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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
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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
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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)
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4 Geofluids
x
y
Boundary pointSource point
(a) A simply connected domain
Boundary pointSource point
∞
(b) An infinite domain with a cavity
Boundary pointSource point
(c) A doubly connected domain
Boundary pointSource point
(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
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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
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6 Geofluids
Inner point
Boundary pointSource 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
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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
Boundary pointSource pointInner point
k jk j
Figure 5: The accuracy of the maximum absolute error versus
thenumber of source points.
50 100 150 200 250 300 350 400The number 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
Boundary pointSource 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
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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
Boundary pointSource pointInner point
k j
k j
Figure 8: The accuracy of the maximum absolute error versus
𝜂.
50 100 150 200 250 300 350 400The number of source points
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
Boundary pointSource pointInner point
k j
k j
Figure 9: The accuracy of the maximum absolute error versus
thenumber of source points.
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.
50 100 150 200 250 300 350 400The number of source points
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
Figure 16: The accuracy of the maximum absolute error versus
𝜂.
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
Figure 17: The accuracy of the maximum absolute error versus
thenumber of source points.
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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