Theoretical derivation of a Brie-like fluid mixing law

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Prediction of the velocity of acoustic waves in partially saturated rocks is very important in geophysical applications. The need to accurately predict acoustic velocities has resulted in a widespread popularity of Brie's effective fluid mixing law. This empirical model together with Gassmann's formula are used routinely in fluid substitution problems in petroleum geophysics and seismic monitoring of carbon capture and storage. Most attempts to justify Brie's model have been focused on interpretation in terms of patchy saturation models and attaching meaning to the Brie parameter in terms of the patch size. In this paper, using a microstructural description of the rock and a parameter relating to capillary pressure, we calculate an effective fluid modulus that is very similar to Brie's law. The fluid mixing law we propose is independent of frequency and has a solid theoretical foundation. This proposed law produces analytically harmonic and arithmetic averaging at the endpoints. Our results indicate that Brie-like behaviour may not necessarily be related to frequency- and patch-size- dependent phenomena.

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