A stable algorithm for simulation of array induction and measurement-while-drilling logging tools Xiang Tian, Ce Liu, and Liang C.

Transcription

1 Radio Science, Volume 33, Number 4, Pages , July-August 1998 A stable algorithm for simulation of array induction and measurement-while-drilling logging tools Xiang Tian, Ce Liu, and Liang C. Shen Department of Electrical and Computer Engineering, University of Houston, Houston, Texas Abstract. A stable numerical method to simulate array induction and measurement-while-drilling (MWD) logs is presented. The algorithm is based on a horizontal eigenmod expansion method. Mode propagation in the vertical direction is described by three-layer modules. Within each module, general transmission and reflection coefficients are obtained by tracing the propagation of the modes. The multilayer cases are treated as a cascade of such modules. Owing to the mode-tracing algorithm, numerical stability and efficiency are superior to the other methods. This numerical algorithm is applied to the simulation of array induction and MWD logs in formations with both vertical and horizontalayers. 1. Introduction be used in two different ways: expand the field into horizontal eigenmodes (in the p direction) or into In well logging, formations with both cylindrical vertical eigenmodes (in the z direction). Chew et al. and planar boundaries are often encountered. Many [1984] proposed to expand the p dependence of the simulation methods to model the induction and field into a series of eigenstates to solve a single measurement-while-drilling (MWD) logs in such vertical boundary problem numerically. Later, this formations have been developed. To reduce the method is developed to analyze complicated borehole complexity, the three-dimensional (3-D) problem is environments with multilayer formations [Chew et al., often simplified to a 2-D problem by assuming that the 1991; Chew andanderson, 1985; Liu et al. 1989]. This formation has the property of axial symmetry. Owing method is much more efficient than the finite element to the absorption of formation to electromagnetic method [Anderson and Chang, 1982]. Pal [1991], Pal waves, the signal propagating in the formation decays et al. [1993], and Li and Shen [1993a, b] solved the exponentially. A simulation method may be unstable if same problem using a vertical eigenmode expansion the wave propagation is not treated properly. Kennett method. In Pai's work, the field is expressed in terms [1983] proposed a recursive algorithm to solve the of vertical eigenmodes, which can be obtained numerical instability problem in the seismic wave analytically. Horizontal modes propagate in the analysis. In his method, he avoided exponentially vertical direction, while vertical modes propagate in growing terms and only taking into account the the horizontal direction. Modes are reflected at layer majority of internal reflections in a layer. In this boundaries. Mathematical description of the wave paper, we present a stable and efficient algorithm to propagation and reflection from layer to layer solve this problem by using a horizontal eigenmode determines the stability of the algorithm. expansion method. We use horizontal eigenstates to solve the 2-D well The eigenmodexpansion method is an extension of logging problem in a formation with both horizontal the method of separation of variables. The method can and verticalayers. The wave propagation from layer to INow at Halliburton Energy Services, Houston, Texas. layer is based on three-layer modules, which guarantee the stability of the computation. Each horizontalayer Copyright 1998 by the American Geophysical Union. is divided into several subregions in the p direction. The eigenmodes are found numerically in each layer, Paper number 98RS00976 and the fields are expressed as the sum of the /98/98RS eigenmodes. 949

2 950 TIAN ET AL.' SIMULATION OF INDUCTION AND MWD LOGGING TOOLS Section 2 reviews the principle of the horizontal eigenmodexpansion method and describes the details of the three-layer module. Section 3 verifies the result by comparing the results with the other publishe data. Simulated array induction and MWD logs in complicated formations are discussed in sections 4 and 5, respectively. Finally, conclusions are given in section Formulations Figure la shows the configuration of an induction tool in a formation with multilayer, multi-invasion zones. The source is a coil at the center of the borehole carrying a uniform current. Owing to the azimuthal symmetry of the formation, the electrical field in the formation satisfies 1 k2,uvx--vxe- E=icot, (]) where p is the magnetic permeability, J is the current density in the transmitting coil, and the time variable e -imt is implied throughout the paper. For a coil carrying a current I having a radius of pt and located at zt, (1) can be written as [ c¾ p Op p2 c3z 2 ] k 2 E z 2c ::::::::::::::::::::::::::::::::::. '!::::i:i:i:i:!:i:::i:::::i:i: :.: i½ 1 R i - i..2b Figure la. A formation with borehole, horizontalayers, and invasion zones. The radii of the transmitting and receiving coils are c, that of the mandrel is a, and that of the borehole is b. layer i+1 Figure lb. Boundary conditions between layer I and layer I+1. =-ico (p-pt) (Z-Zt), (2) where 6 is the Dirac delta function. The wave number k is defined as where k 2 =co21xs, (3) s = s+t--, (4) and e and tr are the dielectric permitfivity and conductivity of the medium, respectively. Generally speaking, both and tr are functions of p and z. In the horizontal layer i, the solution to equation (2) can be expressed as a series of horizontal eigenstates, N [ e_ika z eika z] l E i = Z aai +ba fa/ (P) (5) where aai and bai are unknown constants independent of p and z, k, is the eigenvalue and f, (p) is the eigenfunction of mode a. The k has a positive magmary part. Therefore the terms ao e-ik z and ba i eik, z represent the decaying waves propagating in the negative z and positive z directions, respectively. The function f,,i(p) is defined over a finite region Lom, Pm,] and satisfies the following boundary conditions: fai (Pmin= 0) = O, induction tool, (6a) fa, (Pmin =Pm) =0, MWDtool, (6b) f(xi (Pmax) =0, (6C) where Pm is the radius of the mandrel. The eigenfunctionsatisfy the orthogonality relation... (p)fo aod. lap (7)

3 TIAN ET AL.: SIMULATION OF INDUCTION AND MWD LOGGING TOOLS 951 The finite region Lom Pm,] is divided into where subregions. Triangular functions are used as base (12) functions in the subregions, and eigenfunctions are expressed in a series of these known base functions. The eigenfunctions in each layer can be obtained numerically. In the presence of a horizontal layer boundary at Zo between layer i and layer i+ 1 shown in Figure lb, the Coj and Ca/+ are eigenvectors to tangential electric field E and magnetic field H a eigenvalues kaiand kca+l in layers i and i+1, satisfy the following boundary conditions: (8) Eqi l Z=Zo = l z=zo, 1 OE,l,i l z:zo - 1 OE,l,i+! l z:zo --J'l, [Z=Zo i c3 P. i ' where subscripts i and i+ 1 denote variables in layer i and layer i+ 1, respectively. Substituting (5) into (8) and (9) and taking inner products, the coefficients a i and b in layer i can be expressed as functions of a i+l and b i+l as -ar,z, v 1 { al3ie = 52 a= 2Dpikp aoa+l e-ik" +'z (9) - Pm C0 fp, (p)dp. (13) corresponding respectively. The Ca t is the transpose of C a. The matrix G is called the base matrix whose elements are Gap: I Define wave vectors P. 1 Pmi. p ga (P)gl (p)dp. (14) t t (16) where V(zi) and U(z,) represent the fields propagating the negative and positive z directions, respectively. Equations (10) and (!1) can be expressed in a matrix form: (17) where (lo) b [3i e = 52 a=l N 2Dpi 1 [aai+le- k. z (c,,.),. x[k[ i( 131i+l)t it ad -k[3i+l( Oi+l) t 't+l ] = 2 DI3i k[3i (18) (19) - ' (11) (21) (2O)

4 952 TIAN ET AL.: SIMULA ON OF INDUCTION AND MWD LOGGING TOOLS At each horizontal boundary, the fields are reflected and transmitted. If V (z) represents the incident field and there is no source at the boundary, U (z) is zero. The matrix equation (17) is reduced to (22) Cuu layer i : Zl z2 z3 I Y1 Y2 Y3 Y4 Then Receiver (23) (24) where C ) is the inverse of matrix Caa. Ru+ and T.a+ are defined as reflection and transmission. [ " laver Boundary 3 2 I (a) R laver 2 ' Boundary 1 laver 1 Output '1 I Input Output 2 Figure 2. (a) Wave propagation 'm a three-layer module. (b) The equivalent network for the three-layer module. (b) The layers under the receiver Figure 3. The dynamic network module for computing multilayer formation using three-layer modules. Chew et al, 1991; Chew et al, 1984; Chew and matrices at the vertical boundary i. In general, Ri, +l Anderson, 1985; Liu et al, 1989; Pai, 1991; Pal et al, and T.,i+l are not diagonal except for the case where 1993]. The above method made a success on solving the formations are only cylindrically layered. The this problem. matrix nature of (23) and (24) implies that each As the upgoing or downgoing waves include incident eigenmode will generate a complete set of exponentially increasing and decreasing modes modes in the transmitted and reflected waves. simultaneously, the coefficients may be well beyond Once the eigenmodes in each layer are found, the the limits of the maximum and the minimum wave propagation in the layers is converted to the exponents of floating points in a computer if the waves mode propagation. All modes attenuate in the direction are not treated appropriately. Any kind of truncation they propagate. Higher-order modes decay faster than will restfit in severe instability of the algorithm. lower-order modes. Therefore, after a certain Therefore, after a series of calculations, the result may propagation distance, the lower-order modes are the be beyond the maximum limit of the floating point in a dominant components. Note that transmitted waves computer. An overflow error may occur in such cases. keep the same direction of propagation when passing In this paper, we proposed a new algorithm to avoid through a layer boundary, while reflected waves the instability problem. The waves are traced so that change directions. One way to describe the mode only exponentially decaying waves are involved in the propagating through layers is to categorize the waves computation. Consider a three-layer formation shown into "upgoing" (positive z direction) and "downgoing" in Figure 2. The vector I represents the incident field, (negative z direction) waves, and then match the and the vector R is defined as the reflected field from boundary conditions at layer boundaries to find the boundary 1, which is a summation of reflected unknown coefficients [Anderson and Chang, 1982; fields due to the incident I at the boundary 1 and all transmitted fields of the fields in layer 2. Vector T is defined as the transmitted field from boundary 2, which is a summation of the transmitted fields of all modes in layer 2. This three-layer module can be described as a three-port network shown in Figure 2b. The input, output 1, and output 2 represent/, T, and R, respectively. In the!ossy medium, the field decays in the propagation direction. Therefore, after a certain number of reflections between boundary 1 and 2, the field becomes negligible. The accuracy of the T and R increases as more reflections between boundaries are taken into account. In each of the three-layer modules, the computation of T and R traces the wave until the

5 TIAN ET AL.' SIMULATION OF INDUCTION AND MWD LOGGING TOOLS ,1.0 I [ 110.0, cm 20.3 cm 110.0,1.0!! Figure 4. A three-layer formation. wave is negligible. Using the wave-tracing algorithm, formation has a 25.4 cm borehole, an invasion zone of the computation is guaranteed to be convergent. In multilayer cases, layers are partitioned into a 20.3 c n and a bed thickness of cm. The relative dielectric permittivities and conductivities are 70.0 and series of three-layer modular components. Figure mho/m for the borehole mud, and 1.0 mho/m shows the strategy of simulating a multilayer case for two shoulder beds, 86.0 and 0.5 for the invasion using the three-layer modules. The vector I represents zone, and 38.0 and 0.05 for the thin bed, respectively. the incident field from the transmitter. Compare the Shown in Figure 5 is the computed apparent outputx1 of the module 1 with the incident wave I. If conductivity (denoted by fro ) using a 2C40 tool. The the difference between them is less than a given 2C40 tool is a two-coil tool which has one transmitter tolerance ca, the calculation is stopped. If not, the module 2 is calculated. If the difference between Z1 and I is greater than the e, a third stage is automatically added. This process continues until the -- J i,., Pres'ent Result tolerance is satisfied. Similarly, the reflected field Y [ '... can be calculated by using module 3. If the field X3 is not small enough, module 4 and module 5 may be necessary_. The same is true for modules 6 and 7. Combining Y1, Y2, Y3, and Y4, the field at the position of the receiver can be obtained. For the layers below the receiver, the procedure is the same. Since all the modes decay in the process, this method is very = ff o.?... [.L.d. stable O Depth (m) 3. Verifications Figure 5. Comparison of the computed logs obtained by To verify the algorithm described above, induction using the present method and logs presented by Li and Shen tool responses to a thin layer sandwiched between two [1993b]. the tool is a 2C40, and the formation is shown in shoulder beds as shown in Figure 4 are simulated. The Figure 4.

6 954 TIAN ET AL.' SIMULATION OF INDUCTION AND MWD LOGGING TOOLS 30.5 cm 1, cm :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: i ',,,,,,,,,,,,½ i..,,,,,, - ';'.. '.,..' '. ;: ' 1.0 :: ::;:: : :: ::;::::: :: :: :: :: :: :: ;;:: :: :: : :.:.:.:.,.:.:.:..:.:....: : : :: ': ½½:,,... ::::; ::::;::::::::::::::::::-: :.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:...- ½- --. ;..,... :.:.:.:.: :.:.:::-:::::.: :::::::::::::::::::::::::::::: :,: ½:...:::::::::::::::::::::::::.:-:-:-:-:...-:...-..:...-: '½?.,,',: :, cm.::... ::.:.:.:.:.: [.:.:..:.: :.:::.:...,--!! 25.4 cm 1.0 [ ';' ' ¾" ' ' ' ' '" h' '½' ' "".... :':'=... :... r"" ;' * ';... ;7' '"';... i;'* II il, l i... Figure 6. A formation with multiple layers and invasion zones cm Ra2 Rb2 Rc2 Rd * cm 1', 't, 22.9 cm 30.5 cm [ 45.7 cm 50.8 cm 76.2 cm cm cm Ral Rbl ;::$:;:;;;:;:;:;:;:;:;:;: :;:;:;:;;;:;:;:;;;'-;:;:;: i!._'i:i:i:i:i:.=.:!!½! :! :i:i:-'s..'.: ' '.. i:!_.!:, Rcl Rdl r Number of turns: Transmitter coil: 1 Receiver Coil coils: Ra I I Rb Rc Rd -24, Figure 7. Configuration of an array induction tool.

7 TIAN ET AL.: SIMULATION OF INDUCTION AND MWD LOGGING TOOLS 955 and one receiver. The two coils are separated by 1.01 m. The operating frequency of the tool is 20 khz. It is assumed that the transmitting and receiving coils are all coaxial with the tool and the borehole axis. The computed log is also compared with the one obtained by using a vertical eigenstate method [Li and Shen, 1993b], and the agreement is excellent. 4. Array Induction Logs The second example is the simulation of array induction logs in a formation with multilayer multiinvasion zones shown in Figure 6. The formation has a borehole of 25.4 cm, two shoulder beds, and multiinvasion zones. Some layers have a thickness of cm, and the others are cm thick. The conductivities of each bed and shoulders are also plotted in Figure 6, with the unit of mho per meter. The relative dielectric permittivities of all layers are set to 1.0. The array induction tool in this example has four pairs of receiving coils and one transmitting coil, as shown in Figure 7. Shown in Figure 8 are curves of real and imaginary parts of computed apparent conductivities corresponding to different pairs of receiving coils. The results show that different investigation depths can be obtained by changing the spacing between the transmitter and the receiver. From these curves, more detailed interpretations for the formation can be reached. The coils with shorter distances carry more information about the invasion zones, and the pair with longer distance bears more information about virgin zones. Therefore an array induction tool provides more detailed irfformation about the formation and can be used to image the formation :... I,,.,,, [ o o " R,,. ø,, , t... f , S.OS 3.66 Depth (m) Figure 8. Array induction logs. The tool configuration is shown in Figure 7. o t t -/... -l-1-- Am'plitudel--- /!l,phase k Lu - ' Depth (m) Figure 9. Measurement-while-drilling (MWD) logs in the formation shown in Figure 4. The tool parameters are given in Figure Measurement-While-Drilling Logs The MWD simulation developed in this study is used to calculate the responses of MWD tools in multi- invasion formation. The MWD tool measures the phase difference and the amplitude ratio of the voltages at a pair of receiving coils. These data are then converted to conductivities using the homogeneous medium as a reference. Details of the principle and the conversion scheme are described by Li and Shen [1993a, b] and $hen [1991a]. Figure 9 shows computed MWD logs in a formation shown in Figure 4. The tool has one transmitter and two receivers, shown in Figure 10. The curve with dots is the apparent conductivity convened from the amplitude ratio of the voltages in the two receivers, R a. The other is the apparent conductivity computed from the phase difference, R e. The logging range is from m to m. 6. Conclusion The simulation method presented in this study is based on a horizontal eigenmod expansion method. In order to avoid numerical instability problem associated with any exponentially growing terms, all waves are treated as decaying waves. The three-layer module is used to solve the multilayer problem. Owing to the usage of the three-layer module and elaborately designed numerical calculation strategy, the corresponding computer codes only occupy a very small memory space; and the CPU time is reduced by a factor 5.2 compared with the method w th direct i I

8 956 TIAN ET AL.: SIMULATION OF INDUCTION AND MWD LOGGING TOOLS 16.5 cm References Collar Anderson, B., and S. K. Chang, Synthetic induction logs by the finite element method, Log Anal., 23(6), 17-26, Chew, W. C., and B. Anderson, Propagation of electromagnetic waves through geological beds in a Transmitter geophysical probing environment, Radio Sci., 20, , Chew, W. C., S. Barone, B. Anderson, and C. Hennessy, Diffraction of axisymmetric waves in a borehole by bed cm boundary discontinuities, Geophysics, 49(10), , 1984, Chew, W. C., Z. Nie, Q. H. Liu, and B. Anderson, An efficient solution for the response of electrical well Receiver 1 logging tools in a complex environment, IEEE Trans cm 0.0 Geosci. Remote Sens., 29, , Kennett, B. L. N., Seismic Wave Propagation in Stratified 7.6 cm Receiver 2 Media, Cambridge University Press, New York, Li, J., and L. C. Shen, MWD resistivity logs in invade beds, LogAnal., 34(2), 15-17, 1993a. Li, J., and L. C. Shen, Vertical eigenstate method for simulation of induction and MWD resistivity sensors, 17.8 cm IEEE Trans. Geosci. Remote Sens., 31, , 1993b. Liu, Q. H., W. C. Chew, M. R. Taherian, and K. A. Safinya, Figure 10. AnMWD tool. A modeling study of electromagnetic propagation tool in complicated borehole environments, Log Anal., 30(6), , Pai, D. M., Induction log modeling using vertical eigenstates, solution of the total reflections. Using this method, IEEE Trans. Geosci. Remote Sens., 29, , corresponding computer codes for the mixed-boundary Pai, D. M., J. Ahmad, and W. D. Kennedy, Two-dimensional problem are stable and efficient. induction log modeling using a coupled-mode, multiple- Theoretically speaking, an infinite number of modes reflection series method, Geophysics, 58(4), , are needed. In practice, since the formation is 1ossy, the field decays exponentially in a few skin depths Shen, L. C., Investigation depth of coil-type MWD resistivity from the center of the borehole. Therefore only a finite sensor, Trans. SPWLA Annu. Logging Sympos., 32, Paper number of modes are necessary. In examples shown in C, 1-23, 1991 a. this paper, 25 eigenmodes are used. Smaller grids are Shen, L. C., Theory of a coil-type resistivity sensor for MWD chosen near the borehole, and greater grids are used in application, Log Anal., 32(5), , 199 lb. areas far from the borehole. Comparison the simulated C. Liu and L. C. Shen, Department of Electrical and results with some published data, it is found that the Computer Engineering, University of Houston, 4800 Calhoun developed algorithm is stable and has a satisfactory Road, Houston, TX ( Cliu uh.edu; accuracy. LShen uh.edu) X. Tian, LWD/MWD Department, Halliburton Energy Acknowledgments. The authors are grateful for financial Services, 2135 HWY 6 South, Houston, TX ( support from the Well Logging Consortium at the University mtian@halnet.com) of Houston, composed of 15 oil and service companies. They are also grateful to David Pal for his comments and (Received October 24, 1997; revised March 17, 1998; discussions. accepted March 20, 1998.)