Improving methods of hydrocarbon production and developing new techniques for the creation of natural gas storage facilities are critically important for the petroleum industry. This dissertation focuses on two key topics: (1) mechanisms of sand production from petroleum reservoirs and (2) mechanical characterization of caverns created in carbonate rock formations for natural gas storage. Sand production is the migration of solid particles together with the hydrocarbons when extracted from petroleum reservoirs. It usually occurs from wells in sandstone formations that fail in response to stress changes caused by hydrocarbon withdrawal. Sand production is generally undesirable since it causes a variety of problems ranging from significant safety risks during high-rate gas production, to the erosion of downhole equipment and surface facilities. It is widely accepted that a better understanding of the mechanics of poorly-consolidated formations is required to manage sand production; which, in turn, enables the cost effective production of gas and oil resources. In this work, a series of large-scale laboratory experiments was conducted in fully saturated, cohesionless sand layers to model the behavior of a petroleum reservoir near a wellbore. We directly observed several key characteristics of the sand production phenomenon including the formations of a stable cavity around the wellbore and a sub-radial flow channel at the upper surface of the tested layer. The flow channel is a first-order feature that appears to be a major part of the sand production mechanism. The channel cross section is orders of magnitude larger than the particle size, and once formed, the channel becomes the dominant conduit for fluid flow and particle transport. The flow channel developed in all of our experiments, and in all experiments, sand production continued from the developing channel after the cavity around the borehole stabilized. Our laboratory results constitute a well constrained data set that can be used to test and calibrate numerical models employed by the petroleum industry for predicting the sand production phenomenon. Although important for practical applications, real field cases are typically much less constrained. We used scaling considerations to develop a simple analytical model, constrained by our experimental results. We also simulated the behavior of a sand layer around a wellbore using two- and three-dimensional discrete element methods. It appears that the main sand production features observed in the laboratory experiments, can indeed be reproduced by means of discrete element modeling. Numerical results indicate that the cavity surface of repose is a key factor in the sand production mechanism. In particular, the sand particles on this surface are not significantly constrained. This lack of confinement reduces the flow velocity required to remove a particle, by many orders of magnitude. Also, the mechanism of channel development in the upper fraction of the sample can be attributed to subsidence of the formation due to lateral extension when an unconstrained cavity slope appears near the wellbore. This is substantiated by the erosion process and continued production of particles from the flow channel. The notion of the existence of this surface channel has the potential to scale up to natural reservoirs and can give insights into real-world sand production issues. It indicates a mechanism explaining why the production of particles does not cease in many petroleum reservoirs. Although the radial character of the fluid flow eventually stops sand production from the cavity near the wellbore, the production of particles still may continue from the propagating surface (interface) flow channel. The second topic of the thesis addresses factors affecting the geometry and, hence, the mechanical stability of caverns excavated in carbonate rock formations for natural gas storage. Storage facilities are required to store gas when supply exceeds demand during the winter months. In many places (such as New England or the Great Lakes region) where no salt domes are available to create gas storage caverns, it is possible to create cavities in limestone employing the acid injection method. In this method, carbonate rock is dissolved, while CO₂ and calcium chloride brine appear as products of the carbonate dissolution reactions. Driven by the density difference, CO₂ rises towards the ceiling whereas the brine sinks to the bottom of the cavern. A zone of mixed CO₂ , acid, and brine forms near the source of acid injection, whereas the brine sinks to the bottom of the cavern. Characterization of the cavern shape is required to understand stress changes during the cavity excavation, which can destabilize the cavern. It is also important to determine the location of the mixture-brine interface to select the place of acid injection. In this work, we propose to characterize the geometry of the cavern and the location of the mixture-brine interface by generating pressure waves in a pipe extending into the cavern, and measuring the reflected waves at various locations in another adjacent pipe. Conventional governing equations describe fluid transients in pipes loaded only by internal pressure (such as in the water hammer effect). To model the pressure wave propagation for realistic geometries, we derived new governing equations for pressure transients in pipes subjected to changes in both internal and external (confining) pressures. This is important because the internal pressure (used in the measurement) is changing in response to the perturbation of the external pressure when the pipe is contained in the cavern filled with fluids. If the pressure in the cavern is perturbed, the perturbation creates an internal pressure wave in the submerged pipe that has a signature of the cavern geometry. We showed that the classic equations are included in our formulation as a particular case, but they have limited validity for some practically important combinations of the controlling parameters. We linearized the governing equations and formulated appropriate boundary and initial conditions. Using a finite element method, we solved the obtained boundary value problem for a system of pipes and a cavern filled with various characteristic fluids such as aqueous acid, calcium chloride brine, and supercritical CO₂ . We found that the pressure waves of moderate amplitudes would create measurable pressure pulses in the submerged pipe. Furthermore, we determined the wavelengths required for resolving the cavern diameter from the pressure history. Our results suggest that the pressure transients technique can indeed be used for characterizing the geometry of gas storage caverns and locations of fluid interfaces in the acid injection method.