Abstract
Abstract
The paper investigates the phenomenon of sand production in both axial and radial flow conditions using a coupled particle erosion-fluid transport-stress model that has been developed recently by the authors. We first examine sand production and wormhole growth in a sand pack as heavy oil is being drawn from it through a small orifice. Numerical results are in close agreement with experimental data available for the sand pack experiment. In particular, the computed porosity field and its temporal evolution during wormhole propagation match very well with available CT scan images of the sand pack. Then, we model a sand production hollow cylinder test in which sand is being produced under combined axial and radial flow of light oil in a sandstone specimen. Aspects such as sand flux and porosity evolutions, as well as erosion growth around the inner wall of the cylinder are investigated using the numerical model.
Introduction
Sand is produced in a porous granular material whenever sand particles are being dislodged from the matrix as a result of very high fluid pressure gradients, and thus leaving behind mechanically damaged zones. From a broader perspective, this phenomenon can be viewed as a particle fluidization and erosion process by which a sand matrix is disaggregated due to a combination of stress changes and fluid flow when fluid is aggressively pumped from a porous medium. By virtue of the complexity of the physics of the problem, several challenges are encountered in any numerical modelling endeavour.
Various researchers have attempted to model the above-mentioned physical phenomenon using numerical techniques based on the discrete element method, Jensen et al.1. However, a continuum mechanics approach can also be adopted in which mass balance is applied to a three-phase system comprised of solid, fluid and fluidized solid, see Vardoulakis et al. 2. This approach was subsequently extended by Wan &Wang3,4 in order to include the deformation of the solid matrix and address general boundary value problems. As such, a standard finite element technique combined with Newton-Raphson method was used with some success for the solution of resulting non-linear equations which involve fluidized solid concentration, fluid pressure and porosity as main variables. It was found that numerical results were corrupted with instabilities in the form of node-to-node oscillations or wiggles whenever the solved field variables suffered tremendous distortions with high gradients during sand production. In view of addressing the above-mentioned numerical difficulty, Wan &Wang5 have recently introduced new numerical techniques which are akin to stabilization methods known as Streamline Upwind/Petrov-Galerkin (SUPG) and Galerkin Least Squares (GLS) formulations, see Brooks &Hughes6, and Hughes et al. 7. In Wan &Wang5, local field variables such as density, flux, and stress found in the governing equations are expanded into a Taylor series for a finite size domain. An optimized local mean technique was introduced based on concepts of Finite Increment Calculus8 and 2nd gradient theories9. As such, the original form of the governing equations describing the physics of problem is preserved, while additional terms leading to numerical stabilization naturally emerge during the numerical process. Thus, a fundamental explanation of the ad-hoc terms (numerical diffusion) used in traditional stabilized numerical methods can be given. Numerical solutions pertaining to sand production that are free of oscillations are eventually obtained; see Wan &Wang10.
The Model
The theoretical background relating to the numerical model used in this paper has been extensively covered in a number of publications such as in Wan &Wang3,4,5,10. In order to aid the reader, the essential features of the model are recalled.