Modelling Sand Production and Erosion Growth Under Combined Axial and Radial Flow
Authors: Wan R.G. , Wang J. 
Published: March 2010

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. The motivation for such a study stems from the fact that perforations in a slotted oil well can modify both the local fluid flow and stress regimes, and hence influence sand production. Based on available lab test information, the authors examine sand production in a hollow cylinder test in which sand is being produced under combined axial and radial flow of 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 in view of understanding the physics of sand production.

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, thereby leaving behind mechanically damaged zones. From a broader perspective, this phenomenon can be viewed as a particle fluidization with 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, for instance Jensen et al.(1), have attempted to model the above mentioned physical phenomenon using numerical techniques based on the discrete element method. Alternatively, 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, as was first proposed by Vardoulakis et al.(2). This approach was subsequently extended by Wan and Wang(3, 4) to include the deformation of the solid matrix and address general initial 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 primary variables. It was found that numerical results were corrupted with instabilities in the form of node-tonode 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 and Wang(5) 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 and Hughes(6) and Hughes et al.(7)]. In Wan and Wang(5), local field variables found in the governing equations, such as density, flux, and stress, are expanded into a Taylor series for a finite size domain. An optimized local mean technique was introduced based on concepts of Finite Increment Calculus(8) and second gradient theories(9). As such, the original form of the governing equations describing the physics of the problem is preserved, while additional terms leading to numerical stabilization naturally emerge during the numerical process.