EXACT SOLUTION FOR SPHERICAL FLOW AND TRANSIENT BEHAVIOR OF PRESSURE WITH CONSIDERATION OF EFFECTS OF QUADRATIC-GRADIENT TERM

Conventional models for fluid flow in well tests have not been consistent with material balance. According to the slightly compressible fluid assumption, the quadratic gradient term in the nonlinear partial differential equation has been usually neglected. This approach is questionable for live oil and low permeability reservoirs. We have already known that linearization by neglecting quadratic gradient terms may lead to errors for large values of well-test time. In this paper, a method that is consistent with material balance was proposed on the spherical flow system. All terms in the nonlinear partial eqiation were retained. Exact solution for the resulting nonlinear partial differential equation in an infinite reservoir was obtained by using the Laplace transform considering wellbore storage. Analytical solution for nonlinear partial differential equation are resulted by using orthogonal transforms under both closed and constant-pressure outer boundary conditions. The law of pressure changes for a fluid compressibility α and a storage coefficient C D were discussed.

THE MOTION OF A SPHERICAL PARTICLE IN THE STOKES FLOW OUTSIDE A CIRCULAR ORIFICE

For any study ofa suspension entering a pore, the knowledge of the force and moment exerted on a solute particle in an arbitrary position outside the pore is essential, 'This paper for the first lime presents approximate analytical expressions (in closed form) of all the twelve force and moment coefficienis for a sphere outsied a circular orifice, on the basis of a number of discrete data computed by Yan et al(1987).These coefficients are then applied to calculate the trajectory and angular velocity of a spherical particle approaching the pore at zero Reynolds number. The trajectory is in excellent agreement with the available experimental results. An analysis of the relative importance of the coefficients shows that the rotation effect cannot be neglected near the pore opening or near the wall, and that the lateral force effect must be taken into account in the neighborhood of the edge of the pore opening. It is due to neglecting these factors that previous theoretical results deviate from the experimental ones near the pore opening. The effects of the ratio of the particle to pore radii as well as the influences of the graritytbuoyance on the particle trajectory, velocity distribution and rotation are discnssed in detail. It is pointed out that in the experiments of neutrally-buoyant suspensions, the restriction on the density of the particle is most demanding for a large particle size.The expressions of forces and moments presenled herein are complete, relatively accurate and convenient, thus providing a good prerequisite for further studies of any problems involving the entrance of particles to a pare.

Calculation of terminal velocity in transitional flow for spherical particle

The terminal velocity has been widely used in extensive fields, but the complexity of drag coefficient expression leads to the calculation of terminal velocity in transitional flow （1 〈 Re ≤ 1000） with much more difficulty than those in laminar flow （Re ≤ 1） and turbulent flow （Re ≥ 1000）. This paper summarized and compared 24 drag coefficient correlations, and developed an expression for calculating the terminal velocity in transitional flow, and also analyzed the effects of particle density and size, fluid density and viscosity on terminal velocity. The results show that 19 of 24 previously published correlations for drag coefficient have good prediction performance and can be used for calculating the terminal velocity in the entire transitional flow with higher accuracy. Adapting two dimensionless parameters （w＊, d＊）, a proposed explicit correlation, w＊=-25.68654 × exp （-d＊/77.02069）＋ 24.89826, is attained in transitional flow with good performance, which is helpful in calculating the terminal velocity.

THE GENERAL PROPERTIES OF THE SPHERICAL VORTICES（SV）OF n－TH ORDER AND THE CHAOTIC PHENOMENA AND OF THE ORDERED STRUCTURES OF THE SV OF 3RD ORDER

The general properties of the spherical vortices(SV)of n-th order are discussedin this paper Numerical calculations are carried out in the case of n=3.We find outsome interesting phenomena concerning the chaotic regions and ordered islands on the Poincare sections. Interpretations of these phenomena are also given.

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method.For comparison with experiments of Nakabayashi et al.And Wimmer,a narrow gap and a medium gap with clearance ratioβ=0.06 and 0.18 respectively areconsidered for the Reynolds number range covering the first Hopfbifurcation point.With adequate initial conditions and temporaryimposition of small wave-type perturbation,multiple periodicflow states with three different pair numbers of spiralTaylor-Gortler(TG)vortices have been simulated successfullyforβ=0.06,of which the 1-pair and 2-pair of spiral Tgvortices are newly obtained.Three different periodic flow stateswith shear waves,Stuart vortices or wavy outflow boundary,have been obtained forβ=0.18.Analysis of the numerical resultsreveals these higher flow modes in terms of fundamental frequency,wave number and spatial structure.

An analytical solution for spherical fluid flow modeling in porous media with non-uniform initial pressure considering the supercharging effect

The diffusivity equation is a partial differential equation(PDE)which can be used for fluid flow modeling in porous media.Determining reservoir parameters from pressure data(i.e.,pressure transient analysis)is one of the most important steps in the process of field development.This initial evaluation can be used to make decisions about future developments.Wireline Formation Testing(WFT)is one of the most popular techniques for parameter estimation and has received significant attention in recent years.The main problem plaguing WFT is a phenomenon known as the“supercharging effect,”which essentially refers to mud invasion,and this,in turn,alters pressure distribution across the system.In this study,an analytical solution for fluid flow modeling in spherical coordinates with non-uniform initial pressure is presented.This new procedure takes into account the effect of mud invasion,or,in other words,the supercharging effect.The accuracy of this derivation was validated using previous semianalytical solutions(the Laplace method)in addition to field data.New type curves and dimensionless parameters,which can be used for pressure transient analysis,are also proposed.This procedure is applied to the WFT data that was obtained from an oil field in the south of Iran,and an excellent agreement(less than 10%error)was observed.In addition,there is considerable uncertainty regarding the radius of investigation for spherical flow.This is important as this parameter greatly affects the applicability of WFT.The analytical derivation of this study was used to determine a reasonable value for this parameter as well.