Ningya Cheng

Error analysis of phase screen method in 3‐D

The purpose of the research presented in this paper is to do an error analysis of the phase screen method for forward and inverse wave propagation calculations in 3D heterogeneous acoustic media. The differential operator formalism approach is used, which provides new insight into the phase screen method. Under the assumptions that backscattering and commutator term of operators R and Q are negligible, we derive a partial differential equation and its solution corresponding to the phase screen method. The errors introduced in these derivations are also obtained. For a one-step (δz) phase screen calculation, the leading error term is in the first order of k0Δz and proportional to the error from splitting the square-root operator Q. The propagation angle should be less than 40 degrees to control the error from the splitting operator Q under 5%.

Decomposition And Particle Motion of Acoustic Dipole Log In Anisotropic Formation

For linear wave propagation in anisotropic media, the principle of superposition still holds. The decomposition of the acoustic dipole log is based on this principle. In the forward decomposition inline and crossline acoustic dipole logs at any azimuthal angle the projection of measurements is along the principal direction of the formation. In the inverse decomposition the measurements along the principal direction can be constructed from the orthogonal pair of inline and crossline acoustic dipole log. The analytic formulas for both forward and inverse decompositions of the dipole laaa ssssaog are derived in this paper. The inverse decomposition formula is the solution in the least-square sense. Numerical examples are demonstrated for the acoustic dipole log decomposition in isotropic and anisotropic formations. The synthetic dipole log is calculated by the 3-D finite difference method. The numerical examples also show that the inverse decomposition formula works very well with noisy data. This inverse decomposition formula will be useful to process the field acoustic logging data in anisotropic formations. It can provide the direction of the formation anisotropy as well as the degree of anisotropy. Because acoustic dipole logging is in the near field distance, the particle motion is complicated. The particle motion is linearly polarized only in the principle direction. The initial particle motion with a dipole source at an arbitrary azimuthal angle tends to point in the fast shear wave direction. However, it will be difficult to use this information to find a stable estimation of a fast shear wave direction. 2-1 Cheng and Cheng INTRODUCTION The crust of the Earth is slightly anisotropic, which is related to geological processes. For example, anisotropy can be caused by aligned fractures in the rock. Fine-layered sedimentary rocks also possess transverse isotropy. Acoustic logging provides a technique to measure formation anisotropy from drilled wells. To understand the effects of anisotropy on fluid-filled borehole wave propagation, it is critical to process acoustic logging data in anisotropic formations. Several numerical techniques have been developed to simulate acoustic logging, such as the discrete wavenumber method (White and Tongtaow, 1981; Chan and Tsang, 1983; Schmitt, 1989), the perturbation method (Ellefsen, 1990; Sinha et al., 1994), and the finite difference method (Leslie and Randall, 1992; Cheng et aI., 1995). For a transversely isotropic formation with the symmetry axis aligned with the borehole axis, the solution is wellknown and the different type of waves are well-understood. But when the symmetry axis of the TI formation makes an arbitrary angle with the borehole axis, the borehole wave propagation problem becomes much more complicated. Most of our knowledge comes from the perturbation solution. On the other hand, field data observations in the anisotropic formation are available, e.g., borehole waves generated from a vertical seismic profile (Barton and Zoback, 1988; Leveille and Seriff, 1989) and borehole dipole logging (Esmersoy et al., 1994). The purpose of this paper is try to answer the following three questions: (1) How do we construct the inline and crossline dipole log at any azimuthal angle from the measurement along the principle axis? (2) What dows the particle motion of the dipole log look like in an anisotropic formation? (3) How do we recover the measurements along the principle axis from the inline and crossline logs at an arbitrary azimuthal angle? The first and the third questions are the forward and the inverse problem pair. In order to do numerical experiments, the 3-D time domain finite difference method is used to generate a synthetic dipole log. The formation is assumed to be transversely isotropic with its symmetry axis perpendicular to the borehole axis. FINITE DIFFERENCE METHOD IN AN ANISOTROPIC MEDIUM The wave equation in anisotropic media can be formulated by using velocity and stress. The first-order hyperbolic equations can be written in compact form as: 8Vi Pat = Tij,j (1) and (2) where P is the density, Vi is the velocity vector, and Tij is the stress tensor. Cijkl is the elastic constants and we assume the media is orthorhombic (nine elastic constants). 2-2 (3) Particle Motion of Acoustic Dipole Data A comma between subscripts is used for spatial derivatives. The above equations are discretized on the staggered grid. The finite difference operators are centered. The firstorder time derivative is approximated by the second-order finite difference operator and, the first-order space derivatives are approximated by the fourth-order finite difference operators. The medium parameters Cijkl and p are assigned at the grid point (m+!, n+ ! k). In order to update the velocities, the needed density values are obtained from the average of two nearby assigned densities. In order to update the shear stress, the needed shear moduli are determined by four nearby assigned shear moduli using the harmonic average. This automatically puts the shear modulus zero at the fluid-solid boundary. The stable condition used to determine the time step size is 6 6t < r;; , v 3Vp (IId + 1721) where Vp is the fastest quasi-P wave velocity in the model. 6 is the grid size. II = £and 12 = 214 In order to simulate the infinite medium on a computer with limited memory, we have to eliminate the reflections from the artificial boundaries. Higdons absorbing boundary condition operator is used (Higdon 1990). Because of the anisotropic media, the wave propagates with different velocities in different directions. The velocity in the absorbing boundary condition is chosen according to the direction of the boundary. Finally, the 3-D finite difference scheme is implemented on the nCUBE parallel computer by using the Grid Decomposition Package. A more detailed description of the method can be found in Cheng et al., (1995). The finite difference method is used to compute the synthetic acoustic dipole log in the anisotropic formation. FORWARD DECOMPOSITION OF THE DIPOLE LOG Although we consider wave propagation in anisotropic media, the wave motion is still governed by linear differential equations (Eqs. 1 and 2). The principle of superposition is valid. We assume that the X and the Y axis are aligned with the two principal axes of anisotropy. The dipole source is in the X-Y plane and Gij is the Greens function (index i and j represent x, y). The general form of the dipole measurement is R.; = G ij *Sj, where R represents the receiver and S represents the source. The symmetry of the model requires Gxy = Gyx = O. We define V x = Gxx *S as the inline dipole measurement in the X direction and v y = Gyy * S is the inline dipole measurement in the Y direction. These measurements can be computed by the finite difference method. The same dipole source with azimuthal angle e (counterclockwise from the X axis), can be decomposed into the X and Then the measurement along the e e V x V x cos and the measurement along the Y ·axis will be: e . e v y = vysln 2-3 (4)

Fourth-Order Finite-Difference Acoustic Logs In Transversely Isotropic Formation

In this paper we present a finite difference scheme for seismic wave propagation in a fluid-filled borehole in a transversely isotropic formation. The first-order hyperbolic differential equations are approximated explicitly on a staggered grid using an algorithm that is fourth-order accurate in space and second-order accurate in time. The grid dispersion and grid anisotropy are analyzed. Grid dispersion and anisotropy are well suppressed by a grid size of 10 points per wavelength. The stablility condition is also obtained from the dispersion analysis. This finite difference scheme is implemented on the nCUBE2 parallel computer with a grid decomposition algorithm. The finite difference synthetic waveforms are compared with those generated using the discrete wavenumber method. They are in good agreement. The damping layers effectively absorbed the boundary reflections. Four vertically heterogeneous borehole models: a horizontal layered formation, a borehole with a radius change, a semi-infinite borehole, and a semi-infinite borehole with a layer, are studied using the finite difference method. Snapshots from the finite difference results provide pictures of the radiating wavefields. INTRODUCTION Finite difference method is a very powerful technique in the modelling of seismic wave propagation in inhomogeneous media. There have been some applications of this method to acoustic logging problems. Stephen et al. (1985) directly applied the finite difference method to the second-order displacement formulation of the wave equation. The fluidsolid boundary at the borehole wall was treated explicitly. The velocity-stress finite difference method is widely used after Virieuxs (1986) work. Kostek (1990), using the first order hyperbolic wave equation formulation, discretized on the staggered grid to solve the logging problem with a transducer in the borehole. Randall et al. (1991) al-

Three-dimensional Finite-difference Borehole Wave Propagation Anisotropic Medium

In this paper we developed 3-D finite difference method to simulate using a scaled borehole model in phenolite solid. It also used to wave propagations in an anisotropic medium. The finite difference results agree excellently with the analytic solutions of a point force interprete the field data, which shear wave velocity discrepancy occures. source in the transversely isotropic medium. The finite difference synthetics are also compared with ultrasonic lab measurements in Finite Difference Method in an Anisotrogic Medium a scaled borehole drilled along X axis in an orthorhombic phenolite solid. Both monopole and dipole logs agree well. Wave propagation in an anisotropic elastic medium can be deThe disagreement between the shear wave velocity from the scribed by the equation of motion: Stoneley wave inversion and the direct shear wave log velocity from field data is beyond the errors in the measurements. It is proven that the formation permeability is not the cause of the discrepancy. (1)