scholarly journals Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

2021 ◽  
Vol 2 (3) ◽  
pp. 225-233
Author(s):  
Temitayo Sheriff Adeyemi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Media ◽  
Analytical Solution ◽  
Analytical Approach ◽  
Unsteady State ◽  
Mathematical Relationship ◽  
State Flow ◽  
Infinite Reservoir ◽  
Forchheimer Flow

For several decades, attempts had been made by several authors to develop models suitable for predicting the effects of Forchheimer flow on pressure transient in porous media. However, due to the complexity of the problem, they employed numerical and/or semi-analytical approach, which greatly affected the accuracy and range of applicability of their results. Therefore, in order to increase accuracy and range of applicability, a purely analytical approach to solving this problem is introduced and applied. Therefore, the objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transient in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent condition is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithms approximation (for larger times) of the analytical solution is derived. Moreover, after a rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow. It was observed that this mathematical relationship bears some similarities with that of unsteady-state flow under laminar conditions. Their logarithm approximations also share some similarities. In addition, the results obtained show the efficiency and accuracy of the Boltzmann Transform approach in solving this kind of complex problem. Doi: 10.28991/HEF-2021-02-03-04 Full Text: PDF

Flow of Non-Newtonian Power-Law Fluids Through Porous Media

1979 ◽  
Vol 19 (03) ◽  
pp. 155-163 ◽  
Author(s):  
A.S. Odeh ◽  
H.T. Yang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Porous Media ◽  
Test Data ◽  
Power Law ◽  
Newtonian Fluids ◽  
Unsteady State ◽  
Partial Differential ◽  
State Flow ◽  
Power Law Fluids

Abstract The partial differential equation that describes the flow, of non-Newtonian, power-law, slightly compressible fluids in porous media is derived. An approximate solution, in closed form, is developed for the unsteady-state flow behavior and verified by. two different methods. Using the unsteady-state solution, a method for analyzing injection test data is formulated and used to analyze four injection tests. Theoretical results were used to derive steady-state equations of flow, equivalent transient drainage radius, and a method for analyzing isochronal test data. The theoretical fundamentals of the flow, of non-Newtonian power-law fluids in porous media are established. Introduction Non-Newtonian power-law fluids are those that obey the relation = constant. Here, is the viscosity, e is the shear rate at which the viscosity is measured, and n is a constant. Examples of such fluids are polymers. This paper establishes the theoretical foundation of the flow of such fluids in porous media. The partial differential equation describing this flow is derived and solved for unsteady-state flow. In addition, a method for interpreting isochronal tests and an equation for calculating the equivalent transient drainage radius are presented. The unsteady-state flow solution provides a method for interpreting flow tests (such as injection tests).Non-Newtonian power-law fluids are injected into the porous media for mobility control, necessitating a basic porous media for mobility control, necessitating a basic understanding of the flow behavior of such fluids in porous media. Several authors have studied the porous media. Several authors have studied the rheological properties of these fluids using linear flow experiments and standard viscometers. Van Poollen and Jargon presented a theoretical study of these fluids. They described the flow by the partial differential equation used for Newtonian fluids and accounted for the effect of shear rate on viscosity by varying the viscosity as a function of space. They solved the equation numerically using finite difference. The numerical results showed that the pressure behavior vs time differed from that for Newtonian fluids. However, no methods for analyzing flow-test data (such as injection tests) were offered. This probably was because of the lack of analytic solution normally required to understand the relationship among the variables.Recently, injectivity tests were conducted using a polysaccharide polymer (biopolymer). The data showed polysaccharide polymer (biopolymer). The data showed anomalies when analyzed using methods derived for Newtonian fluids. Some of these anomalies appeared to be fractures. However, when the methods of analysis developed here were applied, the anomalies disappeared. Field data for four injectivity tests are reported and used to illustrate our analysis methods. Theoretical Consideration General Consideration The partial differential equation describing the flow of a non-Newtonian, slightly compressible power-law fluid in porous media derived in Appendix A is ..........(1) where the symbols are defined in the nomenclature. JPT P. 155

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

SOLVING THE ASIAN OPTION PDE USING LIE SYMMETRY METHODS

2010 ◽  
Vol 13 (08) ◽  
pp. 1265-1277 ◽  
Author(s):  
NICOLETTE C. CAISTER ◽  
JOHN G. O'HARA ◽  
KESHLAN S. GOVINDER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Stock Price ◽  
Ad Hoc ◽  
Lie Symmetry ◽  
Asian Option ◽  
Asian Options ◽  
Lie Point Symmetries ◽  
Reduced Order

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

scholarly journals Geometric shape features extraction using a steady state partial differential equation system

2019 ◽  
Vol 6 (4) ◽  
pp. 647-656 ◽  
Author(s):  
Takayuki Yamada
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Analytical Solution ◽  
Geometric Shape ◽  
Geometrical Shape ◽  
Shape Features ◽  
Partial Differential ◽  
Topological Constraint ◽  
Numerical Examples

Abstract A unified method for extracting geometric shape features from binary image data using a steady-state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantage of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method. Highlights A steady state partial differential equation for extraction of geometrical shape features is formulated. The functions for geometrical shape features are formulated by the solution of the proposed PDE. Analytical solution is provided in one-dimension. Numerical examples are provided in two-dimension.

scholarly journals Comparison of Analytical Solution of DGLAP Equations forF2NS(x,t)at Smallxby Two Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Neelakshi N. K. Borah ◽  
D. K. Choudhury ◽  
P. K. Sahariah
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Structure Function ◽  
Method Of Characteristics ◽  
Dglap Equation ◽  
Partial Differential ◽  
Chi Square ◽  
Data Set ◽  
The Method Of Characteristics

The DGLAP equation for the nonsinglet structure functionF2NS(x,t)at LO is solved analytically at lowxby converting it into a partial differential equation in two variables: Bjorkenxandt  (t=ln(Q2/Λ2)and then solved by two methods: Lagrange’s auxiliary method and the method of characteristics. The two solutions are then compared with the available data on the structure function. The relative merits of the two solutions are discussed calculating the chi-square with the used data set.

An evaluation of a dynamic model of the secondary clarifier

1996 ◽  
Vol 34 (5-6) ◽  
pp. 19-26 ◽  
Author(s):  
Ulf Jeppsson ◽  
Stefan Diehl
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Mathematical Modelling ◽  
Dynamic Behaviour ◽  
Linear Partial Differential Equation ◽  
Layer Model ◽  
Important Conclusion ◽  
Secondary Clarifier ◽  
Theoretical Results

The main objective of the paper is to support and illustrate recent theoretical results on the mathematical modelling of the secondary clarifier. A new settler model is compared with a traditional layer model by means of numerical simulations. Emphasis is put on the numerical solution's ability to approximate the analytical solution of the conservation law written as a non-linear partial differential equation. The new settler model is consistent in this respect. Another important conclusion is that a layer model dividing the settler into only ten layers (normally used in settler models) is too crude an approximation to capture the detailed dynamic behaviour of the settler. All simulations presented are performed with the settler models coupled to the IAWQ Activated Sludge Model No. 1.

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