Deriving The Exact Analytical Solution For Deep Bed Filtration With Large Retained Concentrations
Engineering Honours Degree, 2012
University of Adelaide
The process of deep bed filtration occurs frequently in the chemical and upstream petroleum industry. The capability of modelling this behaviour will enable efficient designing of drilling fluids, injection fluids or proppant particles. Currently there exist many deep bed filtration models which take into account the pore size distribution as an important characteristic of a system but none that take into account deposition which is large enough to alter the distribution of pore throats.
An investigation is done on the behaviour of long term deposition in cores due to deep bed filtration. Of the many mechanisms which cause deposition in deep bed filtration only size exclusion is investigated. This allows for a simple mathematical model and a limited amount of empirical data, which must be determined from lab tests. The micro model used is a capillary tube model alternated with mixing chambers. This model characterises the core according to three parameters:
• the mean of the pore size distribution,
• the standard deviation of the pore size distribution and
• the capillary length.
Using this micro model the equations are up scaled for the deep bed filtration equations which describe the process on the macro scale. An Exact Analytical Solution is derived for these equations. Due to the high computation required for the analytical solution a travelling wave solution is also derived as an approximation to the Exact Analytical Solution. A numerical scheme is then developed to evaluate both solutions and to compare them. It could be seen that under long term deposition they converged.
For a core with similar characteristics to the experimental cases, the captured particle solution for the exact and the travelling wave solution converged quickly. However, the suspended particle solution for both methods was quite different. The prime difference which caused this was the suspended particle solution for the exact solution at the core inlet was always the injected suspended concentration, while the travelling wave solution for suspended particle concentration would not reach this unless the inlet had reached the maximum deposited particle concentration.
To test the robustness of the Exact Analytical Solution, effluent concentration profiles are gathered from porous mediums engineered to display purely size exclusion. The effluent concentrations of the model are matched to the experiment by optimizing the selection of the three parameters which characterise the modelled core. Three experimental effluent concentration profiles were matched with an exact solution.
One of the problems encountered while matching was the multiple optimisations found, which caused issues with determining results which were useful beyond effluent curve matching. To resolve this, challenge testing was used to determine the pore size distribution. This reduces the uncertainty of the Pore size distribution. The fitting of the effluent concentration then becomes more efficient.
The sensitivity for each parameter is checked on how it affects the effluent concentration profile. The most sensitive parameter for the tested match was the mean of the pore size distribution. This was due to the dominant control this parameter had over the retention rate.
The derived analytical solution successfully provides a model for the long term behaviour of the particle concentration profiles. It is capable of matching experimental results however the advantage it offered over the short term models could not be seen in the experimental matching due to the low deposition encountered.