Mathematical and numerical modelling of sand production as a coupled geomechanics-hydrodynamics problem

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Sand particles are frequently produced when solids are mobilized from a fluid-saturated porous granular material as a result of sharp fluid pressure and stress gradients, leaving behind mechanically damaged zones. These are commonly called"sand production" and"wormholes" respectively in the Petroleum Geomechanics jargon. When sand is produced from reservoir formations, it can cause a number of problems. These include instability of wellbores, erosion of pipes and pumps, plugging of production liners, subsidence of surface ground, and the disposal of sand in an environmentally acceptable manner. Each year, these issues cost the oil industry hundreds of millions of dollars. Hence, it is imperative to find an efficient computational model which has the predictive capability to assist the field operator to understand this unique process. The ultimate goal is to design a proper production strategy so that sand production and operating costs may be reduced. Modelling such a complex problem is a challenging task since it involves capturing the whole range of material response from bulk to fluid-like behaviours. In this thesis, both the mathematical and numerical descriptions of sand production are explored within the realms of continuum mechanics and finite elements. Attention is given to the physics of sand production and its relation to the interaction between hydrodynamics and geomechanics. Hence, a mathematical model is developed within erosion mechanics theory, while the instability phenomena associated with sand production is treated within the framework of high gradient continuum mixture theory. A Representative Elementary Volume (REV) comprised of three phases namely fluid, fluidized solid, and solid is chosen upon which the particle transport and balance equations are written to reflect the interactions among these phases through mechanical stresses and hydrodynamics. Constitutive laws such as the mass generation law, Darcy's law, and stress-strain relationships are written to describe the fundamental behaviour of sand erosion, fluid flow, and deformation of the solid matrix respectively. Among the various macroscopic field variables that emerge from the mathematical formulation, porosity changes arise from two sources, namely erosion and solid skeleton deformations. When turning to realistic engineering problems, computational challenges are encountered while solving the governing equations as an initial boundary value problem. For instance, numerical instabilities arise since the governing equations contain high convection terms, and field variables also vary drastically with strong gradients in both space and time. A method is developed whereby local field variables such as density, flux, and stress found in the governing equations are enriched with high gradients to account for the effects of the local sharp changes by introducing an Optimized Local Mean Technique (OLMT). As such, the associated node-to-node oscillations encountered in standard numerical schemes are eliminated. It is interesting to note that the developed technique also leads to a framework that establishes a physical explanation for the ad-hoe terms used in traditional stabilized numerical methods, such as the Streamline Upwind/Petrov-Galerkin (SUPG) method. Computational aspects of the proposed technique are investigated such as stability, mesh sensitivity, accuracy, and convergence. Numerical results of sand production afforded by the proposed model are in good agreement with available lab test data. It is found that there is an intimate interaction between sand erosion activity and deformation of the solid matrix. As erosion activity progresses, porosity increases and in turn degrades the material strength. Strength degradation leads to an increased propensity for plastic shear failure that further magnifies the erosion activity. An escalation of plastic shear deformations will inevitably lead to collapse with the complete erosion of the sand matrix. The self-adjusted mechanism enables the model to predict both the volumetric sand production and the propagation of wormholes.
Bibliography: p. 208-229
Wang, J. (2003). Mathematical and numerical modelling of sand production as a coupled geomechanics-hydrodynamics problem (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from doi:10.11575/PRISM/14775