The problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear integral equations of the first kind for a distribution of Stokeslets over the particle surface. The unknown density of Stokeslets is identical with the surface-stress force and can be obtained numerically by reducing the integral equations to a system of linear algebraic equations. This appears to be an efficient way of determining solutions for several external flows past a particle, since it requires that the matrix of the algebraic system be inverted only once for a given particle.The technique was tested successfully against the analytic solutions for spheroids in uniform and simple shear flows, and was then applied to two problems involving the motion of finite circular cylinders: (i) a cylinder translating parallel to its axis, for which the local stress force distribution and the drag were determined; and (ii) the equivalent axis ratio of a freely suspended cylinder, which was calculated by determining the couple on a stationary cylinder placed symmetrically in two different simple shear flows. The numerical results were found to be consistent with the asymptotic analysis of Cox (1970, 1971) and in excellent agreement with the experiments of Anczurowski & Mason (1968), but not with those of Harris & Pittman (1975).

Friction coefficients are calculated numerically for ellipsoids rotating about their principal axes and for a benzene molecule rotating normal to its axis of symmetry under conditions of creeping flow and slip boundary conditions. It has been shown previously, that if a benzene molecule is approximated by an oblate spheroid, the predicted and experimental friction coefficients differ by more than 40%. The present study employs a more realistic shape for benzene and finds a difference of 10%, which is within the limits of the numerical and experimental uncertainty.

Three-Parameter Modification of the Peng-Robinson Equation of State To Peng-Robinson Equation of State To Improve Volumetric Predictions Summary. The Peneloux-Rauzy-Freze (PRF) method of improving volumetric predictions by introducing a third parameter into a two-parameter equation predictions by introducing a third parameter into a two-parameter equation of state (EOS) is applied to the Peng-Robinson EOS (PR-EOS). The modified PR-EOS is evaluated for application to hydrocarbon fluids. A method is PR-EOS is evaluated for application to hydrocarbon fluids. A method is developed for characterizing the third parameter for the heptanes-plus fractions. The usefulness of the modified PR-EOS in improving volumetric predictions is illustrated by applying the equation to several crude-oil predictions is illustrated by applying the equation to several crude-oil and gas-condensate systems from the literature. Introduction In recent years, two-parameter cubic EOS's--e.g., the PR-EOS and the Soave-Redlich-Kwong EOS (SRK-EOS)-have been commonly used by the petroleum industry for predicting the phase behavior and volumetric properties of hydrocarbon fluid mixtures. Once the heptane-plus fraction of the hydrocarbon fluid is properly characterized into a mixture of pseudocomponents, these equations predict the vapor/liquid equilibrium conditions with a reasonable accuracy. However, the volumetric estimates obtained through these two-parameter EOS's are not as accurate. In our experience with the application of the PR-EOS to reservoir fluids, we found that the error in the prediction of gas-phase z factors ranged from 3 to 5 % and the error in the liquid density predictions ranged from 6 to 12 %. Recently, Peneloux et al. developed a method of improving the volumetric predictions by introducing a third parameter into a two-parameter cubic EOS. This method is particularly attractive because the third parameter does not change the vapor/liquid equilibrium conditions determined by the unmodified, two-parameter equation, but modifies the phase volumes by effecting certain translations along the volume axis. Thus, if a given reservoir fluid is already characterized for use in some two-parameter EOS, the application of the PRF method to this fluid requires characterization of only the third parameter. In this work, we apply the PRF method to the PR-EOS. Some background material on the PRF method is given in the next section. For the modified three-parameter PR-EOS, the section Third- Parameter Characterizations for the PR-EOS presents the Parameter Characterizations for the PR-EOS presents the third-parameter values for some lighter hydrocarbons and develops a correlation for characterizing the third parameter for the heptane-plus fractions of reservoir fluids. To apply the modified, three-parameter PR-EOS to calculate the phase and volumetric behavior of reservoir fluids, PR-EOS to calculate the phase and volumetric behavior of reservoir fluids, the section on Applications develops a novel, two-step procedure for characterizing all three parameters for the procedure for characterizing all three parameters for the heptane-plus fractions. The first step characterizes the two parameters of the unmodified PR-EOS with the phase-behavior data derived from the analysis of conventional laboratory experiments. The second step adjusts the correlation coefficients of the third-parameter correlation mentioned earlier, using the heptane-plus density data a standard conditions. Also presented in this section is the application of the modified PR-EOS to several crude-oil and gas-condensate systems. PRF Method PRF Method Consider one mole of a mixture of n components of composition zi, at temperature T and pressure p, obeying an EOS of the formwhere V is the molar volume. Assuming that at the thermodynamic equilibrium, the mixture may split at most into two distinct phases, we can determine the phase properties by solving the following wellposed system of (2n + 3) equations in (2n + 3) unknowns. The equations areandwhere the unknowns are fL, xi, yi, V, and Vg. Here fL denotes the mole fraction of liquid phase. (xi, V ) and yi, Vg) denote the composition and the molar volume of the liquid and gas phases, respectively. Eqs. 2 and 4 apply to each Component i and thus pose 2n equations in total. Subscripts and superscriptsand g denote the liquid and gas phases, respectively. Eq. 4 is the thermodynainic-equilibrium condition for each Component i and equates the fugacity of each component in equilibrium phases. Phi i is the fugacity coefficient for Component i and can be evaluated from the functional form of the Eq. 1 EOS. Eq. 2 is the material-balance equation for each component, while Eq. 3 represents the overall material balance. Eqs. 5 and 6 related the molar volume of each equilibrium phase to its composition, temperature, and pressure through the Eq. 1 EOS. pressure through the Eq. 1 EOS. Two-parameter cubic EOS's mentioned earlier form Eq. 1 as a third-degree polynomial in molar volume V and have two mixture parameters, a and b, related to the component parameters, ai and bi, parameters, a and b, related to the component parameters, ai and bi, through the following mixing rules:andSPERE P. 1033

The method developed previously (Youngren & Acrivos 1975) for obtaining numerical solutions to the Stokes equations for flows past solid particles is extended to problems with free boundaries. This technique is applied to the determination of steady shapes for an inviscid gas bubble symmetrically placed in an extensional flow. For large surface tension the computed bubble shape is found to be in excellent agreement with that obtained analytically by Barthès-Biesel & Acrivos (1973), while for small surface tension it agrees with an expression derived by Buckmaster (1972) using slender-body theory.

Immiscible water-alternating-gas injection (IWAG) has proven to be effective in managing produced gas at Kuparuk River Unit. The process is expected to improve oil recovery due to more efficient waterflooding in the presence of trapped gas. Laboratory data showed that trapped gas reduced water mobility by forcing water to displace oil from smaller pores. This mechanism resulted in lower residual oil saturation. Fully compositional simulation showed that IWAG may get an incremental oil of 1-3% of the original oil-in-place. Several years of operations have also revealed other significant tangible and intangible benefits for IWAG. These benefits are higher production rates, reduced water handling costs, and better reservoir management. Field experience shows that a tapered WAG helps keep produced gas oil ratio manageable. There are ongoing efforts to optimize WAG parameters (WAG ratio and gas slug size) and reduce costs. A potential injection scheme that does both is simultaneous water and gas injection.