We developed a three-dimensional finite-difference codes which can simulate the fully triaxial induction tool in a stratified, cross-bedded anisotropic formation. The sensitivity of the fully triaxial induction measurements in cross-bedded anisotropic formation is studied for the first time. The results show that the tensor measurements are sensitive to the formation resistivity anisotropy, formation structural dip as well as formation stratigraphic dip. A nonlinear least square minimization algorithm has been developed to determine the formation parameters in such a situation.
For many sedimentary rocks, the depositional process will produce fine bedding planes that are parallel to the structural bed boundaries. These are often alternating rock types, such as sandstone and shale. For these rocks, the conductivity tensor will have a well-defined horizontal conductivity (sh) that is parallel to the structural bedding planes, and that is often independent of the direction in that plane. The vertical conductivity (sv) is perpendicular to the bedding planes and will be different from s h. Other depositional environments will produce crossbedding, where the fine bedding is not parallel to the structural bed boundaries. Here the conductivity tensor will be aligned with the crossbedding planes.
Electrical anisotropy in rocks was first discussed by Schlumberger et al. (1934) in the early history of electrical logging. One important result is that many logging tools (including induction tools) only detect the horizontal resistivity in vertical wells. This is called the paradox of anisotropy (Kunz and Moran, 1958). The effect of the vertical resistivity on induction logging will increase when the relative dip angle between the borehole and the formation bed increase (Anderson et. al., 1990, Anderson, et al., 1995). However, the sensitivity is not strong enough to solve for the vertical conductivity using standard induction measurements. The multicomponent induction tool (Kriegshäuser et. al., 2000, Rosthal et al., 2003) has sensitivity to both horizontal and vertical conductivity.
A fully triaixal induction tool consists of a three orthogonal collocated transmitters and three collocated orthogonal receivers. It acquires a 3×3 tensor measurement which is sensitive to resistivity anisotropy and formation geometry. So far, the main application of triaxial induction measurements are finding low resistivity pay and providing formation structural dip (Barber et. al., 2004, Wang et. al., 2006). Most of the past studies on resistivity anisotropy were concentrated on flat-lying sand-shale formations. Anderson et. al. (1998) examined the effect of anisotropic crossbedded formations on standard array induction tools in vertical wells. Wang and Georgi (2004) studied the response of a multi-component tool in an anisotropic crossbedded formation. This was the first work to study the sensitivity of all 9 components of a multicomponent induction array in such a situation. In this paper we will show how to obtain the dip of the crossbeding as well as its resistivity anisotropy by the inversion technique as described in Habashy and Abubakar (2004) and Abubakar et al. (2006).
The mathematical model discussed in this paper consists of a series of anisotropic beds with arbitrary dip (a ) and azimuth ( b ) angles relative to the normal of the formation bed boundaries ( z ) as shown in Figure 1 and 2.