A Novel Boundary-Type Meshless Method for Modeling ...140.121.146.12/wordpress/wp-content/uploads/2019/04/9804291.pdf · collocation,andotherboundarymethods,concludingthat thecollocationTrezmethod(CTM)isthesimplestalgo-

Research ArticleA Novel Boundary-Type Meshless Method for Modeling GeofluidFlow in Heterogeneous Geological Media

Jing-En Xiao, Cheng-Yu Ku , Chih-Yu Liu, and Wei-Chung Yeih

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan

Correspondence should be addressed to Cheng-Yu Ku; [email protected]

Received 3 July 2017; Accepted 18 December 2017; Published 16 January 2018

Academic Editor: Shujun Ye

Copyright © 2018 Jing-En Xiao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. Thenumerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which areexpressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftzmethod and provides a promising solution which integrates the T-Trefftzmethod and F-Trefftzmethod. To deal with the subsurfaceflow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation andthe continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model.The validity of the model is established for a number of test problems. Application examples of subsurface flow problems withfree surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate thatthe proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stabilityfor solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in thehydraulic conductivity.

1. Introduction

Numerical approaches to the simulation of various sub-surface flow phenomena using the mesh-based methodssuch as the finite difference method or the finite elementmethod are well documented in the past [1–5]. Differing fromconventional mesh-basedmethods, the meshless method hasthe advantages that it does not need the mesh generation.The meshless method has attracted considerable attentionin recent years because of its flexibility in solving practicalproblems involving complex geometry in subsurface flowproblems [6–9]. Chen et al. [10] conducted a comprehensivereview of mesh-free methods and addressed that mesh-freemethods have emerged into a new class of computationalmethods with considerable success. Subsurface flow prob-lems are usually governed by second-order partial differentialequations. Problems involving regions of irregular geometryare generally intractable analytically. For such problems,the use of numerical methods, especially the boundary-type meshless method, to obtain approximate solutions isadvantageous.

Several meshless methods have been reported, such asthe Trefftz method [11–16], the method of fundamentalsolutions [7, 17–19], the element-free Galerkin method [20],the reproducing kernel particle method [21, 22], the meshlesslocal boundary integral equation method [23, 24], and themeshless local Petrov-Galerkin approach [25]. Proposed byTrefftz in 1926 [16], the Trefftz method is probably oneof the most popular boundary-type meshless methods forsolving boundary-value problems where approximate solu-tions are expressed as a linear combination of functionsautomatically satisfying governing equations. According toKita and Kamiya [12], Trefftz methods are classified as eitherdirect or indirect formulations. Unknown coefficients aredetermined by matching boundary conditions. Li et al. [14]provided a comprehensive comparison of the Trefftzmethod,collocation, and other boundary methods, concluding thatthe collocation Trefftz method (CTM) is the simplest algo-rithm and provides the most accurate solutions with optimalnumerical stability.

To solve subsurface flow problems with the layered soilin heterogeneous porous media, the domain decomposition

HindawiGeofluidsVolume 2018, Article ID 9804291, 13 pageshttps://doi.org/10.1155/2018/9804291

http://orcid.org/0000-0001-8533-0946http://orcid.org/0000-0002-5077-865Xhttps://doi.org/10.1155/2018/9804291

2 Geofluids

method (DDM) [26] is adopted because the DDM is naturalfrom the physics of the problem to deal with differentvalues of hydraulic conductivity in subdomains. The DDMcan be divided into overlapping domain decomposition andnonoverlapping domain decompositionmethods. In overlap-ping domain decomposition methods, the subdomains over-lap by more than the interface. In nonoverlapping methods,the subdomains intersect only on their interface. One mayneed to use theDDMwhichdecomposes the problemdomaininto several simply connected subdomains and to use thenumerical method in each one. In this study, we adopted thenonoverlapping method to deal with the seepage problemsof layered soil profiles. The problems on the subdomains areindependent, which makes the DDM suitable for describingthe layered soil in heterogeneous porous media.

The subsurface flow problem with a free surface is anonlinear problem in which nonlinearities arise from thenonlinear boundary characteristics [27]. Such nonlineari-ties are handled in the numerical modeling using iterativeschemes [28]. Techniques for solving problems with nonlin-ear boundary conditions have been investigated. Typically,the methods, such as the Picardmethod or Newton’s method,are iterative in that they approach the solution through aseries of steps. Since the computation of the subsurface flowproblem with a free surface has to be solved iteratively, thelocation of the boundary collocation points and the sourcepoints must be updated simultaneously with the movingboundary. Solving subsurface flow with a nonlinear freesurface in layered heterogeneous soil is generally much morechallenging. In addition, the convergence problems oftenarise from nonlinear phenomena. A previous study [28] indi-cates that the Picard scheme is a simple and effective methodfor the solution of nonlinear and saturated groundwater flowproblems. Therefore, we adopted the Picard scheme to findthe solution of the nonlinear free surface.

In this paper, we proposed a novel boundary-type mesh-lessmethod.This pioneering study is based on the collocationTrefftz method and provides a promising solution whichintegrates the T-Trefftz method and F-Trefftz method forconstructing its basis function using one of the particularsolutions which satisfies the governing equation and allowsmany source points outside the domain of interest. To the bestof the authors’ knowledge, the pioneering work has not beenreported in previous studies and requires further research.Two important phenomena in subsurface flowmodelingwereexplored in this study using the proposed method. We firstadopted the domain decomposition method integrated withthe proposed boundary-type meshless method to deal withthe subsurface flow problems of heterogeneous geologicalmedia. The flux conservation and the continuity of pressurepotential at the interface between two consecutive layers canbe considered in the numerical model.Then, we attempted toutilize the proposed method to solve the geofluid flow withfree surface in heterogeneous geological media.

The validity of the model is established for a numberof test problems, including the investigation of the basisfunction using two possible particular solutions and thecomparison of the numerical solutions using different par-ticular solutions and the method of fundamental solutions.

Application examples of subsurface flow problems with freesurface were also carried out.

2. Solutions to the Subsurface Flow Equationin Cylindrical Coordinates

Consider a three-dimensional domain Ω enclosed by aboundary Γ.The steady-state subsurface flow equation can beexpressed as

∇2ℎ = 0 in Ω, (1)

with

ℎ = 𝑓 on Γ𝐷,

ℎ𝑛 =𝜕ℎ𝜕𝑛

on Γ𝑁,(2)

where 𝑛 denotes the outward normal direction, Γ𝐷 denotesthe boundary where the Dirichlet boundary condition isgiven, and Γ𝑁 denotes the boundary where the Neumannboundary condition is given. Equation (1) is also known asthe Laplace equation. In this study, we adopted the cylindricalcoordinate system. In the cylindrical coordinate system, theLaplace governing equation can be written as

𝜕2ℎ𝜕𝜌2

+ 1𝜌𝜕ℎ𝜕𝜌

+ 1𝜌2

𝜕2ℎ𝜕𝜃2

+ 𝜕2ℎ

𝜕𝑧2= 0, (3)

where 𝜌, 𝜃, and 𝑧 are the radius, polar angle, and altitudein the three-dimensional cylindrical coordinate system. ℎis the unknown function to be solved. Considering a two-dimensional domain in the polar coordinate, the Laplacegoverning equation can be written as

𝜕2ℎ𝜕𝜌2

+ 1𝜌𝜕ℎ𝜕𝜌

+ 1𝜌2

𝜕2ℎ𝜕𝜃2

= 0, (4)

where 𝜌 and 𝜃 are the radius and polar angle in thetwo-dimensional polar coordinate system. For the Laplaceequation, the particular solutions can be obtained using themethod of separation of variables.The particular solutions of(4) include the following basis functions:

1, ln 𝜌, 𝜌] cos (]𝜃) , 𝜌] sin (]𝜃) , 𝜌−] cos (]𝜃) , 𝜌−] sin (]𝜃) ,

] = 1, 2, 3, . . . .(5)

The definition of the particular solution in this study isin a wide sense which is to satisfy the homogenous orthe nonhomogenous differential equations with or withoutpart of boundary conditions. If we adopt the solution of aboundary value problem and enforce it to exactly satisfy thepartial differential equation with the boundary conditions ata set of points, this leads to the CTM.

TheCTMbelongs to the boundary-typemeshlessmethodwhich can be categorized into the T-Trefftz method andF-Trefftz method. The T-Trefftz method introduces the T-complete functions where the solutions can be expressed as alinear combination of theT-complete functions automatically

Geofluids 3

r

Source point

(a) A simply connected domain

∞

Source point

(b) An infinite domain with a cavity

Source point

(c) A doubly connected domain

Source point

(d) A multiply connected domain

Figure 1: Illustration of four different types of domain in the CTM.

satisfying governing equations. On the other hand, the F-Trefftzmethod constructs its basis function space by allowingmany source points outside the domain of interest. Thesolutions are approximated by a set of fundamental solutionswhich are expressed in terms of sources located outside thedomain of the problem. The T-Trefftz method and the F-Trefftz method both required the evaluation of a coefficientfor each term in the series. The evaluation of coefficientsmay be obtained by solving the unknown coefficients in thelinear combination of the solutions which are accomplishedby collocation imposing the boundary condition at a finitenumber of points.

The CTM begins with the consideration of T-completefunctions. For indirect Trefftz formulation, the approximatedsolution at the boundary collocation point can be written

as a linear combination of the basis functions. For a simplyconnected domain or infinite domain with a cavity, asillustrated in Figures 1(a) and 1(b), one usually locates thesource point inside the domain or the cavity and the numberof source points is only one for in the CTM [29].

For the doubly and multiply connected domains withgenus greater than one, as illustrated in Figures 1(c) and 1(d),one may locate many source points in the domain. Usually,at least one source point inside the cavity is required. If weconsidered a simply connected domain, the T-complete basisfunctions can be expressed as

ℎ (x) ≈𝑀

∑𝑖=1

b𝑖T𝑖 (x) , (6)

4 Geofluids

x

y

Boundary pointSource point

(a) A simply connected domain


∞

(b) An infinite domain with a cavity


(c) A doubly connected domain


(d) A multiply connected domain

Figure 2: Illustration of four different types of domain in the MFS.

where b𝑖 = [𝐴0 𝐴 𝑖 𝐵𝑖] and T𝑖(x) =[1 𝜌𝑖 cos(𝑖𝜃) 𝜌𝑖 sin(𝑖𝜃)]

𝑇. x ∈ Ω and 𝑀 is the order of

the T-complete function for approximating the solution. Foran infinite domain with a cavity as illustrated in Figure 1(b),one usually locates the source point inside the cavity, and theT-complete functions (negative T-complete set) include

T𝑖 (x) = [ln 𝜌 𝜌−𝑖 cos (𝑖𝜃) 𝜌−𝑖 sin (𝑖𝜃)]𝑇. (7)

The accuracy of the solution for the CTM depends on theorder of the basis functions. Usually, onemay need to increasethe𝑀 value to obtain better accuracy. However, the ill-posedbehavior also grows up with the𝑀 value.

On the other hand, there is another type of the Trefftzmethod, namely, the F-Trefftz method, or the so-calledmethod of the fundamental solutions (MFS) [14]. Insteadof using only one source point and increasing the order ofbasis function, the MFS allows many source points outsidethe domain of interest. The solutions are approximated by aset of fundamental solutions which are expressed in terms ofsources located outside the domain of the problem. Figures2(a), 2(b), 2(c), and 2(d) illustrate the collocation of theboundary and the source points for a simply connecteddomain, an infinite domain with a cavity, doubly connecteddomains, and a multiply connected domain, respectively.

The unknown coefficients in the linear combinationof the fundamental solutions which are accomplished by

Geofluids 5

collocation imposing the boundary condition at a finitenumber of points can then be solved. Due to its singularfree and meshless merits, the indirect type F-Trefftz methodis commonly used. An approximation solution of the two-dimensional Laplace equation using the MFS can also beobtained as

ℎ (x) ≈𝑁

∑𝑗=1

𝑐𝑗𝐹 (x, y𝑗) , (8)

where x ∈ Ω and y𝑗 ∉ Ω and 𝑁 is the number of sourcepointswhich are placed outside the domain.The fundamentalsolution of Laplace equation can be expressed as

𝐹 (x, y𝑗) = −12𝜋

ln (𝜌𝑗) . (9)

𝜌𝑗 is defined as the distance between the boundary point andsource point, and 𝜌𝑗 = |x − y𝑗|. Then we selected a finitenumber of collocation points over the boundary and imposedthe boundary condition at boundary collocation points todetermine the coefficients of b𝑖 and 𝑐𝑗 for the CTM and theMFS, respectively.

For the conventional Trefftz method, the number ofsource points is only one.Theoretically, one may increase theaccuracy by using a larger order of the basis functions [30].Instead of using only one source point and increasing theorder of basis functions, the MFS allows many source pointsbut uses only one basis function, that is, the fundamentalsolution of the differential operator. Onemay be interested toinvestigate a method similar to the MFS which allows manysource points but uses other basis functions.

In the following, we proposed a novel boundary-typemeshless method. This pioneering study is based on thecollocation Trefftzmethod and provides a promising solutionwhich integrates the T-Trefftz method and F-Trefftz methodfor constructing its basis function using one of the particularsolutions which satisfies the governing equation and allowsmany source points outside the domain of interest. Differingfrom the CTM and the MFS, the numerical solutions ofthe proposed method are approximated by a set of basisfunctions which are expressed in terms of source pointslocated outside the domain. An approximation solution of thetwo-dimensional steady-state subsurface flow equation usingthe proposed method can be obtained as

ℎ (x) ≈𝑂

∑𝑗=1

a𝑗P𝑗 (x, y𝑗) , (10)

where x ∈ Ω is the spatial coordinate which is collocated onthe boundary, y𝑗 ∉ Ω is the source point, and𝑂 is the numberof source points which are placed outside the domain. Theunknown coefficients can be expressed as a𝑗 = [𝑎𝑗 𝑏𝑗].P𝑗(x, y𝑗) is the particular solution of Laplace equation. In thisstudy, two different particular solutions of Laplace equationwere adopted as the basis functions. Two possible particularsolutions of Laplace equation can be expressed as

P𝑗 (x, y𝑗) = [𝜌−1𝑗 cos 𝜃𝑗 𝜌

−1𝑗 sin 𝜃𝑗]

𝑇,

P𝑗 (x, y𝑗) = [𝜌−2𝑗 cos 2𝜃𝑗 𝜌

−2𝑗 sin 2𝜃𝑗]

𝑇.

(11)

The determination of the unknown coefficients for the pro-posed method is exactly the same with those in the MFSas described in previous section. We first selected a finitenumber of collocation points x𝑘 over the boundary such that

𝑂

∑𝑗=1

a𝑗P𝑗 (x𝑘, y𝑗) = 𝑔 (x𝑘) , 𝑘 = 1, . . . ,𝑀, (12)

where a𝑗 = [𝑎𝑗 𝑏𝑗] are the constant coefficients to be solved,and 𝑔(x𝑘) is the boundary condition imposed at boundarycollocation points. Considering the boundary conditions, wehave

𝐵ℎ (x) = 𝑔 (x) , (13)

where 𝐵 = 1 represents the Dirichlet boundary condition;𝐵 = 𝜕/𝜕𝑛 represents the Neumann boundary condition.Applying Dirichlet and Neumann boundary conditions, weobtained

ℎ (x𝑘) ≈𝑂

∑𝑗=1

a𝑗P𝑗 (x𝑘, y𝑗) = 𝑔 (x𝑘) ,

𝜕ℎ (x𝑘)𝜕𝑛

≈𝑂

∑𝑗=1

a𝑗𝜕𝜕𝑛

P𝑗 (x𝑘, y𝑗) = 𝑓 (x𝑘) ,

(14)

where 𝑗 = 1, . . . , 𝑂 and x𝑘 ∈ Γ. 𝑓(x𝑘) is the Neumannboundary condition imposed at boundary collocation points.The source points are on the artificial fictitious boundary,which are placed outside the domain to avoid the singularityof the solution at origin. The artificial fictitious boundaryis often chosen as a circle with a radius. However, theposition of source and collocation points may affect theaccuracy. In order to determine the unknowns a𝑗, collocatingthe numerical expansion of (12) at boundary conditionsof (14) at𝑀 boundary collocation points yields the followingequations:

A𝛼 = b, (15)

where A is a matrix which takes values of the solutions at thecorresponding𝑀 collocation points and𝑁 source points,𝛼 =[a1, a2, . . . , a𝑂]𝑇 is a vector of unknown coefficients, and b isa vector of function values at collocation points.

3. Validation of the Proposed Method

3.1. Investigation of the Basis Function. In this example,we adopted two possible particular solutions of Laplaceequation as the basis functions. They are P1𝑗 and P2𝑗 whereP1𝑗(x, y𝑗) = [𝜌

−1 cos 𝜃𝑗 𝜌−1 sin 𝜃𝑗]𝑇

and P2𝑗(x, y𝑗) =[𝜌−2 cos 2𝜃𝑗 𝜌−2 sin 2𝜃𝑗]

𝑇, respectively. In this example, we

verified the accuracy of the proposed method and alsocompared the numerical solution with the MFS. To comparethe results with the analytical solution, we considered thesubsurface flow problem with an exact solution.

For a two-dimensional simply connected domain Ωenclosed by a boundary, the subsurface flow equation can

6 Geofluids

Inner point


kj

kj

Figure 3: The collocation of boundary and source points.

be expressed as Laplace governing equation which can beexpressed as

∇2ℎ = 0 in Ω. (16)

The two-dimensional object boundary under considerationis defined as

Γ = {(𝑥, 𝑦) | 𝑥 = 𝜌 (𝜃) cos 𝜃, 𝑦 = 𝜌 (𝜃) sin 𝜃} , (17)

where 𝜌(𝜃) = 2(𝑒(sin𝜃sin2𝜃)2 + 𝑒(cos𝜃cos2𝜃)2), 0 ≤ 𝜃 ≤ 2𝜋.The analytical solution can be found as

ℎ = 𝑒𝑥 cos𝑦 + 𝑒𝑥 sin𝑦. (18)

TheDirichlet boundary condition is imposed on the amoeba-like boundary by using the analytical solution as shown in (18)for the problem.

Figure 3 shows the collocation point for the boundaryand the source points. To obtain a promising result of thelocation of the source points for the proposed method in thisstudy, a sensitivity study was first carried out. An algorithmsimilar to the study conducted by Chen et al. [31] was adoptedwith scaling of the artificial boundary with the domain size.Assuming the boundary collocation points can be describedas a known parametric representation as follows:

x𝑘 = 𝑟𝑘 (cos 𝜃𝑘, sin 𝜃𝑘) , 𝑘 = 1, . . . ,𝑀. (19)

The source points can also be described as a known paramet-ric representation from the above equation:

y𝑗 = 𝜂𝑟𝑗 (cos 𝜃𝑗, sin 𝜃𝑗) , 𝑗 = 1, . . . , 𝑁, (20)

where 𝜂 is the dilation parameter and is greater than one. 𝜃𝑘and 𝜃𝑗 are the angles of the discretization of the boundaryfor boundary and source points, respectively. 𝑟𝑘 and 𝑟𝑗 arethe radiuses which represent the scale of the domain sizefor boundary and source points, respectively. The sensitivity

2 3 4 5 6 7 8 9 10

Max

imum

abso

lute

erro

r

Boundary pointSource pointInner point

104

103

102

101

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

ＦＨ P1jP2j

k jk j

Figure 4: The accuracy of the maximum absolute error versus 𝜂.

example under investigation is in a simply connected domain.In this example, we investigated the accuracy by choosinglocations of the source points through different 𝜂 values usingthe MFS. Figure 4 shows that 𝜂 = 4 could be the satisfactorylocation of the source points.

Using 𝜂 = 4, we conducted an example to clarify theapproximate number of boundary collocation and the sourcepoints. For simplicity, we took the same number of theboundary points. To examine the accuracy, we collocated1074 uniformed distributed inner points inside the domain,as shown in Figure 5. The maximum absolute error can thenbe found by evaluating the absolute error for each inner point.

Figure 5 depicted the computed results of the maximumabsolute error versus the number of source points. It iswell known that the linear algebraic equation systems maybe ill-conditioned while the global basis functions wereadopted. To clarify this issue, we investigated the conditionnumber versus the number of source points. Figure 6 showsthat the relationship of the condition number versus thenumber of source points for the proposed method and theMFS. For simplicity, we adopted the commercial programMATLAB backslash operator to solve the linear algebraicequation systems. It is found that the proposed methodremains relatively high accuracy compared to theMFS in thisexample. The best accuracy can reach the order of 10−8 whilethe number of source points is greater than 180. On the otherhand, the best accuracy of theMFS can reach only about 10−6in the same example.

3.2. Comparison of the Numerical Results. Similar to theprevious example, we verified the accuracy of the proposed

Geofluids 7

50 100 150 200 250 300 350 400The number of source points

Max

imum

abso

lute

erro

r

104

103

102

101

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

ＦＨ P1jP2j


k jk j

Figure 5: The accuracy of the maximum absolute error versus thenumber of source points.


The c

ondi

tion

num

ber

ＦＨ P1jP2j

1022

1021

1020

1019

1018

1017

1016Boundary pointSource point

Inner point

k jk

j

Figure 6: The condition number versus the number of sourcepoints.

method with the consideration of a complex star-like bound-ary. For a two-dimensional simply connected domain Ωenclosed by a boundary, the subsurface flow equation can

kj

k j

Inner point


Figure 7: The collocation of boundary and source points.

be expressed as Laplace governing equation which can beexpressed as

∇2ℎ = 0 in Ω. (21)

The two-dimensional object boundary under considerationis defined as

Γ = {(𝑥, 𝑦) | 𝑥 = 𝜌 (𝜃) cos 𝜃, 𝑦 = 𝜌 (𝜃) sin 𝜃} , (22)

where 𝜌(𝜃) = 5(1 + (cos(4𝜃))2), 0 ≤ 𝜃 ≤ 2𝜋.The analytical solution can be found as

ℎ = (sinh𝑥 + cosh 𝑥) (cos𝑦 + sin𝑦) . (23)

The Dirichlet boundary condition is imposed on the bound-ary by using the analytical solution as shown in (23) forthe problem. Figure 7 shows the collocation point for theboundary and the source points. A sensitivity study using theMFS was first carried out and 𝜂 = 3 could be the satisfactorylocation of the source points, as shown in Figure 8. Also,to examine the accuracy, we collocated 3250 uniformeddistributed inner points inside the domain, as shown inFigure 9. The maximum absolute error can then be found byevaluating the absolute error for each inner point.

Figure 9 depicted the computed results of the maximumabsolute error versus the number of source points. Figure 10shows the relationship of the condition number versus thenumber of source points for the proposed method and theMFS. It is found that the proposed method remains relativelyhigh accuracy compared to theMFS in this example.The best

8 Geofluids

2 3 4 5 6 7 8

Max

imum

abso

lute

erro

r

104

103

106

105

102

101

100

10−1

10−2

10−3

10−4

10−5

10−6

ＦＨ P1jP2j


k j

k j



Max

imum

abso

lute

erro

r

105106107

104

103

102

101

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

ＦＨ P1jP2j


k j

k j


accuracy can reach the order of 10−6 while the number ofsource points is greater than 150. On the other hand, the bestaccuracy of the MFS can reach only about 10−3 in the sameexample.


The c

ondi

tion

num

ber

1022

1021

1020

1019

1018

1017

1016Boundary pointSource point

Inner point

ＦＨ P1jP2j

k j

k j

Figure 10: The condition number versus the number of sourcepoints.

4. Application of the Proposed Method

4.1. Modeling of Subsurface Flow with Free Surface. The firstapplication under investigation is a free surface seepageproblem of a rectangular dam as depicted in Figure 11.The subsurface flow equation is the Laplace equation. Theexample with the upstream hydraulic head is 24m, thedownstream hydraulic head is 4m, and the width of therectangular dam is 16m. The boundary conditions includesΓ1, Γ2, Γ3, Γ4, and Γ5, as depicted in Figure 11. In Γ2 and Γ5, theDirichlet boundary conditions are given as

ℎ = 𝐻2 on Γ2,

ℎ = 𝐻1 on Γ5.(24)

Based on the Bernoulli equation, we neglected the velocityhead and the total head or the potential can be written as

ℎ = 𝑌 (𝑥) +𝑝𝛾, (25)

where 𝑌(𝑥) is the elevation head, 𝑝 is the pressure head, and𝛾 is the unit weight of fluid. In Γ3 and Γ4, the free surfaceboundaries are given as overspecified boundary conditions as

𝜕ℎ𝜕𝑛

= 0, ℎ = 𝑌 (𝑥) on Γ3 and Γ4. (26)

In Γ1, the no-flow Neumann boundary condition to simulatethe imperious boundary is given as

𝜕ℎ𝜕𝑛

= 0 on Γ1. (27)

Geofluids 9

Seepage domain

Free surface

y

x

Separation point

Seepage face

H1

H2

Γ4

Γ5Γ3

Γ2

Γ1

Ω

Figure 11: Subsurface flow with free surface through a rectangulardam.

Since ℎ = 𝑌(𝑥) is unknown a priori which needs tobe determined iteratively after the initial guess of the freesurface, the proposed method adopted to find the location offree boundary is expressed in the following section.

The subsurface flow with a free surface is a nonlinearproblem in which nonlinearities arise from the nonlinearboundary characteristics. Such nonlinearities are handled inthe numerical modeling using iterative schemes. Typically,the methods, such as the Picardmethod or Newton’s method,are iterative in that they approach the solution through aseries of steps. In this study, the Picard method is adopted.

There are 16 boundary collocation nodes uniformly dis-tributed in the initial guess of the moving boundary withthe spacing of 1m as shown in Figure 12. Figure 12 showsthe computed results using the proposed method. Thereare 132 iterations to reach the stopping criterion using thePicard scheme. The numerical solutions of free surface werethen compared with those obtained from Aitchison et al.[32, 33]. The separation point is the intersection of the freesurface and the seepage face. The location of the separationpoint computed by this study is 13.19m. It is found thatthe computed results agree closely with those from othermethods.

4.2. Free Surface Seepage Flow through Layered HeterogeneousGeologicalMedia. Theprevious examples have demonstratedthat the proposed method can be used to deal with thesubsurface flow with a free surface. Since the appearanceof layered soil in heterogeneous geological media is muchmore common than homogeneous soil in nature, we furtheradopted the proposed method to deal with the subsurfaceflow problems of layered heterogeneous geological mediausing the DDM.

0 2 4 6 8 10 12 14 16

0

2

4

6

8

10

12

14

16

18

20

22

24

y (m

)

x (m)This studyAitchison (1972)Chen et al. (2007)Boundary collocation pointsInitial guess of free surface

Γ4

Γ5 Γ3

Γ2Γ1 4m

16 m

24m

Figure 12: Result comparison of the computed free surface for arectangular dam.

This example under investigation is a rectangular damin layered soil as depicted in Figure 13. We considered theproblem where the upstream hydraulic head is 10m, thedownstream hydraulic head is 2m, and the height and thewidth of the rectangular dam are 10m and 5m, respectively.The boundary conditions including Γ1, Γ2, . . . , Γ10 are shownin Figure 13. At Γ5 and Γ10, the Dirichlet boundary conditionsare given as

ℎ = 10m on Γ5,

ℎ = 2m on Γ10.(28)

At Γ3, Γ4, Γ8, and Γ9, the free surface boundaries are given asoverspecified boundary conditions as

𝜕ℎ𝜕𝑛

= 0, ℎ = 𝑌 (𝑥) on Γ3, Γ4, Γ8, Γ9. (29)

At Γ1 and Γ6, the no-flow Neumann boundary condition tosimulate the imperious boundary is given as

𝜕ℎ𝜕𝑛

= 0 on Γ1, Γ6. (30)

To deal with the geofluid flow through layered heterogeneousgeological media, the domain decomposition method wasadopted. The solution continuity or compatibility between

10 Geofluids

Soil layer 1 Soil layer 2

Soil layer 1 Soil layer 2

Initial guess of free surfaceBoundary points at the interface

(a) (b)

Ω1

Ω1

Ω2

Ω2

Boundary points for Ω1Boundary points for Ω2Source points for Ω1

Source points for Ω2

Γ4

Γ5

Γ3

Γ2

Γ1

Γ7

Γ6

Γ8

Γ9

Γ10

H1H1

H2H2

Figure 13: The collocation of boundary, source points (a) and configuration of boundary condition (b).

different subdomains was assured by remaining equal valuesof the pressure potential and the flux at the interface betweensubdomains. For instance, the free surface seepage flowthrough layered heterogeneous geological media as at Γ2and Γ7, the flux conservation, and the continuity of pressurepotential at the interface between two consecutive layers haveto ensure the solution continuity. Accordingly, the followingadditional boundary conditions must be given:

ℎ|Γ2 = ℎ|Γ7 at Γ2, Γ7,

𝑘1𝜕ℎ𝜕𝑛

Γ2= 𝑘2

𝜕ℎ𝜕𝑛

Γ7at Γ2, Γ7.

(31)

There are two soil layers in this example. The values of thehydraulic conductivity for layer 1 and layer 2 are 𝑘1 and 𝑘2,respectively, and 𝑘1 = 0.1𝑘2 and 𝑘1 = 10−3 cm/s.

In this study, we adopted the nonoverlapping method todeal with the subsurface flowproblems of layered soil profiles.The problems on the subdomains are independent, whichmakes the DDM suitable for describing the layered soil inheterogeneous porous media.

For the modeling of the layered soil, we split the domaininto smaller subdomains in which subdomains were inter-sected only on the interface between soil layers, as shown inFigure 13. For example, there is a problemwith two soil layersas shown in Figure 13.The hydraulic conductivities are 𝑘1 and𝑘2 for soil layer 1 and soil layer 2, respectively. The boundaryand source points were collocated in each subdomain. Atthe interface, the boundary collocation points on left and

right sides coincide with each other. The proposed methodwas then adopted to ensure that flux conservation and thecontinuity of pressure potential at the interface between twoconsecutive layers remain the same.

For the first subdomain, there are a total of 250 boundarycollocation nodes where 50 boundary collocation nodes areuniformly distributed in the initial guess of the movingboundary. For the second subdomain, there are also a totalof 250 boundary collocation nodes where 50 boundarycollocation nodes are uniformly distributed in the initialguess of the moving boundary.

Figure 14 shows the computed results using the proposedmethod.There are 14 iterations to reach the stopping criterionusing the Picard scheme.The numerical solutions of free sur-face were then compared with those obtained from previousstudies [27, 34]. It is found that the computed results agreewell with those from other methods.

4.3. Modeling of Three-Dimensional Subsurface FlowProblem. Because the basis function, 𝑃𝑗(x, y𝑗) =[𝜌−2𝑗 cos 2𝜃𝑗 𝜌

−2𝑗 sin 2𝜃𝑗]

𝑇, is also the particular solution

of the Laplace equation in three-dimensional cylindricalcoordinate system, it implies that the basis function proposedin this study can also be used to solve the three-dimensionalsubsurface flow problems. Accordingly, the last exampleunder investigation is a three-dimensional homogenousisotropic steady-state subsurface flow problem. For a three-dimensional simply connected domain Ω enclosed by a

Geofluids 11

0 1 2 3 4 5x (m)

y (m

)

0

1

2

3

4

5

6

7

8

9

10

This studyLacy and Prevost (1987)Wu et al. (2013)

Figure 14: Comparison of free surface for a rectangular dam inlayered heterogeneous geological media.

boundary as shown in Figure 15, the governing equation isexpressed as

∇2ℎ = 0 in Ω. (32)

The boundary is defined as

Γ = {(𝑥, 𝑦, 𝑧) | 𝑥 = 𝜌 (𝜃) cos 𝜃, 𝑦 = 𝜌 (𝜃) sin 𝜃 sin𝜙, 𝑧

= 𝜌 (𝜃) sin 𝜃 cos𝜙} ,(33)

where 𝜌(𝜃) = 𝑒(sin𝜃sin2𝜃)2 + 𝑒(cos𝜃cos2𝜃)2, 0 ≤ 𝜃 ≤ 2𝜋, and 0 ≤𝑧 ≤ 1.

The analytical solution of the problem is given as

ℎ = 𝑥𝑦𝑧. (34)

The Dirichlet boundary condition is imposed on the bound-ary by using the analytical solution as shown in (34) forthe problem. Figure 15 shows the boundary collocation andthe three-dimensional shape of the problem. A sensitivitystudy was first carried out and 𝜂 = 80 could be the satis-factory location of the source points, as shown in Figure 16.Also, to examine the accuracy, we collocated 889 uniformeddistributed inner points inside the domain. The maximumabsolute error can then be found by evaluating the absoluteerror for each inner point.

Figure 17 depicted the computed results of the maximumabsolute error versus the number of source points. The bestaccuracy of the proposed method can reach the order of 10−9while the number of source points is greater than 350.

04.54 43.5

0.2

3.53 3

X2.5

0.4

Y

2.52 2

Z0.6

1.5 1.51 1

0.8

0.50.5

1

Figure 15: The boundary collocation points of three-dimensionalsubsurface flow problem.

10 20 30 40 50 60 70 80 90 100

Max

imum

abso

lute

erro

r

101

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−10

P2j

0

4.5

4

43.5

0.20.1

3.53

3X

2.5

0.40.3

Y2.52 2

Z0.60.5

1.5 1.51 1

0.80.9

0.7

0.50.5

1

32.5 2


5. Conclusions

This study has proposed a novel boundary-type meshlessmethod for modeling geofluid flow in heterogeneous geo-logical media. The numerical solutions of geofluid flow areapproximated by a set of particular solutions of the subsurfaceflow equation which are expressed in terms of sourceslocated outside the domain of the problem. To deal with thesubsurface flow problems of heterogeneous geological media,the domain decompositionmethodwas adopted.The validityof the model is established for a number of test problems.Application examples of subsurface flow problems with freesurface were also carried out. The fundamental concepts andthe construct of the proposedmethod are addressed in detail.The findings are addressed as follows.

In this study, a pioneering study is based on the col-location Trefftz method and provides a promising solutionwhich integrates the T-Trefftz method and F-Trefftz methodfor constructing its basis function using one of the negative

12 Geofluids

1 × 101

100

1 × 10−1

1 × 10−2

1 × 10−3

1 × 10−4

1 × 10−5

1 × 10−6

1 × 10−7

1 × 10−8

1 × 10−9

1 × 10−10

1 × 10−11

150 200 250 300 350 400 450 500 550 600The number of source points

Max

imum

abso

lute

erro

r

P2j

0

4.5

4

43.5

0.20.1

3.53

3X

2.5

0.40.3

Y2.52 2

Z0.60.5

1.5 1.51 1

0.80.9

0.7

0.50.5

1

32.5 2


particular solutions which satisfies the governing equationand allowsmany source points outside the domain of interest.The proposed method uses the same concept of the sourcepoints in the MFS, but the fundamental solutions can bereplaced by the negative Trefftz functions. It may releaseone of the limitations of the MFS in which the fundamentalsolutions may be difficult to find.

It is well known that the system of linear equationsobtained from the Trefftz method may also become an ill-posed system with the higher order of the terms. In thisstudy, the proposed method integrates the collocation Trefftzmethod and the MFS which approximates the numericalsolutions by superpositioning of the negative particularsolutions as basis functions expressed in terms of manysource points. As a result, only twoTrefftz termswere adoptedbecause many source points are allowed for approximatingthe solution. Meanwhile, the ill-posedness from adopting thehigher order terms for the solution with only one sourcepoint in the collocation Trefftz method can be mitigated. Inaddition, results from the validation examples demonstratethat the proposed method may obtain better accuracy thanthe MFS.

The validity of the model is established for a numberof test problems, including the investigation of the basisfunction using two possible particular solutions and the com-parison of the numerical solutions using different particularsolutions and the method of fundamental solutions. Applica-tion examples of subsurface flow problems with free surfacewere also carried out. Numerical results demonstrate that theproposedmethod is highly accurate and computationally effi-cient. This pioneering study demonstrates that the proposedboundary-type meshless method may be the first successfulattempt for solving the subsurface flow with nonlinear free

surface in layered heterogeneous geological media whichhas not been reported in previous studies. Moreover, theapplication example depicted that the proposed method canbe easily applied to the three-dimensional problems.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This study was partially supported by the Ministry of Scienceand Technology, Taiwan. The authors thank the Ministry ofScience and Technology for the generous financial support.

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