This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is based on a coupled scalar-vector potential formulation for arbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. The discretization of the partial differential equations and the enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. The accuracy and efficiency of the algorithm is assessed with examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic, 2D isotropic, and 3D anisotropic rock formations. The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potential formulation favorably compare in accuracy with results obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electrical conductivity. Our numerical exercises indicate that the coupled scalar-vector potential equations provide a general and consistent algorithmic formulation to simulate borehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex models of electrical conductivity.

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.

A numerical simulation study has been made of borehole sonic measurements that examined shoulder-bed, anisotropy, and mud-filtrate invasion effects on frequency-dispersion curves of flexural and Stoneley waves. Numerical simulations were considered for a range of models for fast and slow formations. Computations are performed with a Cartesian 3D finitedifference time-domain code. Simulations show that presence of transverse isotropy (TI) alters the dispersion of flexural and Stoneley waves. In slow formations, the flexural wave becomes less dispersive when the shear modulus (c 44) governing wave propagation parallel to the TI symmetry axis is lower than the shear modulus (c 66) governing wave propagation normal to the TI symmetry axis; conversely, the flexural wave becomes more dispersive when c 44 > c 66. Dispersion decreases by as much as 30% at higher frequencies for the considered case where c 44 < c 66. Dispersion of Stoneley waves, on the other hand, increases in TI formations when c 44 > c 66 and decreases when c 44 < c 66. Dispersion increases by more than a factor of 2.5 at higher frequencies for the considered case where c 44 < c 66. Simulations also indicate that the impact of invasion on flexural and Stoneley dispersions can be altered by the presence of TI. For the case of a slow formation and TI, where c 44 decreases from the isotropic value, separation between dispersion curves for cases with and without the presence of a fast invasion zone increases by as much as 33% for the flexural wave and by as much as a factor of 1.4 for the Stoneley wave. Lastly, presence of a shoulder bed intersecting the sonic tool at high dip angles can alter flexural dispersion significantly at low frequencies. For the considered case of a shoulder bed dipping at 80 , ambiguity in the flexural cutoff frequency might lead to shear-wave velocity errors of 8%-10%.