scholarly journals Mathematical modeling of well operation in the case of two-dimensional filtration in an anisotropic heterogeneous layer

2020 ◽  
pp. 87-95
Author(s):  
Д.Г. Лекомцев ◽  
В.Ф. Пивень
Keyword(s):  
Mathematical Modeling ◽  
Analytical Solution ◽  
Affine Transformation ◽  
Integral Relation ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
System Of Integral Equations ◽  
Well Production ◽  
Rank Tensor ◽  
Heterogeneous Layer

Поставлена плоская (двумерная) задача о математическом моделировании работы скважины в анизотропном неоднородном пласте грунта с раздельной анизотропией и неоднородностью, когда контур питания произвольный. Рассматривается совершенная скважина, когда она полностью вскрывает пласт своей рабочей частью (фильтром). Проницаемость грунта характеризуется тензором второго ранга, компоненты которого моделируются степенной функцией координат. Гомеоморфным аффинным преобразованием координат эта задача приводится к каноническому виду, что значительно упрощает ее исследование. Получено в конечном виде аналитическое решение задачи о дебите скважины с конкретным эллиптическим контуром питания, а также в случае, когда контур питания удален в бесконечность. В случае произвольного гладкого контура питания задача о дебите редуцирована к системе сингулярного интегрального уравнения и интегрального соотношения, которая решена численно методом дискретных особенностей. Исследовано влияние на дебит анизотропии, неоднородности пласта и формы контура питания. A flat (two-dimensional) problem has been posed on the mathematical modeling of well in an anisotropic inhomogeneous reservoir of soil with separate anisotropy and heterogeneity when the power contour is arbitrary. The considered well completely opens the formation with its working part (filter). Such a well is called perfect. The permeability of the soil is characterized by a second-rank tensor whose components are modeled by a power function of the coordinates. With a homeomorphic affine transformation of coordinates, this problem is reduced to a canonical form which greatly simplifies its study. An analytical solution of the problem of well production with an elliptical power contour is obtained in the final form as well as in the case when the power contour is removed to infinity. In the general case, the problem is reduced to a system of integral equations and the integral relation. The results were obtained in the general case using the discrete singularities method. The influence on the flow rate of anisotropy, heterogeneity of the reservoir and the shape of the power contour was studied.

Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor

1987 ◽  
Vol 54 (4) ◽  
pp. 772-777 ◽  
Author(s):  
R. M. Christensen
Keyword(s):  
Elastic Moduli ◽  
Fiber Composite ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Axial Deformation ◽  
Space Network ◽  
Plane Normal ◽  
Rank Tensor ◽  
Three Space

Symmetry conditions are found that assure isotropy of the fourth rank tensor of elastic moduli. Crystallography provides the answer to this problem in the two-dimensional context, namely one axis of three-fold symmetry assures the isotropy of properties in the plane normal to the axis. The present work provides the answer in the three-dimensional problem: 6 axes of five-fold symmetry are sufficient to give isotropy of the elastic moduli. An important restriction must accompany the present result. The derivation is given in the special form appropriate to low density materials which have a microstructure that transmits load according to the axial deformation of a space network of material distributed into micro-struts. The corresponding fiber composite idealization is that of a fiber dominated system, it therefore follows that if the fibers take the 6 specific orientations in three-space then isotropy is obtained.

On The Theory Of Bending Of A Thick Orthotropic Plate Without Consideration Of Simplifying Hypothesis

2011 ◽  
Vol 3 ◽  
Author(s):  
Makhamatali Koraboyevich Usarov
Keyword(s):  
Analytical Solution ◽  
Orthotropic Plate ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Orthotropic Plates

The problem of bending of a thick orthotropic plate is considered as a three-dimensional problem of the theory ofelasticity. On the basic of the method of expansion of thesolution into the series, a three-dimensional problem isreduced to two independent two –dimensional problems.The theory of thick orthotropic plates free from simplifiedhypothesis is developed: An analytical solution of equationis given. Maximum values of displacements and stressesfor upper, middle and lower surfaces of the plate are calculated.

Numerical-Analytical Solution of Two-Dimensional Problem for Elastic Radially Inhomogeneous Thick-Walled Cylinder

2015 ◽  
Vol 752-753 ◽  
pp. 642-647 ◽  
Author(s):  
Vladimir I. Andreev
Keyword(s):  
Cylindrical Shell ◽  
Analytical Solution ◽  
Axisymmetric Problem ◽  
Infinite System ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Uniform Load ◽  
Radially Inhomogeneous ◽  
Walled Cylinder

In the [1,2] was considered method of separating of variables in three-dimensional problem for radially inhomogeneous cylinder. This problem solves in terms of displacements and reduced to infinite system of ordinary differential equations. The paper deals with the partially case – two-dimensional axisymmetric problem of the calculation of thick-walled cylindrical shell loaded by non-uniform load on the outer surface.

scholarly journals Development of the model to determine the fuel temperature field in a two-dimensional problem statement

2019 ◽  
Vol 5 (3) ◽  
pp. 219-224 ◽  
Author(s):  
Vladimir A. Gorbunov ◽  
Natalya B. Ivanova ◽  
Nikita A. Lonshakov ◽  
Yaroslav V. Belov
Keyword(s):  
Temperature Field ◽  
Analytical Solution ◽  
Numerical Models ◽  
Reactor Core ◽  
Comsol Multiphysics ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Fuel Temperature ◽  
Problem Statement ◽  
Fuel Rods

Water-cooled water-moderated reactors (VVER) are widely used at Russian nuclear power plants. The VVER reactor core is formed by fuel assemblies consisting of fuel rods. The fuel in fuel rods is uranium dioxide. The safety of the reactor operation is ensured through stringent requirements for the maximum nuclear fuel temperature. Calculation of temperature fields within the reactor core requires associated problems to be solved to determine the internal energy release in fuel based on neutronic characteristics. Dedicated software for such calculations is not available to a broad range of users. At the present time, there are numerical thermophysical modeling packages available for training or noncommercial applications which are used extensively, including Elcut, Flow Vision, Ansys Fluent, and Comsol Multiphysics. Verification of the obtained results is becoming an important issue in building models using these calculation packages. An analytical solution was obtained as part of the study for the fuel temperature field determination. A program was developed in MathCAD based on this solution. A model was developed in Comsol Multiphysics to determine the fuel temperature field with constant thermophysical properties in a two-dimensional problem statement. The numerical model was verified using the analytical solution. The influence of the number of the grid nodes on the solution accuracy was established. The analytical solution can be used to determine the fuel temperature field at any radial coordinate of the reactor. The temperature field determination model developed in MathCAD can be used to verify numerical models of the fuel temperature field determination developed in dedicated packages.

Time-Dependent Electrokinetic Flow in Infinitely Extended Rectangular Microchannels With Slip

2003 ◽  
Author(s):  
Jun Yang ◽  
Daniel Y. Kwok
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Slip Flow ◽  
Time Dependent ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Electrokinetic Flow ◽  
Rectangular Microchannel ◽  
Microelectromechanical Devices

Rectangular microchannel is the most popular shape to be widely used in Microelectromechanical devices. However, analytical solutions of flow in this shape of microchannels are seldom reported because of its two dimensional problem. Furthermore, microflows in microchannels are more difficult to describe. In this paper, we addressed analytically the problem of laminar electrokinetic slip flow in microchannels with rectangular cross-section subjected to a time-dependent pressure gradient and a time-dependent electric field. The analytical solution has been determined based on the Debye-Hu¨ckel approximation and its solution is useful to precisely control microflow with slip in rectangular cross-section microchannels.

Modifications of the Godunov method for computing disturbance propagation in an elastic body

2016 ◽  
Vol 11 (1) ◽  
pp. 119-126 ◽  
Author(s):  
A.A. Aganin ◽  
N.A. Khismatullina
Keyword(s):  
Elastic Body ◽  
Godunov Method ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
One Dimensional ◽  
Disturbance Propagation ◽  
Linear Waves ◽  
Longitudinal And Shear Waves ◽  
Contact Discontinuities

Numerical investigation of efficiency of UNO- and TVD-modifications of the Godunov method of the second order accuracy for computation of linear waves in an elastic body in comparison with the classical Godunov method is carried out. To this end, one-dimensional cylindrical Riemann problems are considered. It is shown that the both modifications are considerably more accurate in describing radially converging as well as diverging longitudinal and shear waves and contact discontinuities both in one- and two-dimensional problem statements. At that the UNO-modification is more preferable than the TVD-modification because exact implementation of the TVD property in the TVD-modification is reached at the expense of “cutting” solution extrema.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

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