Incompressible cake formation

Filter cakes may be divided into two classes—incompressible cakes and compressible cakes. In the case of an incompressible cake, the resistance to flow of a given volume of cake is not appreciably affected either by the pressure difference across the cake or by the rate of deposition of material. On the other hand, with a compressible cake, increase of the pressure difference or of the rate of flow causes the formation of a denser cake with a higher resistance. For incompressible cakes e in equation 7.1 may be taken as constant and the quantity e3/[5(l — e)2S2] is then a property of the particles forming the cake and should be constant for a given material. 

The theoretical description of the cake formation in the constant pressure filtration is based on Darcy s Permeation Law, which describes the single-phase laminar flow of an incompressible fluid through a porous incompressible system due to an applied pressure difference. It can be written as follows (2, 3) ... 

For incompressible cakes, the filtration rate is directly proportional to the specific cake resistance and the pressure/vacuum, and inversely proportional to the viscosity and cake thickness. The filtration rate is inversely proportional to the ratio of solids to filtrate, while the rate of cake formation is directly related to this ratio. For compressible cakes, the filtration rate is relatively independent of pressure. The more flocculated the solids, the more compressible will be the filter cake. 

Figure E.l shows cumulative volume of filtrate vs. time data for constant pressure filtrations involving nearly incompressible (calcite) and moderately compressible (talc) cake formations. Both particulate feeds have a mean particle size of —10 pm in aqueous suspension, however, the calcite exhibits a compressibility index (n) less than 0.2 while the talc shows a compressibility index (dependent on the solution environment) in the region of 0.5. The data show that calcite suspensions filter more easily than corresponding talc suspensions (i.e. lower average specific cake resistance) and filtration rates increase as expected with pressure. Also shown on Figure E. 1 are theoretical predictions made using equations (6.4)-(6.7) and (6.12). The excellence of the predictions is indicative of the accuracy of these process design equations. The data are particularly significant as the values for the empirical scale-up constants n, <), Q and were determined on a separate (automated) apparatus to the one used to generate the experimental data.

Wakeman R.J., 1981b. The formation and properties of apparently incompressible filter cakes on downward facing surfaces, Trans IChemE, 59, 260-270. 

By far, the most often cited work is the early paper by Outmans (1963), which applied differential-equation-based filtration methods developed in chemical engineering to static and dynamic invasion in the borehole. In this single-phase flow study, where lineal flow was assumed and the applied differential pressure was completely supported by the mudcake, Outmans derived the well known Vt law, subject to the further proviso of cake incompressibility. (The effects of cake nonlinearity and compaction can be important over time e.g., see Figure 14-7.) Thus, the Vt law cannot be used when the net flow resistance offered by the formation is comparable to that of the mudcake (e.g., thin muds in permeable formations, or thick muds in very impermeable rocks). Also, the law does not apply to slimholes, where the radial geometry is important. Finally, the Vt law does not generally apply to reservoirs with two-phase, immiscible flow, or miscible flow, or both. Only under these restrictive assumptions does Outmans correctly derived law hold. 

Mudcakes in reality may be compressible, that is, their mechanical properties may vary with applied pressure differential, e.g., as in Figure 14-7. We will be able to draw upon reservoir engineering methods for subsidence and formation compaction later. For now, a simple constitutive model for incompressible mudcake buildup, that is, the filtration of a fluid suspension of solid particles by a porous but rigid mudcake, can be constructed from first principles. First, let x (t) > 0 represent cake thickness as a function of the time,... 

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