Approximation Of Pressure Gradient Research Articles

Flow Simulation in Heterogeneous Porous Media With the Moving Least-Squares Method

The two-point-flux finite-volume method is the most widely used technique for discretizing Darcy's law in commercial reservoir simulators. The prevalence of that method within the oil-and-gas industry mostly stems from its simple implementation, intuitive understanding, and high execution speed. However, the method is known to suffer from severe inaccuracies when using non-$K$-orthogonal meshes, which reservoir engineers may need to discretize complex heterogeneous reservoirs. In these cases, the simulation inaccuracies may trigger costly business decisions about facilities and well management based on the erroneous prediction of flow streams. The moving least-squares method, with some novel enhancements presented in this paper to specifically handle discontinuous and anisotropic permeability fields, offers an attractive alternative for two main reasons. First, the reconstruction scheme relies on neighborhoods of points located at cell centers and, therefore, does not directly rely on the mesh topology or whether the mesh is $K$-orthogonal. Second, higher-order approximations can be obtained by increasing the size of neighborhoods. In particular, transported fields can be approximated using a second-order basis, which together with the use of limiters enables a much more accurate resolution of sharp fronts than the resolution obtained with a first-order scheme, as typically encountered in commercial simulators. Numerical experiments are shown that support these claims. Solving Poisson's equation highlights the ability of the new method to use non-$K$-orthogonal meshes and discontinuous tensors. Next, two oil-water problems underline the importance of obtaining an accurate pressure-gradient approximation to drive the flow. Finally, simulations in a complex, anisotropic reservoir focus on the robustness of the new method to several meshes.

A parabolic velocity-decomposition method for wind turbines

An economical parabolized Navier–Stokes approximation for steady incompressible flow is combined with a compatible wind turbine model to simulate wind turbine flows, both upstream of the turbine and in downstream wake regions. The inviscid parabolizing approximation is based on a Helmholtz decomposition of the secondary velocity vector and physical order-of-magnitude estimates, rather than an axial pressure gradient approximation. The wind turbine is modeled by distributed source-term forces incorporating time-averaged aerodynamic forces generated by a blade-element momentum turbine model. A solution algorithm is given whose dependent variables are streamwise velocity, streamwise vorticity, and pressure, with secondary velocity determined by two-dimensional scalar and vector potentials. In addition to laminar and turbulent boundary-layer test cases, solutions for a streamwise vortex-convection test problem are assessed by mesh refinement and comparison with Navier–Stokes solutions using the same grid. Computed results for a single turbine and a three-turbine array are presented using the NREL offshore 5-MW baseline wind turbine. These are also compared with an unsteady Reynolds-averaged Navier–Stokes solution computed with full rotor resolution. On balance, the agreement in turbine wake predictions for these test cases is very encouraging given the substantial differences in physical modeling fidelity and computer resources required.

An Improved Weak Pressure Gradient Scheme for Single-Column Modeling

Abstract A new formulation of the weak pressure gradient approximation (WPG) is introduced for parameterizing large-scale dynamics in limited-domain atmospheric models. This new WPG is developed in the context of the one-dimensional, linearized, damped, shallow-water equations and then extended to Boussinesq and compressible fluids. Unlike previous supradomain-scale parameterizations, this formulation of WPG correctly reproduces both steady-state solutions and first baroclinic gravity waves. In so doing, this scheme eliminates the undesirable gravity wave resonance in previous versions of WPG. In addition, this scheme can be extended to accurately model the emission of gravity waves with arbitrary vertical wavenumber.

Particle velocity estimation based on a two-microphone array and Kalman filter

A traditional method to measure particle velocity is based on the finite difference (FD) approximation of pressure gradient by using a pair of well matched pressure microphones. This approach is known to be sensitive to sensor noise and mismatch. Recently, a double hot-wire sensor termed Microflown became available in light of micro-electro-mechanical system technology. This sensor eliminates the robustness issue of the conventional FD-based methods. In this paper, an alternative two-microphone approach termed the u-sensor is developed from the perspective of robust adaptive filtering. With two ordinary microphones, the proposed u-sensor does not require novel fabrication technology. In the method, plane wave and spherical wave models are employed in the formulation of a Kalman filter with process and measurement noise taken into account. Both numerical and experimental investigations were undertaken to validate the proposed u-sensor technique. The results have shown that the proposed approach attained better performance than the FD method, and comparable performance to a Microflown sensor.

Weak Pressure Gradient Approximation and Its Analytical Solutions

Abstract A weak pressure gradient (WPG) approximation is introduced for parameterizing supradomain-scale (SDS) dynamics, and this method is compared to the relaxed form of the weak temperature gradient (WTG) approximation in the context of 3D, linearized, damped, Boussinesq equations. It is found that neither method is able to capture the two different time scales present in the full 3D equations. Nevertheless, WPG is argued to have several advantages over WTG. First, WPG correctly predicts the magnitude of the steady-state buoyancy anomalies generated by an applied heating, but WTG underestimates these buoyancy anomalies. It is conjectured that this underestimation may short-circuit the natural feedbacks between convective mass fluxes and local temperature anomalies. Second, WPG correctly predicts the adiabatic lifting of air below an initial buoyancy perturbation; WTG is unable to capture this nonlocal effect. It is hypothesized that this may be relevant to moist convection, where adiabatic lifting can reduce convective inhibition. Third, WPG agrees with the full 3D equations on the counterintuitive fact that an isolated heating applied to a column of Boussinesq fluid leads to a steady ascent with zero column-integrated buoyancy. This falsifies the premise of the relaxed form of WTG, which assumes that vertical velocity is proportional to buoyancy.

A method for reducing pressure gradient errors improving the sigma coordinate stretching function: An idealized flow patterned after the Libyan near-shore region with the POM

A technique for reducing the effects of pressure gradient errors in terrain-following models based on the vertical distribution of layers is proposed in this study. The basic idea is to find a vertical distribution that reduces the cumulative pressure gradient error estimated over the whole domain. The technique is applied to the sigma coordinate Princeton Ocean Model (POM), where the vertical distribution of layers is defined by a horizontally uniform stretching function. The cumulative error is used to measure the difference between the z-level-based pressure gradient approximation, used as “ground truth”, and the pressure gradient in sigma coordinates after interpolating the density field to sigma coordinate locations. A vertical stretching function which reduces the cumulative error is sought by iterative methods that need some additional constraints in order to avoid non-useful results (e.g. wildly varying layer thicknesses based on the derived stretching function). Two simple expressions for the cumulative error based on barotropic and baroclinic errors are tested. With the POM model the baroclinic one is found to be the most effective. Improvements are found in the internal current field and on the time evolution of the density distribution. Beneficial effects on the external mode, however, cannot be excluded in the long term. The technique can be used on any pressure gradient scheme adopted; however, iterative methods are applicable only to cases, or to portions of the domain, in which the vertical stretching function is horizontally uniform (sigma, pure ‘quasi’ z-level and hybrid sigma- z level coordinate systems). The application of the technique to generalized vertical coordinate systems (temporarily setting the vertical coordinate as sigma) can be used to establish to which kind of cumulative error the model is sensitive, information useful in building vertical and horizontal adaptive grids that reduce the effects of pressure gradient errors.

Enriched multi-point flux approximation for general grids

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.

Rough‐sea deghosting using a single streamer and a pressure gradient approximation

We present a method for receiver ghost correction of towed streamer data that accounts for the rough sea surface. The method explicitly uses the fact that the pressure is zero at the free (sea) surface to estimate the vertical pressure gradient. Continuous elevation measurements of the wave height directly above the hydrophones are required—a measurement which is currently unavailable. The new deghosting method is fundamentally limited to frequencies below the first ghost notch. The lowest‐order implementation requires that the streamer is towed no deeper than approximately 6 m and a receiver spatial sampling interval of about 3 m or less. Using the lowest‐order and simplest implementation of the new method, the rough‐sea error is reduced from 1.5–2.5 dB to about 1–1.5 dB in amplitude and from 20° to 10° in phase, at 50 Hz in a 4‐m significant wave height sea. Higher‐order terms in the approximation promise to further reduce the error.

A spatial marching technique for the inviscid blunt body problem

A technique has been developed for obtaining approximate solutions of the inviscid, hypersonic flow on a blunt body with a spatial marching scheme. The scheme introduces the Vigneron pressure gradient approximation into the momentum equation in the direction along the body surface. The resulting governing equations are hyperbolic. With a specified shock wave these equations are solved at the stagnation streamline with an iteration procedure and are solved in the downstream direction with a marching scheme. The complete Euler equations are solved with the numerical scheme when the flow is supersonic. A global iteration procedure is required to obtain the shock wave location. The approximate results from the spatial marching technique are compared with the complete solution of the Euler equations for flow over a sphere. The two results are shown to be in approximate agreement and the spatial marching technique provides useful engineering predictions while requiring considerably less computational time.