Swelling model

Further stability models based on surface area, equilibrium water-content-pressure relationships, and electric double-layer theory can successfully characterize borehole stability problems [1842]. The application of surface area, swelling pressure, and water requirements of solids can be integrated into swelling models and mud process control approaches to improve the design of water-based mud in active or older shales. 

Keywords swelling, model predictive control, optimal control. 

This swelling model does not apply to clays with mostly divalent and polyvalent cations on their exchange sites these clays show the hydration stage of expansion but not the osmotic one. The explanation for their behavior is found in the next section. 

Damage mechanics principles have been applied extensively for well stability problems, especially for hard, brittle and fractured shale rocks. Constitutive models using damage concept, coupled with hydro-chemical swelling models, have been developed based on experimental results, and applied for design of drilling fluid density, combined with FEM methods (Liu, 1995). 

Up to now two-phase two-component flow under non-isothermal conditions and coupled THM (one-phase flow) have been implemented in this code and validated against different experimental results. For the modelling of water penetration into unsaturated bentonite or clay, a swelling model is available in the code. If the test sample is confined within a constant volume, then a swelling pressure will build up which causes changes to the pore structure and reduces the porosity. A small change in porosity can, however, create a considerable reduction in permeability. 

For the description of such interactions as well as of polymer swelling, models based on the Flory-Huggins Theory (Flory, 1953 Mulder, 1991) and UNIQUAC are often applied for mixtures in general and, for binary mixtures, also the Solubility Parameter Theory if the feed components are hydrophobic (Hildebrand and Scott,... 

The above die swell model is based on the assumption that the polymer fluid behaves similarly to a rubber-Uke solid. However, it is difficult to measure recoverable shear strain under general conditions. 

Equilibrium swelling models may also be developed for ionic and porous polymeric networks. For example, for an anionic polymer network where the concentration difference of the mobile electrolyte present in the solvent between the inside and the outside of the gel is comparable in magnitude to the concentration of counter-ions in the polymer network, the equilibrium condition may be described [32] as shown in equation (19). 

Tables 3 through 6 present the full swelling models for a number of cases. Tables 3 and 4 are applicable for anionic and cationic polymers where the concentration of ions outside the polymer is much less than the concentration of ions inside the gel. Tables 5 and 6 similarly describe the swelling behavior of anionic and cationic polymer gels but here the Ion concentration in the fluid surrounding the polymer is comparable in magnitude to that of the ion species within the polymer itself. In all these tables, equations A and B are valid for polymers with a Gaussian chain length distribution and equations C and D should be used with polymers exhibiting a non-Gaussian chain length distribution. For polymer crosslinked In the solid state, equations A and C are appropriate. Equations B and C should be used with polymers crosslinked in the presence of solvent.

Equilibrium Swelling Model for Anionic Polymer Gels... 

The Baron model [15], which assumes considerable degradation of thermal conductivity by bumup, is used for the thermal conductivity of MOX fuel. It has been known to be more conservative than MATPRO-11 model [16] in most aspects of the fuel rod thermal behavior such as fuel centerline temperature and fission gas release [17]. The amount of fission gas generation in MOX fuel is assumed to be the same as that in UO2 fuel. Fission gas release is predicted by the White and Tucker-Speight model [18,19]. The Studsvik model [20] is adopted as it is a representative fuel pellet swelling model. Other material properties are taken from the MATPRO-11 model. 

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