A Model for Sand Transport Through a Partially Filled Wormhole in Cold Production
Authors: Yuan J-Y. , Babchin A. , Tremblay B. 
Published: March 2010

Abstract
We propose a simple model of flow in a partially filled wormhole, where layers of oil, slurry and immobile sand can coexist. The slurry at the bottom of the wormhole is assumed to behave as a Bingham material, i.e., it yields when the shear stress exceeds the yield stress. The yield stress is assumed to increase; with depth due to the weight of the overlying sand. A Mohr-Coulomb equation is used to calculate the yield stress within the slurry and immobile sand layers. Linearized Navier- Stokes equations are solved to calculate the oil and mobile sand flow rates. Different characteristic oil and sand flow patterns are studied. Oil and sand flow rates through the wormhole are calculated as functions of pressure gradient and rheological parameters. This simple model could be used to estimate sand transport in a partially filled wormhole, complementing a sand transport model for uniformly filled wormholes(4). These wormhole flow models can be incorporated into a field scale model for cold production.

Introduction
Cold production is a non-thermal primary heavy oil recovery process in which sand production is encouraged. Previous field and laboratory studies(1–3) suggest that high permeability channels (wormholes) develop within the formation starting at the perforations in a cold production well. These wormholes provide high effective fluid mobility, leading to a higher oil recovery as comparedto conventional primary recovery without sand production. According to previous models we developed(4, 5), in the first few months of cold production, wormholes grow rapidly, forming a wormhole network. The produced sand cuts are high during this period. Based on laboratory experiments, during this early stage, most wormholes are uniformly filled with slurry, a mixture of oil and liquefied sand. At later stages of cold production, however, the expansion of the wormhole network zone slows down, resulting in reduced sand cuts. In previous experiments conducted at ARC, we observed the creation and extension of a sand-free zone in the upper part of a wormhole after the wormholes stopped growing(3). The flow behaviour under this circumstance is very different from that in a uniformly filled wormhole.
In this work, we use a simplified description of the layered flow in partially filled wormhole. A comparison with experimental results indicates that this simple model could be used for the estimation of flow properties in a partially filled wormhole, complementing the sand transport model for a fully filled wormhole and the wormhole network model.

Backgroound
The actual geometry of a wormhole is complicated. Therefore, we assumed a plane geometry, as is shown in Figure 1. In the region 0 ≤y ≤h, we assume that sand and oil flow with at the same velocity;
Equation (Available In Full Paper)
The rheology of the slurry is described by the Bingham Equation(8):
Equation (Available In Full Paper)
where, sign(π) = 1 for positive π, and -1 for negative π. The righthand side of this equation is zero if the magnitude of the shear stress is less than the yield stress of the slurry.