Using the theory of volume averaging, we have shown 1 that molecular diffusion, mass tortuosity, and mechanical mixing contribute to the massdispersion coefficient. A series of experiments were conducted on the system Triton x-1O6™ surfactant, n-decane oil, and water to determine the contribution of each mechanism to the total-dispersion matrix for flow through fired Berea sandstone. The dynamics of adsorption and the effect of dead-space volume are considered for the single-phase transport of surfactant through fired Berea. A new dynamic asdorption model is developed which considers both mass transfer to the fluid/solid surface and a kinetic surface-adsorption mechanism. Both kinetic adsorption and mass-transfer rate mechanisms are shown to be important over a wide range of injection rates.
Techniques are developed for using the structural information contained in algebraic equation sets to aid in the study of convergence properties of iterative equation loops.The concept of inverse mapping is applied directly in order to study convergence properties of sequences. For problems in which there exist both a direct and inverse path for the computation of iterative loops, direct substitution techniques will converge via one path and diverge via the other path. When an inverse path existed, the direct substitution method was superior to simultaneous solution techniques for the problems studied. Not all iterative loops have an inverse path. For problems in which inverse paths do not originally exist, rearrangement of the equations of an iterative loop can result in a modified loop for which an inverse path does exist. The digraph representation of an iterative loop aids in deciding whether rearrangements are feasible for a particular problem. When inverse paths cannot conveniently be obtained, direct substitution cannot be successfully used and simultaneous solution techniques such as quasi-linearization are recommended. SCOPEParticularly in the mathematical description of design problems, it is necessary to solve a large set of algebraic equations. Recently much has been Published on how the structural information contained in equation sets can be used to develop algorithms for minimizing and simplifying iterative loops (Ramire2 and Vestal, 1972). However, little has been published on how this structural information influences the convergence properties of systems of algebraic equations. Structural information is important in the study of convergence properties. In this paper we present a direct method for applying the mapping concepts of functional analysis in order to show how structural information relates to, and aids in the study of, solution strategies for algebraic equation sets. The method presented is easy to use and should be of use to practicing engineers.
The objective of this work is an understanding of the dynamic mechanisms associated with the adsorption of surfactants on porous media and the dynamics associated with the lowering of the interfacial tension between oil and a surfactant solution. Most studies on surfactant adsorption have assumed that the adsorption mechanism occurs rapidly as compared with convection and dispersion. Trogus et al. (1977) have proposed a kinetic model which at equilibrium yields the Langmuir isotherm. Results of using this model have shown that the kinetic rate constant increases with an increase in phase velocity. An explanation for this variation is to postulate separate mass transfer and kinetic adsorption mechanisms. A new model is proposed and tested which consists of the solute moving to the rock-fluid interface by a mass transfer mechanism and then adsorbing to the rock via a kinetic mechanism. The mass transfer rate constant has been evaluated and correlated as a function of phase velocity. The correlation obtained for Berea phase velocity. The correlation obtained for Berea sandstone and Triton X-100 (non-ionic) surfactant is in good agreement with existing mass transfer correlations for packed-bed reactors, indicating the validity of our mass transfer model for describing the transport of surfactant through porous media. Using a spinning drop tensiometer, the dynamic response of the interfacial tension between n-decane, and a brine solution of Texas No. 1 surfactant or TRS 10-80 surfactant was obtained. A kinetic adsorption/ desorption rate model for the dynamic behavior was developed. At high surfactant concentrations the normal interfacial tension decrease stops abruptly at a specific plateau value. This plateau breakpoint seems to be due to an abrupt phase change at the interface which prevents the continued adsorption of surfactant monomer molecules. The breakpoint disappears at low surfactant concentrations allowing a normal equilibrium condition to be reached. An increase in surfactant concentration increases the interfacial tension dynamic response rate. Close to the optimal salinity the dynamic behavior is slow with Texas No. 1 and fast with TRS 10-80. We have linked these observations to the Van der Waals energy of desorption and the Gouy surface potential of the double layer. ADSORPTION OF SURFACTANTS In this section the moss transport of a surfactant during miscible displacement in a saturated medium is evaluated. The mechanisms which must be considered are those of convection, dispersion, and adsorption. The convective-dispersion mechanisms have been discussed by Ramirez et al. (1979). Most studies on adsorption have assumed that the adsorption mechanism occurs rapidly as compared with the mechanisms of convective and dispersion. An equilibrium isotherm is therefore assumed usually of the Langmuir type: (1) where Gamma = amount of surfactant adsorbed per surface area (gmoles/cm2) C = the bulk surfactant concentration(gmoles/cc) Trogus et al. (1977) have proposed a kinetic model of the form (2) where k1 = adsorption rate constant k2 = desorption rate constant Qa = total adsorbent capacity on the solid
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