In addition to reliability and stability, the efficiency and expediency of inversion methods have long been a strong concern for their routine applications by well-log interpreters. We have developed and successfully validated a new inversion method to estimate 2D parametric spatial distributions of electrical resistivity from array-induction measurements acquired in a vertical well. The central component of the method is an efficient approximation to Fréchet derivatives where both the incident and adjoint fields are precomputed and kept unchanged during inversion. To further enhance the overall efficiency of the inversion, we combined the new approximation with both the improved numerical mode-matching method and domain decomposition. Examples of application with synthetic data sets show that the new methodis computer efficient and capable of retrieving original model re-sistivities even in the presence of noise, performing equally well in both high and low contrasts of formation resistivity. In thin resistive beds, the new inversion method estimates more accurate resistivities than standard commercial deconvolution software. We also considered examples of application with field data sets that confirm the new method can successfully process a large data set that includes 200 beds in approximately [Formula: see text] of CPU time on a desktop computer. In addition to 2D parametric spatial distributions of electrical resistivity, the new inversion method provides a qualitative indicator of the uncertainty of estimated parameters based on the estimator’s covariance matrix. The uncertainty estimator provides a qualitative measure of the nonuniqueness of estimated resistivity parameters when the data misfit lies within the measurement error (noise).

Presence of near-wellbore damage, resulting from drilling and mud-filtrate invasion, can substantially affect sonic and resistivity borehole logging measurements. Therefore, unbiased log interpretation must account for presence of invasion in order to procure accurate estimates of formation properties. Our objective is to estimate the formation's dry bulk and shear moduli, porosity, and water saturation from the joint inversion of borehole array-induction resistivity and sonic measurements.We assume a radial one-dimensional (1D) model in the inversion, with the formation model described by a radial variation of water and hydrocarbon saturations representative of mud-filtrate invasion. The inversion is guided by the data misfit of both array-induction apparent resistivities and sonic-log flexural and Stoneley wave velocity-frequency dispersion curves. Radial distributions of fluids are converted to distributions of resistivity, density, and bulk modulus, which are input to the simulations of apparent resistivity and sonic logs. We make use of fluid-substitution models to relate bulk density, dry bulk modulus, and dry shear modulus to porosity and fluid saturation. Apparent resistivities are simulated based on a commercially available array-induction logging tool. Sonic measurements are analyzed in the frequency domain via flexural and Stoneley wave mode dispersions, which are calculated directly in the frequency domain.Synthetic cases consider water-base mud filtrate invading a hydrocarbon-bearing sand and oil-base mud filtrate invading a water-bearing sand. Porosities and elastic properties consistent with a soft formation are considered in the models. Sensitivity analysis indicates that sonic flexural and Stoneley mode dispersions naturally complement apparent resistivity measurements in the presence of mud-filtrate invasion. Inversions of synthetic cases produce reliable estimates of dry-rock bulk and shear moduli, porosity, and initial water saturation. Furthermore, these cases show that combining resistivity and sonic measurements reduces ambiguity in the inversion.

We quantify the influence of oil-based mud (OBM)-filtrate invasion and formation-fluid properties on the spatial distribution of fluid saturation and electrical resistivity in the near-wellbore region. The objective is to appraise the sensitivity of borehole resistivity measurements to the spatial distribution of fluid saturation resulting from the compositional mixing of OBM and in-situ hydrocarbons. First, we consider a simple two-component formulation for the oil phase (OBM and reservoir oil) wherein the components are first-contact miscible. A second approach consists of adding water and surfactant to a multicomponent OBM invading a formation saturated with multiple hydrocarbon components. Simulations also include presence of irreducible, capillary-bound, and movable water. The dynamic process of OBM invasion causes component concentrations to vary with space and time. In addition, the relative mobility of the oil phase varies during the process of invasion because oil viscosity and oil density are both dependent on component concentrations. Presence of surfactants in the OBM is simulated with a commercial adaptive implicit compositional formulation that models the flow of three-phase multicomponent fluids in porous media. Simulations of the process of OBM invasion yield 2D spatial distributions of water and oil saturation that are transformed into spatial distributions of electrical resistivity. Subsequently, we simulate the corresponding array-induction measurements assuming axial-symmetric variations of electrical resistivity. We perform sensitivity analyses on field measurements acquired in a well that penetrates a clastic formation and that includes different values of density and viscosity for mud filtrate and formation hydrocarbon. These analyses provide evidence of the presence of a high-resistivity region near the borehole wall followed by a low-resistivity annulus close to the noninvaded resistivity region. Such an abnormal resistivity annulus is predominantly caused by high viscosity contrasts between mud filtrate and formation oil. The combined simulation of invasion and array-induction logs in the presence of OBM invasion provides a more reliable estimate of water saturation, which improves the assessment of in-place hydrocarbon reserves.

We introduce and successfully test an efficient method to simulate triaxial borehole electromagnetic ͑EM͒ induction measurements acquired in axially symmetrical and transversely isotropic ͑TI͒ media. The method uses a Fourier series expansion to express the azimuthal dependence of EM fields and the source term whereby the essentially 3D problem collapses to a series of independent 2D problems. Each 2D problem is solved with a semianalytic method that uses normalized Bessel functions and normalized Hankel functions to express the radial dependence of EM fields, thereby improving numerical stability. In addition, use is made of amplitude and slope basis functions to describe the longitudinal dependence of EM fields to avoid grid refinement in the vicinity of horizontal formation boundaries. For validation, we compare the new simulation method to two 1D analytic methods in horizontally and radially layered formations, and to one 3D finite-difference method ͑3DFD͒ in multilayered formations that include borehole and invasion zones. Numerical results indicate that the method is accurate in formations with high conductivity contrasts compared to 1D methods and is more than ten times more efficient than the 3DFD method in multilayer formations.