A borehole direct-current resistivity boundary value problem for normal and lateral electrode configurations is solved assuming axial symmetry. The borehole mud, a flushed zone, an invaded zone, and an uncontaminated zone are all assumed to be present. A linear transition in resistivity is assumed for the invaded zone. Frobenius extended power series and the method of separation of variables are used to solve the 1-D problem. Single-run borehole resistivity sounding and solution of the inverse problem are suggested for estimating the resistivity of the uncontaminated zone and the radius of invasion. Finite-difference modeling is done to estimate the effect of shoulder beds in borehole sounding. Some of the computed 1-D and 2-D model apparent resistivity curves are compared with the existing scale model data.
The analysis reveals that the mud cake effect is negligible for normal and lateral electrode array and the invasion zone thickness is reflected in the forward models. Apparent resistivity curves with and without a transitional invaded zone are well separated. Resistivity departure curves are well separated for fixed resistivity and variable resistivity invaded zone models. A normal electrode configuration can feel the presence of the shoulder bed in a 2-D model when the bed thickness is about 12 times the electrode separation.
One-dimensional ridge regression inversion with synthetic forward model data is presented to suggest an alternative approach for determining the resistivity of the uncontaminated zone (Rt) and the radius of invasion (ri). We conclude that (1) a single run borehole sounding with 10 or 12 data points from a normal or lateral log may be used, rather than 3 points from a dual laterolog Rxo tool, for better estimation of Rt, and (2) a borehole forward model should include a transitional invaded zone. Finally, an alternative approach for the estimation of the radius of invasion is proposed.