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Richard B Sowers - One of the best experts on this subject based on the ideXlab platform.
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numerical analysis of the stochastic Moving Boundary Problem
Stochastic Analysis and Applications, 2012Co-Authors: Kunwoo Kim, Richard B SowersAbstract:We consider a numerical solution of the stochastic Moving Boundary value Problem, whose existence and uniqueness of solution are proved in [16]. Numerical approximations are based on the transformation, which transforms the stochastic Moving Boundary Problem whose spatial domain is a priori unknown to a nonlinear stochastic partial differential equation which has a fixed spatial domain. We construct a numerical solution of the nonlinear stochastic partial differential equation and investigate the convergence theory.
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a stochastic Moving Boundary value Problem
Illinois Journal of Mathematics, 2010Co-Authors: Kunwoo Kim, Carl Mueller, Richard B SowersAbstract:We consider a stochastic perturbation of a Moving Boundary Problem proposed by Ludford and Stewart and studied by Caffarelli and Vazquez. We prove existence and uniqueness.
Wenchao Liu - One of the best experts on this subject based on the ideXlab platform.
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analytical study on a Moving Boundary Problem of semispherical centripetal seepage flow of bingham fluid with threshold pressure gradient
International Journal of Non-linear Mechanics, 2019Co-Authors: Wenchao LiuAbstract:Abstract It is well known that the Non-Newtonian Bingham fluid flow in porous media does not obey the conventional linear Darcy’s law due to the yield stress for the Bingham fluid: There exists a threshold pressure gradient, which means that the seepage flow only happens when the threshold pressure gradient is overcome. The principle of non-Darcy seepage flow with the threshold pressure gradient is also applicable into the situation of the fluid flow in the low-permeable porous media. Here, a nonlinear Moving-Boundary mathematical model is built for the semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient in a three-dimensional infinite heavy oil reservoir with the type of Bingham fluid; wherein the Moving Boundary conditions are incorporated for describing the effect of the threshold pressure gradient. In consideration of the strong nonlinearity of the model, the similarity transformation method is applied into obtaining the exact analytical solution of the model. In order to keep full self-similarity for the model, the inner Boundary condition is set as variable flow rate that increases linearly with the time. As a result, an exact analytical solution for the nonlinear Moving-Boundary mathematical model of semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient is obtained. The existence and the uniqueness of the exact analytical solution are also strictly proved. It is also theoretically proved that as the threshold pressure gradient tends to zero, the exact analytical solution can be reduced to that of a mathematical model of semispherical centripetal Darcy’s seepage flow. The presented exact analytical solution can be used for strictly verifying the validity of the numerical methods for solving the three-dimensional Moving Boundary models of non-Darcy seepage flow with the threshold pressure gradient in the actual engineering Problems. From the exact analytical solution, it is also revealed that when the threshold pressure gradient exists, the spatial pressure distribution exhibits an instructive feature of compact support; as the threshold pressure gradient tends to zero, the sensitivity of its effect on the transient distance of the Moving Boundary and the transient pressure will grow, which reveals the difficulty in accurately determining the position of the Moving Boundary by the numerical methods and the serious uncertainty Problem in the interpretation of the threshold pressure gradient by the pressure transient analysis method in engineering as the threshold pressure gradient is rather small. Through the comparison of the two different exact analytical solutions that corresponds to the two different models with and without incorporating the Moving Boundary conditions for describing the effect of the threshold pressure gradient, it is demonstrated that when the Moving Boundary conditions are not incorporated in the modeling, the effect of the threshold pressure gradient on the spatial pressure distribution, the transient pressure and the productivity index can be overestimated largely. Therefore, it is very necessary to incorporate the Moving Boundary conditions in the modeling of non-Darcy seepage flow with the threshold pressure gradient. The study in the paper definitely provides solid theoretical basis of fluid mechanics for the relevant engineering applications in the development of heavy oil reservoirs and low-permeable reservoirs in petroleum engineering and in the development of water resources in low-permeable formations in hydraulic engineering.
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an exact analytical solution of Moving Boundary Problem of radial fluid flow in an infinite low permeability reservoir with threshold pressure gradient
Journal of Petroleum Science and Engineering, 2019Co-Authors: Wenchao Liu, Jun Yao, Zhangxin Chen, Weiyao ZhuAbstract:Abstract Many engineering technologies involve the Moving Boundary Problems for the radial seepage flow with a threshold pressure gradient, such as the well testing in the development of low-permeability reservoirs, heavy oil reservoirs and groundwater resources. However, as a result of the strong nonlinearity, an exact analytical solution of the Moving Boundary Problems for the radial seepage flow with a threshold pressure gradient has not been obtained yet. Here, a dimensionless Moving Boundary mathematical model for the radial fluid flow in an infinite low-permeability reservoir with a threshold pressure gradient is developed first. The setting of a variable well production rate for an inner Boundary condition can make the mathematical model exhibit a full self-similarity property. Second, by introducing some similarity transformations, the nonlinear system of PDEs of the model can be equivalently transformed into a closed pseudo-linear system of ODEs, whose exact analytical solution can be easily obtained. What's more, the existence and the uniqueness of the exact analytical solution to the Moving Boundary model are also proved strictly through the mathematical analysis. Third, it is also strictly proved that the exact analytical solution can be degenerated to that of the model of the Darcy's radial fluid flow as the threshold pressure gradient approaches to zero. Finally, by a comparison of model analytical solutions, it is demonstrated that the Moving Boundary conditions must be incorporated in the modeling of the radial seepage flow with a threshold pressure gradient; otherwise, the effect of the threshold pressure gradient on the radial seepage flow can be overestimated largely.
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numerical investigations of the effect of nonlinear quadratic pressure gradient term on a Moving Boundary Problem of radial flow in low permeable reservoirs with threshold pressure gradient
Mathematical Problems in Engineering, 2015Co-Authors: Wenchao Liu, Jun YaoAbstract:The existence of a TPG can generate a relatively high pressure gradient in the process of fluid flow in porous media in low-permeable reservoirs, and neglecting the QPGTs in the governing equations, by assuming a small pressure gradient for such a Problem, can cause a significant error in predicting the formation pressure. Based on these concerns, in consideration of the QPGT, a Moving Boundary model of radial flow in low-permeable reservoirs with the TPG for the case of a constant flow rate at the inner Boundary is constructed. Due to strong nonlinearity of the mathematical model, a numerical method is presented: the system of partial differential equations for the Moving Boundary Problem is first transformed equivalently into a closed system of partial differential equations with fixed Boundary conditions by a spatial coordinate transformation method; and then a stable, fully implicit finite difference method is used to obtain its numerical solution. Numerical result analysis shows that the mathematical models of radial flow in low-permeable reservoirs with TPG must take the QPGT into account in their governing equations, which is more important than those of Darcy’s flow; the sensitive effects of the QPGT for the radial flow model do not change with an increase of the dimensionless TPG.
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analytical solution of a double Moving Boundary Problem for nonlinear flows in one dimensional semi infinite long porous media with low permeability
Acta Mechanica Sinica, 2014Co-Authors: Wenchao Liu, Jun Yao, Zhangxin ChenAbstract:Based on Huang’s accurate tri-sectional nonlinear 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 Moving 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 transformation, the nonlinear partial differential equation (PDE) system is transformed into a linear PDE system. Then an analytical 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 analytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensitive effects of the dimensionless variable on the dimensionless pressure distribution and dimensionless pressure gradient distribution becomemore serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensitive 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.
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Numerical Solution of a Moving Boundary Problem of One-Dimensional Flow in Semi-Infinite Long Porous Media with Threshold Pressure Gradient
Mathematical Problems in Engineering, 2013Co-Authors: Jun Yao, Wenchao Liu, Zhangxin ChenAbstract:A numerical method is presented for the solution of a Moving Boundary Problem of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient (TPG) for the case of a constant flow rate at the inner Boundary. In order to overcome the difficulty in the space discretization of the transient flow region with a Moving Boundary in the process of numerical solution, the system of partial differential equations for the Moving Boundary Problem is first transformed equivalently into a closed system of partial differential equations with fixed Boundary conditions by a spatial coordinate transformation method. Then a stable, fully implicit finite difference method is adopted to obtain its numerical solution. Finally, numerical results of transient distance of the Moving Boundary, transient production pressure of wellbore, and formation pressure distribution are compared graphically with those from a published exact analytical solution under different values of dimensionless TPG as calculated from actual experimental data. Comparison analysis shows that numerical solutions are in good agreement with the exact analytical solutions, and there is a big difference of model solutions between Darcy's flow and the fluid flow in porous media with TPG, especially for the case of a large dimensionless TPG.
Ajay Kumar - One of the best experts on this subject based on the ideXlab platform.
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a Moving Boundary Problem with variable specific heat and thermal conductivity
Journal of King Saud University - Science, 2020Co-Authors: Ajay Kumar, Abhishek K SinghAbstract:Abstract This article presents a Stefan Problem including thermal conductivity and heat capacity as the functions of temperature. At α = β , the exact solutions to the proposed Problem are discussed for two different specific cases, i.e. m = n = 1 and m = n = 2 . For the general case, estimation of the solution to the Problem is deliberated with the help of shifted Chebyshev tau method. To exhibit the accurateness of the obtained approximate solution, the comparison between exact and approximate solution are depicted through tables which shows that the approximate results are in good agreement with the exact solution. We also present the impact of parameters appeared in the considered Problem on temperature profile and location of Moving interface. It is found that the melting of the material effectively enhances when we increase either the value m or[spsbacksalsh]and n or Stefan number.
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an approximate solution to a Moving Boundary Problem with space time fractional derivative in fluvio deltaic sedimentation process
Ain Shams Engineering Journal, 2013Co-Authors: Mohan Singh Kushwaha, Ajay KumarAbstract:Abstract A mathematical model of the movement of the shoreline in a sedimentary ocean basin is discussed. The model includes space–time fractional derivative in Caputo sense and variable latent heat term. An approximate solution of the Problem is obtained by Adomian decomposition method and the results thus obtained are compared graphically with an exact solution of integer order (β = 1, α = 1). Three particular cases, the standard diffusion, the time-fractional and the space-fractional diffusions are also discussed. The model and solution are generalization of previous works.
Viviana Olga Salvadori - One of the best experts on this subject based on the ideXlab platform.
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a Moving Boundary Problem in a food material undergoing volume change simulation of bread baking
Food Research International, 2010Co-Authors: Emmanuel Purlis, Viviana Olga SalvadoriAbstract:This paper presents a mathematical model for describing processes involving simultaneous heat and mass transfer with phase transition in foods undergoing volume change, i.e. shrinkage and/or expansion. We focused on processes where the phase transition occurs in a Moving front, such as thawing, freezing, drying, frying and baking. The model is based on a Moving Boundary Problem formulation with equivalent thermophysical properties. The transport Problem is solved by using the finite element method and the Arbitrary Lagrangian–Eulerian method is used to describe the motion of the Boundary. The formulation is assessed by simulating the bread baking process and comparing numerical results with experimental data. Simulated temperature and water content profiles are in good agreement with experimental data obtained from bread baking tests. The model well describes the stated general Problem and it is expected to be useful for other food processes involving similar phenomena.
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bread baking as a Moving Boundary Problem part 2 model validation and numerical simulation
Journal of Food Engineering, 2009Co-Authors: Emmanuel Purlis, Viviana Olga SalvadoriAbstract:Abstract A simultaneous heat and mass transfer model proposed to describe the bread baking process is validated. The mathematical model is based on a Moving Boundary Problem formulation with equivalent thermophysical properties and includes the Moving evaporation front, the evaporation–condensation mechanism and the development of the crust observed during bread baking. The Problem is solved over an irregular three-dimensional geometry using the finite element method. Variation in temperature and water content of bread during baking is predicted with high accuracy by the model. Parameter estimation procedure and sensitivity analysis are performed for some thermophysical properties. The proposed formulation and analysis can be applied for other bakery products as well as for similar food engineering applications.
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bread baking as a Moving Boundary Problem part 1 mathematical modelling
Journal of Food Engineering, 2009Co-Authors: Emmanuel Purlis, Viviana Olga SalvadoriAbstract:A mathematical model for the bread baking process is developed in this work. Experimental data (temperature, water content, weight loss, crust thickness) obtained during baking is used to well understand the simultaneous heat and mass transfer occurring during the process. The evaporation-condensation mechanism is responsible for the rapid heating of the porous matrix and takes place either in a closed (dough) or open (crumb) structure. The existence of a Moving evaporation front inside bread, which is a determining step of baking, is incorporated in a model applying a Moving Boundary formulation with equivalent thermophysical properties. The approach proposed here can be extended to other similar processes such as baking of other products (e.g. biscuit, cake), high-temperature drying, cooking and roasting.
Kunwoo Kim - One of the best experts on this subject based on the ideXlab platform.
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numerical analysis of the stochastic Moving Boundary Problem
Stochastic Analysis and Applications, 2012Co-Authors: Kunwoo Kim, Richard B SowersAbstract:We consider a numerical solution of the stochastic Moving Boundary value Problem, whose existence and uniqueness of solution are proved in [16]. Numerical approximations are based on the transformation, which transforms the stochastic Moving Boundary Problem whose spatial domain is a priori unknown to a nonlinear stochastic partial differential equation which has a fixed spatial domain. We construct a numerical solution of the nonlinear stochastic partial differential equation and investigate the convergence theory.
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a stochastic Moving Boundary value Problem
Illinois Journal of Mathematics, 2010Co-Authors: Kunwoo Kim, Carl Mueller, Richard B SowersAbstract:We consider a stochastic perturbation of a Moving Boundary Problem proposed by Ludford and Stewart and studied by Caffarelli and Vazquez. We prove existence and uniqueness.