elastic waves in complex radially symmetric media · models. the corresponding acoustic logging...

TG 2.8Elastic waves in complex radially symmetric mediaYouli Quan*, Department of Geophysics, Stanford UniversityXiaofei Chen, Department of Earth Sciences, University of Southern CaliforniaJerry M. Harris, Department of Geophysics, Stanford University

Summary Governing equations

The generalized reflection and transmission coefficientmethod with modal expansion is developed to calculatethe elastic wave field for the acoustic logging simulationand the crosswell seismic simulation in complex radiallysymmetric media. Our model consists of an arbitrarynumber of cylindrical layers and each cylindrical layerhas arbitrary layers in the z-direction. This model can beused to simulate a variety of borehole geophysicsproblems: borehole casing with perforations, faults,source array, open and cased boreholes embedded inlayered media, The simulation based on this methodgives direct waves, reflections, transmissions, tubewaves, tube wave conversions generated at casingperforations and horizontal interfaces, and the radiationpattern for a complicated borehole-formation structure.

A general configuration of the problem considered inthis study is shown in Figure 1, where the formationconsists of several vertically heterogeneous cylinders,the source is located inside the borehole, while receiversare located either inside or outside the borehole.Although this model looks complicated, it still exhibitsnice axial symmetric property. Taking the advantage ofthis symmetry we can derive a set of formulas to simulatethe elastic wave propagation in this complex borehole-formation model. Since the model is radially symmetric,i.e., independent of in cylindrical coordinates, we canwrite the displacement for the jth cylindrical layer as

Introduction

For most borehole simulations (e.g., Tubman et al.,1984; Chen et al., 1993), a borehole is simply modeledas cylindrical homogeneous layers. In a real borehole,however, the medium within a cylindrical layer usuallyare not homogeneous, whose elastic parameters usuallychange with depth. Pai et al. (1993) simulated theelectromagnetic induction logging for this kind ofmodels. The corresponding acoustic logging problemsare more complicated, since we have to solve theelastodynamic equation in which both P and S waves areinvolved in. For elastic wave simulation, Bouchon(1993) and Dong et al. (1995) used the boundary elementmethod for open and cased boreholes embedded in layeredmedia, respectively. In this study, the generalizedreflection and transmission coefficient method isdeveloped to calculate more realistic models, such asmulti-layered boreholes (casing and invaded zones)embedded in multi-layered formation, faults in theformation, and perforation in a casing. This semi-analytical approach is expected to be more flexible andefficient than pure numerical approaches.

(1)

where, and are P wave potential and S wavepotential, respectively. To solve this problem, weexpand these potentials by a set of eigenfunctions,

as

where the eigen-functions and satisfy

I

(3a)

(3b)

and are the corresponding eigen-values to bedetermined. Equation (3) is an eigenvalue problem thatwill be solved in a later section. The general solutionsfor and are

(4a) ( 4 b )

Figure 1. A complex radially symmetric model.

for j > 1, and

(4c)

(4d)

for j = 1 (fluid layer). Here, sign "+" refers to the wavegoing outward, and"-" refers to the wave coming inward.Coefficients are determined by imposing

boundary conditions at r = j =1,2, . . . . J. Using these

1995

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Elastic waves in complex radially symmetric media

potential solutions, we finally obtain the displacementsand stresses in solid cylindrical layers 1) as,

= (5) q=l

= and = and ( p, are given in the’ Appendix.In the fluid-filled borehole

the displacements and stresses are:

= (6) q=l

Using the generalized matrices we can easily obtainthe coefficient vectors aswhere, = = 2 and = + = 2 + ;

and 3 and + are given in the Appendix.

Generalized reflection and transmissionmatrices and frequency domain solution

To effectively determine the unknown coefficients, c(j),we introduce the generalized reflection and transmission

matrices and + , via the following factorizations: for Once having the coefficients we cancalculate the frequency domain solutions (displacementsand stresses) using equations (5) and (6).Thecorresponding time domain solution can be obtained byperforming inverse Fourier transform.

+- +

a

Determination of eigen-functionsfor 1 (solid-solid interface), and

Let us now solve equation (3). First, we expand and

as

=

=

= +s+)

= +s+)

for (liquid-solid interface). By imposing theboundary conditions at each interface,we obtain fhefollowing recursive formulas,

then substituting equation (11) into (3), we obtain

.+ [ , .+-

= 0. (12)

(8)

To solve this eigen-value problem, we assume that thevertically heterogeneity of each cylinder can beapproximately expressed by Fourier series as

forj = N, N-l, . . . . Here, I is the unit matrix; and aregiven by

with

= ,

1996



IHere, k, and is the periodiclength whose value should be larger enough to keep thefinal time domain solution to be correct in the giventime window (Chen et al., 1993). Now equation (12) canbe reduced to

layer in the casing. Tube wave to P and S conversionsclearly show up in Figure 3a.

Finally, eigen-values and and the

corresponded cigen-vectors and can beobtained by using standard linear algebra programs.

Examples

An extreme case is a trivial case in which all cylindricallayers are homogeneous.The more general algorithm inthis study should provide an identical solution to the onewhich is directly solved for this special case (see Chen etal.. 1993). a n d b econstants, then from equation (15) we obtain

Substituting these special solutions into equations (5)and (6), we can obtain identical formulas with thesolutions in Chen et al. (1993).

We present two crosswell profiling examples to showthe applicability of this method for modeling waves invarious complex boreholes. In these examples theseismograms are component. Figure 2 is a commonreceiver gather sorted from a complete crosswell data set.There are 50 sources and 50 receivers. The source intervaland receiver interval are all 2 m. The moveout of the tubewave in a common receiver gather is very different fromthat in a common source gather. In this example weinclude a cased borehole and a layer with fault. Thissynthetic data set can be used to test tomographytechniques and other inversion methods. Tests of thiskind can help us to understand the borehole effects on theinversion. Figure 3 shows an example of cased boreholewith a perforation. To simulate how a perforation in acasing creates tube conversions, we put a low velocity

Conclusions

We have developed a semi-analytical approach tosimulate the haves in a complex borehole-formationmodel in which there exist both cylindrical andhorizontal layers. In this approach, we first calculationthe eigen-functions of horizontal layers for eachcylindrical layer. Then. we expand the wave equationsolutions into these eigen-functions and apply thegeneralized reflection and transmission matrix method tosolve the boundary problem for radial layers. Numericalexamples show that this approach provides an effectivetool to simulate complex models which are found inhorehole modeling.

1997


Figure 3. Test on the effect of casing perforation. Itcan be seen that the tube wave converts to P and Swaves at a perforation.

References

Bouchon. M., 1993. A numerical simulation of theacoustic and elastic wavefields radiated by a source ona fluid-filled borehole embedded in layered medium:Geophysics, 58, No. 4, 475-481.

Chen, X, Quan, Y and Harris, M. J., 1994. Seismogramsynthesis for radially layered media using thegeneralized reflection/transmission coefficientsmethod: Expanded Abstracts of the 64th SEG annualmeeting.

Dong, W., Bouchon. M. and M. N.. 1995,Borehole seismic-source radiation in layered isotropocand anisotropic media: Boundary element modeling:Geophysics, 60. No. 3


Pai, D. M., Ahmad. J. and Kennedy W. D., 1993, Two-dimensional induction log modeling using a coupled-mode, multiple-reflection series method: Geophysics,58, No. 4. 466.474.

Tubman, K. M., Cheng. C. H. and M. N., 1984.Synthetic full waveform acoustic logs in casedboreholes: Geophysics, 49, 1051-1059.

Appendix

For the elements equation (5) are:

the elements in equation (6) are:

1998


elastic waves in complex radially symmetric media · models. the corresponding acoustic logging...

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