Micro and meso descriptions of anelasticity. If subindices 1 and two refer towards the gas-inclusion region and host medium (water), respectively, we’ve got the wet rock moduli K = K 1 – WK (7) (eight)G = Gmd , where K = KG2 (3KG1 4Gmd) 4Gmd (KG1 – KG2)Sg (3KG1 4Gmd) – three(KG1 – KG2)Sg W= Additionally, 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 – 2 Z2)(9) (10)(11)KG2 =(12)are Gassmann moduli, 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 =(2 b 1) (two b – 1) exp[-22 (b – a)] (two b 1)(2 a – 1) – (two b – 1)(2 a 1) – exp[-22 (b – a)]1 = i1 /KEEnergies 2021, 14,five of2 =i2 /KE2 ,(20)where 1 and 2 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 efficient bulk modulus on the gas-water mixture might be calculated from Sg 1 Sw = (25) Kfl K f l1 K f l2 where Sw could be the water saturation. Ultimately, the P-wave phase DBCO-Maleimide Autophagy 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 two is bulk density, and 1 and 2 would be the fluid densities. two.four. Benefits The MFS model is directly applied in partially saturated reservoir rocks, exactly where the gas ater mixture is obtained using the Wood equation (you will discover no gas pockets), and the properties are listed in Table 1. The numerical examples with the qualities of wave Eperisone Autophagy prorogation by the proposed model are shown in Figure 2, as well as the effects of permeability and the outer diameter with the patch around the wave velocity and attenuation are shown in Figures three and four, 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 2.25 0.0022 1000 1.two 0.001 0.00011 0.Energies 2021, 14,Figure 2 compares the P-wave velocity (a) and attenuation (b) on the present model with those on the MFS model, exactly where the quantity among parentheses indicates water saturation. The velocities coincide at low frequencies and increase with saturation, with these from the present model larger at higher frequencies. Two inflection points are clearly observed, corresponding towards the mesoscopic and squirt flow attenuation peaks whenof 18 6 the saturation is 80 , the initial becoming the stronger point. The attenuation of your present model is higher than that with 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 in between parentheses indicates water saturation. Energies 2021, 14, x FOR PEER REVIEW4150 (a) 0.05 (b)7 ofk (10 mD) k (ten mD) Figure 2. P-wave velocityk (a) and attenuation (b) of with the present and MFS (1) The (a) k models. Figure 2.