oilfield_review_res_modeling

Barbara Anderson Gerald Minerbo Michael Oristaglio

Schlumberger-Doll Research Ridgefield, Connecticut, USA T om Barber Bob Freedman Frank Shray

Schlumberger Well Services Houston, Texas, USA

22Oilfield Review

Later, tool responses were studied using mock-up sondes in more realistic environ-ments created by using thin impermeable membranes to separate waters of different salinity. For a number of years, a resistor network was used at the Schlumberger-Doll Research laboratory in Ridgefield, Connecti-cut USA. This network, consisting of tens of thousands of electrical resistors, simulated resistivities in borehole, invaded zone and virgin formation. In addition, theoretical cal-culations of sonde responses to layered and invaded formations generated books of departure curves. This theoretical approach was especially important for tools that had large depths of investigation or were not readily adaptable to laboratory https://www.360docs.net/doc/6115496309.html,rge improvements in computing capa-bility have introduced qualitative changes in

interactive modeling is possible even on personal computers.1Program packages for simple one-dimensional (1D) modeling are commonplace; two-dimensional (2D) and three-dimensional (3D) modeling are practi-cal in many special cases, although they generally require the use of mainframes or supercomputers. T wo-dimensional modeling permits examination of axially symmetric radial variations—for example, treating zero-dip layering and coaxial invasion simultaneously. Three-dimensional model-ing also handles azimuthal variations such as circumferentially irregular caves or inva-sion, sonde eccentering and dipping beds.

Resistivity modeling was used as far back as 1927, when Conrad Schlumberger first rea-soned how current from an electrode spreads out into the formations around a borehole. But he would have called it “the-orizing.” Characteristics were assigned to the formation (the formation model) and the laws of physics, usually in idealized form,were used to predict analytically the response made by some electrode configu-ration (the sonde, or tool, model) to the modeled formation. Both theoretical and experimental modeling have passed through many stages since then.

Early experimental modeling used small electrodes in “infinite” saltwater baths.the role of modeling during the last https://www.360docs.net/doc/6115496309.html,puterized modeling has reduced from weeks to minutes the time required to cal-culate many effects of tool design changes.One can now systematically explore the effects of environmental conditions such as borehole rugosity and caves, mudcake,invasion, dip, shoulder beds, and formation anisotropy on resistivity tool responses.

The latest stage in this evolution is aimed at providing rapid, low-cost log interpreta-tion through the use of fast computers with large, high-speed memories and efficient programs. Recently, even massively parallel processing has been introduced to serve these goals. Some log interpretation by

Resistivity modeling is shortening the learning curve in gaining understanding of the reservoir.

Although almost as old as logging itself, resistivity modeling is an integral part of the latest developments,from steering horizontal wells to investigating the effects of anisotropy.

Modeling Electromagnetic Tool Response

WELL LOGGING

For help in preparation of this article, thanks to Charles Flaum, Services Techniques Schlumberger, Montrouge,France; Stan Gianzero, Austin, Texas, USA; Martin Lüling, Schlumberger-Doll Research, Ridgefield, Con-necticut, USA; Bill MacGregor, Schlumberger Austin Sys-tems Center, Austin, Texas, USA; Richard Rosthal,

Schlumberger Well Services, Houston, Texas, USA; Liang C. Shen, Well Logging Laboratory, University of Hous-ton, Houston, Texas, USA; Julian Singer, Schlumberger,New Delhi, India.

In this article, AIT (Array-Induction Imager tool), CDR (Compensated Dual Resistivity), EPT (Electromagnetic Propagation Tool), Phasor and MicroSFL are marks of Schlumberger. Cray is a mark of Cray Research, Inc.Connection Machine is a mark of Thinking Machines Corp. VAX is a mark of Digital Equipment Corp.1. Georgi DT, Phillips C, Hardman R: “Applications of Digital Core Image Analysis to Thin Bed Evaluation,”Society of Core Analysts Transactions , paper 9206,June 14-17, 1992, Oklahoma City, Oklahoma, USA,(in press).

2.“Vertical Resolution of Well Logs: Recent Develop-ments,” Oilfield Review 3, no. 3 (July 1991): 24-28.The vertical response function describes the character-istic response of the tool as it passes, perpendicularly,an infinitely thin bed.

July 1992Modeling Versus Inversion

The distinction between modeling—fre-quently “forward” modeling—and inversion is sometimes muddied. The latter typically attempts to “back out” true resistivity, R t ,directly from the log with a minimum of assumptions. The best known example of this approach is vertical deconvolution through the use of inverse filters. In its purest form, this method requires only that the vertical response function (VRF) of the tool be known accurately.2In practice, VRFs are usually formation dependent, so approx-imations must be used. Nevertheless,deconvolution has been employed success-fully, running in real time on logging unit computers. Artifacts may appear, however, if inverse filters overreach in trying to achieve fine vertical resolution or if the 2D assump-tions implicit in the filter are violated.

In modeling, on the other hand, the ana-lyst suggests an environmental model. This trial model includes a description of the borehole and formation geometry and “parameter values”—numbers assigned to variables such as borehole diameter and bedding dip, thickness and resistivity. Then,the tool physics —a model in its own right—is used to compute an expected log,which is compared with the field log. If the match isn’t good enough, the initial trial model is altered and the calculation repeated. This process is iterated until the two logs match satisfactorily. Several criteria for the quality of match are used, from sim-ple eyeballing to the more sophisticated least-squares and maximum entropy meth-ods described later. The model’s geometry or parameter changes are executed interac-tively, using the analyst’s intuition and expe-rience, or automatically, if computers and programs of sufficient power are available. Modeling intrinsically yields consistency with the field log, even though the solution isn’t unique. This nonuniqueness is seldom

a serious problem, however, because the range of possible formation models can be severely constrained by local knowledge from cores and logs. An extreme example of this condition shows two grossly different models that predict the same deep induc-tion log (above ). But in practice, almost any additional log with vertical resolution of about 1 foot [30 cm] or less (gamma ray,EPT Electromagnetic Propagation T ool, dip-meter or photoelectric factor, Pe ) would

resolve this ambiguity.

n The lack of uniqueness in forward mod-eling. Different formation models can lead to nearly identical induction logs.The ambiguity is resolved, however, by addition of nearly any log with high verti-cal resolution.

tions include induction and laterologs in multiple horizontal beds with borehole and invasion, and induction and CDR Compen-sated Dual Resistivity logs in multiple dip-ping beds without borehole or invasion.Other tool environments and fast induction codes are being evaluated for addition to the package. Other codes in this program can be used for dipping bed interpretation. In the dipping bed interpretation using the ELMOD program, bed boundaries were pro-vided by the Phasor deep induction log, and an apparent dip of 38°by the dipmeter log.5Discrepancy between the field induction log and initial computed log led to revision of the trial model and a recomputed log (next page, top ). This improved the fit, but one more iteration—fine-tuning the shapes of some beds and adjusting for overcompensa-tion—yielded an excellent visual match (ELMOD Simulation Three). Although the final model is not a unique representation of

Examples of Formation Evaluation

The economic importance of modeling is illustrated by a North Sea reserves calcula-tion based on induction log interpretation carried out with a Schlumberger program called Induction Sonde in Multilayered Media (ISMLM).3,4This is a 1D induction modeling code for layered media that neglects borehole and invasion effects. It handles up to 150 parallel dipping layers.Invasion was considered negligible because the well was drilled with oil-base mud.

The measured deep induction log, initial trial formation model and computed log are shown in the first model (above, left ). High-resolution details of the trial model were provided by an EPT log. The effects of the first model revision are based on the ana-lyst’s experience. The analyst changes the thicknesses of conductive beds, adds layers to the sands, and improves the depth match

24Oilfield Review

between the induction and EPT curves.Since visible discrepancies between the field log and the modeled log remained, fur-ther model revisions were needed to achieve the final results (above, right ). The final model reduced the well’s estimated average water saturation from 9.7% to 7.2%. Because the hole is deviated 56°, the log-measured depths (MD) are greater than the true vertical depths (TVD), and bed thicknesses are similarly magnified. This MD expansion of scale, obvious in highly deviated wells, will be observed again in a later example.

Subsequent to this work, the ISMLM code was made part of the Electromagnetic Mod-eling package, called the ELMOD program.The program consists of 1D and 2D codes that compute the responses of electric log-ging tools to models of downhole environ-ment. The programs can be run at any Schlumberger Data Services Center or on a Schlumberger VAX workstation. Configura-

n The importance of modeling. For a well inclined 56°,an initial trial

model and its com-puted deep induc-tion log are com-pared with the field log (model one).Modification of the model leads to bet-ter agreement

between computed and field logs, but some discrepancies remain (model two). The final model (right) pro-duces nearly per-fect agreement.Depths in models one and two are on a log-measured scale, while the

final results are pre-sented on a true vertical depth scale.The two scales are different because the well is slanted.

n Modeling simulation using the ELMOD program for interpretation of a deep induction Phasor log in a North Sea well with apparent dip of 38°. The initial trial model was refined in two steps, left to right, until agreement was reached between the model-computed and field logs. Simulation Three was consistent with the log analyst’s knowledge of the field.

3. Fylling A and Spurlin J: “Induction Simulation, The Log Analysts’ Perspective,” Transactions of the SPWLA 11th European Formation Evaluation Symposium ,Oslo, Norway, September 14-16, 1988, paper T.

4. Details of the ISMLM program:

Anderson B and Gianzero S: “Induction Sonde Response in Stratified Media,” The Log Analyst 24,no.1 (1983): 25-31.

Anderson B, Safinya KA and Habashy T: “Effects of Dipping Beds on the Response of Induction Tools,”paper SPE 15488, presented at the 61st SPE Annual T echnical Conference and Exhibition, New Orleans,Louisiana, USA, October 5-8, 1986.

Anderson B and Chew WC: “A New High Speed T echnique for Calculating Synthetic Induction and DPT Logs,” Transactions of the SPWLA 25th Annual Well Logging Symposium , New Orleans, Louisiana,USA, June 10-13, 1984, paper HH.

true resistivity, it was consistent with the analyst’s knowledge of the field, and the predicted water saturations were accorded a high degree of confidence.

Many more computer programs for a vari-ety of electrical logging measurements have been developed by service companies, oil companies and universities. Some are avail-able for commercial use, others only for research purposes (see “Summary of Induc-tion Modeling Programs,” above ).

July 1992

ELMOD Simulation One

ELMOD Simulation Two

ELMOD Simulation Three

D e p t h , f t

1000

1050

1100

1150

1.Well Logging Technical Report,No. 7. Houston, Texas, USA: University of Houston Well Log-ging Laboratory, October 30, 1986. These programs are available to supporters of the labora-tory, a consortium including most of the major oil companies.

2. Anderson B and Gianzero S, reference 4.

Anderson B, Safinya KA and Habashy T, reference 4.

3. Anderson B and Chew WC, reference

4.Anderson B and Chang SK, reference 13.4. IBM 3090

5. VAX 11/780

5. Anderson B, Barber TD, Singer J, and Broussard T:“ELMOD—Putting Electromagnetic Modeling to Work to Improve Resistivity Log Interpretation,” Transactions of the SPWLA 30th Annual Logging Symposium , Den-ver, Colorado, USA, June 11-14, 1989, paper M.

Modeling for Bit Guidance in Horizontal Drilling

Forward modeling is serving needs other than conventional log interpretation, such as guiding the bit while drilling deviated or horizontal wells and evaluating the influ-ence of adjacent (shoulder) beds on hori-zontal well logs.

Recent modeling studies in horizontal wells make clear that conventional rules-of-thumb don’t apply. Predictions have been made of responses of induction and focused electrode tools to shoulder beds,6an impor-tant subject when logging thin beds or in horizontal wells exhibiting vertical drift.Because the induction tool is relatively unaffected by the borehole, its calculations were carried out analytically. For the focused electrode tools, however, the bore-hole is an essential part of the problem, so forward modeling was carried out with a 3D finite-element method (FEM) computation, a much larger enterprise. This investigation concluded that the electrode devices are more sensitive to conductive than resistive shoulder beds, indications of which had appeared earlier,7and that the opposite is true for induction tools. This behavior is the reverse of that observed in vertical wells. In the calculated responses of these tools to the boundary between two beds of 1 ohm-m and 10 ohm-m, the characteristic polar-ization horn is clearly visible on the induction curve (above, right ). This horn appears when surrounding beds have high resistivity contrast and the bed boundary dips more than about 45°.8It is created by oscillating polarization charges induced at the bed boundary.

Another modeling study explored the effects of dipping beds and laminated for-mations on induction and CDR tool responses.9Since the CDR model assumes

point dipoles, the code was first qualified by comparison with exact FEM calculations for horizontal multiple beds and with test tank experiments covering dips of 0°to 90°. This study predicted that both dip and shoulders can cause shallow resistivity, R ps , and deep resistivity, R ad ,to separate, with R ps reading closer to R t . Prominent polarization horns appear at high-dip bed boundaries when resistivity contrast is large, and the CDR tool makes resistivity anisotropy apparent (R ps >R ad ), the effect increasing with dip angle.Oxy USA used forward modeling exten-sively while drilling horizontally into the Cruse sand in La Salle Parish, Louisiana,USA.10T o avoid the problem of water con-ing they had observed in vertical wells, Oxy engineers planned to penetrate horizontally into the top 10 feet [3 m] of the 40-foot [12-m] thick Cruse. Resistivity models of marker beds and of the pay sand itself were created

Resistivity, ohm-m Resistivity, ohm-m

-2

-4

D i s t a n c e f r o m b o u n d a r y , f t

n Shoulder-bed effects on induction logs and laterologs in horizontal wells, calculated by modeling. These curves show how traditional heuristic thinking does not apply: in horizontal wells, laterolog tools are more sensitive to conductive than to resistive shoul-der beds, while the opposite is true for induction tools.

26Oilfield Review

from induction logs in two nearby vertical wells. Then, logs expected in the horizontal well were computed for the CDR tool used in Logging While Drilling (LWD). This 2-MHz resistivity tool provides two outputs:R ps from a phase-shift measurement, and R ad from attenuation.11No one knew accu-rately in advance what the logs would look like as the well curved through the markers and into the Cruse. In addition, actual bore-hole inclination is often not exactly as planned. Therefore, prior to drilling,ELMOD’s dipping-bed code was used to compute CDR logs expected at several apparent dips.9Thus, the right modeled log would be available immediately for com-parison with the field log when the actual relative dip became known while drilling. Since the two CDR resistivity measure-ments have different depths of investigation,the curves are predicted to separate just

6. Gianzero S, Chemali R and Su S-M: “Induction, Resis-tivity, and MWD Tools in Horizontal Wells,” Transac-tions of the SPWLA 30th Annual Logging Symposium ,Denver, Colorado, USA, June 11-14, 1989, paper N;also in The Log Analyst 31 (May-June, 1990): 158-170.Burgess T and Voisin B: “Advances in MWD Technol-ogy Improve Real Time Data,” Oil & Gas Journal 90,no. 7 (February 17, 1992): 51-61.

7. Chemali R, Gianzero S and Su SM: “The Dual Lat-erolog in Common Complex Situations,” Transactions of the SPWLA 29th Annual Logging Symposium , San Antonio, Texas, USA, June 5-8, 1988, paper N.

8. Anderson B, Barber TD and Lüling MG: “The Role of Computer Modeling in Log Interpretation,” Transac-tions of the SPWLA 13th European Formation Evalua-tion Symposium , Budapest, Hungary, October 23-25,1990, paper L.

9. Anderson B, Bonner S, Lüling MG, and Rosthal R:“Response of 2-MHz LWD Resistivity and Wireline Induction Tools in Dipping Beds and Laminated For-mations,” Transactions of the SPWLA 31st Annual Logging Symposium , Lafayette, Louisiana, USA, June 24-27, 1990, paper A.

10. Leake J and Shray F: “Logging While Drilling Keeps

Horizontal Well on Small Target,” Oil & Gas Journal 89 (September 23, 1991): 53-59.

11. Clark B, Lüling MG, Jundt J, Ross M and Best D:

“A Dual Depth Resistivity Measurement for FEWD,”Transactions of the SPWLA 29th Annual Logging Symposium , San Antonio, Texas, USA, June 5-8,1988, paper A.

12. “Formation Anisotropy: Reckoning With its Effects,”

Oilfield Review 2, no. 1 (January 1990): 16-23. Anderson et al, reference 8.

n Modeled (left)and measured CDR resis-tivity logs as the tool curves into a pay sand in a horizontal well. The measured R ps and R ad curves (right)show a charac-teristic crossover as the tool crosses from the cap shale into the sand. Just below the boundary, R ps exhibits a polarization horn created by induced oscillating charges at the interface. The logs com-puted from a model assuming an 85°

inclination show the same characteristics.This inclination expands the MD scale by a factor of 11.47 relative to the TVD scale.(At 85°, 1 foot of TVD equals 11.47 feet of MD [30 cm of TVD = 344 cm of MD].)

July 1992How Modeling Calculations Are Carried Out

Prior to starting the modeling process, many analysts apply chartbook corrections to the field log. These corrections make the field log more accurate—closer to R t , for exam-ple—and allow the use of simpler models and faster computer programs for modeling.Then, the corrected field log and all other constraining information are used in setting up the initial trial formation model. Less fre-quently, the uncorrected field log is used,and the burden of accounting for features like borehole, invasion, shoulder and skin effect is borne by the formation model, tool model and computing code. In this case,the environmental corrections are accounted for simultaneously, as preferred,rather than sequentially, as when applying chartbook corrections. Unfortunately, this approach requires large programs and long computer times.

Although further advances in computing power will likely change the picture in the future, current practice usually requires the log analyst to construct an initial trial model and propose changes at each iteration.“Feel” or “intuition” are frequently the basis for doing this, often acquired from field experience or published studies of tool responses to specific environmental fea-tures. In the last five years there has been a surge of such studies that, themselves, were

carried out by modeling (see “T ool Responses to Environmental Features,” below ).

Even when forward modeling strives only for formation description, most of the bur-den is on codes that implement the tool model—a representation of hardware and physics for calculating how the tool responds to its environment. This model may be approximate because it idealizes tool hardware—treating finite-size coils or electrodes as points, for example. Or it may simplify the physics—as by using the geo-metric factor approach rather than Maxwell’s equations in calculating induc-tion tool responses. Alternatively, the hard-ware or the physics, or both, may be treated exactly. The decision is a trade-off between accuracy and computing efficiency. In either case, some particular computation algorithm becomes the vehicle for arriving at the goal. Purely analytical methods, using exact mathematical solutions, employ codes that run rapidly and require only modest com-puter memories. This makes them well suited to the small computers readily avail-able to most log analysts, but they are intrin-sically limited to simple geometries such as invasion with no layering or layering with no invasion.

From this standpoint, numerical methods are ideal. They break intractable mathemati-cal problems into smaller, more manageable pieces. Numerical methods, such as 2D-and 3D-FEM codes, can solve differential equations in almost any geometry. The FEM is widely used in research and engineering,from the design of automobile bodies to the study of diffusion over corrugated surfaces.A typical logging application is the numerical solution of Maxwell’s equations for induction tools. This problem eventually

reduces to solving a (usually) large number of simultaneous linear equations by matrix methods.13The immediate objective is to find the electromagnetic field’s vector poten-tial at the nodes, or intersections, of a 3D grid, in the most general case. Simpler grids (above ) can be used in solving axially sym-metric (2D) problems. The complete grid may extend hundreds of feet vertically and radially to adequately cover the electromag-netic field. In one modeling exercise, the grid was terminated where the vector poten-tial had fallen 15 orders of magnitude—to zero for practical purposes—from the start-ing point near the transmitter. Grid size increases with distance from the transmitter in regions where both vector potential and generalized geometric factor are falling slowly. This increases computational effi-ciency, with negligible loss in accuracy. In 1982, a CDC CYBER 750 computer typically took five hours to compute 25 feet [7.6 m] of induction log.14T oday, the whole log takes only 15 minutes on a Cray supercomputer.Most pure FEM codes need a fast vector pro-cessor or a Cray unit to run with reasonable turnaround times. Unfortunately, this means interfacing from a remote site, a capability with only limited availability at present.

Hybrid techniques that retain the advan-tages of both purely analytic and purely numerical methods have been developed.They typically break the problem into two

28Oilfield Review

n A small section of a simple grid used in axially symmetric finite-element model-ing calculations of induction logs. In the calculation, the electromagnetic field is determined at each node, or intersection,on the grid. In some cases the complete grid must extend hundreds of feet radially

and vertically.

5040302010

0-10-20-30-40-50

z , i n .

ρ, in.

1. Anderson B: “The Analysis of Some Unresolved Induction Interpretation Problems Using Computer Modeling,” The Log Analyst 27 (September-October 1986): 60-73.

2. Chemali R, Gianzero S and Su SM: “The Effect of Shale Anisotropy on Focused Resistivity Devices,”Transactions of the SPWLA 28th Annual Logging Symposium,London, England,June 29-July 2, 1987, paper H.

3. Anderson B and Barber T: “Strange Induction Logs—A Catalog of Environmental Effects,”The Log Analyst 29 (July-August 1988): 229-243.

4. Reference 9, main text.

5. Grove GP and Minerbo GN: “An Adaptive Borehole Correction Scheme for Array Induction Tools,” Transactions of the SPWLA 32nd Annual Logging Symposium,Midland, Texas, USA,June 16-19, 1991, paper P .

6. Hunka JF , Barber TD, Rosthal R, Minerbo GN, Head EA, Howard AQ Jr, Hazen GA and Chandler RN: “A New Resistivity Measurement System for Deep Formation Imaging and High-Resolution Formation Evaluation,” paper SPE 20559, presented at the 65th SPE

Annual Technical Conference and Exhibition, New Orleans, Louisiana, USA, September 23-26, 1990.7. Reference 7, main text.

8. Strickland R, Chemali R, Su SM, Gianzero S, Walker M, Klein J and Sakurai S: “New Devel-opments in the High Resolution Induction Log,” Transactions of the SPWLA 32nd Annual Logging Symposium,Midland, Texas, USA, June 16-19, 1991, paper ZZ.9. Anderson B, Liu Q-H, Taherian R, Singer J, Chew WC, Freedman B and Habashy T: “Inter-preting the Response of the Electromagnetic Propagation Tool in Complex Borehole Envi-ronments,” Transactions of the SPWLA 32nd Annual Logging Symposium,Midland, Texas,USA, June 16-19, 1991, paper XX.10. Habashy T and Anderson B: “Reconciling Differences in Depth of Investigation Between 2-MHz Phase Shift and Attenuation Resistivity Measurements,” Transactions of the SPWLA 32nd Annual Logging Symposium, Midland, Texas, USA, June 16-19, 1991, paper E.

n Hybrid modeling calculation of deep and shallow lat-erologs in 25 beds.Calculating time increased to 12minutes (compare with figures above),but most of the additional time was used in seg-menting the log and recombining the pieces, rather than in the compu-tation itself.

n Comparison of hybrid and finite-element modeling calculations for deep and shallow laterologs. The two methods yield nearly identical logs, but the

hybrid calculation uses only one-eighth as much computer time

(21/2minutes on an IBM 3081)as the

complete finite-ele-ment calculation.

parts, one of which is attacked analytically and the other numerically. This leads to codes that can quickly solve, even on exist-ing workstations, some problems in fairly complicated geometries. The axially sym-metric problem of the laterolog in a bore-hole through horizontal, layered formations provides a simple example of a hybrid approach. The 2D Laplace’s equation that governs the potential distribution around the tool is reduced to two 1D problems in radial (r ) and vertical (z ) planes, respectively, by the customary separation-of-variables tech-nique. Then, the vertical distribution of the potential can be treated numerically—by the FEM, for example—and the radial distri-bution treated analytically—by modal anal-ysis—or vice versa.14Compare the dual lat-erolog results of this method with a complete FEM calculation in a 5-foot [1.5-m] thick noninvaded bed (left ). The logs are essentially identical, but the hybrid calcula-tion uses only one-eighth as much computer time as the complete FEM calculation. If the hybrid calculation is extended to 25 beds (below, left ), the hybrid time is increased from 2.5 minutes to 12 minutes on an IBM 3081. Most of the additional time, however,was used in segmenting the log into man-ageable portions and then recombining them, rather than in the computation itself.Earlier studies of hybrid techniques describe other methods for modeling induc-tion tool responses.15Numerous codes and

July 1992R e s i s t i v i t y , o h m -m

100

0.1R e s i s t i v i t y , o h m -m

100

0.1

-120

-90

-60

-30

120

Depth, in.

R e s i s t i v i t y , o h m -m

Depth, in.

1000

100

0.1

13. Anderson B and Chang SK: “Synthetic Induction

Logs by the Finite Element Method,”Transactions of the SPWLA 23rd Annual Logging Symposium , Cor-pus Christi, Texas, USA, July 6-9, 1982, paper M;The Log Analyst 23 (November-December 1982):17-26. Also published with more mathematical detail as “Simulation of Induction Logging by the Finite Element Method,” Geophysics 49 (November 1984): 1943-1958.

14. Gianzero S, Lin YY and Su SM: “A New High-Speed

Hybrid Technique for Simulation and Inversion of Resistivity Logs,” SPE Formation Evaluation 3 (March 1988): 55-61.

15. Kaufman AA: “Theory of Induction Logging,”

Siberian Dept. of Nauka Press, Novosibirsk, 1965.Anderson and Chew, reference 4.

computer times exist for various types of modeling (next page ). Dashed boundaries indicate problems solved with geometric factor theory; solid boundaries indicate solutions developed using exact electromag-netic theory. The times shown are all for 50-foot [15-m] sections of log, where bed boundaries exist, and for a single-point cal-culation in the two cases where one infinitely thick bed is indicated (in this case the log is constant as the tool moves verti-cally). These codes may be used for study-ing tool responses to specific environmental perturbations, as well as in modeling for for-mation evaluation.

Least-Squares and Maximum Entropy Matching Criteria

Subjective eyeballing can usually evaluate the fit between the field log and the mod-eled log. But two other matching criteria,least squares and maximum entropy (MEM),have received attention. They are of interest,however, more because they use algorithms that converge automatically to their respec-tive best-fitting models than because of their inherent objectivity. No human intervention is needed to alter the model at each itera-tion. In application, these criteria lead to computational methods that are different from one another and from the manual interactive approach.

One application of the least-squares crite-rion picks the set of formation parameters that minimizes the sum of the squares of dif-ferences (SSD) between the field log and the log derived from the assumed formation model.16In the language of statistics, the mean square deviation, or variance,between the two logs is minimized. For example, with initial trial values of thickness and resistivity assigned to each bed in the model, the program computes the predicted log and finds the SSD between it and the

30Oilfield Review

n A sampling of codes and computer times used in modeling induction tool responses.Dashed boundaries indicate cases for which geometric factor theory was used;solid boundaries indicate use of Maxwell’s equations. Times shown are for 50-foot sections of log, where boundaries exist,and for a single-point calculation where one infinitely thick bed is indicated.

1. Doll HG: “Introduction to Induction Logging and Applica-tion to Logging of Wells Drilled with Oil-Base Mud,”Transactions, AIME 186 (1949): 148-16

2.2. Gianzero S and Anderson B: “Mathematical Theory for the Fields Due to a Finite H.C. Coil in an Infinitely Thick Bed with an Arbitrary Number of Co-Axial Layers,” The Log Analyst 25, no. 2 (March-April 1984): 25-32.

3. Gianzero SC: “Effect of Sonde Eccentricity on Responses of Conventional Induction-Logging Tools,” IEEE Transac-tions on Geoscience Electronics GE-16, no. 4 (October 1978): 332-339.

field log. Then, the model’s bed thicknesses and resistivities are automatically changed to reduce the SSD. This procedure repeats until the SSD converges to a minimum. Computing time depends, as in other iter-ative schemes, on how close the initial model log is to the field log. To keep this time short, the user frequently minimizes the number of parameters to be optimized since computing time increases with this number. Also, if some parameters “interfere”with one another—produce similar effects on the log—their precision is adversely affected even though they are optimally determined in the least-squares sense.

Consider an application of the least-squares method to a set of simulated logs (right ). The initial trial model used parame-ter values estimated by a standard interpre-tation. Because flushed zone resistivity, R xo ,values were assumed known from the MicroSFL resistivity tool, the modeling pro-gram was called upon to determine the pay zone thickness in addition to R t and inva-sion diameter in each of the three beds. This early-1980s calculation took 20 minutes on an IBM 360, using FEM. Today it can be done on a workstation in under 60 seconds.The maximum entropy criterion comes from Shannon’s information theory and has been used for reconstruction of blurred satellite photographs and for extracting sig-nals from noisy data. It leads to a different method of calculation 17although the method bears some similarity to the least-squares approach—differences between field log and modeled log are used in calcu-lation of χ2, a quantity related to the SSD.With respect to the use of χ2, the maximum entropy method can be considered an extension of the least-squares approach. The actual algorithms employed reduce χ2itera-tively, while maintaining entropy close to its maximum at each iteration. This procedure

selects the unique least-squares solution having maximum entropy.

There is an important difference between the MEM and least-squares methods. The final model selected by the MEM is the smoothest possible profile rather than the one that yields a minimum SSD. Of all pos-sible models, it is the one that has minimum information content (in the information-the-ory sense) consistent with the field log. Con-sequently, this criterion inhibits the appear-ance of artifacts that sometimes show up when using the least-squares method.Unfortunately, the MEM is computationally expensive. The cost is reduced if a priori knowledge of bed boundaries is included in the initial model, as with other methods, but cost remains the main impediment to wider MEM use at present.

16. Lin Y-Y , Gianzero S and Strickland R: “Inversion of

Induction Logging Data Using the Least Squares T echnique,” Transactions of the SPWLA 25th Annual Logging Symposium , New Orleans,Louisiana, USA, June 10-13, 1984, paper AA.17. Dyos CJ: “Inversion of Induction Log Data By the

Method of Maximum Entropy,” Transactions of the SPWLA 28th Annual Logging Symposium , London,England, June 29-July 2, 1987, paper T.

Freedman R and Minerbo GN: “Maximum Entropy Inversion of Induction Log Data,” paper SPE 19608,presented at the 64th SPE Annual Technical Confer-ence and Exhibition, San Antonio, Texas, USA,October 8-11, 1989. Also published with minor changes in SPE Formation Evaluation 6 (June 1991):259-268.

July 1992

n Forward model-ing by applying the least-squares method to a set of simulated logs.Synthetic induction logs and a lat-erolog (lower left)were computed for the formation model shown.Then, using the dotted-line param-eter values (esti-mated by a stan-dard interpretation)as the initial guess,the least-squares method converged to the solid-line “final iteration”model (lower right)

Nevertheless, the MEM can provide a for-mation model from an induction log, with-out assuming knowledge of bed boundaries (left ). The synthetic “field” log was calcu-lated by applying geometric factor theory (for simplicity) to the R t distribution shown,and geometric factor theory was thus used in representing tool response in the forward modeling. Oscillations visible in the thicker beds are not a consequence of the MEM,but result from so-called blind frequencies in the Fourier spectrum of the deep induc-tion’s VRF. These oscillations are readily suppressed by adding information from the medium induction. Addition of the medium induction contributes a smoothing effect. An actual example shows how well an MEM-predicted log matches its corresponding field-measured induction (below, left ).

The Future

Further developments in modeling are already in the pipeline. Generally, these take two forms, depending on the computa-tion size envisioned. One effort is aimed at providing a computed log in about one minute, using interfaces that are suitable for workstations. These should make the pro-cess more analyst-friendly, through features like entry of parameters in graphical format and windows that display the field log, the computed log and the model, including its parameters, dips and boundaries.

The other development arena involves extending the capabilities of large comput-ers. Primarily intended for use in tool design, these studies are deep into the 3D domain, using the FEM for modeling com-putations that take hours even on a super-computer. Still more exploratory are investi-gations of advanced algorithms and automatic code generation for speeding up complex calculations. Some algorithms are being applied in commercial seismic pro-cessing using parallel computers like the CM-2 Connection Machine.—JT

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n Forward model-ing of an induction log by the maxi-mum entropy method (MEM)(left)and using information from the medium induc-tion log to remove oscillations

(below). Geometric factor theory was used to calculate the synthetic deep induction log from the assumed R t pro-file. An advanced version of the max-imum entropy

method, assuming no bedding struc-ture in the initial trial model, then converged to the final R t profile.Oscillations in the thicker beds result from the Fourier spectrum of the deep induction tool’s vertical reso-lution function,and are unrelated to use of the maxi-mum entropy mod-eling approach.

n A deep induction field example com-paring the log computed by the maximum entropy method with the measured log. The predicted final-model profile is also shown.