The Algorithm To Compute Groundwate Velocity Field And The Streamlines Visualization Based On The Continuous Galerkin Finte Element Method

Groundwater flow field is a mixing field with the groundwater potential field and the groundwater velocity field.In the practical applications,the groundwater flow filed is usually obtained with numerical simulation.The groundwater 3-D visualization is an important part in groundwater researches.It means to represent the groundwater clearly and straightly for researchers,engineers and even lay persons.The local conservation property of groundwater flow field computed based on the numerical algorithm is of important.It would affect the accuracy of the corresponding computation of solute transport,groundwater management,delineation of wellhead protection zone,multi-phase flow and visualization of groundwater flow field.In this paper,we discuss the three methods to construct the finite element equation that are the Galerkin method,the Ritz method and the method based on conservation for vertexes.The finite element equations obtained by these three methods are indeed the same.Because of the strict theory and simple implementation,the Galerkin method is widely used.To be distinguished from the method,discontinuous Galerkin method,the traditional Galerkin method is called as the continuous Galerkin method.It is always considered as a non-locally conservative numerical method.Here,we discuss the viewpoint in theory and compare that perspective to these locally conservative theories for continuous Galerkin method proposed in the last years.We emphasize the importance of the domain and the corresponding fluxes simultaneously.The last locally conservative theories are not widely accepted in terms of the post-processing way to compute fluxes.In this paper,we propose another new criterion to judge the numerical method locally conservative or not.It is whether a general kind of domain exists that is locally conservative for which the fluxes can be computed directly by the hydraulic gradient.According to the new criterion,we analyze the physical meaning of each term in the discretized governing equation for triangular element and triangular prism element and find out the conditions for locally conservative domain.To depict that domain precisely is a hard work.In this research,we provide a mathematical approach to prove the existence of the direct conservative domain and draw the conclusion that the continuous Galerkin finite element method is locally conservative.The groundwater potential field could be obtained by the continuous Galerkin method and the velocity field could be computed with the potential field by Darcy's Law.This procedure is called the direct method which makes the velocity field with big errors.The errors represent as the discontinuity of vertical velocity across element edges and the non-local conservation for a single element.To circumvent those errors,the mixed finite element and the discontinuous Galerkin method is developed which are more complex and takes more computational efforts.Another way is to correct the velocity field by a post-processing procedure.We analyze some popular algorithm such as CK method,Diresch method and FM method.These three methods could deal with the discontinuity of vertical velocity across element edges,but introduce the new error,the discontinuity of weighted tangential velocity along material interface.These three kinds of errors are considered as the most error for velocity field obtained based on the continuous Galerkin method.And they are become the prime content to evaluate the post-procedure algorithms.Based on the locally conservative theory for continuous Galerkin method,considering the continuity of weighted tangential velocity along material interface,we proposed the new algorithm to computed more accurate velocity field.The new method could eliminate the discontinuity of vertical velocity across edges and conserve the mass balance for a local domain.It also could satisfy the continuity of weighted tangential velocity along material interface extremely.The local domain in triangular mesh is the sub-triangle which is obtained by connecting the midpoints of three edges of a triangular element.In triangular prism mesh,the local domain is a sub-prism which is one-eighth of an original prism.To check the accuracy of the new method,we introduce the classical 2-D model,Mose model,and the 3-D model,three layer model.Mose model simulate the lenses with aquitard bars.The three layer model simulates the unconfined aquifer,the aquitard aquifer and the confined aquifer simultaneously with a pumping well and a window.The two models represent most of the situations in practical engineering.Comparing to Feflow,Modflow,CK methods and FM method,the new algorithm is demonstrated to be more superior.It belongs to the vector filed visualization and to scientific visualization.The technique for visualization mainly includes direct visualization,texture based visualization,geometry visualization and feature based visualization.The geometry visualization is the most popular technique which employs the streamlines,path lines,streak lines,streamlet and pathlet to represent the flow field.It is also the main pattern of feature based visualization.There are two procedures to construct streamlines,stream function method and particle tracking method.We choose the more flexible way,particle tracking method,as the procedure to construct streamlines and discuss the primary two problems which are the delineation of flow field and the perception.Too little streamlines make the flow field not clear and too many would make the clutter and visual complexity or even occlusion.According to the character of particle tracking method,the arrangement of streamlines is indeed the seeding strategy of initial points of streamlines.In this paper,the global principle to seed points around the geologic feature and then to draw thematic map is proposed.In local domain,the seeding strategy includes two methods,equidistance and equal fluxes.The selecting between them is depends on the goal for visualization.Based on the above analysis,we develop an independent visualization system and an embedded visualization plug-in module.The independent system could supply more flexible function for visualization which is more appropriate to lay persons.The embedded plug-in module could exchange data with Feflow which is more suitable to professional researchers.At last,we apply these visualization systems to the practical hydrogeological condition,Yingjun No.1 mine area in Inner Mongolia.We first analyze the geologic aspects and hydrogeological conditions of the research area and in the mine area.These data include stratum conditions,structural conditions,surface water,the main aquifers,the main aquitards and the groundwater flow regularity.Then,the conceptional hydrogeological model and the numerical model are established.Considering the less hydrogeological data,we calibrate the numerical model with P-H0 method.After the calibration,we simulate the groundwater flow regularity in the above aquifers with the mine water inrush simply.The two visualization systems are then employed to represent the flow filed.The application on practical hydrogeological conditions verifies the reliability of the two visualization systems.