Methods using dc electrical arrays to measure formation resistivity through casing have relied on approximate forms for the current and potential distributions to derive a simple relationship between the formation resistivity and the transverse resistance calculated from measurements of the potential and its second derivative inside the casing. We have derived a numerical solution for the potentials and their derivatives to examine the accuracy of the approximate forms for casing of finite-length, annular zones of varying radius, and for vertical discontinuities such as layers or abrupt changes in annular zone radius. For typical conductivity contrasts between the casing and formation, the approximate relationships may be off by as much as 60 percent for long casing and may show variations of 20 to 30 percent as the electrode array moves along the casing. In principle an iterative scheme could be devised to correct the readings if high accuracy was required. The numerical results show that to first order the current flow from the casing is radial, and that all the analytic expressions based on this assumption for evaluating layer resolution and the effects of annular layers are valid. An interesting byproduct of this study has been the discovery that the distortion of the potentials in a nearby well by an annular disk (e.g., an injected steam zone) surrounding the current injection well is greater if the injection well is cased. Crosswell resistivity surveys appear feasible if one of the wells is cased.

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