Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.

Linear Complementary Method for Moving Boundary Problems with Phase Transformation

In this paper, the linear complementary method for moving boundary problems with phase transformation is presented, in which a pair of unknown vectors of heat source with phase transforming and the temperature field can be solved exactly, and a large amount of iterative calculations can be avoided.

MOVING BOUNDARY PROBLEM FOR DIFFUSION RELEASE OF DRUG FROM A CYLINDER POLYMERIC MATRIX

An approximate analytical solution of moving boundary problem for diffusion release of drug from a cylinder polymeric matrix was obtained by use of refined integral method. The release kinetics has been analyzed for non-erodible matrices with perfect sink condition. The formulas of the moving boundary and the fractional drug release were given. The moving boundary and the fractional drug release have been calculated at various drug loading levels, mid the calculated results were in good agreement with those of experiments. The comparison of the moving boundary in spherical, cylinder, planar matrices has been completed. An approximate formula for estimating the available release time was presented. These results are useful for the clinic experiments. This investigation provides a new theoretical tool for studying the diffusion release of drug from a cylinder polymeric matrix and designing the controlled released drug.

Closed-form Solution for a Moving Boundary Problem

This paper considers a moving boundary problem with Neumann boundary conditions and a general initial value, which occurs in an unsaturated flow with extraction. The closed form solution for the moving boundary problem is obtained using a Laplace transform boost. This method has been successfully applied to solve moving boundary problems with Dirichelet boundary conditions, but not to the case with Neumann boundary conditions.

Solving Two-Dimensional Moving-Boundary Problems with Meshless and Level Set Method

During the past decade, increasing attention has been given to the development of meshless methods using radial basis functions for the numerical solution of Partial Differential Equations （PDEs）. A level set method is a promising design tool for tracking, modelling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. In the conventional level set methods, the level set equation is solved to evolve the interface using a capturing Eulerian approach. The solving procedure requires an appropriate choice of the upwind schemes, reinitialization, etc. Our goal is to include Multiquadric Radial Basis Functions （MQ RBFs） into the level set method to construct a more efficient approach and stabilize the solution process with the adaptive greedy algorithm. This paper presents an alternative approach to the conventional level set methods for solving moving-boundary problems. The solution was compared to the solution calculated by the exact explicit lime integration scheme. The examples show that MQ RBFs and adaptive greedy algorithm is a very promising calculation scheme.

One-dimensional non-Darcy flow in a semi-infinite porous media:a multiphase implicit Stefan problem with phases divided by hydraulic gradients

One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.

Computational Techniques of 2D Tidal Flow in Estuaries and Bays

Some problems in the numerical calculation of tidal flow in estuaries and bays, such as distribution bed roughness, relationship between model moving boundary and water volume balance, remedy of certain shortage under open boundary conditions, smooth transfer of the controlling message for the inner boundaries of nested model, can not be solved usually by means of the fundamental equations and computing mode adopted in the numerical model, but can be done by the measures which not only satisfy the physical features but also are convenient for operation. Based on the calculated samples of some areas of Modaomen, Lingdingyang and Huangmaohai of the Pearl River Estuary, and Shuidong Bay (a typical barrier- lagoon tidal channel) in Guangdong Province, method and process of calculation for the above mentioned problems are briefly presented in this paper.

Moving computational multi-domain method for modelling the flow interaction of multiple moving objects

This study proposes a method for modelling the flow interaction of multiple moving objects where the flow field variables are communicated between multiple separate moving computational domains.Instead of using the conventional approach with a single fixed computational domain covering the whole flow field,this method advances the moving computational domain(MCD)method in which the computational domain itself moves in line with the motions of an object inside.The computational domains created around each object move independently,and the flow fields of each domain interact where the flows cross.This eliminates the spatial restriction for simulating multiple moving objects.Firstly,a shock tube test verifies that the overset implementation and grid movement do not adversely affect the results and that there is communication between the grids.A second test case is conducted in which two spheres are crossed,and the forces exerted on one object due to the other’s crossing at a short distance are calculated.The results verify the reliability of this method and show that it is applicable to the flow interaction of multiple moving objects.