An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

Some Approximate Solutions of Radial Flow Problems Associated With Production at Constant Well Pressure

1967 ◽  
Vol 7 (01) ◽  
pp. 31-42 ◽  
Author(s):  
M.W. Clegg
Keyword(s):  
Porous Media ◽  
Infinite Series ◽  
Analytical Solutions ◽  
Laplace Transformation ◽  
Radial Flow ◽  
Approximate Solutions ◽  
Time Interval ◽  
Pressure Buildup ◽  
Flow Problems ◽  
The Times

Abstract The application of the Laplace transformation to problems in the flow of compressible fluids in porous media has provided a large number of exact solutions. For plane radial flow, however, these solutions are either complex integrals or infinite series and are of little value to the field engineer. !n the case of production at constant well pressure, the available approximate solutions are valid for large times only. In this paper it is shown that an approximate inversion formula for the Laplace transform, developed for the solution of viscoelastic problems, is applicable to radial flow problems and provides simple analytical solutions to constant terminal pressure problems. The method may be used to obtain approximate solutions to many problems, including media with radial permeability discontinuities, multi-layer formations and pressure buildup in wells after shut-in. The results are compared with the few available computer solutions as well as the large time solutions, and it is shown that this approximate method greatly extends the time interval over which a simple analytical solution is acceptable. INTRODUCTION The study of transient problems in the flow of fluids through porous media has benefited greatly from the application of transform methods. The use of the Laplace transformation for solving parabolic equations has been widely discussed in the field of heat conduction and diffusion as well as in the petroleum literature. Removal of the time variable with the Laplace transformation generally reduces the problem to a boundary value problem which may be solved by standard techniques. A much more formidable problem then faces the engineer, however, for frequently the transform does not possess a simple inverse. The result is that the general inversion integral must be used and this leads to either an infinite integral or an infinite series, both of which are difficult to handle from a computational standpoint. Asymptotic approximations for the inverse have been known for some time and these yield approximate inverse functions that are valid for very large or very small times - but frequently the times of interest lie somewhere between these two extremes. Therefore, some acceptable approximation valid over a larger interval of time is desirable. During the past few years a number of methods for achieving this have been developed and some of these are discussed briefly in this paper. The relative merits of the various methods are not evaluated here, but some general conclusions reached by other authors are given. One of these methods has been applied to problems associated with the radial flow of compressible liquids to producing wells. In the case of production at constant well pressure, the method leads to simple analytical solutions for a number of standard problems; e.g., homogeneous formation, permeability discontinuities, pressure buildup. These solutions greatly extend the range of validity of the asymptotic ones (valid for large times only) and should be of value in studying the behavior of wells producing under constant pressure conditions.

Magnetoencephalography inverse problem in the spheroid geometry

2019 ◽  
Vol 27 (2) ◽  
pp. 159-169 ◽  
Author(s):  
Petr I. Karpov ◽  
Tatyana Zakharova
Keyword(s):  
Inverse Problem ◽  
Exact Solution ◽  
Magnetic Fields ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Approximate Analytical Solutions ◽  
Wide Range ◽  
Ill Posed ◽  
The Brain

AbstractThe inverse problem of magnetoencephalography is ill-posed and difficult for both analytical and numerical solutions. Additional complications arise from the volume (passive) currents and the associated magnetic fields, which strongly depend on the brain geometry. In this paper, we find approximate analytical solutions for the forward and the inverse problems in the spheroid geometry. We compare the obtained results with the exact solution of the forward problem and deduce that for a wide range of parameters our approximation is valid. The analysis sheds new light on the role of the volume magnetic fields for solving the inverse problem of magnetoencephalography.

scholarly journals Perturbation Solutions For Magnetohydrodynamics (Mhd) Flow of in a Non-Newtonian Fluid Between Concentric Cylinders

2019 ◽  
Vol 24 (1) ◽  
pp. 199-211
Author(s):  
M. Yürüsoy ◽  
Ö.F. Güler
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Variable Viscosity ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Mhd Flow ◽  
Model Space ◽  
Viscosity Model ◽  
Concentric Cylinders ◽  
Approximate Analytical Solutions

Abstract The steady-state magnetohydrodynamics (MHD) flow of a third-grade fluid with a variable viscosity parameter between concentric cylinders (annular pipe) with heat transfer is examined. The temperature of annular pipes is assumed to be higher than the temperature of the fluid. Three types of viscosity models were used, i.e., the constant viscosity model, space dependent viscosity model and the Reynolds viscosity model which is dependent on temperature in an exponential manner. Approximate analytical solutions are presented by using the perturbation technique. The variation of velocity and temperature profile in the fluid is analytically calculated. In addition, equations of motion are solved numerically. The numerical solutions obtained are compared with analytical solutions. Thus, the validity intervals of the analytical solutions are determined.

A Mathematical Model Development for Simulating Nitrate Pollutant Transport along a River

2021 ◽  
Vol 57 ◽  
pp. 149-168
Author(s):  
Kayode O. Olowe ◽  
Muthukrishnavellaisamy Kumarasamy
Keyword(s):  
Mathematical Model ◽  
Water Quality ◽  
Surface Water ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Nitrate Concentration ◽  
Pollutant Transport ◽  
Spatial And Temporal Variation ◽  
Wide Range ◽  
Natural Rivers

Contamination of surface water bodies by a wide range of organic and inorganic pollutants has been a serious problem in the recent time, these have an effect on human and aquatic animals. The water quality deterioration calls for regular monitoring of the water quality in order to maintain the health and sustainability of the aquatic ecosystems. Accurate monitoring of discharged pollutants into the rivers may be time taking and labour intensive. Water quality models are significant tools for simulating water quality and controlling the surface water pollution. The purpose of this study is to develop a simplified mathematical model which is hybrid cells in series model (HCIS) to simulate the spatial and temporal variation of nitrate concentration in natural rivers. The HCIS model was formulated to serve as an alternative method to the Fickian based models. Analytical solutions for the first order reaction kinetics of nitrate with the advection and dispersion process were derived using Laplace transformation technique. The model considered the effect of nitrate concentration at several points along the river downstream by considering the transformation of nitrite to nitrate through nitrification process. In addition, the uptake of nitrate by algae for its growth and conversion of nitrate to nitrogen gas due to denitrification process were considered. The HCIS-NO3 model was applied to uMgeni River, South Africa to investigate the nitrate concentration along the river. Furthermore, the quantitative measures based on the coefficient of determination (R2) and standard errors (SE) were used to evaluate the performance of the model. The result shows that the simulated values agreed with the measured values of nitrate concentration in the river which resulted in a R2 value of 0.72 and a low standard error. Analytical solutions of HCIS - NO3 model were compared with the numerical solutions of the Fickian based ADE model for hypothetical problems. Comparison of the responses indicates that the HCIS - NO3 and ADE- NO3 models were in good agreement. The study shows that the hybrid model is a simple and effective tool for simulating pollutant transport in natural rivers.

Aspects of de Wit's Equations Governing the Apparent Spontaneous Yaw of a Ship

1989 ◽  
Vol 42 (1) ◽  
pp. 144-150
Author(s):  
J. O. Flower
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linearized Equations ◽  
First Order ◽  
Approximate Analytical Solutions

de Wit has produced an analysis of the apparent spontaneous yaw of a ship when undergoing combined rolling and pitching. This analysis produces a set of four first-order simultaneous differential equations which govern the motion. In de Wit the numerical solutions of these equations for a couple of representative examples are given, as well as the corresponding analytical solutions to the linearized equations. In this communication it is shown how two of the four equations can be solved analytically; these solutions can be used to obtain approximate analytical solutions to the remaining two equations.

A Simple Numerical Simulation Method for Lorenz System Families

2013 ◽  
Vol 444-445 ◽  
pp. 786-790
Author(s):  
Cheng Li Zhang ◽  
Yun Zeng
Keyword(s):  
Analytical Solutions ◽  
Lorenz System ◽  
Numerical Solutions ◽  
Simulation Method ◽  
Chen System ◽  
Similar System ◽  
Recursion Method ◽  
Small Segment ◽  
Approximate Analytical Solutions ◽  
Recursion Formulas

Lorenz system families contain Lorenz system, Chen system and Lu system, their accurate analytical solutions are not yet obtained now. The segmenting recursion method was put forward in this paper, the equations of Lorenz system families were reasonably linearized within small segment, the recursion formulas were obtained by solving the approximate analytical solutions within small segment, and all numerical solutions were got by the recursion formulas. The chaotic motion of Lorenz system families were numerically simulated by means of the segmenting recursion method, the simulation results were compared with Runge-Kutta method. The comparative results show that the segmenting recursion method is very effective to numerically simulate Lorenz system families, not only method is simple, programming is easy, but result is accurate. this method is a universal new method to numerically simulate similar system.

scholarly journals The free compressible viscous vortex

1991 ◽  
Vol 230 ◽  
pp. 45-73 ◽  
Author(s):  
Tim Colonius ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Mach Number ◽  
Radial Velocity ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Axisymmetric Flow ◽  
Tangential Velocity ◽  
Uniform Density ◽  
Heat Conducting

The effects of compressibility on free (unsteady) viscous heat-conducting vortices are investigated. Analytical solutions are found in the limit of large, but finite, Reynolds number, and small, but finite, Mach number. The analysis shows that the spreading of the vortex causes a radial flow. This flow is given by the solution of an ordinary differential equation (valid for any Mach number), which gives the dependence of the radial velocity on the tangential velocity, density, and temperature profiles of the vortex; estimates of the radial velocity found by solving this equation are found to be in good agreement with numerical solutions of the full equations. The experiments of Mandella (1987) also report a radial flow in the vortex, but their estimates are much larger than the analytical predictions, and it is found that the flow inferred from the iexperiments violates the Second Law of Thermodynamics for two-dimensional axisymmetric flow. It is speculated that three-dimensionality is the cause of this discrepancy. To obtain detailed analytical solutions, the equations for the viscous evolution are expanded in powers of Mach number, M. Solutions valid to O(M2), are discussed for vortices with finite circulation. Two specific initial conditions – vortices with initially uniform entropy and with initially uniform density – are analysed in detail. It is shown that swirling axisymmetric compressible flows generate negative radial velocities far from the vortex core owing to viscous effects, regardless of the initial distributions of vorticity, density and entropy.

Progress in Analytical Modeling of Water Hammer

2021 ◽  
Author(s):  
Kamil Urbanowicz ◽  
Haixiao Jing ◽  
Anton Bergant ◽  
Michał Stosiak ◽  
Marek Lubecki
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Dynamic Pressure ◽  
Complete Solution ◽  
Water Hammer ◽  
Inverse Transformation ◽  
Mathematical Form ◽  
Dimensionless Time ◽  
Wide Range ◽  
Analytical Formulas

Abstract In this paper analytical formulas of water hammer known from the literature are simplified to the shortest possible mathematical form based on dimensionless parameters: dimensionless time, water hammer number, etc. Novel formulas are determined, for example for the flow velocity and wall shear stress in the Muto and Takahashi solution. A complete solution in the Laplace domain is presented and the problem of its inverse transformation is discussed. A series of comparative studies of analytical solutions with numerical solutions and the results of experimental research were carried out. The compared analytical solutions, taking into account the frequency-dependent nature of the hydraulic resistances, show very good agreement with the experimental results in a wide range of water hammer numbers, in particular when Wh ≤ 0.1. On the other hand, it turned out that the analytical model based on the quasi-steady friction in great detail simulates dynamic pressure response in systems characterized by a high value of the water hammer number Wh ≥ 0.5.

Derivation of an approximate analytical solution for understanding the response characteristics of the EBS

2012 ◽  
Vol 1475 ◽  
Author(s):  
T. Ohi ◽  
T. Chiba ◽  
T. Nakagawa ◽  
T. Takase ◽  
T. Nakazawa ◽  
...  
Keyword(s):  
Analytical Solutions ◽  
Safety Assessment ◽  
Numerical Solutions ◽  
Geological Disposal ◽  
One Dimensional ◽  
Response Characteristics ◽  
Excavation Damaged Zone ◽  
Approximate Analytical Solutions ◽  
Target Values ◽  
Damaged Zone

ABSTRACTTo perform a safety assessment for the geological disposal of radioactive waste, it is important to understand the response characteristics of the disposal system. In this study, approximate analytical solutions for steady-state nuclide releases from the engineered barrier system (EBS) of a repository were derived for an orthogonal one-dimensional diffusion model. In these approximate analytical solutions, inventory depletion, decay during migration and the influence of groundwater flow in the excavation damaged zone (EDZ) were considered. These solutions were simplified by the Taylor theorem in order to clearly represent the response characteristics of the EBS. The validity of these solutions was shown by comparison with numerical solutions. The response characteristics of the EBS are useful for identifying target values for important parameters that would have the effect of improving the robustness of system safety. The robustness of the geological disposal system and the reliability of the safety assessment can thus potentially be improved using the approximate analytical solutions.

scholarly journals Analytical solutions for dense, inclined, granular flow over a rigid, bumpy base

2021 ◽  
Vol 249 ◽  
pp. 03039
Author(s):  
James Jenkins ◽  
Diego Berzi
Keyword(s):  
Analytical Solutions ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Mean Velocity ◽  
Velocity Fluctuations ◽  
Gravitational Flow ◽  
Approximate Analytical Solutions ◽  
The Mean

We first phrase a boundary-value problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over in inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow. We compare these with those obtained in numerical solutions of the exact equations.

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