## 1 Introduction
Their approach was successfully implemented on an actual sandstone well-log data set when inverting rock porosity and fluid saturation from P-velocity, electrical conductivity, and density measurements. They proved that there are ambiguities when looking at the different domains separately, but ambiguities were removed by exploiting the existence of rock cross-properties that influence different borehole measurements. [PERSON] et al. [11] developed a petrophysical joint inversion (PJI) algorithm and tested it on a carbonate reservoir in Saudi Arabia. They utilized Archie [1] and Gassmann [12] models to simulate electrical resistivity and sonic well logs. Porosity and fluid saturation were the petrophysical properties retrieved in PJI by inverting density, resistivity, and sonic well logs. They showed that the PJI resulted in porosities that were in good agreement with independently measured neutron porosity. [PERSON] and [PERSON] [13] constructed another joint inversion algorithm and tested it on sandstone core and well-log measurements. They utilized the Raymer model to simulate P- and S-wave velocities, Archie model to compute electrical resistivity, and linear average equations to express density and gamma ray responses of the composite medium. The outputs of their workflow were porosity, volume concentration of clay and quartz, and fluid saturations. [PERSON] et al. [14] applied the same PJI as [PERSON] et al. [11] on a carbonate reservoir in Saudi Arabia. In their Bayesian inversion approach, prior knowledge of both solid matrix and fluid properties were the key factors to reduce ambiguity. Total dissolved solids (TDS) measured in the laboratory showed high variability amongst the wells. Due to the fact that water salinity changed water resistivity, any change in salinity affected the estimated hydrocarbon saturation. To enhance the performance of PJI in predicting porosity and saturation, they utilized an offline statistical workflow to estimate water resistivity. More recent papers have also addressed new algorithms for joint optimization and inversion of acoustic measurements [15; 16]. In this research, we integrate multiphysics modeling techniques to construct resistivity, elastic, and density borehole-derived well logs for complex sandstones. We develop an inversion workflow to jointly convert electrical resistivity, density, and sonic well logs into petrophysical properties. Similar to [PERSON] et al. [17] we utilize a global optimization machine called very fast simulated annealing (VFSA). Primary porosity (termed intergranular herein), secondary porosity (termed micropores herein), water saturation, matrix density, matrix bulk and shear moduli, salinity, critical porosity, resistivity lithology exponents, and elastic length scales are the petrophysical, matrix, fluid, and model parameters recovered. Note that this novel workflow is discriminated from previous studies by the following salient features:  
* Forward modeling algorithms to simulate electrical resistivity and elastic moduli are based on pore combination modeling (PCM) and staged differential effective medium (SDEM) theories, while previous studies either utilize very simple models, e.g. [Raymer], or mostly use inclusion models, e.g., [6; 7]. PCM and SDEM are superior to other models because they naturally simulate the depositional and diagenesis histories of reservoir formations. * Optimization algorithms in previous studies are mostly local in nature (e.g., conjugate gradient, Levenberg-Marquardt) while VFSA is naturally global. When significant nonlinearity and discontinuities exist in the model parameter space, local optimization methods can become trapped in the local optimums of the objective function (see Appendices A and B for details). * Optimization algorithms in previous studies are mostly deterministic while VFSA optimization utilized here is stochastic in nature.