We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postula...We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we can tell) be expressed as examples of the O’Shaughnessy and Procaccia equations. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. We illustrate the ideas both analytically and with a detailed computational example for a non-trivial choice of point transformation. Finally, we summarize our results.展开更多
Due to the nanometer scale pore size and extremely low permeability of a shale matrix,traditional Darcy＇s law can not exactly describe the combined gas transport mechanisms of viscous flow and Knudsen diffusion.Three...Due to the nanometer scale pore size and extremely low permeability of a shale matrix,traditional Darcy＇s law can not exactly describe the combined gas transport mechanisms of viscous flow and Knudsen diffusion.Three transport models modified by the Darcy equation with apparent permeability are used to describe the combined gas transport mechanisms in ultra-tight porous media,the result shows that Knudsen diffusion has a great impact on the gas transport and Darcy＇s law cannot be used in a shale matrix with a pore diameter less than 1 μm.A single porosity model and a double porosity model with consideration of the combined gas transport mechanisms are developed to evaluate the influence of gas transport mechanisms and fracture parameters respectively on shale gas production.The numerical results show that the gas production predicted by Darcy＇s law is lower than that predicted with consideration of Knudsen diffusion and the tighter the shale matrix,the greater difference of the gas production estimates.In addition,the numerical simulation results indicate that shale fractures have a great impact on shale gas production.Shale gas cannot be produced economically without fractures.展开更多
The current research of nonlinear seepage theory of shale-gas reservoir is still in its infancy. According to the characteristics of shale gas in adsorption-desorption, diffusion, slippage and seepage during accumulat...The current research of nonlinear seepage theory of shale-gas reservoir is still in its infancy. According to the characteristics of shale gas in adsorption-desorption, diffusion, slippage and seepage during accumulation, migration and production, a mathematical model of unstable seepage in dual-porosity sealed shale-gas reservoir was developed while considering Knudsen diffusion, slip-flow effect and Langmuir desorption effect. By solving the model utilizing the Stehfest numerical inversion and computer programming in Laplace space, several typical curves of bottomhole pressure were obtained. In this paper, we discussed the effects of several parameters on the pressure dynamics, i.e. storativity ratio, Langmuir volume, Langmuir pressure, adsorption-desorption, tangential momentum accommodation coefficient, flow coefficient, boundary. The results show that the desorbed gas extends the time for fluid to flow from matrix system to fracture system;the changes of Langmuir volume and Langmuir pressure associated with desorption and adsorption effect are the internal causes of the storativity ratio change;when the tangential momentum accommodation coefficient decreases, the time for pressure wave to spread to the border reduces;interporosity flow coefficient determines the occurrence time of the transition stage;boundary range restricts the time for pressure wave to spread to the border.展开更多
文摘We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we can tell) be expressed as examples of the O’Shaughnessy and Procaccia equations. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. We illustrate the ideas both analytically and with a detailed computational example for a non-trivial choice of point transformation. Finally, we summarize our results.
基金supported by the National Natural Science Foundation of China (No. 51234007, No. 11072268)Program for Changjiang Scholars and Innovative Research Team in University (IRT1294)+5 种基金the Major Programs of Ministry of Education of China (No. 311009)Specialized Research Fund for the Doctoral Program of Higher Education (No. 20110133120012)the National Natural Science Foundation of Shandong Province (No. 11072268)the Fundamental Research Funds for the Central Universities (No. 11CX05007A)the Fundamental Research Funds for the Central Universities (No. 11CX04022A)Introducing Talents of Discipline to Universities (B08028)
文摘Due to the nanometer scale pore size and extremely low permeability of a shale matrix,traditional Darcy＇s law can not exactly describe the combined gas transport mechanisms of viscous flow and Knudsen diffusion.Three transport models modified by the Darcy equation with apparent permeability are used to describe the combined gas transport mechanisms in ultra-tight porous media,the result shows that Knudsen diffusion has a great impact on the gas transport and Darcy＇s law cannot be used in a shale matrix with a pore diameter less than 1 μm.A single porosity model and a double porosity model with consideration of the combined gas transport mechanisms are developed to evaluate the influence of gas transport mechanisms and fracture parameters respectively on shale gas production.The numerical results show that the gas production predicted by Darcy＇s law is lower than that predicted with consideration of Knudsen diffusion and the tighter the shale matrix,the greater difference of the gas production estimates.In addition,the numerical simulation results indicate that shale fractures have a great impact on shale gas production.Shale gas cannot be produced economically without fractures.
文摘The current research of nonlinear seepage theory of shale-gas reservoir is still in its infancy. According to the characteristics of shale gas in adsorption-desorption, diffusion, slippage and seepage during accumulation, migration and production, a mathematical model of unstable seepage in dual-porosity sealed shale-gas reservoir was developed while considering Knudsen diffusion, slip-flow effect and Langmuir desorption effect. By solving the model utilizing the Stehfest numerical inversion and computer programming in Laplace space, several typical curves of bottomhole pressure were obtained. In this paper, we discussed the effects of several parameters on the pressure dynamics, i.e. storativity ratio, Langmuir volume, Langmuir pressure, adsorption-desorption, tangential momentum accommodation coefficient, flow coefficient, boundary. The results show that the desorbed gas extends the time for fluid to flow from matrix system to fracture system;the changes of Langmuir volume and Langmuir pressure associated with desorption and adsorption effect are the internal causes of the storativity ratio change;when the tangential momentum accommodation coefficient decreases, the time for pressure wave to spread to the border reduces;interporosity flow coefficient determines the occurrence time of the transition stage;boundary range restricts the time for pressure wave to spread to the border.
