Micro and meso descriptions of anelasticity. If subindices 1 and two refer for the gas-inclusion area and host medium (water), respectively, we’ve the wet rock moduli K = K 1 – WK (7) (8)G = Gmd , where K = KG2 (3KG1 4Gmd) 4Gmd (KG1 – KG2)Sg (3KG1 4Gmd) – 3(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 – 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) – (2 b – 1)(two a 1) – exp[-22 (b – a)]1 = i1 /KEEnergies 2021, 14,5 of2 =i2 /KE2 ,(20)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 line with Wood [29], the effective bulk modulus with the gas-water mixture can be calculated from Sg 1 Sw = (25) Kfl K f l1 K f l2 where Sw is definitely the water saturation. Lastly, 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, exactly where = (1 -)s Sg 1 Sw two is bulk density, and 1 and two would be the fluid densities. 2.four. Benefits The MFS model is directly applied in partially saturated reservoir rocks, exactly where the gas ater mixture is obtained with the Wood equation (you can find no gas pockets), plus the (±)-Leucine Epigenetic Reader Domain properties are listed in Table 1. The numerical examples of your characteristics of wave prorogation by the proposed model are shown in Figure 2, along with the effects of permeability as well as the outer diameter of the patch on 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 12.6 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.two 0.001 0.00011 0.Energies 2021, 14,Figure two compares the P-wave velocity (a) and attenuation (b) with the present model with these with the MFS model, exactly where the quantity amongst parentheses indicates water saturation. The velocities coincide at low frequencies and increase with saturation, with these on the present model larger at higher frequencies. Two inflection points are clearly observed, corresponding for the mesoscopic and squirt flow attenuation peaks whenof 18 six the saturation is 80 , the first becoming the stronger point. The attenuation from the present model is higher than that of your MFS a single.Energies 2021, 14, x FOR PEER REVIEW7 ofFigure 2. P-wave velocity (a) and attenuation (b) of your present and MFS Piperlonguminine Technical Information models. The quantity involving parentheses indicates water saturation. Energies 2021, 14, x FOR PEER REVIEW4150 (a) 0.05 (b)7 ofk (10 mD) k (ten mD) Figure two. P-wave velocityk (a) and attenuation (b) of on the present and MFS (1) The (a) k models. Figure two.
