For a two-electron system the Kohn-Sham potential of density-functional theory is equal to the effective local potential V,.~(x, ) occurring in the one-electron Schrodinger equation that is satisfied by the square root of the exact many-electron density, p' -'(x, ). Making use of the theory of marginal and conditional probability amplitudes, it is shown that V,,il-(x, ) is the sum of three potentials, each of which has a clear physical interpretation and will be studied in detail. The correlation part of the Kohn-Sham potential in a two-electron system can then be obtained by subtraction of the Coulomb and exchange potential, and it is shown how we can express this correlation potential as the sum of three physically meaningful contributions. The connection between the Kohn-Sham potential in a many-electron system and V"& is also discussed. Calculations of the various potentials from highly accurate configuration-interaction wave functions are presented for the helium atom and for the hydrogen molecule at various distances of the two hydrogen nuclei.

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The nondynamical correlation error in first row transition metal complexes is studied through calculations on the permanganate ion. The source of the error is the well-known Hartree-Fock failure in the weak-interaction limit, which is shown to exist for both the metal-ligand and the ligand-ligand bonds: the metal-ligand and the ligand-ligand distances are large compared to the size of the metal 3d and ligand 2p atomic orbitals (AO's). Pauli repulsion between ligand orbitals and 3s/3p orbitals prevent the metal-ligand and ligand-ligand distances to become small enough for efficient overlap and bonding. In multiply bonded systems the Hartree-Fock error does not show up in excessive electron repulsion, but leads to localization of the bonding orbitals (which sometimes requires symmetry breaking), resulting in a loss of covalent character. It is shown how, in the Mn0 4 -ion, the bonding electrons of E symmetry are localized on the oxygens while the T2 electrons are localized on the metal. The mechanism behind this (unphysical) localization is studied in detail, making use of a simple model system. The covalent character is reintroduced in configuration interaction or multiconfiguration selfconsistent-field calculations: density is transferred from the ligand to the metal in the E bonds and vice versa in the T2 bonds. The total metal 3d occupation, however, remains unchanged. Several configuration selection schemes in the space of bonding, nonbonding, and anti bonding orbitals are tested with the purpose to recover a large fraction of the nondynamical correlation error but still retain a manageable wave function. It is shown that the "nonbonding" 02P orbitals play an important role in the correlation process and cannot be excluded (kept closed) in a correlated calculation if quantatively correct results are required.

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Summary In this paper, acid wormholing in carbonate formations is studied. The wormhole growth rate and the geometry of the final wormhole pattern depend on the combined effect of acid spending and fluid flow. Acid spending is studied by modeling the wormhole as a cylindrical pore and numerically solving the convection-diffusion equations. A finite acid-rock reaction rate is assumed, allowing calculation and study of spending profiles in both the diffusion-controlled and the reaction-controlled regime. Flow properties such as fluid loss from wormhole to formation and fluid distribution in a multiple wormhole geometry are studied through numerical simulations. It is shown how wormhole growth properties are affected by the length and distance of neighboring wormholes. The effect of injection rate and diffusion is studied with a simple model. This model explains several experimentally observed phenomena, such as the existence of an optimum injection rate and a reduced wormhole efficiency at higher rates. Introduction The physics of acidizing is complex, and often only poorly understood, due to the coupling of mechanical and chemical processes. In fracture acidizing, rock mechanical properties play a dominant role in fracture initiation and fracture growth, while the chemistry of the acid-rock reaction determines the final fracture conductivity. In matrix treatments, formation properties such as permeability and porosity determine the direction and magnitude of fluid flow, but these properties are continously altered as a result of acid-rock dissolution. For a proper understanding of the acidizing process it is essential to study the combined effect of acid reaction and fluid flow. When acid is injected into a carbonate formation, highly conductive channels or "wormholes" are usually formed.1,2 The success of acid treatments in carbonates both above and below fracturing pressure depends on the characteristics of the wormhole pattern. Wormholing has a negative effect on acid-fracturing treatments because it will increase fluid loss from fracture to formation. In matrix acid treatments, wormholes are favorable because they can bypass the damaged near-wellbore region and decrease the skin. Wormholing is the result of two, physically distinct, but intrinsically connected processes:the chemistry of acid reaction and acid spending in the rock pores and in wormholes, andthe physics of fluid loss from wormhole to formation and fluid distribution in a multiple wormhole geometry. Acid reaction and spending in a wormhole is largely a chemical problem. The characteristics of the acid spending process determine whether wormholing occurs and how much reactive acid is available for length growth at the wormhole tip. In this work, acid spending is studied in two ways:modeling the wormhole as a cylindrical pore, andnumerically solving the mass balance equations that connect acid transport by convective diffusion and acid reaction at the pore wall. Models based on acid spending in a pore have been used before in relation to wormholing. However, the mathematical problem was simplified because an infinite acid-rock reaction rate and diffusion-controlled spending were assumed.2,3 This assumption excludes the study of situations in which the acid spending rate is (partly) reaction-rate controlled, such as low-temperature dolomites or low-reactivity acids. In this work, a finite acid-rock reaction rate is assumed, given by the power relation J=krateCn. If a finite reaction rate is used, the transition from a reaction-controlled spending rate to a diffusion-controlled spending rate can be studied. With this model, wormholing can be explained on a qualitative level. The difference between reaction-controlled and diffusion-controlled spending will be discussed and it will be shown why wormholing only occurs when the spending rate is diffusion controlled. The Levich approximation, based on the definition of a mass-transport coefficient KMT is often used to calculate the acid concentration in a pore when the spending rate is controlled by mass transport.4 Some useful formulas, based on this approximation, are derived in the Appendix. However, as with every approximation, the Levich approximation must be used with caution and can be a source of errors when used in a situation where it is no longer accurate.5,6 For example, there is no theoretical basis for the assumption in some wormhole models that the acid penetration depth is proportional to the Peclet number to the power ?1/3.7 In the model discussed in this paper, the equations are solved numerically through a finite-difference scheme with (in principle) exact results. The use of a numerical scheme to calculate spending properties is a relatively complex procedure. To simplify and accelerate the calculations, an accurate one-formula fit was derived, based on the numerical results. In the derivation of the necessary equations, dimensionless numbers are introduced: Peclet, Damkohler, and kinetic number.4,8 These dimensionless numbers simplify the mathematical equations and make them more transparent. Interestingly, the expression of these dimensionless numbers in terms of parameters such as diffusion rate, reaction rate, acid concentration, and injection rate directly explain several experimentally observed properties of wormholing, for example, its dependence on acid strength. The study of acid spending in a single wormhole gives insight into the effects of the acid-rock chemistry on the wormhole pattern. For example, the effect of acid concentration, acid reaction rate, and acid diffusion can be explained, at least on a qualitative level. However, the chemistry alone cannot explain other properties such as wormhole growth rate, the number of wormholes per m2 and the final geometry of the wormhole pattern. These properties also depend on the details of fluid flow in the wormholes and surrounding porous medium, such as fluid loss from wormhole to formation and fluid distribution (wormhole competition). Calculations of the fluid-loss profile indicate that only a relatively small percentage of the acid reaches the wormhole tip. Therefore, fluid loss seriously limits the wormhole propagation rate. Fluid distribution in a multiple wormhole geometry is analyzed with calculations in a two-dimensional (2D) permeability grid. These calculations show how wormhole growth properties are affected by the length and distance of neighboring wormholes. Wormhole competition explains why side branches die out quickly and it determines the final wormhole density.