Conventional crosswell direct-arrival traveltime tomography solves for velocity in a 2-D slice of the subsurface joining two wells. Many 3-D aspects of real crosswell surveys, including well deviations and out-of-well-plane structure, are ignored in 2-D models. We present a 3-D approach to crosswell tomography that is capable of handling severe well deviations and multiple-profile datasets.
Three-dimensional pixelized models would be even more seriously underdetermined than the pixelized models that have been used in 2-D tomography. We, therefore, employ a thinly layered, vertically discontinuous 3-D velocity model that greatly reduces the number of model parameters. The layers are separated by 2-D interfaces represented as 2-D Chebyshev polynomials that are determined using a priori structural information and remain fixed in the traveltime inversion. The velocity in each layer is also represented as a 2-D Chebyshev polynomial. Unlike pixelized models that provide limited vertical resolution and may be overparameterized horizontally, this 3-D model provides vertical resolution comparable to the scale of wireline logs, and reduces the degrees of freedom in the horizontal parameterization to the expected in-line and out-of-well-plane horizontal resolution available in crosswell traveltime data.
Ray tracing for the nonlinear traveltime inversion is performed in three dimensions. The 3-D tomography problem is regularized using penalty constraints with a continuation strategy that allows us to extrapolate the velocity field to a 3-D region containing the 2-D crosswell profile. Although this velocity field cannot be expected to be accurate throughout the 3-D region, it is at least as accurate as 2-D tomograms near the well plane of each 2-D crosswell profile. Futhermore, multiple-profile crosswell data can be inverted simultaneously to resolve better the 3-D distribution of velocity near the profiles.
Our velocity parameterization is quite different from pixelized models, so resolution properties will be different. Using wave-modeled synthetic data, we find that near horizontal raypaths have the largest mismatch between ray-traced traveltimes and traveltimes estimated from the data. In conventional tomography, horizontal raypaths are essential for high vertical resolution. With our model, however, the highest resolution and most accurate inversions are achieved by excluding raypaths that travel nearly parallel to the geologic layering. We perform this exclusion in both a static and model-based manner. We apply our 3-D method to a multiple-profile crosswell survey at the Cymric oil field in California, an area of very steep structural dips and significant well trajectory deviations. Results of this multiple-profile 3-D tomography correlate very well with the independently-processed single profile results, with the advantage of an improved tie at the common well.