Micro and meso descriptions of anelasticity. If subindices 1 and two refer for the gas-inclusion area and host medium (water), respectively, we have the wet rock moduli K = K 1 – WK (7) (8)G = Gmd , exactly where K = KG2 (3KG1 4Gmd) 4Gmd (KG1 – KG2)Sg (3KG1 4Gmd) – 3(KG1 – KG2)Sg W= In addition, KG1 = K0 – Kmd Kmd K0 /K f l1 – 1 1 – – Kmd /K0 K0 /K f l1 K0 – Kmd Kmd K0 /K f l2 – 1 1 – – Kmd /K0 K0 /K f l2 3ia ( R1 – R2)( F1 – F2) . b3 (1 Z1 – two Z2)(9) (ten)(11)KG2 =(12)are Gassmann moduli, exactly where K f l1 and K f l2 are fluid moduli, R1 =(KG1 – Kmd)(3KG2 4Gmd) (1 – Kmd /K0) KG2 (3KG1 4Gmd) 4Gmd (KG1 – KG2)Sg (KG2 – Kmd)(3KG1 4Gmd) (1 – Kmd /K0) KG2 (3KG1 4Gmd) 4Gmd (KG1 – KG2)SgF1 = F2 = Z1 =(13)R2 =(14) (15) (16) (17) (18) (19)(1 – Kmd /K0)K A1 KG1 (1 – Kmd /K0)K A2 KG1 – exp(-21 a) (1 a – 1) (1 a 1) exp(-21 a)Z2 =(two b 1) (two b – 1) exp[-22 (b – a)] (two b 1)(two a – 1) – (two b – 1)(2 a 1) – exp[-22 (b – a)]1 = i1 /KEEnergies 2021, 14,5 of2 =i2 /KE2 ,(20)exactly where 1 and two are fluid viscosities, and K f l1 (1 – KG1 /K0)(1 – Kmd /K0) K A1 KE1 = 1 – KG1 1 – K f l1 /K0 KE2 = 1 – K f l2 (1 – KG2 /K0)(1 – Kmd /K0) KG2 1 – K f l2 /K0 1 – Kmd – two K f l1 K0 K0 1 – Kmd – 2 . K f l2 K0 K0 K A(21)(22)1 = K A1 1 = K A(23)(24)In accordance with Wood [29], the successful bulk modulus of the gas-water mixture can be calculated from Sg 1 Sw = (25) Kfl K f l1 K f l2 exactly where Sw could be the water saturation. Finally, the P-wave phase velocity and attenuation are Vp = Q -1 = p Re(K 4G/3) , Im(K 4G/3) , Re(K 4G/3) (26)(27)respectively, where = (1 -)s Sg 1 Sw 2 is bulk density, and 1 and two would be the fluid densities. two.four. Final results The MFS model is directly applied in partially saturated reservoir rocks, exactly where the gas ater mixture is obtained with all the Wood equation (there are no gas pockets), as well as the properties are Tropinone site listed in Table 1. The numerical examples on the qualities of wave prorogation by the proposed model are shown in Figure two, along with the effects of permeability as well as the outer diameter of the patch around the wave velocity and attenuation are shown in Figures 3 and 4, respectively.Table 1. Rock physical properties. Mineral density (kg/m3) Mineral mixture bulk modulus (GPa) Dry rock bulk modulus (GPa) Dry rock shear modulus (GPa) Permeability (mD) Squirt flow length (mm) High-pressure modulus (GPa) Crack porosity 2650 38 17 12.six 1 0.01 22 0.02 Porosity Water bulk modulus (GPa) Gas bulk modulus (GPa) Water density (kg/m3) Gas density (kg/m3) Water viscosity (Pa) Gas viscosity (Pa) External diameter (m) ten two.25 0.0022 1000 1.two 0.001 0.00011 0.Energies 2021, 14,Figure two compares the P-wave velocity (a) and attenuation (b) of your present model with those with the MFS model, exactly where the number among 2-Hydroxyethanesulfonic acid Autophagy parentheses indicates water saturation. The velocities coincide at low frequencies and raise with saturation, with those in the present model larger at high frequencies. Two inflection points are clearly observed, corresponding for the mesoscopic and squirt flow attenuation peaks whenof 18 6 the saturation is 80 , the initial getting the stronger point. The attenuation with the present model is larger than that on the MFS one particular.Energies 2021, 14, x FOR PEER REVIEW7 ofFigure two. P-wave velocity (a) and attenuation (b) from the present and MFS models. The quantity between parentheses indicates water saturation. Energies 2021, 14, x FOR PEER REVIEW4150 (a) 0.05 (b)7 ofk (10 mD) k (10 mD) Figure 2. P-wave velocityk (a) and attenuation (b) of with the present and MFS (1) The (a) k models. Figure two.