN FI is high 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 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 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 AT/AT9 is high THEN FI is high 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 AT/AT9 is high and ILD/ILM is high THEN FI is high 3