Brooks-Corey

To use Equation 6.4, one must measure or calculate hc dr for a given aquifer material. A mathematical formula for the water pressure-saturation curve during drainage is needed to calculate the average water pressure head. The Brooks-Corey model of the drainage curve can be used as follows (Corey, 1986) ... 

Now that hcii is expressed solely in terms of the Brooks-Corey parameters for a given soil, Equation 6.4 becomes... 

In the above equations, Pc is the capillary pressure, /, is the Brooks-Corey pore-size distribution index, Sr is the residual saturation of the free product phase, a is the van Genutchen fluid/soil parameter, and the superscript n is the van Genutchen soil parameter. 

Although the approach is theoretically sound, both the proposed relationships between capillary pressure and saturation (Equations 6.23 and 6.24) are highly nonlinear and limited in practicality by the requirement of multiparameter identification. In addition, due to the inherent soil heterogeneities and difference in LNAPL composition, the identified parameters at one location cannot be automatically applied to another location at the same site, or less so at another site. For example, Farr et al. (1990) has reported the Brooks-Corey and van Genutchen parameters, X, ii, and o.a0, for seven different porous media based on least-square regression of laboratory data. The parameters are found to vary about one order of magnitude and do not show any specific correlation for a particular soil type. 

Closed-form expressions for pore-scale saturation (Eq.[16] and [17]) for a unit cell with triangular central pore were fitted to measured saturation data of Hygiene Sandstone (Brooks Corey, 1964). A triangular central pore was selected to emphasize capillary contributions expected for sand pore space. An elementary unit c ell with dimensions l. = 0.033 mm, ex = 0.0012, p = 0.0001 was found to yield best lit to the measured dala as depleted in I ig. I 13a. flic resulting saturation curve is... 

Figure 8. Two-dimensional infiltration photo and simulation result of PCE saturation at elapsed time = 60 minutes using Brooks-Corey/Burdine capillary pressure/relative permeability/saturation constitutive relationships.

We note, that the Brooks-Corey function in pc = Pe and the van Genuchten function with n<2 for pc = 0 are not differentiable, which may cause trouble depending on the numerical realization. On the other hand, there appears to be no physical reason for having a discontinuity in the derivatives. 

On one hand, it is possible to fit quite well the behavior of the Brooks-Corey and van Genuchten curves (Figure 8.1), using this relationship for a regularized pressure pressure formulation. On the other hand, fits to measurements for pore spaces of mixed wettability can be performed quite well using this ansatz [1]. 

To the van Genuchten and Brooks-Corey parametrizations correspond relationships for the relative permeability [9, 14]. However, in many cases, a simple power law is sufficient to describe the behavior, an approach which we preferred in [1]. 

Brooks-Corey equation (Brooks and Corey, 1964 Corey, 1994) This method also uses a dimensionless equation with normalized saturation terms,... 

Relative permeability curves are also estimated via the Brooks-Corey equations (see Salazar et ah, 2006) ... 

Li (2004) shows that the empirical Brooks-Corey capillary pressure model can theoretically be derived from a fractal model of the porous media, where the pore-size distribution coefficient increases with the decrease of the fractal dimension. 

The Thomeer model is similar to the Brooks-Corey model. The difference between the two is the definition of the normalized saturation. 

Li, K., 2004. Theoretical development of the Brooks-Corey capillary pressure modeling of porous media. SPE 89429, In Paper Presented at the 2004 SPE 14-th Symposium on Improved Oil Recovery, Tulsa/OK. 

Brooks, R. H. and Corey, A. T., 1966, Properties of Porous Media Affecting Fluid Flow Journal of Irrigation Drainage Division, ASCE, Vol. 92, No. IR2, pp. 61-88. 

Brooks, D. A. Claisen Isoxazole Synthesis In Name Reactions in Heterocyclic Chemistry, Li, J. J. Corey, E. J., Eds. Wiley Sons Hoboken, NJ, 2005, 220-224. (Review). 

Brooks RH, Corey AT (1966) Properties of porous media affecting fluid flow. J Irtig Drain Div Am Soc Civil Eng 92 61-88... 

Retro-Brook rearrangement of the [l,3]-variant will readily take place in sp and sp carbanion systems. Kuwajima and Takeda and Corey and Rticker have developed the [l,3]-retro-Brook rearrangement of silyl enol ether anions which provide a-silyl ketones (equation 100 and 101). 

Scheme 12.9. Synthesis of <5-araneosene using DMF = dimethylformamide, TMS = trimethyl-a three-component nucleophilic addition/Brook silyl, Tf = trifluoromethanesulfonyl, rearrangement/alkylation on acylsilanes, by Hu dba = dibenzylideneacetone, dppf = and Corey [30]. TBS = t-butyldimethylsilyl, TAS- 1,1 -bis(diphenylphosphino)ferrocene, DME = F = tris(dimethylamino)sulfur (trimethylsilyl)- 1,2-dimethoxyethane. difluoride, THF = tetrahydrofuran,...

Scheme 12.10. Synthesis of a-onocerin using butyldimethylsilyl, THF = tetrahydrofuran, a three-component nucleophilic addition/Brook Tf = trifluoromethanesulfonyl, DME = 1,2-rearrangement/alkylation on acylsilanes, by dimethoxyethane, TMS = trimethylsilyl, Corey and co-workers [33]. TBS = t- TBAF = n-tetrabutylammonium fluoride.

The total synthesis of (+)- -onocerin via four-component coupling and tetracyclization steps was achieved in the laboratory of E.J. Corey. The farnesyl acetate-derived acyl silane was treated with vinyllithium, which brought about the stereospecific formation of a (Z)-silyl enol ether as a result of a spontaneous Brook rearrangement. In the same pot, the solution of I2 was added to obtain the desired diepoxide via oxidative dimerization. 

Karam R, Marcello S, Brooks RR, Corey AE, Moore A. 1998. Azimilide dihydrochloride, a novel antiarrhythmic agent. Am. J. Cardiol. 8L40DM6... 

Kuwajima et al. developed an efficient route to (Z)-y-siloxyallylmagnesium bromides utilizing the Brook rearrangement of a-silylalkoxides derived from acylsi-lanes and vinylmagnesium bromide [594], Corey et al. recently applied this method to the generation of (Z)-y-siloxyallyllitliiums to achieve rapid syntheses of natural products [595],... 

This section uses the Corey-type (Brooks and Corey, 1966) equation (Eq. 7.125) to describe relative permeabilities. We first use this type of equation to see the effect of IFT on the k, ratio in two-phase flow. Figure 7.37 compares the three k, ratios normal water/oil k miscible k and k between. Their parameters are presented in Table 7.10. The figure clearly shows that as the IFT is increased from the normal waterflooding high IFT to the intermediate IFT to the miscible flooding low IFT, the ratio of kj to k increases in the low Sw... 

For the hydrolysis step, for example (13) - (14), Corey used mercuric chloride-mercuric oxide in aqueous methanol, while Brook used a modification of Wolfrom s procedure (1, 654, ref. 14), using acetone-benzene-water as solvent. For volatile ketones, Brook prefers DMSO-water as the solvent system. 

---