Research On Fractional Derivative Modelling Of Non-Darcian Flow And Solute Transport

The theoretical modeling of fluid flow and diffusive transport in low-permeability porous media is a common problem within engineering,such as the exploitation of deep resources,the underground storage of combat readiness energy,the geological disposal of C02 and nuclear wastes,and the exploitation of tight gas reservoirs.Tight rocks feature low permeability,minute passages of fluid flow,high capillary resistances and strong fluid-solid interactions.The coaction of these mentioned factors leads the flow behaviors to deviate from the classical Darcy's and Fick's laws.Consequently,the fluid flow and diffusive transport in porous media do not satisfy normal diffusion but appear anomalous characteristics,and the linear constitution law of flow and diffusion are not applicable anymore.Therefore,it is of great scientific significance to carry out the research on theoretical modeling of non-Darcian flow and diffusive transport in low-permeability media.With the non-Darcian flow and anomalous diffusion in low-permeability porous media as research subjects,this dissertation synthetically investigated the low and high velocity non-Darcian flow as well as diffusive transport in low-permeability by invoking the fractional calculus theory.At first,the non-Darcian flow behaviors are depicted by fractional derivative approaches.A fractional Darcian model is proposed for the description of high velocity non-Darcian flow in porous media.Meanwhile,the Swartzendruber model of low velocity non-Darcian flow is generalized by different fractional derivative approaches.Additionally,in order to characterize the non-Darcian flow and solute transport phenomena in porous media,the normal diffusion model is developed to fractional diffusion and advection-diffusion model.The proposed fractional derivative models are analytical solved by applying several integral transform methods.Furthermore,the applicabilities of the presented fractional derivative models are illustrated by fitting experimental data for fluid flows and solute transport in porous media.Ultimately,takeing into account the memory effectiveness non-Darcian flow induced by fluid-solid interaction,a fractional transient pulse method is proposed to determine the permeability of tight rocks.Moreover,based on the fractional transient pulse method,flow tests are carried out by passing water through triaxial stressed cylindrical granite specimens under high water pressures and temperatures.In conclusion,the major research results of this dissertation are drawn as follows.1,On the basis of Darcy's law,the memory effectiveness deduced by solid-fluid interaction is depicted by fractional derivative approach,and a fractional Darcian model is proposed for the description of high velocity non-Darcian flow in porous media.Meanwhile,the Swartzendruber model of low velocity non-Darcian flow is generalized by applying different fractional derivative approaches.All the fractional derivative flow models are analytically addressed,and parameter sensitivity analyses are adopted.Additionally,all the relevant parameters of fractional derivative flow models are determined by fitting analyses on the basis of experimental data using nonlinear least square method.The fitting results indicate that the presented fractional derivative flow models can describe the non-Darcian flow in porous media with high flexibility and accuracy.Furthermore,the memory effectiveness of fractional calculus is also discussed,indicating that fractional derivative models can be used to describe the non-Darcian flow in porous media as a non-Markov process.2.The non-Darcian flow and solute transport phenomena are characterized via the newly Caputo-Fabrizio fractional derivative approach.A Caputo-Fabrizio fractional diffusion model(CFFD)is proposed and analytically solved.Moreover,the error model is a special case of the CFFD model with the fractional derivative order of a=1.Comparisons with the classical diffusive transport models involving memory based on experimental data show that the presented CFFD model performs well at describing the characteristics of non-Darcian flow and diffusive transport in porous media.The discussion about the memory effect of Caputo-Fabrizio fractional derivative document that the Caputo-Fabrizio fractional derivative models can be used to characterize the physical processes which feature the short memory effectiveness.3.The normal diffusion equation is developed by using the conformable derivative approach to modelling the anomalous diffusion in porous media.The analytical solutions and the power law of the mean square displacement for the presented conformable fractional diffusion model are derived.Additionally,On the basis of the experimental data of chloride ions transport in reinforced concrete,the parameters of the conformable diffusion model are determined.Meanwhile,the relationship between the conformable derivative order and the duration time is also investigated.The fitting results indicate that the conformable diffusion model is in better agreement with experimental data than the normal diffusion equation in terms of Error model.Moreover,the conformable derivative order can be obtained by the short-term experimental data,consequently,the concentration distributions of the long-term subdiffusion process are effectively predicted by using the proposed conformable derivative model.The conformable advection-diffusion model is also analytically derived and validated on the basis of experimental data about advective-diffusive transport in porous media.The results of fitting analysis present a good agreement with experimental data.Furthermore,the potential application in numerical simulations of thermal-hydro-mechanical(THM)coupling processes in complex media using conformable derivative is further investigated.4.Motivated by the fact that non-exponential decay of pressure drop has been observed in the transient pulse test for determining the permeability of rocks,the transient pulse method is developed by invoking the fractional derivative approach.A fractional relaxation equation is developed via the fractional governing equation of fluid flow,leading to a Mittag-Leffler law to depict the non-exponential decay of pressure drop in transient pulse test.Meanwhile,the physical meaning of the Mittag-Leffler law is analysed in the framework of continuous time random walk.Furthermore,on the basis of the proposed fractional transient pulse method,flow tests are carried out by passing water through triaxial stressed cylindrical granite specimens under high water pressures and temperatures,and the permeability of Beishan granite specimens is accurately determined.The stress-strain curves and permeability evolutions of granite specimens under different temperatures,confining pressures and water pressures are obtained.