A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs

Horizontal wells are commonly used in bottom water reservoirs,which can increase contact area between wellbores and reservoirs.There are many completion methods used to control cresting,among which variable density perforation is an effective one.It is difficult to evaluate well productivity and to analyze inflow profiles of horizontal wells with quantities of unevenly distributed perforations,which are characterized by different parameters.In this paper,fluid flow in each wellbore perforation,as well as the reservoir,was analyzed.A comprehensive model,coupling the fluid flow in the reservoir and the wellbore pressure drawdown,was developed based on potential functions and solved using the numerical discrete method.Then,a bottom water cresting model was established on the basis of the piston-like displacement principle.Finally,bottom water cresting parameters and factors influencing inflow profile were analyzed.A more systematic optimization method was proposed by introducing the concept of cumulative free-water production,which could maintain a balance（or then a balance is achieved）between stabilizing oil production and controlling bottom water cresting.Results show that the inflow profile is affected by the perforation distribution.Wells with denser perforation density at the toe end and thinner density at the heel end may obtain low production,but the water breakthrough time is delayed.Taking cumulative free-water production as a parameter to evaluate perforation strategies is advisable in bottom water reservoirs.

DISAPPEARANCE OF INTERFACES FOR DEGENERATE PARABOLIC EQUATIONS WITH VARIABLE DENSITY AND ABSORPTION

In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.

Effi cient modeling of the gravity anomaly caused by a sedimentary basin with lateral variable density contrast and its application in basement relief estimation

The forward calculation of gravity anomalies is a non-negligible aspect contributing to the time consumption of the entire process of basement relief estimation.In this study,we develop a fast hybrid computing scheme to compute the gravity anomaly of a basement.We use the vertical prism source equation in a given region R centered at a certain gravity observation point and the vertical line source equation outside R to derive the gravity anomaly.We observe that the computation with the vertical line source equation is much faster than that of the vertical prism source equation,but the former is slightly inaccurate.Therefore,our method is highly effi cient and able to avoid the errors caused by the low accuracy of the vertical line source equation near the observation point.We then derive the general principle of choosing the size of R via a series of prism model tests.Our tests on the gravity anomaly over the Los Angeles Basin confirm the correctness of our proposed forward strategy.We modify Bott’s method with an accelerating factor to expedite the inversion procedure and presume that the density contrast between the sediments and the basement in a sedimentary basin varies laterally and can be obtained using the equivalent equation.Synthetic data and real data applications in the Weihe Basin illustrate that our proposed method can accurately and effi ciently estimate the basement relief of sedimentary basins.

Properties of Solutions of Kolmogorov-Fisher Type Biological Population Task with Variable Density

In this paper, we discussed population model of two competing populations with non-linear double diffusion and variable density which described by nonlinear system of competing individuals. We identify new properties, such as finite speed of propagation, and localization of the outbreaks in a specific area.

Effects of Reynolds number and Schmidt number on variable density mixing in shock bubble interaction

Effects of Reynolds(Re)number and Schmidt(Sc)number on the flow structures and variable density mixing are numerically investigated through the canonical shock cylindrical bubble interaction.By determining the viscosity and diffusivity within a wide range,the controlling parameters,total vortex circulation,and compression rate,are conservative under a broad range of Re and Sc numbers(Re≈10^(3)-10^(5)and Sc≈0.1-5)in the same shock Mach(Ma)number condition(Ma=2.4).As for the Re number effect,the circulation of secondary baroclinic vorticity(SBV),induced by the main vortex centripetal acceleration,is observed to be higher in high Re number and vice versa.Based on the vorticity transport equation decomposition,a growth-inhibition vorticity dynamics balance mechanism is revealed:the vorticity viscous term grows synchronously with baroclinic production to inhibit SBV production in low Re number.By contrast,the viscous term terminates the baroclinic term with a time lag in high Re number,leading to the SBV production.Since the SBV reflects the local stretching enhancement based on the advection-diffusion equation,mixing is influenced by the Sc number in a different behavior if different Re numbers are considered.The time-averaged variable density mixing rate emerges a scaling law with Sc number asχ^(∗)=β·Sc^(−α),where the coefficientβ∼Re−0.2 and the scaling exponentα∼Re−0.385.The understanding of Re number and Sc number effect on variable density mixing provides an opportunity for mixing enhancement from the perspective of designing the viscosity and diffusivity of the fluid mixture.

A High-Order Direct Discontinuous Galerkin Method for Variable Density Incompressible Flows

In this work,we develop a novel high-order discontinuous Galerkin(DG)method for solving the incompressible Navier-Stokes equations with variable density.The incompressibility constraint at cell interfaces is relaxed by an artificial compressibility term.Then,since the hyperbolic nature of the governing equations is recovered,the simple and robust Harten-Lax-van Leer(HLL)flux is applied to discrete the inviscid term of the variable density incompressible Navier-Stokes equations.The viscous term is discretized by the direct DG(DDG)method,the construction of which was initially inspired by the weak solution of a scalar diffusion equation.In addition,in order to eliminate the spurious oscillations around sharp density gradients,a local slope limiting operator is also applied during the highly stratified flow simulations.The convergence property and performance of the present high-order DDG method are well demonstrated by several benchmark and challenging numerical test cases.Due to its advantages of simplicity and robustness in implementation,the present method offers an effective approach for simulating the variable density incompressible flows.

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Rurthermore, the instability of the bounded shear layer is studied in two cases： （i） the density which takes two different constant values in two layers and （ii） the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.

Cylindrical effects in weakly nonlinear Rayleigh Taylor instability

The classical Rayleigh–Taylor instability（RTI） at the interface between two variable density fluids in the cylindrical geometry is explicitly investigated by the formal perturbation method up to the second order. Two styles of RTI, convergent（i.e., gravity pointing inward） and divergent（i.e., gravity pointing outwards） configurations, compared with RTI in Cartesian geometry, are taken into account. Our explicit results show that the interface function in the cylindrical geometry consists of two parts： oscillatory part similar to the result of the Cartesian geometry, and non-oscillatory one contributing nothing to the result of the Cartesian geometry. The velocity resulting only from the non-oscillatory term is followed with interest in this paper. It is found that both the convergent and the divergent configurations have the same zeroth-order velocity, whose magnitude increases with the Atwood number, while decreases with the initial radius of the interface or mode number. The occurrence of non-oscillation terms is an essential character of the RTI in the cylindrical geometry different from Cartesian one.

Numerical simulation on influence of jet angle on jet's characteristics in flowing ambient fluid

Based on the stress-algebraic model, the turbulent buoyant jet with variable density was studied by the relation between density and concentration. A simple expression for buoyancy coefficient was proposed. The governing equations of turbulent buoyant jet with variable density were closed by introducing the expression of β and the relation between density and concentration. Numerical results for the jet axis with density difference agree well with experimental ones. By finite volume method, the 2 - D vertical jet＇s flow field with different jet angles was studied. The analysis of the relation among the vortex center, the position of separation point and jet angles shows that the circumfluenee field is the largest when the jet angle is 90°. The area turbulent kinetic energy ka is proposed and the relationship between mixing intensity and jet angles is analyzed based on it. Results show that the jet angle of is the optimum condition for jet water mixing with environment water;and the reduced rate of difference between the centerline density of jet and the density of ambient water is the largest at the jet angle of 90°.

The relation between the Mindanao Current and Mindanao Undercurrent on seasonal scale

The ocean general circulation model for the earth simulator（OFES） products is applied to estimate the transports of the Mindanao Current（MC） and the Mindanao undercurrent（MUC） and explore the relation between them on seasonal scale. In general, the MUC is composed of the lower part of the Southern Pacific Tropical Water（SPTW）and Antarctic Intermediate Water（AAIW）. While the deep northward core below 1 500 m is regarded as a portion of MUC. Both salinity and potential density restrictions become more reasonable to estimate the transports of MC/MUC as the properties of water mass having been taken into consideration. The climatological annual mean transport of MC is（37.4±5.81）×10~6 m^3/s while that of MUC is（23.92±6.47）×10~6 m^3/s integrated between 26.5 σ_θ and 27.7 σ_θ, and（17.53±5.45）×10~6 m^3/s integrated between 26.5 σ_θ and 27.5 σ_θ in the OFES. The variations of MC and MUC have good positive correlation with each other on the seasonal scale： The MC is stronger in spring and weaker in fall, which corresponds well with the MUC, and the correlation coefficient of them is 0.67 in the OFES.The same variations are also appeared in hybrid coordinate ocean model（HYCOM） results. Two sensitive experiments based on HYCOM are conducted to explore the relation between MC and MUC. The MUC（26.5〈σ_θ〈27.7） is strengthening as the MC increases with the enhancement of zonal wind field. It is shown,however, that the main part of the increasement is the deeper northward high potential density water（HPDW）,while the AAIW almost remains stable, SPTW decreases, and vice versa.

Efficient Variable-Coefficient Finite-Volume Stokes Solvers

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variablecoefficient Stokes equations on a uniform staggered grid.Building on the success of using the classical projection method as a preconditioner for the coupled velocitypressure system[B.E.Griffith,J.Comp.Phys.,228(2009),pp.7565–7595],as well as established techniques for steady and unsteady Stokes flow in the finite-element literature,we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems.We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems.Contrary to traditional wisdom,we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems,making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow,even for steady flow and in the presence of large density and viscosity contrasts.Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size,making them suitable for large-scale problems.Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Study on damage characteristics of iron ore under variable line density charge structure blasting

Blasting test research was conducted on iron ore specimens with variable line density charging structures.Computer tomography(CT),digital image processing,and three-dimensional model reconstruction techniques were used to analyze the damage characteristics of iron ore specimens after blasting based on the calculated number of box dimensions.The results show that increasing the variable line density section charge uncoupling coefficient reduces the overall damage to the specimen by up to 1.73%,indicating that the overall damage size negatively correlates with the size of the variable line density section charge uncoupling coefficient.The damage characteristics of iron ore specimens from different layers(uncoupled charging section,transition section,coupled charging section)have some variability;when the uncoupling coefficient of the uncoupled charging section was reduced,the uncoupled section of the center of the damaged layer increased and then reduced.In contrast,the transition section shows a trend of increase,and the coupled section shows a minor difference,fully demonstrating the change in the variable line density section of the uncoupling coefficient of the specimen blasting damage effects.This study concludes that in the actual blasting project,choosing a reasonable variable line charge density structure can make the release of explosive blast energy more uniform to efficiently and thoroughly use explosive power to improve the iron ore crushing effect.

WAVE DISSIPATING PERFORMANCE OF AIR BUBBLE BREAKWATERS WITH DIFFERENT LAYOUTS

The wave dissipating performance of air bubble breakwaters with different layouts is studied by experimental and numerical methods in this article. Based on the assumpation that the mixture of air and water is regarded as a variable density fluid, the mathematical model of the air bubble breakwater is built. The numerical simulation results are compared with the experimental data, which shows that the mathematical model is reasonable for the transmission coefficient Ct m. The influencing factors are studied experimentally and numerically, including the incident wave height H i, the incidentt wave period T , the air amount Qm , the submerged pipe depth D and the single or double air discharging pipe structure. Some valuable conclusions are obtained for further research of the mechanism and practical applications of air bubble breakwaters.

The productivity calculation model of perforated horizontal well and optimization of inflow profile

Aiming at the large error in productivity predication and incomplete consideration in completion parameters design of perforated horizontalwell,a model which coupled the relationship of pressure and flow rate in reservoir seepage,near-wellbore inflow and wellbore flow was established.The impact of near-wellbore heterogeneity,wellbore flow pressure drop and completion parameters on the inflow profile of horizontalwell is analysed.Studies showed that with a stronger near-wellbore heterogeneity,the inflowprofilewouldfluctuatemore seriously.Perforationdensityhada great influence onthe inflow profile and local changes of it would bring a shunt effect.Completion design of variable density perforated horizontalwell with an optimized inflowprofile whichwas close to a standard profilewould improve the horizontal well development effect.The achievement can provide directive meanings to productivity predication and completion parameters design of horizontal wells in oilfield.

SPLITTING SCHEMES FOR A NAVIER-STOKES-CAHN-HILLIARD MODEL FOR TWO FLUIDS WITH DIFFERENT DENSITIES

In this work, we focus on designing efficient numerical schemes to approximate a ther- modynamically consistent Navier-Stokes/Cahn-Hilliard problem given in [3] modeling the mixture of two incompressible fluids with different densities. The model is based on a diffuse-interface phase-field approach that is able to describe topological transitions like droplet coalescense or droplet break-up in a natural way. We present a splitting scheme, decoupling computations of the Navier-Stokes part from the Cahn-Hilliard one, which is unconditionally energy-stable up to the choice of the potential approximation. Some nu- merical experiments are carried out to validate the correctness and the accuracy of the scheme, and to study the sensitivity of the scheme with respect to different physical pa- rameters.

Novel Mesh Technique and Its Application in the Wind Field Simulation for Flexible Spatial Structure

In this paper,novel mesh techniques are proposed for wind field simulation of flexible spatial structure.For mesh generation,an interpolation strategy is presented to obtain a mesh system with variable density.Two spatial structure examples are used to examine the efficiency and applicability of this technique.Then based on the structured mesh system generated by the technique,the mesh nodal coordinates are updated to adapt the moving boundary conditions by means of the mapping interpolation functions and some examples are given to verify the effectiveness.Furthermore,the constrained counterforce distribution technique and projection interpolation strategy are developed to implement the data exchange on the interaction surface of wind and structure.Finally,the computational accuracy is numerically validated.