A New High-Speed Hybrid Technique for Simulation and Inversion of Resistivity Logs

Abstract

Summary. The inversion of resistivity logs with multiple bed boundaries is of vital interest to log analysts, but previously published methods have been optimized for cases of many radial boundaries with a serious loss of speed when many bed boundaries are considered. This paper describes a hybrid solution (analytic radially and numeric vertically) that can simulate a 100-ft [30.4-m] log with 25 beds in less than 12 minutes on an IBM 3081. The model assumes axial symmetry with no dip and with the tool centered in the borehole. Decomposition of the measured potential into the product of two independent functions describing its radial and axial product of two independent functions describing its radial and axial dependence allows us to choose the best method for evaluating each function. For the case dominated by bed boundaries, the most efficient choice is to obtain the radial solution analytically by means of a finite number of modes and to obtain the axial solution using piecewise continuous functions. Several simulations are presented of the dual laterolog (DLL) in multibed cases with invasion, and the results are combined in a least-squares inversion technique to demonstrate the derivation of true resistivity profiles from real logs. Introduction Before proceeding with the details of the hybrid method, a brief description of the essence of the approach is warranted. We will confine our discussion to axisymmetric configurations (e.g., non-dipping beds with centered tools). In the specific context of resistivity logging, Laplace's equation describes the electrostatic voltage as a function of the radial and axial coordinates. In the presence of both radial boundaries (borehole, invasion, etc.) and horizontal bed boundaries, there are no analytic solutions for this voltage. Unto recently, the standard solutions to such problems were -achieved totally numerically by the finite-element method (FEM) or integral-equation methods, One exception to this is where a semianalytic solution was obtained but was limited to mild resistivity contrasts between the various media and therefore will not be discussed. The name hybrid implies that the method is partially numerical and partially analytic in nature. The hybrid method is a separation-of-variables approach where the radial dependence is treated numerically and the axial dependence is treated analytically by assuming that the voltage can be written as a discrete set of modes whose modal amplitudes describe the radial dependence of the voltage. The modal phases, of course, are described analytically with complex exponentials in the axial coordinates. If this assumed solution is inserted into the governing equation (i.e., the equation of continuity with a radially varying conductivity), then the modal am-plitudes satisfy a Stun-n-Liouville equation with a variable profilin radial conductivity. Specifically, a one-dimensional (ID) Bessel equation where all of the radial boundaries, except at infinity, are internal to the problem. Because the radial boundaries separating problem. Because the radial boundaries separating the media with different conductive properties are internal, the solutions to the Sturm-Liouville equation implicitly satisfy the continuity of voltage and normal current flux. Thus, all of the radial heterogeneities are treated as if there were one continuous inhomogeneous medium. The modal amplitudes are determined by expanding them in terms of a convenient set of basis functions (i.e., piecewise continuous functions). Because the modes form a discrete set of functions that vanish at infinity, we might expect that the basis functions themselves would satisfy an eigenvalue problem. Indeed, this is the case. The eigenvalues themselves turn out to be the modal phases for the original expansion. The attendant eigenvectors are nothing more than the expansion coefficients of the basis functions that determine the modal amplitudes. At this point in the discussion, the radial (numerical) portion of our solution is specified entirely. If we now specify appropriate representations for the modes in each horizontal layer, including the presence of the source electrode, we can match these analytically at each horizontal boundary for the completion. Up to now we have concentrated entirely on a discussion of the "forward" or "direct" solution. The other half of the solution is the inversion of the logging data to obtain the electrical parameters of the formation. We have chosen the least-squares technique, which essentially minimizes the sum of squares of errors between the forward model and the logging data. It should be emphasized that the rapidity of convergence to the "true" electrical parameters in the inversion technique is a strong function of speed of the forward solution. It is for this reason that we have developed the hybrid method described here. Formulation Nothing prevents us from reversing the mathematics described in the Introduction. Specifically, the axial part of the solution for the voltage can be solved numerically and the radial part analytically. Indeed, we shall show for the most common cases encountered in the field (i.e., a multiplicity of noninvaded beds) that this reversal is computationally advantageous. The physical configuration for this case is depicted in Fig. 1. As indicated in Fig. 1, we will emphasize the axial variation in conductivity and specify the governing equation of continuity for the voltage Vi(r, z), as ..........................................(1) where i = 1,2, and denote the borehole and formation regions, respectively. Specifically, the conductivity profiles in the respective regions are ..........................................(2) ..........................................(3) P. 55

An efficient numerical modeling of the DC resistivity logging: Theory and applications

“…Results f~om Ref. [12]; 2. Results obtained by NMM method in this paper is l•m, and the resistivity of the two beds is 10f~m and 1000~m, respectively.…”

“…The logging response of dual laterolog sonde in four bed formation with invasion in middle two beds (a) The result in this paper, Sign 1,,,2 represent the shallow and deep log.~ng response, respectively;(b) The result in Ref [12],. Sign 1,,,4 represent the shallow and deep logging response by hybrid method, the shallow and deep logging response by FEM,…”

“…-~ ~ppa ~pg~, = -< g,~, g,~ >where the prime stands for the derivative with respect of p. From the above equation and Eq (14). we can see that both B,,~ and Gn,~ is symmetrical matrix.Eq (12). can be rewritten as the eigenvalue equation with matrix form:Solving this eigenvalue equation, one can get N eigenvalues k2z and N eigenvectors ba which represent N numerical eigenfunctions .f~(p), respectively.…”

Fast Induction-Log Inversion Using Quasi-Newton Updates

“…3 An alternative approach to enhance induction-tool resolution is to carry out a full inversion. Least-squares inversion based on the iterative minimization of the sum of squared errors between log data and forward predictive model results were reported by Lin et al 4 and Gianzero et al 5 in the past. However, least-squares solutions are often questionable because of their high sensitivity to noise.…”