CN106202746A - The Yeh multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity - Google Patents

Abstract

The invention discloses a kind of Yeh multi-level finite element modeling method simulating Water in Porous Medium stream Darcy velocity, use Galerkin method by Solve problems variation；It is coarse grid cell by study area subdivision, all coarse grid cell are carried out subdivision, obtain refined net unit；Each coarse grid cell solves degenerate elliptic equation, obtains basic function；Use basic function to solve variational form, obtain global stiffness matrix；Source sink term according to study area and boundary condition obtain right-hand vector；Simultaneous obtains head equation group；Effective Numerical Methods Solve equation group is used to obtain the node head of study area；Gal the Liao Dynasty gold FEM (finite element) model in conjunction with Yeh, use the basic function and the head value of study area constructed, at survey region direct solution Darcy's equation, obtain continuous print darcy seepage velocity on thick yardstick node, recycle basic function linear expression thin yardstick darcy seepage velocity.Compared with prior art, the method has close precision and higher efficiency.

Description

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity

Technical field

The invention belongs to hydrodynamic technology field, be specifically related to a kind of simulate two dimension flow Darcy velocity in porous media Yeh-multi-level finite element modeling method.

Background technology

Subsoil water is mainly distributed in porous media, is the important component part of water resource, in the flowing of simulated groundwater During with solute transfer, flow velocity and the flow of accurate simulated groundwater are particularly significant.Therefore, to the darcy seepage velocity of subsoil water The research of computational methods and numerical simulation, be of great significance for investigating groundwater occurrence and solute transfer state tool.

The gal the Liao Dynasty gold finite element model [Yeh 1981] of Yeh is the side that a kind of classics solve subsoil water darcy seepage velocity Method, the method is directly used Finite Element to solve Darcy's equation on study area, is had higher computational accuracy [Zhang 1994], quite varied [Boufadel et al.1999 is applied；Jang et al 2007；Park et al.2007；Park et al.2008].Many studies demonstrate that the method ensure that darcy seepage velocity seriality on node, thus ensured Equal by the inflow and outflow amount in cross section.But, during problem of groundwater in solving large scale or nonisotropic medium, the method Finite element Property requirements unit internal penetration coefficient be constant, it is necessary to survey region use fine dissection, solve darcy and ooze Thoroughly need substantial amounts of calculating time and space consuming during flow velocity, inefficient.

Last century Mo, solve the computational efficiency of water flow problems in non-homogeneous porous medium to promote Finite Element, number Learn and hydrologic research worker proposes Multi-scale remote sensing [Hou, T.Y., and X.H.Wu (1997)].The method is led to Cross and make basic function meet infinitesimal operator on coarse grid cell, catch thin dimensional information.Therefore, ooze inside the method unit Coefficient is variable thoroughly, it is not necessary to fine dissection can solve the problem of groundwater in large scale or nonisotropic medium, has the highest meter Calculate efficiency.Many mathematics, hydrologist have been proven that Multi-scale remote sensing than Finite Element computational efficiency more Height, and restrain, accurately [Hou, T.Y., and X.H.Wu (1997), X.H.Wu, and Z.Cai (1999), W.Deng et al.(2008),Ye,S.,Y,Xue,and C.Xie(2004)].But, Multi-scale remote sensing shortage solves darcy infiltration The means of flow velocity, it is impossible to ensure speed and flow seriality on node.

In order to solve the problems referred to above, the gal the Liao Dynasty gold model of Yeh is organically combined by the present invention with Multi-scale remote sensing, combines The advantage having closed two kinds of methods, it is proposed that the mode of a kind of brand-new calculating darcy seepage velocity.

Summary of the invention

It is directed to above-mentioned the deficiencies in the prior art, it is an object of the invention to provide a kind of simulation Water in Porous Medium stream and reach The Yeh-multi-level finite element modeling method of western speed, the result obtained by numerical simulation is coincide with analytic solutions, at same unit number Purpose situation, calculates the darcy seepage velocity efficiency method higher than prior art.

For reaching above-mentioned purpose, a kind of Yeh-multi-scale finite simulating Water in Porous Medium stream Darcy velocity of the present invention Unit's method, comprises the following steps that

(1) determine boundary condition according to survey region to be simulated, set grid cell yardstick, subdivision survey region, Obtain coarse grid cell, all coarse grid cell subdivisions are obtained refined net unit；

(2) according to infiltration coefficient and the boundary condition of basic function, each coarse grid cell solves the ellipse of degeneration Type problem, determines basic function, forms Finite Element Space；

(3) calculating the element stiffness matrix of coarse grid cell, add up to obtain global stiffness matrix；Border according to survey region, Source sink term, calculates right-hand vector, forms the head equation group of multi-level finite element modeling；

(4) use cholesky decomposition method, try to achieve the head of thick yardstick node in survey region, in conjunction with basic function linear list Show thin yardstick head；

(5) combine the gal the Liao Dynasty gold model of Yeh, use in the basic function constructed in above-mentioned steps (2) and step (4) and obtained The head value of study area；Multi-scale remote sensing is directly used to solve Darcy's equation in institute's survey region；Obtain thick yardstick joint The equation group of the darcy seepage velocity on point；

(6) use cholesky decomposition method, try to achieve the darcy seepage velocity on the thick yardstick node of study area；

(7) by the basic function constructed in above-mentioned steps (2) and above-mentioned steps (6) are obtained thick yardstick darcy osmotic flow Darcy seepage velocity on speed linear expression thin yardstick node.

Preferably, the subdivision forming coarse grid cell in above-mentioned steps (1) uses triangular element subdivision.

Preferably, the subdivision forming refined net unit in above-mentioned steps (1) uses triangular element subdivision.

Preferably, in above-mentioned steps (2), (5), the permeability coefficient on refined net unit takes all summits of this unit On the meansigma methods of infiltration coefficient.

Preferably, in above-mentioned steps (3), the infiltration coefficient source sink term value on refined net unit takes all tops of this unit The meansigma methods of the source sink term on point.

Preferably, described Yeh-multi-level finite element modeling method the most also includes: first passes through Multi-scale remote sensing and asks Solve the head numerical value in survey region, then the basic function using Multi-scale remote sensing to be constructed is being studied with obtained head value Direct solution Darcy's equation in district, it is assumed that K_xy=K_yx=0, then the Darcy's equation in x direction is:

$V_x=-K_x\frac{\∂H}{\∂x}---\left(1\right)$

Assume Ψ_IFor the basic function of the Multi-scale remote sensing corresponding to I point, being multiplied by (1) formula two ends can obtain:

$\∫{\∫}_{\mathrm{\Ω}}{V}_x{\mathrm{\Ψ}}_{I}dxdy=-\∫{\∫}_{\mathrm{\Ω}}{K}_x\frac{\∂H}{\∂x}{\mathrm{\Ψ}}_{I}dxdy,I=1,2,\mathrm{...}{N}_n---\left(2\right)$

Wherein N_nIt it is the total interstitial content on study area；

Gal the Liao Dynasty gold model according to Yeh and the basic theories of Multi-scale remote sensing, at any coarse grid cell Δ_ijk On, darcy seepage velocity can be by basic function linear expression:

V_x=V_x(i)Ψ_i+V_x(j)Ψ_j+V_x(k)Ψ_k (3)

(3) formula is substituted into (2) formula, then by discrete on coarse grid cell for (2) formula, obtains the table on each coarse grid cell Reach formula, owing to the basic function of Multi-scale remote sensing can represent by the linear fundament function on each refined net unit, then will Expression formula on each coarse grid cell is discrete on refined net, can obtain on coarse grid about V_xElement stiffness matrix and Right-hand vector, is added to obtain the general equation group of darcy seepage velocity, solves the darcy infiltration that can obtain on the thick yardstick node of study area Flow speed value, more i.e. can get thin yardstick darcy seepage velocity value by (3) formula.

Beneficial effects of the present invention:

The present invention proposes a kind of mode calculating darcy seepage velocity, uses Multi-scale remote sensing to improve Yeh's Gal the Liao Dynasty gold FEM (finite element) model calculates the efficiency of darcy seepage velocity, and the model inheriting Yeh ensure that darcy seepage velocity and The successional feature of flow.Compared with existing classic algorithm, when study area mesh generation is identical, the present invention has higher Computational efficiency, higher precision；Compared with the gal the Liao Dynasty gold finite element of the Yeh of fine dissection, when refined net number of unit is identical, two Method precision is close, but calculation consumption is less than the 1% of Yeh method.The present invention can efficiently, accurately solve multiple in the case of non-all Matter porous media subsoil water darcy seepage velocity field, when processing unsteady fluid flow and non-linear contour computational problem, the method Odds for effectiveness becomes apparent from.

By to the two-dimensional oscillations medium steady flow model under porous media, two-dimensional graded medium transient flow models, two Dimension vibration head steady flow model, two dimension alluvial plain steady flow model, the numerical simulation of Two-dimecnsional steady flow diving dielectric model, Eligible result of the present invention is fine with what analytic solutions were coincide.Compared with the method that several classics solve darcy seepage velocity, in research When district's grid node number is identical, the precision of the thick yardstick Darcy velocity that the present invention obtains is higher；(include thick at total interstitial content Node on grid) equal time, the present invention calculates the in hgher efficiency of thin yardstick Darcy velocity.

Accompanying drawing explanation

Fig. 1 is the study area subdivision schematic diagram of Yeh-multi-level finite element modeling method.

Fig. 2 is the coarse grid cell subdivision schematic diagram of Yeh-multi-level finite element modeling method.

Fig. 3 is two-dimensional oscillations medium steady flow model, each numerical method head value at the y=0.6 rice of cross section.

Fig. 4 is two-dimensional oscillations medium steady flow model, and each numerical method obtainsRelatively missing at the y=0.6 rice of cross section Differential is intended to.

Fig. 5 is two-dimensional graded medium transient flow models, and each numerical method obtainsAbsolute at the y=0.6 rice of cross section Error schematic diagram.

Fig. 6 is two-dimensional graded medium transient flow models, and each numerical method obtainsAbsolute at the y=0.6 rice of cross section Error schematic diagram.

Fig. 7 is the inventive method flow chart.

Detailed description of the invention

For the ease of the understanding of those skilled in the art, the present invention is made further with accompanying drawing below in conjunction with embodiment Bright, that embodiment is mentioned content not limitation of the invention.

Shown in reference Fig. 7, a kind of Yeh-multi-level finite element modeling simulating Water in Porous Medium stream Darcy velocity of the present invention Method, comprises the following steps that

(1) determine boundary condition according to survey region to be simulated, set grid cell yardstick, subdivision survey region, Obtain coarse grid cell, all coarse grid cell subdivisions are obtained refined net unit；

(2) according to infiltration coefficient and the boundary condition of basic function, each coarse grid cell solves the ellipse of degeneration Type problem, determines basic function, forms Finite Element Space；

(3) calculating the element stiffness matrix of coarse grid cell, add up to obtain global stiffness matrix；Border according to survey region, Source sink term, calculates right-hand vector, forms the head equation group of multi-level finite element modeling；

(4) use cholesky decomposition method, try to achieve the head of thick yardstick node in survey region, get final product line in conjunction with basic function Property represents thin yardstick head；

(5) combine the gal the Liao Dynasty gold model of Yeh, use in the basic function constructed in above-mentioned steps (2) and step (4) and obtained The head value of study area；Multi-scale remote sensing is directly used to solve Darcy's equation in institute's survey region；Obtain thick yardstick joint The equation group of the darcy seepage velocity on point；

(6) use cholesky decomposition method, try to achieve the darcy seepage velocity on the thick yardstick node of study area；

(7) by the basic function constructed in above-mentioned steps (2) and above-mentioned steps (6) are obtained thick yardstick darcy osmotic flow Darcy seepage velocity on speed linear expression thin yardstick node.

Wherein, the subdivision forming coarse grid cell in above-mentioned steps (1) uses triangular element subdivision.

Wherein, the subdivision forming refined net unit in above-mentioned steps (1) uses triangular element subdivision.

Wherein, in above-mentioned steps (2), (5), the permeability coefficient on refined net unit takes on all summits of this unit The meansigma methods of infiltration coefficient.

Wherein, in above-mentioned steps (3), the infiltration coefficient source sink term value on refined net unit takes all summits of this unit On the meansigma methods of source sink term.

The present invention first passes through the head numerical value that Multi-scale remote sensing solves in survey region, then uses multi-scale finite The basic function that elements method is constructed and obtained head value direct solution Darcy's equation on study area, it is assumed that K_xy=K_yx=0, then x The Darcy's equation in direction is:

$V_x=-K_x\frac{\∂H}{\∂x}---\left(1\right)$

Assume Ψ_IFor the basic function of the Multi-scale remote sensing corresponding to I point, being multiplied by (1) formula two ends can obtain:

$\∫{\∫}_{\mathrm{\Ω}}{V}_x{\mathrm{\Ψ}}_{I}dxdy=-\∫{\∫}_{\mathrm{\Ω}}{K}_x\frac{\∂H}{\∂x}{\mathrm{\Ψ}}_{I}dxdy,I=1,2,\mathrm{...}{N}_n---\left(2\right)$

Wherein N_nIt it is the total interstitial content on study area；

Gal the Liao Dynasty gold model according to Yeh and the basic theories of Multi-scale remote sensing, at any coarse grid cell Δ_ijk On, darcy seepage velocity can be by basic function linear expression:

V_x=V_x(i)Ψ_i+V_x(j)Ψ_j+V_x(k)Ψ_k (3)

(3) formula is substituted into (2) formula, then by discrete on coarse grid cell for (2) formula, obtains the table on each coarse grid cell Reach formula, owing to the basic function of Multi-scale remote sensing can represent by the linear fundament function on each refined net unit, then will Expression formula on each coarse grid cell is discrete on refined net, can obtain on coarse grid about V_xElement stiffness matrix and Right-hand vector, is added to obtain the general equation group of darcy seepage velocity, solves the darcy infiltration that can obtain on the thick yardstick node of study area Flow speed value, more i.e. can get thin yardstick darcy seepage velocity value by (3) formula.

Below in conjunction with specific embodiment, the present invention will be further explained, is directed to some shorthand notations, is below Explain:

N: the halved number in border, study area；

V_x: the darcy seepage velocity on x direction；

Darcy seepage velocity V on the grid node of study area_x, i.e. thick yardstick darcy seepage velocity；

Darcy seepage velocity V on thick unit grid node_x, i.e. thin yardstick darcy seepage velocity；

FEM: conventional finite elements method；

The conventional finite elements method of FEM-F: fine dissection；

MSFEM-Y:Yeh-multi-level finite element modeling method；

The linear gal the Liao Dynasty gold model of Method-Yeh:Yeh, uses FEM to solve head；

Linear gal the Liao Dynasty gold model (fine dissection) of Method-Yeh-F:Yeh, uses FEM-F to solve head；

Method-Zhang: Cubic Spline Method, uses LFEM to solve head；

The basic function of MSFEM-Y uses oscillating edge movement condition.Additionally, calculating V_xRelative error time, if at certain node V_x, and numerical solution V_x＜ 10^-10, then the relative error of this node is designated as 0%；Otherwise it is designated as 100%.

Embodiment 1: two-dimensional oscillations dielectric model

Study area is a square shaped cells: Ω=[0,1m] × [0,1m], and current equation is:

$-\frac{\∂}{\∂x}\left(K\frac{\∂H}{\∂x}\right)-\frac{\∂}{\∂y}\left(K\frac{\∂H}{\∂y}\right)=W,---\left(4\right)$

Infiltration coefficient is:

$K=\frac{1}{2+P_1\sin \[\mathrm{\π}(x+y)\·P_2\]}---\left(5\right)$

Wherein P₁=1.965, P₂=1.99, source sink term W and determine head boundary and determined by analytic solutions H=xy (1-x) (1-y).

MSFEM-Y, Method-Yeh, Method-Zhang and Method-Yeh-F is used to solve this model.In order to ensure Coarse grid nodes number is identical, and every limit subdivision of study area is all identical by MSFEM-Y, Method-Yeh and Method-Zhang 30 parts (N=30), totally 1800 (2NN) individual triangular element (Fig. 1).Each element subdivision is 25 fine-structure meshes by MSFEM-Y again Lattice unit (Fig. 2).In order to ensure that unit number is identical with the refined net number of unit of MSFEM-Y, Method-Yeh-F is by study area Every limit subdivision is 150 parts (N=150), totally 45000 refined net unit.

Thick yardstick darcy seepage velocity:

The head value of MFEM-Y, Method-Yeh (Method-Zhang) and Method-Yeh-F respectively by MSFEM, FEM, FEM-F tries to achieve, wherein each method cross section y=0.6 rice head value as shown in Figure 3.It can be seen that MSFEM and The head of FEM-F is all sufficiently close to analytic solutions, and precision is higher than the obtained head of FEM.

After trying to achieve head, above-mentioned for use four kinds of methods are solved Darcy velocity by us, wherein MSFEM-Y, Method- Yeh and Method-Zhang calculatesCross section y=0.6 rice average relative error as shown in Figure 4.Can from figure Going out, the curve shape of MSFEM-Y is consistent with Method-Yeh, but precision is higher than Method-Yeh and Method-Zhang.Work as x= When 0.5 meter, the velocity amplitude of analytic solutions is 0, can produce bigger relative error.Now, MSFEM-Y only has one relatively at x=0.5 rice High level error, and Method-Yeh and Method-Zhang is respectively provided with bigger error at two.In the present embodiment, Method-Yeh and Method-Zhang calculate H andCPU time be respectively 3 seconds and 2 seconds.

Thin yardstick darcy seepage velocity

Meanwhile, MSFEM-Y is possible not only to calculateCan also calculateAs in figure 2 it is shown, each coarse grid cell has 21 Individual thin yardstick node, then MSFEM-Y needs to calculate 37800 thin yardstick nodes.Owing to the segment boundary of coarse grid cell is mutual Overlap.In study area only have 22801 unduplicated thin yardstick nodes, therefore we the most only compare MSFEM-Y and Method-Yeh-F is in the value of these 22801 thin yardstick nodes.

Table 1

As shown in table 1, the precision of MSFEM-Y is slightly below Method-Yeh-F.Because MSFEM-Y'sIt is to pass through( Solve on the grid of 30 × 30) and basic function (solving on 5 × 5 grids) obtain, Method-Yeh-F be then 150 × Solve on the grid of 150Meanwhile, the required CPU time of MSFEM-Y is much smaller than Method-Yeh-F, its calculated water head Time is only the 1% of Method-Yeh-F, and the time calculating darcy seepage velocity is only the 0.04% of Method-Yeh-F, aobvious Show that MSFEM-Y has higher computational efficiency.

Embodiment 2: two-dimensional graded medium transient flow models

Study area is a square shaped cells: Ω=[0,1m] × [0,1m], and current equation is:

$S\frac{\∂H}{\∂t}-\frac{\∂}{\∂x}\left(K\frac{\∂H}{\∂x}\right)-\frac{\∂}{\∂y}\left(K\frac{\∂H}{\∂}\right)=W,---\left(6\right)$

Coefficient of permeability K (x, y)=(1+x) (1+y) m/d.Water storage coefficient is 0.1/m, and water-bearing layer thickness is 1 meter, time step A length of 1 day, total time was 5 days.Source sink term W and determine head boundary by analytic solutions: H=xy (1-x) (1-y) e^-tDetermine.

MSFEM-Y, Method-Yeh, Method-Zhang and Method-Yeh-F is used to solve this model.Subdivision is with upper State embodiment one identical, i.e. study area subdivision is 1800 unit by MSFEM-Y, Method-Yeh and Method-Zhang, Each coarse grid subdivision is 25 thin unit by MSFEM-Y, and Method-Yeh-F is by 45000 unit of study area subdivision.

Thick yardstick darcy seepage velocity

Identical with above-described embodiment one, MSFEM-Y and Method-Yeh-F obtains more accurate head than additive method Value, MSFEM-Y, Method-Yeh and Method-Zhang are at y=0.6 riceAbsolute error as shown in Figure 5.Can see Going out, the curve of MSFEM-Y is positioned at below the curve of Method-Yeh and Method-Zhang, and absolute error is much smaller.Display MSFEM-Y has higher computational accuracy than Method-Yeh with Method-Zhang when study area subdivision is identical.

Thin yardstick darcy seepage velocity

Calculated by MSFEM-Y and Method-Yeh-FAt the y=0.6 rice of cross section, absolute error is as shown in Figure 6.Permissible Find out the precision of MSFEM-Y and Method-Yeh-F two method closely, and error amount is the lowest.Due to MSFEM'sIt is ByObtain, comparison diagram 5, Fig. 6, it appeared that MSFEM-Y is similar at the curve shape of two figures.MSFEM-Y in the present embodiment Use 246 seconds calculated water heads, within 3 seconds, calculateWithMethod-Yeh-F then needs 24020 seconds calculated water heads, within 441 seconds, calculatesIdentical with above-described embodiment one, the time needed for MSFEM-Y is less than the 1% of Method-Yeh-F.This result demonstrates MSFEM-Y has higher computational efficiency and precision.In conjunction with above-described embodiment one, it appeared that MSFEM-Y solve astable During flow problem, advantage becomes apparent from.

The concrete application approach of the present invention is a lot, and the above is only the preferred embodiment of the present invention, it is noted that for For those skilled in the art, under the premise without departing from the principles of the invention, it is also possible to make some improvement, this A little improvement also should be regarded as protection scope of the present invention.

Claims (6)

1. the Yeh-multi-level finite element modeling method simulating Water in Porous Medium stream Darcy velocity, it is characterised in that include step Rapid as follows:

(1) determine boundary condition according to survey region to be simulated, set grid cell yardstick, subdivision survey region, obtain All coarse grid cell subdivisions are obtained refined net unit by coarse grid cell；

(2) according to infiltration coefficient and the boundary condition of basic function, the ellipse solving degeneration on each coarse grid cell is asked Topic, determines basic function, forms Finite Element Space；

(3) calculating the element stiffness matrix of coarse grid cell, add up to obtain global stiffness matrix；Border according to survey region, Yuan Hui , calculate right-hand vector, form the head equation group of multi-level finite element modeling；

(4) use cholesky decomposition method, try to achieve the head of thick yardstick node in survey region, thin in conjunction with basic function linear expression Yardstick head；

(5) combine the gal the Liao Dynasty gold model of Yeh, use obtained research in the basic function and step (4) constructed in above-mentioned steps (2) The head value in district；Multi-scale remote sensing is directly used to solve Darcy's equation in institute's survey region；Obtain on thick yardstick node The equation group of darcy seepage velocity；

(6) use cholesky decomposition method, try to achieve the darcy seepage velocity on the thick yardstick node of study area；

(7) by the basic function constructed in above-mentioned steps (2) and above-mentioned steps (6) are obtained thick yardstick darcy seepage velocity line Property represents the darcy seepage velocity on thin yardstick node.

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity the most according to claim 1, its Being characterised by, the subdivision forming coarse grid cell in above-mentioned steps (1) uses triangular element subdivision.

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity the most according to claim 1, its Being characterised by, the subdivision forming refined net unit in above-mentioned steps (1) uses triangular element subdivision.

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity the most according to claim 1, its Being characterised by, in above-mentioned steps (2), (5), the permeability coefficient on refined net unit takes oozing on all summits of this unit The meansigma methods of coefficient thoroughly.

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity the most according to claim 1, its Being characterised by, in above-mentioned steps (3), the infiltration coefficient source sink term value on refined net unit takes on all summits of this unit The meansigma methods of source sink term.

The Yeh-multi-level finite element modeling method of simulation Water in Porous Medium stream Darcy velocity the most according to claim 1, its Being characterised by, described Yeh-multi-level finite element modeling method the most also includes: first passes through Multi-scale remote sensing and solves research Head numerical value on region, then use basic function and obtained head value that Multi-scale remote sensing constructed on study area directly Connect and solve Darcy's equation, it is assumed that K_xy=K_yx=0, then the Darcy's equation in x direction is:

$V_x=-K_x\frac{\∂H}{\∂x}---\left(1\right)$

Assume Ψ_IFor the basic function of the Multi-scale remote sensing corresponding to I point, being multiplied by (1) formula two ends can obtain:

$\∫{\∫}_{\mathrm{\Ω}}{V}_x{\mathrm{\Ψ}}_{I}dxdy=-\∫{\∫}_{\mathrm{\Ω}}{K}_x\frac{\∂H}{\∂x}{\mathrm{\Ψ}}_{I}dxdy,I=1,2,...N_n---\left(2\right)$

Wherein N_nIt it is the total interstitial content on study area；

Gal the Liao Dynasty gold model according to Yeh and the basic theories of Multi-scale remote sensing, at any coarse grid cell Δ_ijkOn, reach Western seepage velocity can be by basic function linear expression:

V_x=V_x(i)Ψ_i+V_x(j)Ψ_j+V_x(k)Ψ_k (3)

(3) formula is substituted into (2) formula, then by discrete on coarse grid cell for (2) formula, obtains the expression on each coarse grid cell Formula, owing to the basic function of Multi-scale remote sensing can represent by the linear fundament function on each refined net unit, then will be every Expression formula on individual coarse grid cell is discrete on refined net, can obtain on coarse grid about V_xElement stiffness matrix and the right side End item, is added to obtain the general equation group of darcy seepage velocity, solves the darcy osmotic flow that can obtain on the thick yardstick node of study area Speed value, more i.e. can get thin yardstick darcy seepage velocity value by (3) formula.