Application of the differential effective medium theory to determine fluid saturation and crack density from measured P- and S-wave velocities

Abstract

Summary The differential effective medium (DEM) theory studied in this article describes elastic moduli of a fractured medium with help of differential equations, where crack density is the independent variable and fluid saturation is a parameter. The effective medium is isotropic for randomly oriented flat ellipsoidal cracks and thus fully characterized by two elastic constants. In this article we derive an analytical solution of the equation for Poisson’s ratio and we transform the differential equation for Young’s modulus into a non-linear algebraic equation. Fluid saturation and crack density can then be determined from measured wave propagation velocities by a simple algorithm. We also derive approximate solutions for elastic moduli as a function of crack density and saturation, which allows to quantify the uncertainty of the result due to measurement errors. The DEM theory leads to higher crack densities than the self-consistent (SC) method and to lower crack densities than the non-interacting (NI) theory for measured elastic moduli, while all three methods give similar fluid saturation fractions. As an example of application of our theoretical results, we study weathered granite in the Strengbach water catchment in the Vosges mountains in France. We have performed full waveform sonic logging measurements in an 86 m deep borehole located at an altitude of 1130 m above sea level, which is used for hydro-geophysical and geochemical studies of a granitic aquifer. The logging data allows us to investigate P and S waves in the depth range between 40 and 80 m. The P and S wave propagation velocities take average values of 5.0 km/s and 2.7 km/s, respectively, with the highest values of 5.8 km/s and 3.2 km/s at 75–80 m depth. From these velocities we obtain a water saturation of 75 ± 25 per cent. The crack density describes the degree of weathering of the granite, which generally decreases with depth, but takes high values near layers of strongly weathered granite. Crack density is on average 0.5, with the highest value of 1.0 at 65 m and the lowest value of 0.2 at 75 m depth. The analysis of the full waveform logging data by the DEM method supports results from previous geochemical and hydrological studies in the Strengbach catchment which concluded that water is stored in deeper layers of the granitic aquifer.

Hydrogeological structure of a granitic mountain catchment inferred from multi‐method electrical resistivity datasets

“…Zillmer et al. (2021) computed the crack density from full waveform sonic logging measurements on F1, describing the crack density as the number of cracks multiplied by the mean volume occupied by an ellipsoidal crack, per unit volume. It is showed that the crack density has average values of 0.5 from 38 m to 55 m depths, reaches a maximum of 0.8–1.0 around 64 m depth and decreases to 0.3 below, until depths of 80 m, suggesting that the unweathered granitic basement is not reached in the deepest borehole.…”

“…Moreover, Zillmer et al. (2021) showed that most cracks between 40 m and 80 m depth are water‐saturated.…”