An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for ...An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.展开更多
A Janus particle has two distinct areas on its surface.Denoting the two areas as P(patch)and N(nonpatch),when two particles come close to each other,the strength of the PP interaction is usually different from that of...A Janus particle has two distinct areas on its surface.Denoting the two areas as P(patch)and N(nonpatch),when two particles come close to each other,the strength of the PP interaction is usually different from that of the NN interaction.Recently the interplay between a rotational-symmetry-breaking continuous phase transition and percolation has been explored for an equilibrium system of asymmetrically interacting(i.e.,attractive PP interaction,zero NN and PN interactions)Janus disks on the triangular lattice.By Monte Carlo simulation and finite-size scaling analysis,in this work we study an equilibrium system of symmetrically interacting(i.e.,attractive PP and NN interactions with the same strength,zero PN interaction)Janus disks on the same lattice.By definition,the phase diagram in the T-θplane is symmetric for systems with patch sizesθbelow and above 90°.We determine the phase diagram and compare it with that of the asymmetric system.Similar to the latter system,for 60°<θ<90°,a rotational-symmetry-breaking continuous phase transition and an anisotropic percolation transition are found in the symmetric system,though the transition points in the two systems are quite different.Phase crossover curves are found to be different,e.g.,a continuous varying crossover line extends betweenθ=0°and 90°for the symmetric model;and in the range 0°<θ≤30°,along the crossover lines of the two models,the trends of 1/T vs.θare opposite in the two systems.We understand the latter by analytically solving the models with two particles in 0°<θ≤30°.These results are helpful for understanding close-packed systems of Janus disks with more complex interactions.展开更多
Percolation theory deals with the numbers and properties of the clusters formed in the different occupation probability. In this Paper, we study the calculation method of small clusters. We calcu-lated the small clust...Percolation theory deals with the numbers and properties of the clusters formed in the different occupation probability. In this Paper, we study the calculation method of small clusters. We calcu-lated the small cluster density of 1, 2 and 3 in the percolation model with the exact method and the numerical method. The results of the two methods are very close, which can be verified by each other. We find that the cluster density of all three kinds of small clusters reaches the highest value when the occupation probability is between 0.1 and 0.2. It is very difficult to get the analytical formula for the exact method when the cluster area is relatively large (such as the area is more than 50), so we can get the density value of the cluster by numerical method. We find that the time required calculating the cluster density is proportional to the percolation area, which is indepen-dent of the cluster size and the occupation probability.展开更多
The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathema...The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.展开更多
Making use of modern nonlinear physics theory and earthquake focus theory, combined with seismicity characteristics, the percolation model of earthquake activity is given in this paper. We take the seismogenic process...Making use of modern nonlinear physics theory and earthquake focus theory, combined with seismicity characteristics, the percolation model of earthquake activity is given in this paper. We take the seismogenic process of alarge earthquake as a phase transition process of percolation and apply the renormalization method to phase transition of percolation. The critical property of the system, which is like percolation probability exponential andcorrelative length exponential, etc, can be calculated under the fixed point as which in the renormalization transformation infinite correlative length in percolation phase transition is taken. The percolation phase transition process of two large earthquakes, which are Haicheng and Tangshan event occurred in 1975 and 1976 respectively, hasbeen discussed by means of seismicity data before and after two shocks.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11171193 and11371229)the Natural Science Foundation of Shandong Province(No.ZR2014AM033)the Science and Technology Development Project of Shandong Province(No.2012GGB01198)
文摘An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.
基金supported by the National Natural Science Foundation of China(Grant No.11905001)。
文摘A Janus particle has two distinct areas on its surface.Denoting the two areas as P(patch)and N(nonpatch),when two particles come close to each other,the strength of the PP interaction is usually different from that of the NN interaction.Recently the interplay between a rotational-symmetry-breaking continuous phase transition and percolation has been explored for an equilibrium system of asymmetrically interacting(i.e.,attractive PP interaction,zero NN and PN interactions)Janus disks on the triangular lattice.By Monte Carlo simulation and finite-size scaling analysis,in this work we study an equilibrium system of symmetrically interacting(i.e.,attractive PP and NN interactions with the same strength,zero PN interaction)Janus disks on the same lattice.By definition,the phase diagram in the T-θplane is symmetric for systems with patch sizesθbelow and above 90°.We determine the phase diagram and compare it with that of the asymmetric system.Similar to the latter system,for 60°<θ<90°,a rotational-symmetry-breaking continuous phase transition and an anisotropic percolation transition are found in the symmetric system,though the transition points in the two systems are quite different.Phase crossover curves are found to be different,e.g.,a continuous varying crossover line extends betweenθ=0°and 90°for the symmetric model;and in the range 0°<θ≤30°,along the crossover lines of the two models,the trends of 1/T vs.θare opposite in the two systems.We understand the latter by analytically solving the models with two particles in 0°<θ≤30°.These results are helpful for understanding close-packed systems of Janus disks with more complex interactions.
文摘Percolation theory deals with the numbers and properties of the clusters formed in the different occupation probability. In this Paper, we study the calculation method of small clusters. We calcu-lated the small cluster density of 1, 2 and 3 in the percolation model with the exact method and the numerical method. The results of the two methods are very close, which can be verified by each other. We find that the cluster density of all three kinds of small clusters reaches the highest value when the occupation probability is between 0.1 and 0.2. It is very difficult to get the analytical formula for the exact method when the cluster area is relatively large (such as the area is more than 50), so we can get the density value of the cluster by numerical method. We find that the time required calculating the cluster density is proportional to the percolation area, which is indepen-dent of the cluster size and the occupation probability.
基金Project supported by the National Program on Key Basic Research Project(973 Program)(No.2013CB228002)
文摘The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.
文摘Making use of modern nonlinear physics theory and earthquake focus theory, combined with seismicity characteristics, the percolation model of earthquake activity is given in this paper. We take the seismogenic process of alarge earthquake as a phase transition process of percolation and apply the renormalization method to phase transition of percolation. The critical property of the system, which is like percolation probability exponential andcorrelative length exponential, etc, can be calculated under the fixed point as which in the renormalization transformation infinite correlative length in percolation phase transition is taken. The percolation phase transition process of two large earthquakes, which are Haicheng and Tangshan event occurred in 1975 and 1976 respectively, hasbeen discussed by means of seismicity data before and after two shocks.
