Analytical Solutions for Partitioned Diffusion in Laminates: II. Harmonic Forcing Conditions

The migration of organic compounds in stratified media is of fundamental concern in environmental and chemical engineering research. The diffusive transport of volatile organic compounds through laminate systems is characterized by partitioning, i.e., the development of concentration discontinuities at the interfaces between the individual laminae. If the transport is governed by cyclic transients, the relevant equations can be written in terms of coupled systems of diffusion equations subject to sinusoidal boundary conditions. This paper solves these systems of equations to present new algebraic solutions for propagation of the sinusoidal modes through arbitrary (finite) numbers of contiguous one-dimensional laminae. Both Cartesian and radial coordinate systems are considered. Two independent formulations of the lamina interface matching conditions are considered, corresponding to (1) an instantaneous partitioning model and to (2) a mass-limited partitioning model. It is shown that sinusoidal components of the concentration solutions propagate dispersively throughout the laminates. This is manifested, for example, in changes in shape of concentration pulses as measured at different points in the laminate system. Algorithms for generating exact dispersion relations for partitioning laminates are given, and an experimental technique for studying interfacial dynamics via frequency domain measurements is proposed.     

Impact of Rock Heterogeneity on Interactions of Microbial-Enhanced Oil Recovery Processes

Residual oil saturation reduction and microbial plugging are two crucial factors in microbial-enhanced oil recovery (MEOR) processes. In our previous study, the residual saturation was defined as a nonlinear function of the trapping number, and an explicit relation between the residual oil saturation and the trapping number was incorporated into a fully coupled biological (B) and hydrological (H) finite element model. In this study, the BH model is extended to consider the impact of rock heterogeneity on microbial-enhanced oil recovery phenomena. Numerical simulations of core flooding experiments are performed to demonstrate the influences of different parameters controlling the onset of oil mobilization. X-ray CT core scans are used to construct numerical porosity-permeability distributions for input to the simulations. Results show clear fine-scale fingering processing, and that trapping phenomena have significant effects on residual oil saturation and oil recovery in heterogeneous porous media. Water contents and bacterial distributions for heterogeneous porous media are compared with those for homogenous porous media. The evolution of the trapping number distribution is directly simulated and visualized. It is shown that the oil recovery efficiency of EOR/MEOR will be lower in heterogeneous media than in homogeneous media, largely due to the difficulty in supplying surfactant to unswept low-permeability zones. However, MEOR also provides efficient plugging along high-permeability zones which acts to increase sweep efficiency in heterogeneous media. Thus, MEOR may potentially be more suited for highly heterogeneous media than conventional EOR.     

Analytical Solutions for Partitioned Diffusion in Laminates: I. Initial Value Problem with Steady Cauchy Conditions

Studies of the transport of contaminants and nutrients in industrial and environmental systems are complicated by the heterogeneous nature of the supporting porous or permeable media, and by the numerical problems associated with high Peclet number advection and sharp interface models. In order to provide independent theoretical checks of numerical transport theories, this set of papers presents analytical solutions to diffusive transport equations in simplified (one-dimensional) laminate systems subject to partitioning interactions. Here, in Part I, a standard separation of variables technique is used to develop analytical eigenfunction expansions of the concentration solution in an N-laminate system subject to steady Cauchy (third-type) nonhomogeneous boundary conditions. Both Cartesian and radial (axisymmetric) coordinate systems are considered. The solutions are developed for two different interface partitioning formulations, allowing the partitioning processes to be described by instantaneous equilibration mechanisms, or in terms of gradual equilibration mediated by mass transfer coefficients. Worked examples are presented and limitations of the approach discussed.     

Two-center molecular repulsion integrals over slater functions

For the general two-electron two-center integral over Slater functions, use of the Neumann expansion for the electron-electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantum numbers and exponents and the internuclear separation. Hence all two-electron hydrid, coulomb, exchange, and one-center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral.