In this article a system of equation to describe the steady state external filter cake thickness profile along the cuboid crossflow filtration setup geometry as well as the well bore geometry are introduced. The first equation is obtained from the analysis of forces acting on a deposited particle at the outermost limit of the external cake. The second equation is obtained from the volume conversation of the transporting fluid in the given geometry. Coupling the two equations yields an implicit solution of the cake profile. Boundary conditions are required as well as the quantification of an empirical factor that can be quantified experimentally.
Experiments conducted were analysed and the empirical friction coefficient was quantified and presented. Simulations based on the developed solution are also presented.
Serious produced water reinjection (PWRI) trials date back to the early 1980s. The impetus behind such trials was the increasing amounts of water being produced as the fields matured as well as the evolution of the environmental regulation to more stringent levels. Thus, PWRI presented itself as a viable waste water management solution as well as a secondary recovery driving fluid. Indeed, some studies4 indicated that PWRI was both the Best Practicable Environmental Option (BPEO) and the most economical option. Sea water injection (SWI) was used as a secondary recovery mechanism earlier than PWRI, and suffers from problems such as water incompatibility leading to scaling.
Suspended solid particles are a common constituent of both injection fluids, causing injectivity decline. This is attributed to the penetration of suspended particles (and droplets in the case of PW) into the formation (deep-bed filtration) and the associated formation damage; as well as the buildup of an external filter cake at the injection face. For typical injection schemes and operating conditions it is believed that the following sequence of particle deposition phenomena takes place: initial surface coverage, followed by internal bridge formation at the pore throats until a critical non-percolation threshold is reached in the vicinity of the injection face, followed by severe internal filtration that transitions into external filter cake. This concept of transition time has also been proposed by other authors and is further defined as the point in time after which no further internal filtration takes place.
Wennberg et. al. postulated that transition time coincides with the filling up of a critical fraction of the porosity, c.a 50%. Da Silva et. al. adopted the critical porosity hypothesis in their impedance based formation damage model, the so called three-point-method. Their analysis of published experimental data[13, 14] recovered an average critical porosity value of 10%.
The study of external filter cake in the field of petroleum engineering dates back to the 1940s and continues with the works of and Khatib, Civan and Sharma's group9. More fundamental research into the buildup of external filter cake can be found in the fields of colloid science and membrane science, where phenomenological models are replaced with fundamental models incorporating force analysis.
The principal of filter cake buildup is based on the multi-layer deposition of the suspended particles carried by the permeate flux. Yet, experimental observation indicate that not all of the particles transported by the permeate flux to the interface/membrane are deposited. Different authors attempted to explain the back transport of particles using different physical models among which are: concentration polarisation of reverse osmosis based on Stokes-Einstein diffusion; also based on shear induced diffusion; and inertial lift. Song and Elimelech introduced a general model that can account for both the concentration polarisation layer and external cake formation by considering both the hydrodynamic and thermal energies of the system and identifying an appropriate critical filtration number.