Analysis of particle-laden fluid flow near an individual perforation (and assuming the fracture is only being fed by this one perforation) showed that the efficiency of particle transport to the fracture (ratio of particle concentration in the fracture and in the wellbore) depends on two dimensionless parameters—the particle stokes number St and parameter Λ, which is the ratio of flow velocity in the wellbore and to the fracture. In a simplified geometry approach, the entrance to the fracture is modelled as a short segment of a circular pipe. Using a detailed 3D CFD model of fluid-particle flow to the fracture (built in a commercial software), the efficiency of particle transport to the fracture was calculated for various combinations of parameters, St and Λ, and tabulated. Using the obtained tables, the proppant delivery efficiency can be very efficiently and accurately calculated by linear interpolation of tabulated data. The calculations presented in the paper have been carried out using water as the fracturing fluid with low proppant concentrations.

This paper presents an innovative method for the identification of multiphase flow patterns and the estimation of pressure drop. This method uses a data-driven classification model based on a fuzzy inference system (FIS). The paper discusses the specific problem of two-phase, gas-liquid regime identification and pressure drop prediction for pipe flow and demonstrates the ability of the fuzzy-mechanistic (fuzzynistic) model to accurately identify the flow regime. In addition, the fuzzynistic model is computationally efficient, and can help to provide real-time monitoring and control of production and pipeline equipment. The fuzzynistic model is demonstrated for two-phase, gas-liquid pipe flow over a wide range of superficial velocities and pipe inclination angles. Seven flow regimes are identified. Historically, regime predictions for multiphase flows have been based on either empirical maps created from experimental observations or on mechanistic models, which consider the physical mechanisms that cause variations of fluid phase distribution. Both have disadvantages in that they are either tied to limited experimental data or are prone to discontinuities at the regime transitions, or both. An innovative contribution of the fuzzynistic method is that it classifies flow patterns with associated weights. These flow patterns allow the aggregation of pressure drop functions of classified flow patterns to help estimate the corresponding pressure drop automatically across flow regime transitions. Flow regime maps generated using the fuzzy logic approach accurately mimic those generated by mechanistic models, but have “fuzzy” transitions between regimes because of the partial degree of membership to adjacent and neighboring regimes. Thus, along the flow path, partial degrees of membership to adjacent and neighboring flow regimes are assigned to account for the pressure drop prediction at the regime transitions. Furthermore, because the fuzzynistic method is computationally efficient, it could be used for real-time monitoring and control of equipment in wells, pipelines, and downstream processing.

Mechanistic regime-transition functions, where one or more physical arguments are used to describe transitions, are often used to identify equilibrium multiphase flow regimes. The mechanistic models for two-phase, gas-liquid pipe flow rely on one or more closure relations, most of which have been fit to experiments of air and water systems. However, these models are often applied to problems in the oil and gas industry, which span a very broad parameter space of fluid properties. Sensitivities of the one-dimensional, two-phase, gas-liquid, mechanistic models for dispersed-bubble pipe flow are investigated over a parameter space typical of that observed in the oil and gas industry. This spans several orders of magnitude in gas density, gas and liquid viscosities, and surface tension, in addition to large ranges in superficial gas and liquid velocities, pipe diameter, and pipe inclination angle. Dispersed-bubble regime identification is most sensitive to the superficial velocities, with secondary sensitivities to densities and pipe-inclination angle in special cases.

The prediction of flow patterns transition is performed for oil-water, two-phase flows over a wide range of oil viscosities. Theoretical models of oil-water systems are studied in horizontal and near-horizontal pipe flows. The equilibrium flow pattern is dependent on the oil and water properties, pipe parameters, and flow rates. Four flow patterns are considered in the mechanistic model, including stratified (ST), core-annular (CA), oil-in-water (O/W), and water-in-oil dispersions (W/O). A mechanistic criterion, proposed by Zhang and Sarica (2006) for the same purpose, is used to identify stratified flow. The Brinkman (1952) model is used to distinguish the phase inversion of oil-in-water from water-in-oil emulsions. Boundaries of core-annular flow are based on the critical core diameter given by Brauner (2003). Comparisons between the mechanistic model predictions and published experimental measurements show good agreement in regime identification.The importance of regime identification for correct pressure drop predictions is demonstrated by comparing the mechanistic model with a simple mixture model. The values of pressure drop predicted by both models are calculated and compared to existing experimental data. The mechanistic model shows significant improvement in pressure drop predictions.Dimensionless groups from Buckingham Pi theory are used to investigate the sensitivity to the input parameters. The use of dimensionless groups reduces the number of dependencies from nine input parameters to six dimensionless groups. This reduces the complexity in the optimum design of pipeline systems. From most to least sensitive, for ranges typical of pipeline flow and fluids, the pressure drop prediction depends on the superficial Reynolds number of oil, Eotvos number, pipe inclination angle, superficial velocity ratio, density ratio, and viscosity ratio.

One-dimensional (1D), equilibrium-based mechanistic model predictions are compared to three-dimensional (3D) transient computational fluid dynamics results for horizontal two-phase, gas-liquid pipe flow. The 3D regions of interest include both those expected to be in equilibrium conditions and those where transitions between flow regimes occur. Equilibrium simulations, such as those for stratified flow in a horizontal pipe, allow crucial validation of the equilibrium-based closure relations by means of numerical experiments. In the transitional regions, fully 3D, time-dependent numerical simulations provide a means to estimate the error in the equilibrium-based models and suggest how reasonable approximations can be made in these regions.