JP2019212049A - Numerical analysis device, numerical analysis method, numerical analysis program - Google Patents

Abstract

To provide a numerical analysis device, a numerical analysis method, and a numerical analysis program capable of maintaining and improving analysis accuracy of a point source model regardless of a method of element division.SOLUTION: A device for numerically analyzing a physical phenomenon by a finite element method comprises: analysis means which divides an analysis area around a point source part modeled by a node string into a finite number of elements to numerically analyze physical amount of the analysis area by using the finite element method; correction means which corrects physical properties of a predetermined element around the point source part; and calculation means which, based on the physical amount obtained by numerically analyzing by the analysis means by using the physical properties corrected by the correction means and the physical properties before correcting by the correction means, calculates another physical amount.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a numerical analysis device, a numerical analysis method, and a numerical analysis program for numerically simulating groundwater flow during water injection / pumping using a borehole or a well, for example, using a point source model of a finite element method.

  Conventionally, in the evaluation of groundwater flow (e.g., evaluation of the results of inter-hole permeability tests) in the planning and design of large-scale underground structures (for example, rock stockpiling and radioactive waste disposal), numerical simulations such as the finite element method and the difference method are used. However, if you try to model boreholes and wells with small diameters in the analysis model with actual dimensions, it will not only require a lot of labor for mesh division, but it will also be an unrealistic analysis, so ignore the hole diameter Thus, it is often modeled as a series of point sources (a sequence of point sources).

  Even when water injection / pumping holes are modeled not as point source arrays but as actual sized cavities, in order to accurately analyze the flow field around the well, it is necessary to divide the vicinity of the holes very finely.

  When modeling water injection / pumping holes as point sources, it is necessary to make the elements and grids around the point sources have a specific size that is equivalent to the actual hole diameter. Therefore, it is often necessary to divide the analysis area without making it equivalent to the actual hole diameter.

  Many methods have been proposed in the finite difference method / finite volume method in order to avoid accuracy degradation by the point source sequence model of pumping / water injection holes (see, for example, Non-Patent Documents 1 to 8), but FIG. As shown in (3) to (3), it is limited to a structured grid (regular grid) or a mapping mesh according to it.

  Even in the finite element method, there is research for avoiding accuracy degradation by the point source sequence model of pumping / water injection holes (see, for example, Non-Patent Documents 9 to 13), but as shown in FIGS. In addition, the condition is that the point source is regularly divided by congruent elements.

  The present inventor has shown that, in the finite element method, the accuracy of analysis can be remarkably improved by correcting the hydraulic conductivity of elements around the point source array of the pumping / water injection holes (Non-Patent Document). 14 and 15), as shown in FIGS. 14 (1) to (2), the element shapes around the point sources are all the same (congruent).

  On the other hand, for example, methods disclosed in Patent Documents 1 and 2 are known as conventional numerical simulation techniques for groundwater.

Peaceman, D. W .: Interpretation of Wellblock Pressures in Numerical Reservoir Simulation, SPEJ (June), Trans., AIME, 253., pp.183-194, 1978. Peaceman, D. W .: Interpretation of Wellblock Pressures in Numerical Reservoir Simulation With Nonsquare Gridblocksand Anisotropic Permeability, SPEJ (June), pp.531-543, 1983. Peaceman, D. W .: Interpretation of Wellblock Pressure in Numerical Reservoir Simulation Part 3-0ff Center and Multiple Wells Within a Wellblock, SPERE (May), Trans., AIME, 289., pp.227-232, 1990. Peaceman, D. W .: Representation of a Horizontal Well in Numerical Reservoir Simulation, SPE Adv.Technology Series, Vol.1, No.1, pp.7-16, 1993. Hiroyuki Tosaka, Shinji Kojima, Akio Miki, Takeshi Chino: Development of 3D inland water simulation method combining surface and groundwater flow, Journal of Groundwater Society, Vol.38, pp.253-267, 1996. Satoshi Yamaishi, Hitoshi Kobayashi, Yoshiro Tanito, Akio Okamoto, Hiroyuki Tosaka, Junji Kojima: Numerical simulation of hydrological and hydraulic behavior associated with the construction of underground oil storage bases, Journal of Groundwater Society, Vol.40, pp.167-183, 1998. Ding, Y .: A Generalized 3D Well Model for Reservoir Simulation, SPE Journal., December, No.SPE30724, pp.437-450, 1996. Wolfsteiner, C., Durlofsky, L. J., and Aziz, K .: Calculation of well index for nonconventional wells on arbitrary grids, Computational Geosciences., Vol.7, pp.61-82, 2003. Kono, I .: The equivalent radius of a source in numerical models of groundwater flow, Proc. Of JSCE, No.218, pp.103-107, 1973. Keiji Uemura, Toshihiro Tsuji, Yoshihiro Tanaka: A Consideration on Modeling of Wells in Infiltration Flow Analysis, Proceedings of the 28th Geotechnical Engineering Conference, E-7, No.841, pp.2245-2246, 1993. Toshihiro Tsuji, Keiji Uemura, Yoshihiro Tanaka: A Study on Modeling of Wells in Two-Dimensional Permeable Flow Analysis, Proceedings of the 28th Geotechnical Research Conference, E-7, No.842, pp.2247-2248 , 1993. Keiji Uemura, Toshihiro Tsuji, Yoshihiro Tanaka: A Study on Modeling of Partial Penetration Holes in 3D FEM Infiltration Flow Analysis, Proceedings of the 49th Annual Conference of Japan Society of Civil Engineers, III-84, pp.162-163 , 1994. Chen, Z. and Zhang, Y .: Well Flow Models for Various Numerical Methods, Int. J. Numer.Analysis and Modeling., Vol.6, No.3, pp.375-388, 2009. Toshiko Yamada, Hideyuki Togashi, Makoto Suzuki: Practical simple model of water injection and pumping holes in seepage flow analysis using finite element method, Transactions of the Japan Society of Civil Engineers C (Geosphere Engineering), Vol.71, No.4, pp .407-417, 2015. Toshiko Yamada, Hideyuki Togashi, Makoto Suzuki: Code verification of simple finite element model of water injection / pumping holes, JSCE C (Geosphere Engineering), Vol.73, No.4, pp.450-459, 2017.

JP 2002-286860 A JP 2001-323477 A

  As mentioned above, in the groundwater seepage flow analysis by the conventional finite element method or the difference method, when modeling with the water injection / pump well as a point source, the size of the surrounding elements is equivalent to the actual pore size. There is a problem that an accurate analysis result cannot be obtained unless the specific size is set.

  Further, as shown in FIGS. 14 (1) to (3), the conventional method for avoiding accuracy deterioration of the water injection / pumping well by the point source model is limited to the structural grid and the mapping type mesh corresponding thereto. There is.

  Therefore, there is a need for a method that can maintain and improve the accuracy of groundwater flow analysis of water injection / pumping wells using a point source model, regardless of the element division method.

  The present invention has been made in view of the above, and provides a numerical analysis device, a numerical analysis method, and a numerical analysis program capable of maintaining and improving the analysis accuracy of a point source model regardless of the element division method. For the purpose.

  In order to solve the above-mentioned problems and achieve the object, a numerical analysis device according to the present invention is a device for numerical analysis of a physical phenomenon by a finite element method, and is a point source unit modeled by a nodal sequence An analysis unit that divides an analysis region around a finite number of elements and numerically analyzes a physical quantity of the analysis region using a finite element method, and a correction unit that corrects physical characteristics of a predetermined element around the point source unit And a calculating means for calculating another physical quantity based on the physical quantity obtained by numerical analysis by the analyzing means using the physical characteristics corrected by the correcting means and the physical characteristics before being corrected by the correcting means. It is characterized by.

  In addition, in another numerical analysis apparatus according to the present invention, in the above-described invention, the correction unit assumes that the physical quantity of a node of a predetermined element around the point source unit is a constant value, and the predetermined unit around the point source unit. It is characterized by correcting the physical characteristics of the element.

  Another numerical analysis device according to the present invention is the above-described invention, wherein the analyzing means numerically analyzes the groundwater flow around the water injection / pumping well modeled by the nodal sequence, and the correcting means includes: Assuming that the total head value of the nodes around the node sequence of the injection / pump well is equal to their average value, the hydraulic conductivity of the elements connected to the node sequence of the injection / pump well is corrected. The calculating means is characterized in that the flow velocity is calculated by applying a hydraulic permeability coefficient before correction to the total head distribution as an analysis result.

  The numerical analysis method according to the present invention is a method for numerical analysis of a physical phenomenon by a finite element method, and divides an analysis region around a point source modeled by a nodal sequence into finite elements. An analysis step for numerical analysis of the physical quantity in the analysis region using a finite element method, a correction step for correcting physical characteristics of a predetermined element around the point source unit, and a physical characteristic corrected by the correction step A calculation step of calculating another physical quantity based on a physical quantity obtained by numerical analysis in the analysis step and a physical characteristic before correction in the correction step is provided.

  In addition, in another numerical analysis method according to the present invention, in the above-described invention, the correction step assumes that the physical quantity of a node of a predetermined element around the point source unit is a constant value, and the predetermined step around the point source unit. It is characterized by correcting the physical characteristics of the element.

  Further, in another numerical analysis method according to the present invention, in the above-described invention, the analysis step numerically analyzes the groundwater flow around the water injection / pumping well modeled by the node sequence, and the correction step includes: Assuming that the total head value of the nodes around the node sequence of the injection / pump well is equal to their average value, the hydraulic conductivity of the elements connected to the node sequence of the injection / pump well is corrected. The calculating step is characterized in that the flow velocity is calculated by applying the permeability coefficient before correction to the total head distribution as the analysis result.

  A numerical analysis program according to the present invention causes a computer to execute the numerical analysis method described above.

  The numerical analysis device according to the present invention is a device for numerical analysis of a physical phenomenon by a finite element method, and divides an analysis region around a point source modeled by a nodal sequence into a finite number of elements. An analysis unit that numerically analyzes a physical quantity in the analysis region using a finite element method, a correction unit that corrects a physical characteristic of a predetermined element around the point source unit, and a physical characteristic that is corrected by the correction unit Since it has a calculation means for calculating other physical quantities based on the physical quantity obtained by numerical analysis by the analysis means and the physical characteristics before correction by the correction means, the point source model is provided regardless of the element division method. The analysis accuracy can be maintained and improved.

  According to another numerical analysis apparatus of the present invention, the correction means assumes that the physical quantity of the node of the predetermined element around the point source unit is a constant value, and determines the predetermined element around the point source unit. Since the physical characteristics are corrected, there is an effect that the physical characteristics can be corrected efficiently.

  Further, according to another numerical analysis apparatus according to the present invention, the analysis means numerically analyzes the groundwater flow around the water injection / pumping well modeled by the nodal sequence, and the correction means includes the water injection / pumping water. Assuming that the total head value of the nodes around the well node sequence is equal to their average value, the hydraulic conductivity of the elements connected to the water injection / pump well node sequence is corrected, and the calculation means Is to calculate the flow velocity by applying the permeability coefficient before correction to the total water head distribution, which is the analysis result. Therefore, the infiltration flow analysis of the groundwater around the water injection / pumping well modeled by the node sequence There is an effect that the accuracy can be maintained and improved.

  The numerical analysis method according to the present invention is a method for numerical analysis of a physical phenomenon by a finite element method, wherein an analysis region around a point source unit modeled by a nodal sequence is limited to a finite number of elements. The analysis step of numerically analyzing the physical quantity of the analysis region using the finite element method, the correction step of correcting the physical characteristics of a predetermined element around the point source unit, and the physical characteristics corrected in the correction step Using a physical quantity obtained by numerical analysis in the analysis step and a calculation step for calculating another physical quantity based on the physical characteristics before correction in the correction step. The analysis accuracy of the source model can be maintained and improved.

  According to another numerical analysis method of the present invention, the correction step assumes that the physical quantity of the node of the predetermined element around the point source unit is a constant value, and the predetermined element around the point source unit is corrected. Since the physical characteristics are corrected, there is an effect that the physical characteristics can be corrected efficiently.

  According to another numerical analysis method of the present invention, the analysis step numerically analyzes the groundwater flow around the water injection / pumping well modeled by the node sequence, and the correction step includes water injection / pumping. Assuming that the total head value of the nodes around the well node sequence is equal to their average value, the hydraulic conductivity of the elements connected to the node sequence of the water injection / pump well is corrected and the calculation step Is to calculate the flow velocity by applying the permeability coefficient before correction to the total water head distribution, which is the analysis result. Therefore, the infiltration flow analysis of the groundwater around the water injection / pumping well modeled by the node sequence There is an effect that the accuracy can be maintained and improved.

  In addition, according to the numerical analysis program according to the present invention, since the computer executes the numerical analysis method described above, there is an effect that the numerical analysis method described above can be efficiently executed using the computer.

FIG. 1 is a diagram showing an embodiment of a numerical analysis device, a numerical analysis method, and a numerical analysis program according to the present invention, wherein (1) is an arbitrarily shaped element group (view in the well axis direction) connected to a well node, (2) is a single hole model in a confined aquifer, (3) is a tetrahedral element (N ^e = 4), and (4) is a schematic diagram of element division along the well axis. FIG. 2 is a schematic flowchart showing an embodiment of a numerical analysis device, a numerical analysis method, and a numerical analysis program according to the present invention. (1) is a processing routine based on the algorithm of the present invention incorporated in a normal FEM analysis program. In the case of (2), the input data is corrected by the algorithm of the present invention. FIG. 3 is a flowchart of a processing routine according to the algorithm of the present invention. FIG. 4 is a diagram illustrating a verification target model divided by uniform tetrahedral elements. FIG. 5 is a comparison diagram of the analysis results of FIG. 4, where (1) is a comparative example and (2) is an example. FIG. 6 is a diagram illustrating a verification target model divided by non-uniform tetrahedron elements. FIG. 7 is a comparison diagram of the analysis results of FIG. 6, where (1) is a comparative example and (2) is an example. FIG. 8 is a diagram illustrating a verification target model divided by uniform hexahedron elements. FIG. 9 is a comparison diagram of the analysis results of FIG. 8, where (1) is a comparative example and (2) is an example. FIG. 10 is a diagram illustrating a verification target model divided by unequal pentahedron elements. FIG. 11 is a comparison diagram of the analysis results of FIG. 10, where (1) is a comparative example and (2) is an example. FIG. 12 is a diagram illustrating a verification target model divided by unequal hexahedron elements. FIG. 13 is a comparison diagram of the analysis results of FIG. 12, where (1) is a comparative example and (2) is an example. FIG. 14 is an example of element division around a conventional pumping / filling hole, where (1) is a congruent triangular element group connected to a well node (field of view in the well axis direction), and (2) is a congruent connection connected to a well node. A rectangular element group (field of view in the well axis direction), (3) is a mapping type mesh (Source: Non-Patent Document 8).

  Hereinafter, embodiments of a numerical analysis device, a numerical analysis method, and a numerical analysis program according to the present invention will be described in detail with reference to the drawings. Note that the present invention is not limited to the embodiments.

  In the present embodiment, an algorithm for analyzing groundwater flow around a water injection / pump well (hereinafter referred to as a well) modeled by a nodal sequence in the finite element method is used with high accuracy.

  According to the present embodiment, as shown in FIG. 1 (1), the present invention can be applied to arbitrary mesh division. Further, as a mesh element, it is possible to cope with a general two-dimensional element (triangle, quadrangle), three-dimensional element (tetrahedron, pentahedron (triangular prism), hexahedron (tetragonal prism)) and the like.

  Further, since only the hydraulic conductivity around the point source is corrected, as will be described later, it can be easily incorporated into an existing finite element code (see FIG. 2 (1)). As will be described later, it is also possible to correct the hydraulic conductivity coefficient of the input data instead of incorporating it into the existing finite element code (see FIG. 2 (2)). When calculating the flow velocity from the total water head distribution of the analysis result, the permeability coefficient before correction may be used.

  In calculating the correction coefficient for correcting the hydraulic conductivity, it is assumed that the total head value of the nodes close to the point source is equal to their average value. However, the actual head value in the vicinity of the point source is not equal because it is obtained by finite element analysis.

  Next, applying the theoretical solution of the single hole model problem in the pressurized aquifer shown in FIGS. 1 (1) and (2), correcting the hydraulic conductivity of the finite element connected to the point source sequence An example will be described.

Where Q _theory is the flow rate of the well, r _w is the radius of the well, φ _w ^ is the total head value of the well, k is the hydraulic conductivity of the ground, D is the length of the well, R is the distance from the well center, φ _R ^ is the total head value at the position of distance R. ^ (Hat) represents a known amount.

  The method of correcting the hydraulic conductivity using equation (1) has already been published by the inventor in a paper (see Non-Patent Documents 14 and 15), which is shown in FIGS. 14 (1) and (2). Thus, it is limited to the case where the periphery of the well is divided by elements having a congruent shape. On the other hand, the present embodiment is an algorithm for dealing with an arbitrary mesh division as shown in FIG. Although it is an arbitrary mesh, normally, in order to avoid deterioration in analysis accuracy, the well is not divided into element groups having extremely different sizes. Since the osmotic flow field near the well is dominated by the pressure and flow rate of the well, in this embodiment, the head value of the node group near the well is corrected to be equal to the average value thereof. Deriving method. Hereinafter, a specific procedure of the present embodiment will be described.

The equation of each element obtained by the finite element method can be written as follows.

Where the index e represents the quantity related to the element,
C ^e _ij : Coefficient matrix of element e,
N ^e : Number of nodes constituting element e φ ^e _j : Head value of node j constituting element e
q ^e _i : Flow rate value of component node i of element e

The permeability tensor is isotropic, constant within the element, and the permeability coefficient of element ^e is expressed as k ^e

Here, when modeling a well as a point source, the hydraulic conductivity k ^e of an element (hereinafter referred to as a connected element) connected to a well node (nodes constituting the well, see FIG. 1 (3)) is corrected. Think about what to do. Assuming that one element is not connected to two different wells, in the case of an element having a node only at the vertex of the element (in the present invention, for example, a three-node triangular element, a four-node square element, a four-node tetrahedron Element, 6-node triangular prism element, 8-node hexahedron element, etc.), and among the element constituent nodes, only one or two nodes are connected to the well. There can be more than two, but it becomes an extremely distorted element and is not actually used. In the following, description will be made separately on the case where one node coincides with the well node and the case where two coincide with each other among the constituent nodes of the connected element.

(A) When one constituent node of a connecting element corresponds to a well node For simplicity of explanation, among the connecting nodes of a certain connecting element, the first node corresponds to a well node, and the head value φ _w ^ Is given. This is always the case for two-dimensional planar analysis.

From the equation for element i = 1

  In general, in order to avoid deterioration of the solution, the finite element is divided so as not to be distorted as much as possible. Therefore, the constituent nodes excluding the well nodes of the connecting elements (hereinafter referred to as the nearest nodes) are arranged at approximately the same distance from the well. It can be regarded as. In the present embodiment, it is assumed that there is no great difference in the head values of the closest nodes of all elements connected to a certain well node, and the approximation is as follows.

Here, N _w is a set of nearest nodes of a connected element group having one well node as a constituent node, and φ _s (bar) is an average head value of the nodes belonging to N _w . Therefore, Formula (4) can be approximated as follows.

Here, the property of the following coefficient matrix is used.

For general description, if w representing a well node is used instead of the index of the first node,

(B) When two constituent nodes of a connected element correspond to a well node For convenience, it is assumed that the first and second constituent nodes of an element correspond to a well node, and a head value φ _w ^ is defined. From the equation for the first node

Here, the closest node is considered in the same manner as described above, and Expression (5) is used.

From equation (7)

Here, in order to make the description general, c ^e _ww = (c ^e ₁₁ + c ^e ₁₂ ) and q ^e _w = q ^e ₁ are the same as in the case of only one node above.

(C) Calculation of flow rate collected at nodes When a set of finite elements (hereinafter referred to as nearest elements) having a well node w as a constituent node is represented by E _w , the flow rate Q _w collected at the well node _w is
Here, using the theoretical solution of equation (1) and _assuming that φ _w ^ ← φ _R ^, R _s (bar) ← R, Q _{theory in} equation (1) and Q _{w in} equation (13) are equal. When,

Here, _Dw is the length of the well occupied by the well node w, and is 1/2 of the distance between the upper and lower well nodes connected to a certain well node, as shown in FIG. In the case of the well node at the end, it is ½ of the distance to the adjacent well node.

Here once as the permeability of the coupling element groups is well nodes are equal, equation _{^{(15) k e w (~}} ) instead of the left side of the k ^e, the use of k ^e instead of the right side of k, the following become that way.

k ^e is the permeability coefficient before correction given as input data, k ^e _w (~) is the permeability coefficient of the connected element corrected to reproduce the flow field near the well node w, and B _w is the correction of k ^e Corresponds to the coefficient. Also, φ in equation (18) corresponds to the average distance from the well node to the nearest node group, and one of the following is used.

Here, R _j is the distance between the nearest node j and the well node w, R _s (bar) is the average distance to the nearest node at the well node w, and m _w is the number of the nearest nodes belonging to N _w (FIG. 1). (See (1)).

Expressions (19) and (20) have the same value when R _j is all equal, such as when all connected elements are congruent.

(D) m ^e _w = 1 if the calculated element e is linked to only one of the well node w of permeability after correction of each element, any element with its front and rear two wells nodes (wells node w )), Set m ^e _w = 2 and calculate the hydraulic conductivity of each element by either of the following.

Here, N ^e _w is a set of well nodes connected by the element e.

(Example 1)
Next, Example 1 of the present invention will be described.
As described above, since the present invention only corrects the hydraulic conductivity around the point source, it can be easily incorporated into an existing finite element code (ordinary FEM analysis program). Moreover, it is also possible to correct the permeability coefficient of the input data instead of incorporating it into the existing finite element code. These examples will be described with reference to FIG.

<Example when incorporating into existing finite element code>
FIG. 2A is an example in which the above algorithm is incorporated into an existing finite element code (FEM analysis program). As shown in this figure, first, the next input data is prepared in steps S101 and S102.
[Data set 1] Input data required for normal finite element analysis [Data set 2] Well data consisting of point source rows of the following specifications: Well node row of each well (from well start point to well end point start point)
The total head (or pressure head) value φ _w ^ of the well. However, it is possible to quote from [Data set 1]. Well radius r _w
・ An angle at which the well is modeled (If the well is located at the boundary of the analysis region due to a symmetry condition and is not modeled at 360 °, it can be obtained by calculation, but it is more efficient to give it)

  Next, the FEM analysis program to which the above algorithm is added is executed. This program includes steps S103 to S107 corresponding to a normal FEM analysis program, and steps S108 and S109 corresponding to the above algorithm.

  First, the above data sets 1 and 2 are read (step S103). On the other hand, the hydraulic conductivity of the elements around the point source row is corrected by the above algorithm (step S108). Next, FEM simultaneous equations are constructed using the data sets 1 and 2 and the corrected hydraulic conductivity as input values (step S104), and a solution is obtained (step S105). On the other hand, the correction of the water permeability coefficient is canceled (step S109). Next, post-processing such as calculating a flow velocity vector is performed using the corrected water permeability coefficient (permeability coefficient before correction) (step S106), and the result is output (step S107). In this way, an analysis result with high analysis accuracy can be obtained regardless of the element division method (step S110).

<Example of correcting the hydraulic conductivity of input data>
FIG. 2 (2) is an example of correcting the hydraulic conductivity of input data. As shown in this figure, first, in steps S201 and S202, the above-described input data (data sets 1 and 2) are prepared.

  Next, a program for correcting the hydraulic conductivity is executed based on the above algorithm (step S203). Thereby, FEM input data in which the hydraulic conductivity is corrected is obtained (step S204). In addition, correction data for the hydraulic conductivity of the elements around the point source array is obtained (step S212).

  Next, a normal FEM analysis program is executed. In this case, first, the data sets 1 and 2 and FEM input data are read (step S205). Next, FEM simultaneous equations are constructed (step S206), and a solution is obtained (step S207). Next, post-processing such as calculating a flow velocity vector is performed (step S208), and the result is output (step S209).

  Thereafter, when the physical quantity inverse correction program related to the hydraulic conductivity such as the flow velocity vector is executed using the correction data of the hydraulic conductivity of the element (step S211), the calculation result of the physical quantity after the reverse correction is obtained (S213). In this way, an analysis result with high analysis accuracy can be obtained regardless of the element division method.

(Example 2)
Next, a second embodiment of the present invention will be described.
As a specific procedure of the above algorithm, a flowchart for a computer program as shown in FIG. 3 can be used. Hereinafter, these processing flows will be described.

  As shown in FIG. 3, first, the data sets 1 and 2 are read (step S301). Next, the following steps S302 to S313 are repeated from the well number iw to 1 to the number of wells nw.

In step S302, it calculates the occupied length D _w of the well nodes constituting a well iw. Next, the processes of steps S303 to S311 below are repeated from the well node number inw of each well to 1 to the number of well nodes nnw. Here, the processing of steps S303 to S308 is repeated from element number ie to 1 to the total number of elements ne.

  In step S303, it is determined whether the element ie is a connected element of the well node inw. If it is a connected element, the process proceeds to step S304 (Yes), and if it is not a connected element, the next element is determined (No). Next, in step S304, the element ie connected to the well node inw is stored as a set Ew of connected elements. Next, in step S305, it is determined whether the element ie is also connected to the well nodes before and after the well node inw. As a result, when the element is connected only to the well node inw, the flow rate value is calculated from the above equation (8) (step S305). On the other hand, when the element is connected to any one of before and after the well node inw, the flow rate value is calculated from the above equation (12) (step S307). After the calculation, the nodes other than the well node among the constituent nodes of the element ie are stored as a set Nw of the closest nodes of the well node inw (step S308).

  Next, the distance between the well node inw and its nearest node group is calculated using Nw (step S309), and the average distance is calculated using the above equation (19) or equation (20) (step S310). Then, the corrected hydraulic conductivity at the well node inw is calculated and stored by the above equation (17) (step S311).

Next, the well node numbers INW each well to 1 well clause number nnw, it repeats the processing of counting the wells clause number m ^e _w connecting element utilizing Ew regarding wells node INW (step S312). Subsequently, the element number ie. 1 to up to the total number of elements ne, if m ^e _w is not zero, the above equation (21), equation (22), calculates the corrected permeability in any of the formulas (23) The process is repeated (step S313). As a result, the corrected hydraulic conductivity data of the element is obtained (step S314).

(Verification of the effect of the present invention)
[When divided by equal tetrahedron elements]
As shown in FIG. 4, a numerical analysis by a finite element method is performed on a verification model in which an analysis region is divided by a large number of uniform tetrahedron elements, and the result is obtained as a numerical analysis method (example) of the present invention and a conventional FEM. Comparison was made with the numerical analysis method (comparative example). In the verification model, the radius of the well r _w = 0.05 m and the distance from the well center R = 60 m. Further, the at _{r = r w φ w ^ =} 100m ( total hydraulic head value of the well), and the r = 60m φ _R ^ = a 0 m (total hydraulic head value at the position of the distance R). As shown in FIG. 4, the exact solution and the example are in good agreement, and there are portions that cannot be distinguished by overlapping on the paper surface, but it can be seen that the comparative example is far from both.

As shown in FIG. 5A, in the case of the comparative example, it can be seen that the plots indicating the analysis results at all the node positions are deviated from the exact solution, and the variation is large. The flow rate of well Q _FEM (flow rate of numerical analysis result) is 1.38 times the result of Q _exact (flow rate of exact solution), and it can be seen that there is a deviation from the exact solution.

As shown in FIG. 5 (2), in the case of the example, it can be seen that the plot showing the analysis results at all the node positions is in good agreement with the exact solution and the variation is small. The well flow rate Q _FEM (flow rate of the numerical analysis result) is also 0.99 times Q _exact (flow rate of the exact solution), which indicates that it is very close to the exact solution.

[When dividing by non-uniform tetrahedron elements]
The analysis result was compared with the Example and the comparative example about the verification model by non-uniform tetrahedral element division as shown in FIG. The boundary conditions are the same as in the verification model of FIG. As shown in FIG. 6, the exact solution and the example are in good agreement, and there are portions that cannot be distinguished by overlapping on the paper surface, but it can be seen that the comparative example is far from both. The reason why the isolines are strangely distributed in the rough connected elements below the center is that the isolines are drawn by linear interpolation within the elements.

As shown in FIG. 7A, in the case of the comparative example, it can be seen that the plot showing the analysis results at all the node positions deviates from the exact solution, and the variation is large. The well flow rate ratio Q _FEM (flow rate of the numerical analysis result) is 1.66 times the Q _exact (flow rate of the exact solution), which shows that it is deviated from the exact solution.

As shown in FIG. 7 (2), in the case of the example, it can be seen that the plots indicating the analysis results at all the node positions are in good agreement with the exact solution and the variation is small. The flow rate Q _FEM (flow rate of the numerical analysis result) of the well is also 0.98 times Q _exact (the flow rate of the exact solution), which indicates that it is very close to the exact solution.

[When divided by equal hexahedron elements]
The analysis result was compared with the Example and the comparative example about the verification model by equal hexahedron element division | segmentation as shown in FIG. The boundary conditions are the same as in the verification model of FIG. As shown in FIG. 8, the exact solution and the example are in good agreement, and there are portions that cannot be distinguished by overlapping on the paper surface, but it can be seen that the comparative example is far from both.

As shown in FIG. 9 (1), in the case of the comparative example, it can be seen that the plots indicating the analysis results at all the node positions are deviated from the exact solution, and the variation is large. The flow rate of well Q _FEM (flow rate of numerical analysis result) is 1.32 times the result of Q _exact (flow rate of exact solution), which shows that it is deviated from the exact solution.

As shown in FIG. 9 (2), in the case of the example, it can be seen that the plot showing the analysis results at all the node positions is in good agreement with the exact solution and the variation is small. The flow rate Q _FEM (flow rate of the numerical analysis result) of the well is 1.01 times Q _exact (the flow rate of the exact solution), which is very close to the exact solution.

[When dividing by non-uniform pentahedron elements]
The analysis results were compared between the example and the comparative example for the verification model based on the uneven pentahedral element division as shown in FIG. The boundary conditions are the same as in the verification model of FIG. As shown in FIG. 10, the exact solution and the example are in good agreement, and there are portions that cannot be distinguished by overlapping on the paper surface, but it can be seen that the comparative example is far from both.

As shown in FIG. 11 (1), in the case of the comparative example, it can be seen that the plots indicating the analysis results at all the node positions are deviated from the exact solution, and the variation is large. The flow rate of well Q _FEM (flow rate of numerical analysis result) is 1.53 times the result of Q _exact (flow rate of exact solution), and it can be seen that there is a deviation from the exact solution.

As shown in FIG. 11 (2), in the case of the example, it can be seen that the plots showing the analysis results at all the node positions are in good agreement with the exact solution and the variation is small. The flow rate Q _FEM (flow rate of the numerical analysis result) of the well is 0.94 times Q _exact (the flow rate of the exact solution), and it can be seen that it is close to the exact solution. In this embodiment, the variation is larger than in the case of the tetrahedron division, but this is considered to be an effect of intentionally distorting the element shape.

[When dividing by non-uniform hexahedral elements]
The analysis results of the verification model based on the uneven hexahedral element division as shown in FIG. 12 were compared between the example and the comparative example. The boundary conditions are the same as in the verification model of FIG. As shown in FIG. 12, the exact solution and the example are in good agreement, and there are portions that cannot be distinguished by overlapping on the paper surface, but it can be seen that the comparative example is far from both.

As shown in FIG. 13 (1), in the case of the comparative example, it can be seen that the plots indicating the analysis results at all the node positions deviate from the exact solution, and the variation is large. The flow rate of well Q _FEM (flow rate of numerical analysis result) is 1.44 times the result of Q _exact (flow rate of exact solution).

As shown in FIG. 13 (2), in the case of the example, it can be seen that the plot showing the analysis results at all the node positions is in good agreement with the exact solution and the variation is small. The flow rate Q _FEM (flow rate of numerical analysis result) of the well is also 0.99 times Q _exact (flow rate of exact solution), and it can be seen that it is close to the exact solution. In this embodiment, the variation is larger than in the case of the tetrahedron division, but this is considered to be an effect of intentionally distorting the element shape.

  Therefore, according to the present embodiment, in the groundwater flow analysis using an arbitrarily divided mesh by the finite element method, the flow field around the hole can be analyzed with high accuracy even if the water injection / pumping hole is modeled by a point source sequence. For this reason, if the present Example is applied in the identification of the permeability coefficient of the ground rock based on the permeability test between holes, a reliable result will be obtained.

  In the above embodiment, the case of applying to the osmotic flow problem of groundwater has been described as an example. However, the present invention is not limited to this, and is not limited to the osmotic flow problem, and heat conduction that can be described by the same partial differential equation. Can be diverted to problems, diffusion problems, etc.

  As described above, the numerical analysis device according to the present invention is a device for numerical analysis of a physical phenomenon by a finite element method, and an analysis region around a point source unit modeled by a nodal sequence is obtained. Dividing into a finite number of elements and analyzing the physical quantity in the analysis area using the finite element method, correcting means correcting the physical characteristics of the predetermined elements around the point source, and correcting by the correcting means The method of element division is provided with a calculation means for calculating other physical quantities based on the physical quantity obtained by numerical analysis by the analysis means using the physical characteristics and the physical characteristics before correction by the correction means. Regardless, the analysis accuracy of the point source model can be maintained and improved.

  According to another numerical analysis apparatus of the present invention, the correction means assumes that the physical quantity of the node of the predetermined element around the point source unit is a constant value, and determines the predetermined element around the point source unit. Since the physical characteristics are corrected, the physical characteristics can be corrected efficiently.

  Further, according to another numerical analysis apparatus according to the present invention, the analysis means numerically analyzes the groundwater flow around the water injection / pumping well modeled by the nodal sequence, and the correction means includes the water injection / pumping water. Assuming that the total head value of the nodes around the well node sequence is equal to their average value, the hydraulic conductivity of the elements connected to the water injection / pump well node sequence is corrected, and the calculation means Is to calculate the flow velocity by applying the permeability coefficient before correction to the total water head distribution, which is the analysis result. Therefore, the infiltration flow analysis of the groundwater around the water injection / pumping well modeled by the node sequence The accuracy can be maintained and improved.

  The numerical analysis method according to the present invention is a method for numerical analysis of a physical phenomenon by a finite element method, wherein an analysis region around a point source unit modeled by a nodal sequence is limited to a finite number of elements. The analysis step of numerically analyzing the physical quantity of the analysis region using the finite element method, the correction step of correcting the physical characteristics of a predetermined element around the point source unit, and the physical characteristics corrected in the correction step Using a physical quantity obtained by numerical analysis in the analysis step and a calculation step for calculating another physical quantity based on the physical characteristics before correction in the correction step. The analysis accuracy of the source model can be maintained and improved.

  According to another numerical analysis method of the present invention, the correction step assumes that the physical quantity of the node of the predetermined element around the point source unit is a constant value, and the predetermined element around the point source unit is corrected. Since the physical characteristics are corrected, the physical characteristics can be corrected efficiently.

  Further, according to another numerical analysis method according to the present invention, the analysis step numerically analyzes the groundwater flow around the water injection / pumping well modeled by the node sequence, and the correction step includes water injection / pumping. Assuming that the total head value of the nodes around the well node sequence is equal to their average value, the hydraulic conductivity of the elements connected to the node sequence of the water injection / pump well is corrected and the calculation step Is to calculate the flow velocity by applying the permeability coefficient before correction to the total water head distribution, which is the analysis result. Therefore, the infiltration flow analysis of the groundwater around the water injection / pumping well modeled by the node sequence The accuracy can be maintained and improved.

  Moreover, according to the numerical analysis program according to the present invention, since the computer executes the numerical analysis method described above, the numerical analysis method described above can be efficiently executed using the computer.

  As described above, the numerical analysis device, the numerical analysis method, and the numerical analysis program according to the present invention are useful for finite element analysis of heat conduction problems and diffusion problems that can be described by partial differential equations. Regardless of the division method, it is suitable for maintaining and improving the analysis accuracy of the point source model.

Claims (7)

An apparatus for numerical analysis of physical phenomena by the finite element method,
An analysis means for dividing the analysis region around the point source section modeled by the node sequence into a finite number of elements, and analyzing the physical quantity of the analysis region using the finite element method;
Correction means for correcting physical characteristics of a predetermined element around the point source unit;
It comprises a calculation means for calculating another physical quantity based on a physical quantity obtained by numerical analysis by an analysis means using a physical characteristic corrected by a correction means and a physical characteristic before correction by a correction means. Numerical analysis device.   The correction means corrects the physical characteristics of the predetermined element around the point source unit on the assumption that the physical quantity of the node of the predetermined element around the point source unit is a constant value. Numerical analysis device. The analysis means is to numerically analyze the groundwater flow around the water injection / pump wells modeled by the node sequence.
The correction means corrects the hydraulic conductivity of the elements connected to the node sequence of the water injection / pump well, assuming that the total head values of the nodes around the node sequence of the water injection / pump well are equal to their average value. Is what
The numerical analysis device according to claim 1 or 2, wherein the calculation means calculates a flow velocity by applying a permeability coefficient before correction to the total head distribution as an analysis result. A method for numerical analysis of a physical phenomenon by a finite element method,
An analysis step of dividing the analysis region around the point source part modeled by the node sequence into a finite number of elements and numerically analyzing the physical quantity of the analysis region using the finite element method;
A correction step for correcting physical characteristics of a predetermined element around the point source unit;
A calculation step of calculating another physical quantity based on a physical quantity obtained by numerical analysis in the analysis step using the physical characteristic corrected in the correction step and a physical characteristic before correction in the correction step is provided. Numerical analysis method.   The correction step corrects the physical characteristics of the predetermined element around the point source unit, assuming that the physical quantity of the node of the predetermined element around the point source unit is a constant value. Numerical analysis method. The analysis step is a numerical analysis of the groundwater flow around the water injection / pump wells modeled by the node sequence.
The correction step corrects the hydraulic conductivity of the elements connected to the water injection / pump well column array, assuming that the total head values of the nodes around the water injection / pump well column are equal to their average value. Is what
6. The numerical analysis method according to claim 4, wherein the calculating step calculates a flow velocity by applying a permeability coefficient before correction to the total head distribution as an analysis result.   A numerical analysis program for causing a computer to execute the numerical analysis method according to any one of claims 4 to 6.