## 1 Introduction  
For clean formations, \"[PERSON]'s equation\" [1] has been employed as the fundamental method for evaluating the presence of hydrocarbons for the last 75 years. For rocks with multiple pore systems, [PERSON] [2] proposed pore combination modeling (PCM) methodology, which explicitly quantifies the contribution of various pore systems and their saturations in fluid flow permeability and electrical resistivity. PCM is consistent with [PERSON]'s equation, including the saturation dependence, but represents an extension allowing for the effect of multiple pore systems. Similar to resistivity modeling for dual-porosity rocks, [PERSON] and [PERSON] [3] developed a staged differential effective medium (SDEM) theory to model elastic moduli. In fact, PCM is a special case of SDEM, and these are general techniques for modeling the effect of mineralogy and texture on permeability, resistivity, and acoustic measurements. To demonstrate the advantages of multiphysics inversion, [PERSON] et al. [4] designed an inversion algorithm to jointly convert resistivity, density, and sonic well logs into the petrophysical properties of dual-pore carbonates. For doing so, the joint modeling of resistivity and velocities was performed using PCM and SDEM techniques. To fine tune the model parameters, they utilized very fast simulated annealing (VFSA), a global optimization search engine [5]. In this research study, we integrate multiphysics modeling techniques to construct resistivity, sonic, density, and gamma-ray well logs for shaly sandstones. An inversionalgorithm is developed to jointly convert well logs into petrophysical properties. To retrieve model parameters, VFSA is employed. Sandstone porosity, microporosity, volume concentration of shale, water saturation (total, bound, and sandstone), sand density, sand bulk and shear moduli, salinity, critical porosity, resistivity lithology exponents, and elastic length scales are the petrophysical quantities, sand and fluid properties, and model parameters recovered. [PERSON] et al. [6] proposed a technique for quantitative estimation of secondary porosity (e.g., cracks or micro) using sonic and ultrasonic well-log and core measurements. In their model, a double-porosity rock consists of grains, primary porosity, and secondary porosity. Effective medium approximation (EMA) was applied to calculate P- and S-wave velocities. Spheroids with different aspect ratios, i.e., inclusion, were utilized to model all the rock components. Matrix P- and S-wave velocities were first calculated as functions of the primary porosity. Elastic moduli of skeleton were computed using the volume concentration of minerals or defined by laboratory data. They determined aspect ratios for grains and primary pores by the inversion of published P- and S-wave velocities. Then, the aspect ratio of secondary pores was estimated by minimizing the difference between simulated velocities and the measured P- and S-velocities using joint inversion. The type of secondary pore system (cracks or micro) was indicated by the aspect ratio of inclusions obtained. [PERSON] et al. [7] utilized a unified microstructure model to implement the joint inversion of elastic velocities and electrical conductivity. They used self-consistent approximation (SCA) to replicate elastic and electrical properties. Three-axial ellipsoids with various aspect ratios approximated the grains and primary and secondary pores. Micro-pores and cracks were modeled with different aspect ratios. Conducting a case study in southern Mexico, they designed a joint inversion algorithm to evaluate the vuggy and cracked carbonate reservoirs using experimental core and well-log data. [PERSON] et al. [8] proposed a Bayesian joint inversion procedure to describe model and data uncertainties in the problem of well-log interpretation. They utilized the Raymer model [9] to simulate P-wave velocity of the equivalent medium and complex refractive index model (CRIM) to compute electrical resistivity of the composite medium [10].