## 7 Multiphysics Modeling to Construct Dual-Porosity Sandstones  
As mentioned earlier, PCM is a special case of SDEM, and these are general techniques for modeling the effect of mineralogy and texture on permeability, resistivity, and elastic measurements. On core scale measurements of resistivity and velocities, PCM and SDEM have been found satisfactory to model pore structure of dual-porosity carbonates [20]. They made three independent porosity measurements (Archimedes, \(\mu\)CT, and NMR) on twelve carbonate core plugs from the northern Niagaran reef. They modeled two pore types, micropore and vugs. Resistivity, P- and S-wave ultrasonic measurement of the same brine saturated core plugs were made. The joint modeling of resistivity and velocities was performed using PCM and SDEM techniques. The parameters in the resistivity and velocity models were optimized using the VFSA algorithm [5; 23]. The optimization algorithm was able to fine tune the model parameters of resistivity and velocities to provide vuggy and microporosities close to independently measured porosities using NMR and \(\mu\)CT. In this paper, we extend this technique from core data to well logs and demonstrate it using constructed logs from an actual carbonate formation. Figure 3 illustrates dual-pore shaly sandstone with 24 m thickness that is partially saturated with oil in the upper 10 m of the formation. The residual water saturation in the upper portion is about 3%. The lower 9 m is fully water saturated. The middle (\(\sim\)4 m) is  
Figure 2: Illustration of staged differential effective medium (SDEM) technique for dual-porosity sandstone. the transition zone. The salinity of the formation water is 120 Kppm. To model the density log, quartz with density (g/cm\({}^{3}\)) of 2.65, feldspar of 2.52, illite of 2.52, brine of 1.05, and oil with density of 0.75 are used. To replicate gamma-ray responses, gamma-ray values of 0.0, 220, and 275 API are considered for quartz, feldspar, and illite, respectively. To simulate the resistivity using PCM, the lithology exponents of shale and sandstone are assumed to be 0.75 and 1.75, respectively. Temperature is 60 degrees Celsius. P- and S-wave slowness (sonic travel times) is modeled using SDEM. Bulk moduli (in GPa) are set to 37, 29, and 25 for quartz, feldspar, and illite, respectively. Shear moduli (in GPa) are set to 44, 15, and 16 for quartz, feldspar, and illite, respectively. Bulk moduli (in GPa) are set to 2.5 and 0.75 for brine and oil, respectively. The critical porosity is 0.45. Length scales for shale and sandstone are set to 1.0 and 0.2, respectively. In the first track from the left, intergranular porosity is displayed in red and water-filled intergranular porosity is shown in blue. In the middle track, total porosity is in black and microporosity is in blue. In the right track, total, intergranular, and bound water saturations are shown in black, red, and blue, respectively. Figure 4 displays the histograms of the true petrophysical properties including intergranular porosity, microporosity, intergranular water saturation, and micropore water saturation. Figure 5 displays the simulated well-log responses over the dual-porosity sandstone formation shown in Figure 3. Bulk density (\(\rho_{b}\)) is modeled using mass conservation, P-wave slowness (DTCO) and S-wave slowness (DTSM), modeled using SDEM, and electrical resistivity (\(R_{t}\)) using PCM. Model parameters are summarized in Table 1. As expected, distinct fluid zones are clearly separated on the resistivity log in logarithmic scale and one can determine oil column, transition zone, and water leg from reservoir top to bottom, respectively. This trend can be seen on density and sonic logs in some degree, but it is not as clear as the resistivity log due to the higher combination effect of porosity and saturation on sonic and density logs. Figure 3: Plots of the true petrophysical properties over a dual-porosity sandstone formation containing intergranular pores and micropores.