Robin P. Gardner* and. Lianyan Liu. Computalog U.S.A., Inc., 501 Winscott Road, Fort Worth, Texas 76126

Transcription

1 NUCLEAR SCIENCE AND ENGINEERING: 133, ~1999! Monte Carlo Simulation of Neutron Porosity Oil Well Logging Tools: Combining the Geometry-Independent Fine-Mesh Importance Map and One-Dimensional Diffusion Model Approaches Robin P. Gardner* North Carolina State University, Department of Nuclear Engineering Box 7909, Center for Engineering Applications of Radioisotopes Raleigh, North Carolina and Lianyan Liu Computalog U.S.A., Inc., 501 Winscott Road, Fort Worth, Texas Received September 8, 1998 Accepted January 4, 1999 Abstract The generation of first estimate geometry-independent fine-mesh three-dimensional importance maps with simple one-dimensional diffusion models is demonstrated for the Monte Carlo simulation of the neutron porosity oil well logging tool response benchmark problem. By combining the approach of using simple one-dimensional steady-state diffusion models for calculating neutron adjoint flux with the geometry-independent fine-mesh-based Monte Carlo importance approach previously developed, an automated and efficient variance reduction method is obtained for this specific problem. A surprising result is that the converged figures of merit after iteration are consistently larger when the initial importance map is based on the one-dimensional diffusion model rather than that obtained from an analog Monte Carlo simulation. I. INTRODUCTION The present work is based on combining the authors previous work on the development of a geometryindependent fine-mesh importance map approach 1 with the approach developed by Mickael 2 of using a simple onedimensional diffusion model for importance. This combined approach is used for simulating the neutron porosity oil well logging tool response and is demonstrated with the benchmark problem 3 for that application. The authors previously developed and demonstrated 1,4,5 a geometry-independent fine-mesh importance map * gardner@ncsu.edu Current address: North Carolina State University, Department of Nuclear Engineering, Box 7909, Center for Engineering Applications of Radioisotopes, Raleigh, North Carolina approach that can be used adaptively for implementing the weight windows ~splitting and Russian roulette! variance reduction scheme in MCNP ~Refs. 6 and 7!. This new approach has two demonstrated improvements over the normal MCNP physical geometry-based importance map: ~a! it is much more user friendly because the actual complex geometrical cells need not be split into smaller matching cells for importance mapping, and ~b! it converges to computational figures of merit ~! that are four to six times larger for the benchmark neutron and gamma-ray nuclear oil well logging problems, 3 respectively. The is defined in the usual way as follows: 1 TE R 2, ~1! where T is CPU time in minutes and E R is the calculated response relative error ~precision! achieved. 80

2 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 81 This work is specifically designed for simulation of the neutron porosity oil well logging tool and the benchmark problem 3 for that tool. Briefly, that problem consists of a logging tool with an 241 Am-Be neutron source and two 3 He proportional counters at source-to-detector-center distances of and 50.8 cm. The tool has a 7.62-cm diameter and is placed on the side of a cm-diam borehole that contains pure water and is surrounded by an infinite homogeneous limestone formation that contains either 20% ~by volume! pure water or 1% pure water. The difficulty in implementing this approach when one first begins to work on a specific problem ~this is the same for the usual MCNP weight windows approach! is in obtaining a good first estimate for the importance map. This has normally been done 6 by using analog or near analog Monte Carlo for the first iteration. However, a recent contribution by Van Riper et al. 8 uses a threedimensional deterministic adjoint calculation with the code THREEDANT to obtain the initial importance map. While this approach is very accurate, it requires access and use of both the THREEDANT and MCNP codes, which is a difficult undertaking. It is our understanding 9 that this approach has not been enthusiastically received so far and that a simpler approach might be preferable in many cases. For the very low yield nuclear oil well logging problems ~10 4 for neutron and 10 8 for gamma-ray!, when the initial importance map is to be determined by analog or near analog calculation, it is found that analog calculation requires a large number of histories for even a poor first estimate and leads to requiring five or six iterations for convergence. The authors previously showed 1 that when a better first estimate of the importance map is available ~as that from a similar problem!, convergence requires only one or two iterations. This work reports on our use of a previously developed approach by Mickael 2,10 and Mickael and Towsley 11 for the neutron porosity problem of using a simple one-dimensional, three-energy-group diffusion model for solving the adjoint neutron transport problem. It appears that this approach essentially eliminates the convergence problem since very efficient calculations can be made with the initial importance map in this case. This paper describes the combining of the Mickael approach with the authors fine-mesh approach to obtain an automated, userfriendly, efficient variance reduction method for the MCNP4B code for the simulation of the neutron porosity oil well logging tool response problem. A recent paper by Evans and Hendricks 12 investigates five different state-of-the-art approaches for using the MCNP Monte Carlo code with two different weight window generators and the AVATAR code to solve the neutron porosity oil well logging benchmark problem. We will compare the present combined geometryindependent fine-mesh importance generator and onedimensional simple diffusion model approach to these five approaches. II. GENERATION OF ONE-DIMENSIONAL ADJOINT NEUTRON FLUX USING DIFFUSION MODELS Use of the adjoint solution to radiation transport problems for Monte Carlo simulation has been recognized 13 for many years. As far as we know, the TRIPOLI ~Refs. 14 and 15! and MORSE ~Ref. 16! Monte Carlo codes have not been used for the specific neutron porosity oil well logging tool problem treated here, but they do use biasing schemes in general that require importance functions based on the adjoint solution to the problem of interest. The MCBEND Monte Carlo code is used for this problem, and it makes use of geometric importance splitting and Russian roulette ~this is similar to the MCNP weight windows approach! based on energy-dependent importances on either xyz or rzu meshes that overlay the problem geometry. Importances are calculated by adjoint diffusion models that are executed as an automatic part of a MCBEND Monte Carlo calculation. Recent improvements have been the use of point energy data for adjoint calculations 21 and a method of making selfadjusting improvements 22 to the diffusion model importance maps. Mickael and Towsley 11 and Mickael 2 originally developed a diffusion model approach for generating onedimensional adjoint neutron fluxes for a specific purpose Monte Carlo code that they developed named MCNL. Subsequently that approach was adapted to make the calculation with the general purpose Monte Carlo code MCNP ~Ref. 10!. The approach essentially consists of two parts: ~a! solution of the appropriate one-dimensional steady-state multigroup adjoint diffusion equations and ~b! estimation of the group parameters using analog Monte Carlo simulation. The one-dimensional steady-state multigroup adjoint diffusion equations for n groups are given by Mickael 2 ~from Bell and Glasstone 23! as follows: D i 1 r 2 and D n 1 r 2 d dr r 2 d dr f i ~r! S i f i ~r! S iri 1 f i 1 ~r! d dr r d 2 dr f n ~r! S n f n ~r! 0 with the source condition: for i 1,2,...,n 1 ~2! for i n, ~3! 4pr 2 D n ~r!0dr# rr0 R ~4! and the boundary condition: f i ~r! rr` 0 for i 1,2,...,n, ~5!

3 82 GARDNER and LIU where r distance from the source f i adjoint flux of group i D i diffusion coefficient of group i S i removal cross section of group i S irj transfer cross section from group i to group j R detector response function. In the present case, the detector response function is zero for all but the thermal neutron group. This formulation assumes that significant transfers between energy groups occur only for adjacent energy-decreasing groups. Note that group energies change from high to low as the group index i goes from 1 to n. Mickael 2 gives the solution for a four-group system. It is the following: f 4 ~r! exp~ r0l 4!, ~6! 4pD 4 r f 3 ~r! S 3r4 4pS 3 S 4 r exp~ r0l 3! exp~ r0l 4! L L 4, ~7! f 2 ~r! S 2r3S 3r4 L 4pS 2 S 3 S 4 r 2 2 exp~ r0l 2! ~L 2 2 L 2 3!~L 2 2 L 2 4! L2 3 exp~ r0l 3! ~L 2 3 L 2 2!~L 2 3 L 2 4! L2 4 exp~ r0l 4! ~L 2 4 L 2 2!~L 2 4 L 2 3!, ~8! and f 1 ' ~r! S 1r2S 2r3 S 3r4 4pS 1 S 2 S 3 S 4 r f 1 '' ~r! S 1r2 S 2r3 S 3r4 4pS 1 S 2 S 3 S 4 r L 1 L 3 4 exp~ r0l1! ~L 2 1 L 2 2!~L 2 1 L 2 3!~L 2 1 L 2 4! L 4 2 exp~ r0l2! ~L 2 2 L 1 2!~L 2 2 L 3 2!~L 2 2 L 4 2! 4 exp~ r0l3! ~L 2 3 L 2 1!~L 2 3 L 2 2!~L 2 3 L 2 4! L 4 4 exp~ r0l4! ~L 4 2 L 1 2!~L 4 2 L 2 2!~L 4 2 L 3 2! f 1 ~r! f 1 ' ~r! f 1 '' ~r!. ~9!,, The required L i are given by L i %~D i 0S i!, ~10! and the neutron importance I i of group i is taken as: I i f i 0S i. ~11! The group parameters required in the solution are obtained ~all S i, D i, and L i! by using an analog Monte Carlo simulation of the problem. 2,24 First, the group fluxes are estimated from m 1 f i ~10m! j 1 ( S~E j! for E i 1 E j E i, ~12! where E j neutron energy before collision j S~E j! total macroscopic cross section at neutron energy E j m total number of collisions in the simulation. The group cross sections are computed as and S i k i mf i ~13! S irj k irj mf i, ~14! where k i and k irj are the total number of removal ~absorption and scattering out of the group! and scattering to group j events in group i after m collisions, respectively. The diffusion length L i of group i is estimated as L 2 i 1 ( N d 2 l, ~15! 6N l 1 where d l is the distance from the point where the neutron entered group i either by source emission or scattering from another group to the point where the neutron leaves the group either by scattering to another group

4 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 83 or absorption. The summation is over N histories. The diffusion coefficient D i is then computed from D i S i L i 2. ~16! A specific purpose Monte Carlo code named McDNL, originally developed by Mickael 25 ; Mickael, Gardner, and Verghese 26 ; Little et al. 27 ; Gardner, Mickael, and Verghese 28 ; and Gardner et al. 29 to calculate the neutron porosity log responses without using the importance map splitting and Russian roulette variance reduction approach, was rewritten as a package of codes that does use this approach by Prettyman. He used the same approach as reported for an epithermal neutron logging tool by Prettyman, Gardner, and Verghese 30 and proposed by Mickael. 2 The McDNL code package was written for three energy groups and the IDNL code of that package was used in the present work to generate the one-dimensional steady-state adjoint flux. III. CONVERTING ONE-DIMENSIONAL ADJOINT NEUTRON FLUXES TO THREE-DIMENSIONAL IMPORTANCE MAPS The one-dimensional adjoint neutron flux provided by the IDNL code of the McDNL code package is the groupwise radial distribution for a point detector located at the center of the assumed spherically symmetric system. To convert this one-dimensional distribution to the three-dimensional mesh format required by the modified MCNP code, in addition to converting the onedimensional radial distribution to a three-dimensional x-y-z mesh distribution, one needs to consider the finite detector size, which is not a point detector as is assumed in calculating the one-dimensional adjoint fluxes. To accommodate the adjoint flux format required by the fine-mesh weight window approach, a superimposed x-y-z mesh system that is independent of the actual geometry is established. For simplicity, the position of each mesh is represented by the center point of the mesh. For the sample neutron problem, a uniform mesh structure is adopted, with the ranges of 50, x 60, 60 y, 60, and 30, x, 90 ~all units in centimetres!. The selection of this mesh structure is intended to adequately cover the important space required as the scoring particles travel from the source at ~0,0,0! and reach the farthest 3 He detector centered at ~0, 0, 50.8! with a 2.54-cm radius and a 25.4-cm height. Because of the size of the detector, large errors might occur in treating the detector as an effective point detector, especially considering that the adjoint fluxes tend to change dramatically near the detectors. Instead, the authors opted to use a flux-averaging method. The detector is divided into small cells according to a cylindrical geometry system, so each detector cell can be treated as a point detector. The adjoint flux of a spatial mesh with respect to the actual detector is obtained by averaging the adjoint fluxes of the mesh with repect to all the detector cells ~point detectors!. A computer code named MAPCVT was designed for this computation. IV. MONTE CARLO RESULTS FOR THE NEUTRON POROSITY OIL WELL LOGGING TOOL BENCHMARK PROBLEM The method outlined in Secs. II and III for using the one-dimensional steady-state three-group diffusion model to provide the initial importance map for Monte Carlo simulation of the neutron porosity benchmark problem 3 was used ~case 2! and is compared to the method previously reported 1 ~case 1! of using an analog Monte Carlo calculation to obtain the initial importance map. The results for four subcases of case 1 are given in Table I and the same four subcases for case 2 in Table II. Six iterations of histories for each iteration for each subcase were used in case 1 while three iterations of histories each were used in case 2. The calculations were made on a personal workstation ~Alpha 433! by Digital running Digital UNIX and took ;40 min per iteration per subcase. Calculation of the adjoint neutron fluxes with the McDNL code took ;3 min on the same computer. The results indicate that about five iterations are required for convergence when the analog initial importance map is used and about one when the diffusion model initial importance map is used. In addition to the values given in Tables I and II for each iteration, the average yield and its actual and average Monte Carlo standard deviations are given. The yield is defined as the number of neutrons detected per neutron emitted by the source. The actual standard deviation of the yield is that obtained by using the general estimator on the individual ~unweighted! yield values. The average Monte Carlo standard deviation is the average of the standard deviations predicted by the Monte Carlo program for each iteration. Importance maps for the sum of all three energy groups are shown in Figs. 1 through 8 for the onedimensional steady-state diffusion model cases and the Monte Carlo-generated converged cases for comparison. The maps shown are for the y 0 plane, which is the plane of symmetry for this problem. This plane passes through the centers of the detectors, the source, and the borehole. The diffusion model cases are smooth and almost symmetric about both x and z as expected. The Monte Carlo-generated maps are not as smooth especially far from the detector. The Monte Carlogenerated maps are also not symmetric in x they tend to fall off more sharply on the back side of the detector ~away from the formation and into the borehole!. Monte Carlo tool response calculations were also made with the specific purpose code McDNL. This code was found to have converged values about

5 84 GARDNER and LIU TABLE I Case 1 Relative Results with Initial Importance Map Provided by an Analog Monte Carlo Calculation Relative to the Analog Result for the 20 PU Far Detector* Subcase Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6 Average Yield Y ~ 10 4! Actual s~y! ~%! Average Monte Carlo s~y! ~%! 1PU a near detector 1PUfar detector 20 PU near detector 20 PU b far detector *The average CPU time per iteration was 44.3 min with a standard deviation of 9.9 min and a range from 32.2 to 63.9 min. a PU refers to porosity unit and is water volume percentage in the sample. b The actual for the 20 PU far detector analog ~iteration 1! case was TABLE II Case 2 Relative Results with Initial Importance Map Provided by the One-Dimensional Steady-State Three-Group Diffusion Model Adjoint Flux Map Compared to the Analog Result for the 20 PU Far Detector Subcase* Subcase Iteration 1 Iteration 2 Iteration 3 Average Yield Y ~ 10 4! Actual s~y! ~%! Average Monte Carlo s~y! ~%! 1PU a near detector PU far detector PU near detector PU b far detector *The average CPU time per iteration was 38.4 min with a standard deviation of 6.9 min and a range from 29.7 to 49.4 min. a PU refers to porosity unit and is water volume percentage in the sample. b The actual for the 20 PU far detector analog ~iteration 1! case was one-fourth of those for the fine-mesh importance map MCNP code. This means that McDNL had about the same converged as the weight windows version of MCNP. It appears that the MCNP code is more efficient in particle tracking and0or cross-section accessing than the McDNL code. The McDNL code could probably be modified to obtain the same values as MCNP by improving the efficiency of these aspects of the code, but there would be little advantage for this code at present since MCNP when used with the fine-mesh importance map approach is almost as user-friendly as McDNL, has a more general geometry package, and now has correlated sampling capability like the McDNL code. However, the diffusion model approach could be added directly to the fine-mesh importance map version of MCNP so that one would not have to run two different computer codes that have different geometry input files to take full advantage of the present proposed combined approach. V. DISCUSSION AND CONCLUSIONS In many cases, a number of Monte Carlo calculations of the same general type are of interest to a particular user. For example, the nuclear oil well logging practitioner usually fits this case. This type of Monte Carlo user normally keeps a file of converged ~accurate! previously obtained importance maps for the range of problem parameters of interest. Therefore, this type

6 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 85 Fig. 1. One-dimensional diffusion adjoint flux of near detector for 1 PU formation. Fig. 2. Three-dimensional Monte Carlo adjoint flux of near detector for 1 PU formation.

7 86 GARDNER and LIU Fig. 3. One-dimensional diffusion adjoint flux of far detector for 1 PU formation. Fig. 4. Three-dimensional Monte Carlo adjoint flux of far detector for 1 PU formation.

8 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 87 Fig. 5. One-dimensional diffusion adjoint flux of near detector for 20 PU formation. Fig. 6. Three-dimensional Monte Carlo adjoint flux of near detector for 20 PU formation.

9 88 GARDNER and LIU Fig. 7. One-dimensional diffusion adjoint flux of far detector for 20 PU formation. Fig. 8. Three-dimensional Monte Carlo adjoint flux of far detector for 20 PU formation.

10 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 89 of Monte Carlo user would rarely need a completely new initial importance map estimate. That type of user would only need to use the approach described here when just beginning to build up his or her library of converged importance maps or whenever a sufficiently new and different problem surfaces. Whenever a new initial estimate importance map is needed for a neutron porosity oil well logging problem, the approach described here should prove useful. It is very user-friendly and efficient. In some cases, when the problem yield is very low ~say,10 4!, it may prove impossible to obtain a good initial estimate by analog calculation thereby making this approach indispensible. It is pertinent to compare the results of the present approach with the approaches investigated by Evans and Hendricks 12 for the neutron porosity oil well logging benchmark problem. They examined five different cases ~six including analog Monte Carlo!. The first case ~ANA! was the use of the normal MCNP4B code in analog mode with the problem divided into eight geometric cells. The second ~MWWG! was the use of the normal MCNP4B code with the problem divided into 231 geometric cells. Weight windows are calculated in each cell using the existing MCNP generator. Three iterations are made in which the original importance map is updated by the next Monte Carlo calculation. The third ~GDWWG! was the same except that corrections to the weight window generator were made. The fourth ~GIWWG! divided the problem into only eight geometric cells with a geometryindependent x, y, z grid with dimensions for calculating the weight windows. The fifth ~AVA- TAR! also divides the problem into eight geometric cells, anda cell importance map generated by the three-dimensional code THREEDANT is used by the code AVATAR for weight windows. No iteration is made in this case. The sixth ~AVRWWG! is the same as the previous one except that one iteration using the geometryindependent weight window generator is added on a grid. All of the variance reduction options used five energy groups. The results are shown in Table III. The GIWWG approach is the one that corresponds to the present case. The value of 105 compares with the three values given in Table II of 109, 110, and 122. The present approach has the advantages that ~a! no iterations are necessary for converging the importance map and, and ~b! the required external computation time is not for another large code like THREE- DANT, but rather for a simple analog version of the Monte Carlo code that is being used combined with a simple code for calculating the resulting adjoint flux at any position and energy. The external computation time required in this case is,3 min. Thus, the present approach retains the user-friendliness of the geometry-independent importance map and eliminates the need for making iterations to converge the importance map and. This approach may be useful for other similar neutron transport problems. This might be true in those cases where the neutrons travel primarily through a single homogeneous medium or through several media that have similar neutron properties and where the phenomenon of primary interest is neutron thermalization. The results in Table II indicate that there is little improvement in from the one-dimensional diffusion model to the converged three-dimensional Monte Carlo importance maps for the 20 PU far detector subcase. This must mean that the one-dimensional diffusion model importance map is accurate for this case. This is probably because in this case the formation represents a large fraction of the important neutron transport space and the formation also has neutron properties close to those of the borehole. Both nearer detectors and formation neutron properties that are farther from those of the borehole cause larger variations of the Monte Carlo three-dimensional importance map from that of the one-dimensional diffusion model. It was a surprise to the authors that the converged values were consistently larger for this new combined method. It appears that the one-dimensional TABLE III Relative Results for Six Cases for the 20 PU* Far Detector Subcase Approach External a Time ~min! Iteration 1 Time ~min! Iteration 2 Time ~min! Iteration 3 Time ~min! Converged ANA MWWG GDWWG GIWWG AVATAR AVRWWG *PU refers to porosity unit and is water volume percentage in the sample. a The external time refers to the time required to run the code THREEDANT. Five-hundred-thousand histories were run for each iteration.

11 90 GARDNER and LIU diffusion model somehow provides an initial importance map that allows convergence to slightly better values than those from an analog Monte Carlo run. VI. FUTURE WORK There is also a need for a method similar to the present one for gamma-ray or photon transport in nuclear oil well logging. For those nuclear oil well logging tools that are not highly for example, the prompt gamma-ray carbon-oxygen ~C0O! logs, the formation absorption cross-section logs, and the natural gamma-ray logs#, a simple one-dimensional model may also be appropriate. For the highly collimated logs like the gamma-ray density-lithology logging tools, it is likely that a more complex model will be required. The authors intend to investigate models for this purpose and additional variance reduction approaches including those based on zero-variance principles. ACKNOWLEDGMENTS The authors gratefully acknowledge the following: ~a! the financial support of the Associates Program-Nuclear Oil Well Logging, which presently consists of ARCO, Amoco, Halliburton Energy Services, and WesternAtlas Logging Services and ~b! participation in a cooperative research and development agreement ~CRADA! ~entitled Computer Simulation in Support of Nuclear Well Logging! with Los Alamos National Laboratory and industrial partners Chevron, Computalog, Halliburton Energy Services, and Western Atlas Logging Services. REFERENCES 1. L. LIU and R. P. GARDNER, A Geometry-Independent Fine-Mesh-Based Monte Carlo Importance Generator, Nucl. Sci. Eng., 125, 188 ~1997!. 2. M. W. MICKAEL, Importance Estimation in Monte Carlo Modelling of Neutron and Photon Transport, Nucl. Geophys., 6, 3, 341 ~1992!. 3. R. P. GARDNER and K. VERGHESE, Monte Carlo Nuclear Well Logging Benchmark Problems with Preliminary Intercomparison Results, Nucl. Geophys., 4, 4~1991!. 4. L. LIU, Self-Optimizing Monte Carlo Method for Nuclear Well Logging Simulation, PhD Thesis, North Carolina State University, Nuclear Engineering Department ~1997!. 5. L. LIU and R. P. GARDNER, A Mesh-Based Weight Window Approach for Monte Carlo Simulation, Trans. Am. Nucl. Soc., 77, 169 ~1997!. 6. T. E. BOOTH and J. S. HENDRICKS, Importance Estimation in Forward Monte Carlo Calculations, Nucl. Technol.0 Fusion, 5, 90~1984!. 7. J. F. BRIESMEISTER, MCNP A General Monte Carlo N-Particle Transport Code, Version 4B, LA M, Los Alamos National Laboratory ~1997!. 8. K. A. VAN RIPER, T. J. URBATSCH, P. D. SORAN, D. K. PARSONS, J. E. MOREL, G. W. McKINNEY, S. R. LEE, L. A. CROTZER, F. W. BRINKLEY, T. E. BOOTH, J. W. AN- DERSON, and R. E. ALCOUFFE, AVATAR AutomaticVariance Reduction in Monte Carlo Calculations, Proc. Joint Int. Conf. Mathematical Methods and Superconducting for Nuclear Applications, Saratoga Springs, New York, October 5 9, 1997, Vol. 1, p. 661, American Nuclear Society ~1997!. 9. G. W. McKINNEY, Personal Communication ~1998!. 10. M. W. MICKAEL, A Fast, Automated, Semideterministic Weight Windows Generator for MCNP, Nucl. Sci. Eng., 119, 34~1995!. 11. M. W. MICKAEL and C. W. TOWSLEY, Development of an Efficient Self-Optimized Monte Carlo Code for Neutron Porosity Logs, IEEE Trans. Nucl. Sci., 38, 2, 501 ~1991!. 12. T. M. EVANS and J. S. HENDRICKS, An Enhanced Geometry-Independent Mesh Weight Window Generator for MCNP, Proc. Radiation Protection and Shielding Division Topl. Conf., Nashville, Tennessee, April 19 23, 1998, Vol. I, p. 167, American Nuclear Society ~1998!. 13. G. GOERTZEL and M. H. KALOS, Monte Carlo Methods in Transport Problems, Progress in Nuclear Energy Series I: Vol. 11, Physics and Mathematics, Pergamon Press, New York ~1958!. 14. J. P. BOTH, H. DERRIENNIC, B. MORILLON, and J. C. NIMAL, A Survey of TRIPOLI-4, Proc. 8th Int. Conf. Radiation Shielding, Arlington, Texas, April 24 28, 1994, Vol. 1, p. 373, American Nuclear Society ~1994!. 15. J. P. BOTH, J. C. NIMAL, and T. VERGNAUD, Automated Importance Generation and Biasing Techniques in TRIPOLI- 3, Prog. Nucl. Energy, 24, 1~1990!. 16. J. S. TANG and T. J. HOFFMAN, Monte Carlo Shielding Analyses Using an Automated Biasing Procedure, Nucl. Sci. Eng., 99, 329 ~1988!. 17. E. SHUTTLEWORTH and S. J. CHUCAS, Linked Monte Carlo and Finite-Element Diffusion Methods for Reactor Shield Design, Proc. 6th Int. Conf. Radiation Shielding, Tokyo, Japan, May 16 20, 1983, Vol. I, p. 180, Japan Atomic Energy Research Institute ~1983!. 18. P. C. MILLER, G. A. WRIGHT, C. B. BOYLE, and S. W. POWER, The Use of an Inbuilt Importance Generator for Acceleration of the Monte Carlo Code MCBEND, Proc. Int. Conf.

12 MONTE CARLO SIMULATION OF NEUTRON POROSITY OIL WELL LOGGING TOOLS 91 Physics of Fast Reactors: Operation, Design and Computation, Marseille, France, p. 124, Organization for Economic Cooperation and Development ~1990!. 19. S. CHUKAS, I. CURL, T. SHUTTLEWORTH, and G. MORRELL, Preparing the Monte Carlo Code MCBEND for the 21st Century, Proc. 8th Int. Conf. Radiation Shielding,Arlington, Texas, April 24 28, 1994, Vol. 1, p. 381, American Nuclear Society ~1994!. 20. S. CHUCAS and M. GRIMSTONE, The Acceleration Techniques used in the Monte Carlo Code MCBEND, Proc. 8th Int. Conf. Radiation Shielding, April 24 28, 1994, Arlington, Texas, Vol. 2, p. 1126, American Nuclear Society ~1994!. 21. M. GRIMSTONE, Extension of the MCBEND Monte Carlo Code to PerformAdjoint Calculations Using Point Energy Data, Proc. Radiation Protection and Shielding Division Topl. Conf., April 19 23, 1998, Nashville, Tennessee, Vol. I, p. 143, American Nuclear Society ~1998!. 22. E. SHUTTLEWORTH, Self-Adjusting Importances for the Acceleration of MCBEND, Proc. Radiation Protection and Shielding Division Topl. Conf., April 19 23, 1998, Nashville, Tennessee, Vol. I, p. 151, American Nuclear Society ~1998!. 23. G. I. BELL and S. GLASSTONE, Nuclear Reactor Theory, Robert E. Krieger Publishing Company, Huntington, New York ~1979!. 24. M. E. T. ORABY, suggestion made in a Center for Engineering Applications of Radioisotopes group meeting, probably in M. W. MICKAEL, Monte Carlo Simulation of Dual- Spaced Neutron Porosity Well-Logging Tool Responses, PhD Thesis, North Carolina State University, Nuclear Engineering Department ~1988!. 26. M. MICKAEL, R. P. GARDNER, and K. VERGHESE, McDNL: A New Specific Purpose Monte Carlo Code for Simulation of Dual-Spaced Neutron Porosity Logs, presented at Society of Professional Well Log Analysts 29th Annual Logging Symp., June R. C. LITTLE, M. MICKAEL, K. VERGHESE, and R. P. GARDNER, Benchmark Neutron Porosity Log Calculations: A Comparison of MCNP and the Specific Purpose Code McDNL, IEEE Trans. Nucl. Sci., 36, 1, 1223 ~1989!. 28. R. P. GARDNER, M. W. MICKAEL, and K. VER- GHESE, Complete Composition and Density Correlated Sampling in the Specific Purpose Monte Carlo Codes McPNL and McDNL for Simulating Pulsed Neutron and Neutron Porosity Logging Tools, Nucl. Geophys., 3, 3, 157 ~1989!. 29. R. P. GARDNER, M. W. MICKAEL, C. W. TOWSLEY, C. M. SHYU, and K. VERGHESE, Correlated Sampling in the McDNL and McPNL Codes for Neutron Porosity and Neutron Lifetime Log Corrections, IEEE Trans. Nucl. Sci., 37, 3, 1360 ~1990!. 30. T. H. PRETTYMAN, R. P. GARDNER, and K. VER- GHESE, The Specific Purpose Monte Carlo Code McENL for Simulating the Response of Epithermal Neutron Lifetime Well Logging Tools, IEEE Trans. Nucl. Sci., 40, 4, 933 ~1993!.