Traditional methods of calculating densities from borehole gravimeter (BHGM) data produce poor results when used at small station spacing, which limits the depth resolution of the method. This paper describes a new method for calculating densities using an inversion technique. Using this method, useful densities can be recovered at station spacings as small as one meter. The inversion method has been sucessfully spplied to a multi-well BHGM survey from the U.S. Department of Energy Hanford Waste Treatment Plant in Washington state. Approximately 4200 feet was logged in three wells, with a station spacing of 10 feet. The inversion results have excellent correlation with gamma-gamma density logs, and an uncertainty of ± 0.02 g/cm3 or better.
Borehole gravity data can provide density logs with advantages over conventional gamma-gamma density logs (McCulloh (1966); LaFehr (1983)): • A large radius of investigation around the borehole • Relative insensitivity to borehole conditions (washouts, casing, fluid invasion, etc.) • A straightforward relation between the gravity measurements and rock density One of the drawbacks of BHGM logging compared to other methods is the inability to calculate useful densities at high depth resolution, usually taken as less than 30 feet. The culprit here is the division by Dg, which amplifies the effect of errors in Dg as Dz becomes small. For a station spacing of 10 feet and a Dg measurement uncertainty of 0.0071 milliGals (mGals), the density uncertainty is ± 0.051 g/cm3 . If we want densities accurate to ±0.02 g/cm3 or better using this method, we would need to have gravity reading errors of 0.002 mGals or better.
Given the problem for small station spacing inherent in equation 2, we can list some desireable features of an improved method of calculating densities. • Since we know that the gravity data has errors, we should not expect to fit the data exactly: if we do, we know we are fitting noise. • We would like to avoid the explicit division by Dz, since that amplifies the noise in the gravity data. • There is obviously some redundancy in the problem, which should stabilize it in the presence of noise: every pair of gravity stations which bracket a given interval tells us something about the density over that interval, albeit possibly averaged together with other intervals. • Potential field theory tells us that there is an infinite number of models which fit the data: of that infinte set, we would like to find a unique model which has some physically reasonable property. All of these criteria can be met using inversion techniques. In order to write this as an inverse problem, we need to take another look at the forward problem: given a set of interval densities, calculate the BHGM data. If we view the densities as a stack of infinite horizontal slabs, assigned the average density between each consecutive set of stations, we can see that all the slabs a given station act to decrease the gravity at the station.
