Theoretical prediction of elastic modulus at different states and squirt-flow-related attenuation: extension of Cracks-Pores Effective Medium model

Squirt flow plays an essential role in elastic modulus dispersion and attenuation for fluid-saturated cracked porous rocks. The Mavko–Jizba model and relevant modified models can describe the squirt flow well based on the related elastic moduli, such as dry/drained bulk modulus. However, when these elastic moduli are challenging to attain, it is impossible to model the squirt-flow-related elastic moduli and attenuations with the models. On the other hand, the effective medium theory (EMT) model can estimate these elastic moduli, but cannot predict the undrained/relaxed and partially relaxed saturated elastic moduli and the squirt-flow-related attenuations. This paper extended an EMT model—Cracks–Pores Effective Medium (CPEM) model—to cover the undrained/relaxed and partially relaxed states following the elastic–viscoelastic correspondence principle. The proposed model [i.e. frequency-dependent CPEM (CPEMF) model] can thus estimate the elastic moduli over the different states (dry/drained, undrained/relaxed, partially relaxed and unrelaxed) and associated attenuations. It agrees well with the prediction of the modified Mavko–Jizba–Gurevich model (MJGZ-HF) at unrelaxed state and is precisely consistent with the prediction of Gassmann at undrained/relaxed state. Also, it analytically shows good consistency with the modified Mavko–Jizba–Gurevich model (MJGZ-MF) at partially relaxed state. The numerical simulations of CPEM/CPEMF models and MJGZ-HF/MJGZ-MF models show good agreement at the different states. Furthermore, we interpreted the experimental data on a basaltic sample and a sandstone sample with the CPEM/CPEMF models. The CPEMF model's predictions of elastic modulus at different states and associated modulus dispersion/attenuation are in good agreement with the corresponding measured ones, suggesting that the proposed CPEMF model can efficiently predict the elastic moduli at different states (dry/drained, undrained/relaxed, partially relaxed and unrelaxed) and quantify the squirt-flow-related elastic modulus dispersion and attenuation among different states well.