Micro and meso descriptions of anelasticity. If subindices 1 and 2 refer towards the gas-inclusion

Micro and meso descriptions of anelasticity. If subindices 1 and 2 refer towards the gas-inclusion region 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= Moreover, 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) (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 =(2 b 1) (two b – 1) exp[-22 (b – a)] (2 b 1)(2 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 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 – 2 K f l1 K0 K0 1 – Kmd – two . K f l2 K0 K0 K A(21)(22)1 = K A1 1 = K A(23)(24)As outlined by Wood [29], the efficient bulk Fexinidazole Autophagy modulus on the gas-water mixture could be calculated from Sg 1 Sw = (25) Kfl K f l1 K f l2 where Sw would 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 two is bulk density, and 1 and 2 would be the fluid densities. 2.4. 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), along with the properties are listed in Table 1. The numerical examples of the traits of wave Emixustat Inhibitor prorogation by the proposed model are shown in Figure 2, plus the effects of permeability plus the outer diameter of the patch on the wave velocity and attenuation are shown in Figures three 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) 10 2.25 0.0022 1000 1.2 0.001 0.00011 0.Energies 2021, 14,Figure two compares the P-wave velocity (a) and attenuation (b) of your present model with these of your MFS model, where the number between parentheses indicates water saturation. The velocities coincide at low frequencies and enhance with saturation, with these of your 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 very first being the stronger point. The attenuation in the present model is greater than that with the MFS one particular.Energies 2021, 14, x FOR PEER REVIEW7 ofFigure two. P-wave velocity (a) and attenuation (b) with the present and MFS models. The number among parentheses indicates water saturation. Energies 2021, 14, x FOR PEER REVIEW4150 (a) 0.05 (b)7 ofk (10 mD) k (10 mD) Figure two. P-wave velocityk (a) and attenuation (b) of with the present and MFS (1) The (a) k models. Figure 2.