Darcy's Law Applied to Drilling Fluid Filtration
Authors: Chelton H.M. 
Published: December 2007

INTRODUCTION AND DISCUSSION
The study of high temperature-high pressure filtration was brought on by the trend toward drilling deeper, hotter holes. Although in drilling wells the filter cake is initially laid down under dynamic conditions, static filtration is still important from both an experimental and a practical point of view. Deep holes require long periods of time to make trips-when it is necessary to change drilling bits. During these trips the mud is not being circulated, and static filtration occurs. Static filtration should take place about one-half of the time in wells drilled below about 16,000 ft.
A number of experiments were run using the standard high temperature-high pressure filter press and the procedure described in API RP-13B [Nov., 1962]. The results from these tests appeared to have some unknown variable operating to alter results in an unpredictable way. The data in Fig. 1 illustrate this point. It is inconsistent that the volume of filtrate is lower at 500 Psi than at the lower pressures. This was frequently, though not always, the case. If cake compressibility were the cause, consistent results would be expected when using the same clay slurry in the same mud systems. A series of tests did not yield consistent results when using a low pH mud prepared by treating a 32 per cent Panther Creek clay slurry with 2 lb/bbl of a lignosulfonate. Panther Creek clay is a product of the American Colloid Co.
The cap of the filter press was modified to determine whether the lower filtrate volume at 500 Psi was the result of pressure forcing the filter paper and/or its retaining screen against the cap. This could restrict flow of filtrate from the outside edge towards the center. The screen was removed from an API cap and a pattern of concentric rings and radial grooves was milled 1/16 in. into the cap. The support screen was then replaced. In a series of experiments, the pattern cut into the cap was repeatedly reproduced on the filter paper. Fig. 2 shows the pattern impressed on a filter paper that has been removed from the press and rinsed with water. The grooves appear as the light colored bands, and the lands as the dark colored bands. This indicates solid particles were forced into the filter paper supported by the lands possibly restricting flow.
These results led to further alteration of the cap. A depression was milled into the cap 1/16 in. deep and 2-1/8 in. in diameter, which was the diameter of the filter paper exposed to the mud sample for filtration. The depression was then packed with layers of monel screen, and the original screen replaced. Fig. 3 is a sketch of the modification.
A series of filtration tests was run using this recessed cap. The modification was made to remove restrictions to flow. However. the unexpected result was that the filtrate volume was lower when using the recessed cap. One explanation for this is that because of the depression below the filter paper, some filtration takes place before the sample comes to temperature. This would increase the original resistance to flow. The data from the tests using the recessed cap consistently showed that the filtrate volume increased with increasing temperature and pressure.
To determine if the data obtained using the modified cap fit filtration theory, an adaptation of Darcy's law was used. Darcy's equation' for the flow of fluids through porous media can be adapted for cake filtration as follows:
qulK =Ap
where k = permeability q = flow rate, volume/unit time, V/tu = viscosity of the flowing fluid L = length A = cross sectional area
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