A Single Equation to Depict Bottomhole Pressure Behavior for a Uniform Flux Hydraulic Fractured Well

Pressure transient analysis for a vertically hydraulically fractured well is evaluated using two different equations, which cater for linear flow at the early stage and radial flow in the later stage. However, there are three different stages that take place for an analysis of pressure transient, namely linear, transition and pseudo-radial flow. The transition flow regime is usually studied by numerical, inclusive methods or approximated analytically, for which no specific equation has been built, using the linear and radial equations. Neither of the approaches are fully analytical. The numerical, inclusive approach results in separate calculations for the different flow regimes because the equation cannot cater for all of the regimes, while the analytical approach results in a difficult inversion process to compute well test-derived properties such as permeability. There are two types of flow patterns in the fracture, which are uniform and non-uniform, called infinite conductivity in a high conductivity fracture. The study was conducted by utilizing an analogous study of linear flow equations. Instead of using the conventional error function, the exponential integral with an infinite number of wells was used. The results obtained from the developed analytical solution matched the numerical results, which proved that the equation was representative of the case. In conclusion, the generated analytical equation can be directly used as a substitute for current methods of analyzing uniform flow in a hydraulically fractured well.

Static Performance of Smooth Liquid Annular Seals in the Transition and Turbulent Regimes

Static test results are presented for smooth annular seals with a length-to-diameter ratio of 0.50, radius R = 51.00 mm, at the nominal radial clearance Cr = 0.2032 mm. Tests were conducted for angular shaft speeds; ω = 2, 4, 6, 8 krpm, axial pressure drops; ΔP = 2.1, 4.13, 6.21, 8.27 bars, and eccentricity ratios ϵ0 = e0/Cr = 0.00, 0.27, 0.53, 0.8 where e0 is the static eccentricity. Three pre-swirl inserts were used to target zero, medium, and high (0., 0.4, and 0.8) pre-swirl ratios for a set of pre-determined operating conditions with ISO VG 2 oil at 46.1°C. Pitot tubes measured the circumferential velocity at separate upstream and downstream seal locations and were used to calculate pre-swirl ratio, PSR = vinlet/Rω, and outlet-swirl ratio, OSR = voutlet/Rω. For all tested pre-swirl inserts, PSR tended to converge to 0.4∼0.5 as ω increased. PSR and OSR were poorly correlated. Volumetric leakage rate Q ˙ versus pressure differential ΔP was measured. The measured vector Reynolds number Re, combining the axial and circumferential Reynolds numbers ranged from ∼1000 to ∼3500. Based on Zirkelback and San Andrés 1996 publication, almost all of the flow regime is predicted to lie in the transition regime, with fewer points in the turbulent regime. Generally, the seals’ static centering properties were obtained by applying a static load Fs and measuring the resulting displacement vector e0. At many low-speed, low-ΔP test conditions, the seal would not remain in the desired centered or near-centered position and had to be forced into place with a centering force Fs. The authors believe that the observed de-centering effects resulted from test operations in the transition flow regime where the friction factor λ does not drop with increasing ΔP and increasing Re. A positive centering Lomakin effect requires that λ drop with increasing axial Reynolds number. The seals had positive centering effects over a large portion of the predicted transition flow regime, supporting the view that the shift from transition-to-turbulent flow regularly occurred at lower Re values than the Re = 3000 boundary used by Zirkleback and San Andrés.

A Practical Gas Apparent Permeability Model: Multi-Scale Simulations of Rarefied Gas Flow in Matrix

A common practice in gas-shale reservoir simulation, which arbitrarily increases intrinsic matrix permeability to match the production data, has been proven inefficient and unreliable. Alternatively, accurate estimations of gas apparent permeability (AP) in matrix is desired. This work presents an analytical AP model considering rarefaction in nanopores and coupling experimentally confirmed mechanisms in shale matrix for theoretical completeness. Meanwhile, physical terms in AP model are simplified with semi-empirical correlations for the practicability in large-scale field simulation. Compared with other gas transport models in nanopores, the newly-developed analytical model has been successfully validated against molecular dynamic (MD) simulation, direct simulation Monte Carlo (DSMC), Lattice Boltzmann (LB) simulation, and experimental flux results for five types of gases (i.e., methane, nitrogen, helium, argon, and oxygen) with the minimum deviation. It is observed that analytical models excluding Knudsen diffusion mechanism cannot fully characterize rarefaction effect. Next, Knudsen diffusion cannot be explained as the only underlying mechanism of rarefaction because the mass flux is largely underestimated in transition flow regime. However, the weighted superposition of second-order slip boundary and Knudsen diffusion can provide the satisfactory fitting with data. This work provides an analytical model which not only considers non-negligible multi-physics in shale reservoirs (i.e., rarefaction effect, multilayer adsorption, surface diffusion and confinement effect) but also simplifies non-linear physical terms using semi-empirical linear correlations to facilitate AP calculations in core-scale simulations.

Experimental Study on the Effect of Localized Blockages on the Friction Factor of a 61-Pin Wire-Wrapped Bundle

The thermal-hydraulic behavior of the flow in rod bundles has motivated numerous experimental and computational investigations. Previous studies have identified potential for accumulation of debris within the small subchannels of typical wire-wrapped assemblies with subsequent total or partial blockage of subchannel coolant flow. A test campaign was conducted to study the effects of localized blockages on the bundle averaged friction factor of a tightly packed wire-wrapped rod bundle. Blockages were installed within the bundle, and fluid pressure drop was measured across one wire pitch for a Reynolds number range of 500–17,200. The Darcy–Weisbach friction factor of the perturbed rod bundle geometry was compared with that of the unblocked bundle, as well as with the predictions of a well-established friction factor correlation. Differing effects based on blockage size and location for various flow regimes were studied. A number of conclusions can be made about the effects of the blockages on the friction factor, such as an increasing effect of the blockage on friction factor with an increase in Reynolds number, a change in flow behavior in the turbulent transition flow regime near Reynolds number 3000, differences in effect on friction factor for different types of subchannel blockage, and a nonlinear trend in friction factor variation with flow area impeded for edge subchannels. To this end, all data and quantified uncertainty produced in this study are made available for comparison and validation of advanced computational tools.

Thermal-Hydraulic Performance Analysis of Fluid Flow in Tube With Helical Screw Inserts in Transition Flow

An experimental analysis is arranged to evaluate thermal hydraulic performance analysis on fluid flow in helical screw inserts in tube with number of strips and different twist ratios in Transition flow regime. Single strip insert is also compared with double strip inserts of helical screw inserts with three values of twist ratios. Heat transfer enhancement is achieved with fluid flow in double strip as compared to single strip helical screw insert at decreases values of twist ratios and increases values of Reynolds number (Re). Maximum enhancement in the value of Nusselt number is achieved with double strip inserts at low value of twist ratio and Reynolds number as compared to Single strip inserts. Common correlations of Nusselt number and friction factor are generated. Thermal performance factor (TPF) is achieved maximum values with double strip insert at all flow rates at 2.5 of twist ratio than single strip inserts. Double strip inserts show suitability of helical screw insert in heat exchangers to compact the size of thermal applications.

Preswirl and Mixed-Flow (Mainly Liquid) Effects on Rotordynamic Performance of a Long (L/D = 0.75) Smooth Seal

Measured results are presented for rotordynamic coefficients and mass leakage rates of a long smooth annular seal (length-to-diameter ratio L/D = 0.75, diameter D = 114.686 mm, and radial clearance Cr = 0.200 mm) tested with a mixture of silicone oil (PSF-5cSt) and air. The test seal is centered, the seal exit pressure is maintained at 6.9 bars-g while the fluid inlet temperature is controlled within 37.8–40.6 °C. It is tested with three inlet-preswirl inserts, namely, zero, medium, and high (the preswirl ratios (PSRs), i.e., the ratio between the fluid's circumferential velocity and the shaft surface's velocity, are in ranges of 0.10–0.18, 0.30–0.65, and 0.65–1.40 for zero, medium, and high preswirls, respectively), six inlet gas-volume fractions GVFi (0%, 2%, 4%, 6%, 8%, and 10%), four pressure drops PDs (20.7, 27.6, 34.5, and 41.4 bars), and three speeds ω (3, 4, and 5 krpm). The targeted test matrix could not be achieved for the medium- and high-preswirl inserts at PD ≥ 27.6 bars due to the test-rig stator's dynamic instability issues. Spargers were used to inject air into the oil, and GVFi values higher than 0.10 could not be consistently achieved because of unsteady surging flow downstream from the sparger mixing section. Leakage mass flow rate m˙ and rotordynamic coefficients are measured, and the effect of changing inlet preswirl and GVFi is studied. The test results are then compared with predictions from a two-phase, homogeneous-mixture, bulk-flow model developed in 2011. Generally, both measurements and predictions show little change in m˙ as inlet preswirl changes. Measured m˙ remains unchanged or slightly increases with increasing GVFi, but predicted m˙ decreases. Measured m˙ is comparable to predicted values but consistently lower. Dynamic-stiffness coefficients are measured using an ensemble of excitation frequencies and curve-fitted well by frequency-independent stiffness Kij, damping Cij, and virtual mass Mij coefficients. Planned tests with the medium- and high-preswirl inserts could not be accomplished at PD = 34.5 and 41.4 bars because the seal stator became unstable with any finite injection of air. The test results show that the instability arose because the seal's direct stiffness K became negative and increased in magnitude with increasing GVFi. The model predicts a drop in K as GVFi increases, but the test results dropped substantially more rapidly than predicted. Also, the model does not predict the observed strong tendency for K to drop with an increase in preswirl in moving from the zero-to-medium and medium-to-high preswirl inserts. The authors believe that the observed drop in K due to increasing GVFi is not explained by either (a) a reverse Lomakin effect from operating in the transition flow regime or (b) the predicted drop in K at higher GVFi values from the model. A separate and as yet unidentified two-phase flow phenomenon probably causes the observed results. The negative K results due to increasing GVFi and moving from the zero to medium, and medium to high preswirl observed here could explain the instability issue (sudden subsynchronous vibration) on a high-differential-pressure helico-axial multiphase pump (MPP), reported in 2013. Effective damping Ceff combines the stabilizing effect of direct damping C, the destabilizing effect of cross-coupled stiffness k, and the influence of cross-coupled mass mq. As predicted and measured, increasing inlet preswirl significantly increases k and decreases Ceff, which decreases the seal's stabilizing properties. Ceff increases with increasing GVFi—becomes more stable.

Numerical Modelling of Microchannel Gas Flows in the Transition Flow Regime Using the Cascaded Lattice Boltzmann Method

In this article, a lattice Boltzmann (LB) method for studying microchannel gas flows is developed in the framework of the cascaded collision operator. In the cascaded lattice Boltzmann (CLB) method, the Bosanquet-type effective viscosity is employed to capture the rarefaction effects, and the combined bounce-back/specular-reflection scheme together with the modified second-order slip boundary condition is adopted so as to match the Bosanquet-type effective viscosity. Numerical simulations of microchannel gas flow with periodic and pressure boundary conditions in the transition flow regime are carried out to validate the CLB method. The predicted results agree well with the analytical, numerical, and experimental data reported in the literature.

Grid criteria for numerical simulation of hypersonic aerothermodynamics in transition regime

Grid is an important factor in numerical simulation of hypersonic aerothermodynamics. This paper introduces three criteria for determining grid size in the transition flow regime when using the computational fluid dynamics (CFD) method or the direct simulation Monte Carlo (DSMC) method. The numerical relationship between these three criteria sizes is deduced according to the one-dimensional fluid theory. Then, the relationship is verified using the CFD method to simulate the flow around a two-dimensional cylinder. At the same time, the dependence of simulation accuracy on grid size in the CFD and DSMC methods is studied and the mechanism is given. The result shows that the simulation accuracy of heat flux especially depends on the normal grid size next to surfaces, where the $Re_{\mathit{cell},w}$ criterion and the $\unicode[STIX]{x1D706}_{w}$ criterion based on local parameters are applicable and equivalent, while the $Re_{\mathit{cell},\infty }$ criterion based on the free-stream parameter is only applicable under the assumption of constant viscosity coefficient and constant temperature wall conditions. On the other hand, the trend of the heat flux changing with grid size obtained by CFD and DSMC is exactly the opposite. Therefore, the grid size must be strictly satisfied with the grid criteria when comparing CFD with DSMC and even the hybrid DSMC with Navier–Stokes method.