Groundwater modeling using the Radial BasisFunctions method: the drawdown problem in

heterogeneous media

CECILE PIRET∗, NADUN DISSANAYAKE†, JOHN GIERKE ‡

Michigan Technological University, Houghton, MI, USA

AND

BENGT FORNBERG§

University of Colorado,Boulder, CO, USA

April 4, 2018

Abstract

This study develops the use of the Finite Difference-based Radial Basis Func-tions (RBF-FD) method to solve interrelated physical processes represented bypartial differential equations (PDEs) in heterogeneous systems. The focus is ongroundwater flow, representing physicochemical work in a complex geological en-vironment. We wish to combine RBFs’ strengths in representing complex ge-ometries with its ability to accurately discretize differential operators. In the lastdecade, the RBF approach has become a method of choice for the latter, com-peting very successfully against other techniques, such as the finite differencemethod (FDM), the finite element method (FEM) or the finite volume method(FVM), as demonstrated most convincingly for some large-scale, well-establishedtest problems in the geosciences. The most commonly used methods for solv-ing the groundwater flow and contaminant transport equations are: the classicalFDMs, FEMs, FVMs, and the analytic element method (AEM). All of these meth-ods have severe weaknesses in representing commonly occurring heterogeneities.This study has lead to the development of a robust modeling approach that by-passes those issues. RBF-FD, drawdown problem, groundwater flow.

∗Corresponding author. Email: cmpiret@mtu.edu†Email: nldissan@mtu.edu‡Email: jsgierke@mtu.edu§Email: bengt.fornberg@colorado.edu

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1 IntroductionProblem Statement, Pros and Cons of Current Methods and Need for a NewMethod

Konikow started his provocatively titled paper “The Secret to Successful Solute Trans-port Modeling” [28] with an obvious statement: “Modeling subsurface solute transportis difficult ...” He follows this with an outline of the mathematical, theoretical, andpractical reasons for the difficulties. His highly cited paper further concludes that nosingle numerical method works well for all conditions nor works optimally everywherein a particular site setting. The premise of his arguments focuses only on conventionalnumerical methods. We are proposing a new method that will better represent hetero-geneities and anisotropy and lead to better numerical solutions of flow and transport insuch complex systems.

In [4], Bianchi et al. cite several papers that highlight how ”... misrepresenting orignoring connectivity of highly permeable sediments can lead to flawed interpreta-tions of groundwater circulation and contaminant migration.” Like many researcherswho recognize the challenges of including small-scale, but important, heterogeneitiesespecially in contaminant transport simulations, they advocate stochastic approachesfor representing the overall effects of heterogeneities. Stochastic analyses are quitevaluable in gaining insight into the importance of heterogeneities and for characteriz-ing probabilities of forecasted outcomes. The sheer number of scenarios needed forstochastic simulations, however, makes them less practical for exploring remediationstrategies or altering operational configurations of water pumping systems to avoidimpacts. The other most common strategies for solving groundwater equations arethe finite difference method (FDM) ([22, 25, 38]), the finite element method (FEM)([41, 45, 7]), and the analytic element method (AEM) ([13, 21]). All three have ad-vantages and drawbacks. FDM and FEM are versatile and can be used in the presenceof heterogeneity and anisotropy, but both methods give rise to low order accuracy andnecessitate uniformy dense meshes. Majumder et al. report the first application ofthe AEM for simulating contaminant transport, [?]. AEM has been used for over threedecades for simulating groundwater flow. They reiterate how FDM and FEM are ”oftenplagued with artificial oscillations and numerical dispersion.” AEM has the advantageof continuous solutions for velocity and heads but is not conducive to simulating highlyheterogeneous systems [?]. Furthermore, the code is not easily generalizable to threedimensions; in fact, the available 3D models make restrictive simplifying assumptionsin the vertical direction. The current state of the art seems to lie in hybrid techniquesthat combine the analytic element method for modeling the groundwater flow, and ei-ther the finite difference or the finite element method to model the solute transport.

Watson et al. [42] articulate how the ability to extract properties from site investigationdata to be used in the construction of conceptual and mathematical models is criti-cal. The Radial Basis Functions method (RBF) circumvents the issues related to theclassical approaches for building both conceptual and mathematical models, and thedevelopments in this work will likewise address the shortcomings in converting con-

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ceptual models to numerical grids. De Marsily et al. provide in [9] a comprehensivereview on how to account for heterogeneity in subsurface flow and transport model-ing. They argue that facies models are better for representing ”connectivity” of stratawith different permeabilities, ”... which greatly affect[s] flow and, above all, transportin aquifers.” They recommend that future work should focus on improving the faciesmodels. We propose to develop an RBF method that better represents mathematicallynew and detailed facies models in terms of both the conceptual definition of facies ge-ometries and the numerical approximation of the flow of water and transport of solutes.

The RBF Finite difference (RBF-FD): A Numerical Advancement

The radial basis functions method is a meshfree technique that was introduced as aninterpolation scheme in the 1970s before being used to solve partial differential equa-tions in the 1990s. The RBF method has recently taken a new turn with the RBF-FDmethod, the subject of the recent book by Fornberg and Flyer, [18]. This method pro-duces a finite difference-like discretization for scattered nodes, an easy approach forspatial node refinement, and a flexible control over the order of accuracy. The RBF-FD method has been shown to be competitive, sometimes superior to its commonlyused counterparts (including the FDM and the FEM) in some key applications in thegeosciences [18, 17, 31, 19]. We will show that using the RBF-FD constitutes a funda-mental numerical advancement on the current techniques.

Issues the FDM and FEM compared to the RBF framework

• The FEM, like the present RBF-FD approach, achieves great geometric flexi-bility, including the ability for local refinement, as it does not need its nodes tobe lattice based. However, a main difference is that FEM needs its nodes to beconnected into elements, which is quite straightforward in 2-D (e.g. Delaunaytriangularization), but its counterparts in 3-D (ensuring these elements all havefavorable aspect ratios, i.e. never form thin slivers) are very complex and costly.RBF-FD eliminates this bottleneck step. The very fast kd-tree algorithm identi-fies to each node its nearest neighbors, which is all that is needed for its accuratePDE approximations.

• Especially when using the FDM on a regular mesh, the different scalings of theproblem result in the necessity for very small global mesh sizes in order to pre-serve a certain level of accuracy. The differentiation matrices that are producedvia RBFs don’t need a uniform scaling. The RBF discretization thus completelybypasses this issue.

• Constituent elements such as wells can be modeled with logarithmic singulari-ties, which cannot be well approximated by smooth functions such as polynomi-als. However, since the positions of the wells as well as their respective abstrac-tion rates are known, we can account for them a priori in an RBF expansion.

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Issues with the AEM compared to the RBF framework

• The AEM is not available in three spacial dimensions. It is trivial, however, togeneralize RBFs to higher dimensions.

• The AEM relies on the linear superposition of analytic elements in the solution.There are hundreds of these elements of which to keep track. The RBF methoddoes not rely on such a library, although one can equip the RBF expansion withsingularities associated with elements that are present in the problem, such aslogarithmic singularities to account for wells.

• The AEM is limited to simplistic models and performs poorly with heterogeneitiesand anisotropy. The RBF method suffers no such limitation.

2 The Radial Basis Functions Method

2.1 The State of the Art: The RBF-FD MethodThe first suggestion of an RBF-FD type approach appears to be a conference presen-tation by A.I. Tolstykh in 2000 [39]. It was mentioned ’in passing’ in [11], and it wasexplored in more depth in three independent studies in 2003 [37, 40, 43]. During thelast decade, the literature has grown rapidly. The RBF-FD approach offers:

• accuracy levels comparable to high order FD (or FEM) methods,

• particularly straightforward opportunities for local (dynamically adaptive) noderefinements,

• conditional stability of high order explicit time stepping methods for most convection-diffusion as well as for purely convective PDEs, even on irregular node layoutsand when using large (i.e. very accurate) stencils [20],

• excellent opportunities for parallel computing, on hardware ranging from GPUboards to large distributed memory systems [6].

While there have been many RBF-FD related papers in the last decade, the first survey-type articles have appeared only very recently:

• Chapter in Springer’s 2014 edition of Handbook of Geomathematics [17],

• Survey article in Acta Numerica 2015, [19],

• Monograph [18] (2015).

Both global RBF and local RBF-FD approximations are remarkably straightforwardto program. For example about 20-30 lines of MATLAB typically suffice for solvingan elliptic PDE over an irregular 2-D or 3-D domain, and about 40 lines for solvingconvective PDEs over the surface of a sphere. More important than convenience are ofcourse accuracy, robustness, and cost-effectiveness.

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2.2 Common Misconceptions Surrounding RBF-based MethodsThere are still relatively few computational scientists who have had first-hand expe-rience with either RBF or RBF-FD discretizations for large scale PDE applications.Another contributing factor to misconceptions might be that articles in the area stilloften focus on small scale PDE (or ODE) test problems. Although no misconceptionlist can ever be complete, the items noted below appear to be somewhat pervasive. Asis clarified thoroughly in [18], they all lack justification:

• Unclear competitiveness for real-world problems (incl. cost vs. accuracy);

• Lack of fast solvers for resulting algebraic systems;

• General lack of necessary supporting analysis;

• Inherent instability, requiring extended precision arithmetic;

• Numerical PDEs is a solved problem; computer science forefronts are nowadaysis other areas.

2.3 Some Large-Scale Applications of RBFs for PDEsRBFs have in the last decade been applied to a wide range of PDEs, as these arise forexample in fluid mechanics, wave motions, astro- and geosciences, mathematical biol-ogy, elasticity, flame propagation, etc. (cf. for ex. [3, 14, 34]). In particular, thanks tothe research effort led by Dr. Natasha Flyer at NCAR (closely affiliated with the presentPIs and coPI at CU Boulder and at MTU), the RBF approach has advanced from beingmainly just another method that can be made to work on small academic toy problemsto one that can compete successfully against the very best previous approaches on somewell-established large-scale benchmark problems (most strikingly in the geosciences).Several direct accuracy and cost-effectiveness comparisons of RBF-based implemen-tations against all the previously published competitive approaches in the literature forsuch problems are summarized in [14, 16] (containing further references).

In one particularly striking example [2, 44], a fully 3-D time dependent RBF calcu-lation (carried out on a regular PC) revealed a mantle flow instability in a parameterregime for which this was not expected by the geoscience community. A Japaneseteam, using the Earth Simulator (at the time the largest parallel computer system inthe world), then repeated the same case, and were able to confirm the phenomenon.Even though global RBFs were used (which we here advocate to be replaced by lo-calized RBF-FD discretizations), the computational effectiveness was nevertheless sohigh that the PC calculation more than held its own against the industry-standard FDmethod on the Earth Simulator.

Developments in both hardware and numerical methods are critical for further advanc-ing modeling capabilities in the geosciences. In particular, this synergy now suggeststhe implementation of RBF-FD methods on GPUs. The two first RBF implementationson GPUs were carried out by the present MTU PI, with researchers affiliated with thepresent CU Boulder PI’s group, on tsunami modeling [35], and for spherical flows [6].

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Name of RBF Abbreviation DefinitionMultiquadric MQ

√1+(εr)2

Gaussian GA e−(εr)2

Polyharmonic Spline PHS rk (k odd)rk logr (k even)

Table 1: Definitions of some common (global) radial functions.

2.4 A Brief Introduction to the RBF MethodologyThe basic RBF interpolant takes the form

s(~x) =N

∑i=1

λi φ(‖~x−~xi‖) (2.1)

where || · || denotes the Euclidean norm. In order for it to take the values fi at centers~xi, i = 1,2, . . .n, the expansion coefficients λi need to satisfy

A~λ = ~f , (2.2)

where the entries of the collocation matrix A are Ai, j = φ(||~xi−~x j||). We denote thenumerical use of (2.2) followed by (2.1) as the ‘RBF-Direct’ method. The most com-monly used radial functions φ(r) are listed in Table 1, although for this project, wewill concentrate our attention on the polyharmonic spline radial function. The param-eter ε, included in most of these radial functions, is known as the shape parameterand is often seen as an obstacle to using RBFs because it greatly affects accuracy andstability, and because its optimal value, which is problem dependent, can be difficultto estimate. We instead choose a polynomial augmented polyharmonic spline, whoseintrinsic parameter k only has a small impact on the quality of the interpolant. TheRBF method is easily generalizable to higher dimensions. It also has the advantage ofbeing meshfree, allowing the rendition of very complicated domains and boundaries,and an improved accuracy with a node refinement.

The RBF methodology was introduced by R. Hardy around 1970 in connection witha cartography application that required multivariate scattered-node interpolation [23].Although unconditional non-singularity of the RBF linear system matrices was knownlong before in some special cases [5, 36] (where the same matrices arose in other con-texts), the proof in 1986 of guaranteed non-singularity also for MQ RBFs [32] greatlyaccelerated the development of RBFs. Extensive analysis of RBFs on infinite latticeswere also carried out in the 1990s. Two papers by E.J. Kansa in 1990 [26, 27] intro-duced the idea of analytically differentiating (spatial) RBF interpolants, and therebyobtain a novel numerical approach for solving both steady state and time dependentPDEs. This approach is described below.

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2.5 RBFs for PDEsKansa’s original idea [27] consisted in solving time dependent PDEs using the methodof lines (MOL). The spatial differential operators are discretized via RBFs, by develop-ing differentiation matrices, and the resulting system of ODEs is solved in time usinga standard ODE solver.

Computation of a Global Differentiation Matrix

We wish to find a matrix D that discretizes, via an RBF representation, the continuousdifferential operator L. Assuming that the solution takes the function values f , werequire that

f (~xi) =N

∑j=1

λ j φ(∥∥~xi−~x j

∥∥), (2.3)

for all~xi, and it leads to the matrix equation A~λ = f . Analytically applying the differ-ential operator to the radial function gives

g(~xi) =N

∑j=1

λ j Lφ(∥∥~xi−~x j

∥∥), (2.4)

where g(~xi) is the value of the underlying function’s derivative at each ~xi. Thus inmatrix form, B~λ =~g, where Bi, j = Lφ(||~x−~x j||)~x=~xi . The collocation matrix A is un-conditionally nonsingular. This allows us to eliminate the expansion coefficient vector~λ , leading to ~g = BA−1~f . The differentiation matrix D = BA−1 gives an RBF dis-cretization of L.

The RBF-FD Technique for Computing a Differentiation Matrix

Weights in traditional FD formulas for 1-D are determined to provide exact results forpolynomials of as high degree as the stencil size permits [18]. Grid-based formulas inhigher-D are then formed as tensor products of 1-D approximations. For a number ofreasons, this approach generally fails for scattered nodes in more than 1-D. The RBF-FD idea is to then replace multivariate polynomials by either (1), or its generalization

s(~x) =n

∑k=1

λk φ(‖~x−~xk‖)+pm(~x). (2.5)

Here, pm(~x) are multivariate polynomials up to some degree m, and the RBF coef-

ficients obey the constraintsn

∑k=1

λk pm(~x) = 0. It transpires that the resulting RBF-FD

weights then can be obtained by solving a similar linear system to (2) [18, 30]. Forexample, in the case of 2-D and when including only linear polynomials, the weightsw1, ...,wn for approximating an operator L are obtained by solving

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1 x1 y1

A...

......

1 xn yn1 · · · 1x1 · · · xn 0y1 · · · yn

w1...

wnwn+1wn+2wn+3

=

Lφ(||~x−~x1||)~x=~xc...

Lφ(||~x−~xn||)~x=~xc

L1~xc

Lx~xc

Ly~xc

. (2.6)

Here A is the same matrix as in (2), and ~xc is the stencil’s center point. Also systemsof this form are commonly guaranteed to be non-singular, no matter how the n stencilnodes are scattered in d dimensions. In 2-D, RBF-FD stencils might feature n around30-50, and use polynomials of degree m up to around 4. When refining mesh-free nodesets, the polynomial part in (5) will eventually take over from the RBF part. However,the RBFs remain even then essential for providing non-singularity and stability [15].

Thus, for each node ~xi, a cluster of n neighboring points is chosen amongst all thenodes and equation (6) is solved to obtain the finite difference weights associated with~xi. These weights will be entered on the ith row of the global, now sparse, differentia-tion matrix. Building a differentiation matrix with the RBF-FD method is comparableto discretizing a differential operator with the finite difference method. The biggestdifference between the two strategies is the underlying interpolating basis: radial basisfunctions for the first and polynomials for the second. Using radial basis functionsallows us to scatter nodes away from a definite grid, but rather on locations that arepertinent to the geometry of the domain.

3 Quasi-realistic Test Case Example and the Limita-tions of the Current Techniques

Test Case Example No single (or suite of) test case(s) exist for comparable evalua-tions of different groundwater modeling and numerical approximation approaches (C.Langevin, personal communication). The need for common bases for testing has beenrecognized by the National Science Foundation (NSF) and the Consortium of Univer-sity for the Advancement of Hydrological Science, Inc. (CUAHSI) through a seriesof three workshops, the last held in 2011 [12]. It is reasonable to argue that past andnear-future testing of numerical methods for improved approximations of the solutionsfor the groundwater flow equations will always entail comparisons to the MODFLOWfamily of programs (e.g., [10], [33]). In the absence of a well-recognized test case, wehave selected a large realistic problem used to test new grid-structure advancementsfor MODFLOW [29]. Their test cases were configured so as to reflect a quasi-realisticgroundwater aquifer system that included interactions with surface water (i.e., a river)and a municipal-scale pumping well in a heterogeneous one-layer aquifer. They tested

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Figure 1: Model configuration for the proposed test case, obtained from [29]. Theinitial hydraulic head is shown on the left. The logarithmic distribution of the hydraulicconductivity field for the model domain is displayed on the right. In natural systems,hydraulic conductivities will vary within the same geological formation by orders ofmagnitude. The well is located at the center of the domain.

their unstructured grid configuration with areas of local refinement to structured grids.The modeled region is shown in Figure 1. Heads were specified constant on the westand east boundaries at 48 and 40m elevation, respectively, above an arbitrary datum andcausing a natural groundwater flow from west to east with local variations due to theheterogeneous hydraulic conductivity field. No-flow boundaries (default) were usedfor the northern and southern edges of the model. The lone pumping well was locatedin the center of the domain and the withdrawal rate is 1000m3/day (this is equivalentto a pumping well for a community of approximately 2000 people). A uniform aquiferbottom was set at an elevation of −100m, so heterogeneities in the transmissivity fieldwere directly proportional to heterogeneities in the hydraulic conductivity field, whichis depicted in Figure 1. All of the groundwater flow that was satisfying the demandof the pumping well came from aquifer storage and the constant-head boundaries. Norecharge is applied to the model. The domain is subdivided into a matrix of 751×775squared cells of length 25.6m.

Limitations of Current Techniques This test case is ... designed to be a challengingtest for local refinement methods [29]. Grid-based methods are not ideal to representspatial variables that vary abruptly or irregularly. A pumping well causes the poten-tiometric surface (represented by the variable, head, which is the potential energy perunit weight) to change abruptly in the well vicinity. A typical production well mightbe less than 0.5m in diameter and depending on the the aquifer properties, the majorityof the overall head changes due to pumping might occur within meters of the well, yetalso causing significant effects (and being affected by conditions) thousands of metersdistant. Grid-based models, like MODFLOW, might employ mesh refinement near thepumping well cell, allowing for the cells to grow in size with distance from the well.For a single pumping well, this is not often an issue, but in many situations, where thereare more than one pumping well, each of which requiring refinement, the mesh can be

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Figure 2: Drawdown steady-state solution modeled by MODFLOW. The well is locatedat the center of the domain and the thick horizontal and vertical lines crossing at thecenter illustrate the grid refinement needed across the full domain for the algorithm toconverge.

challenging. Moreover, geological heterogeneities and hydrological boundaries (suchas a river) might necessitate grid refinement. Quantitative rules for guiding the meshsize and kind of refinement do not exist for complex groundwater and solute transportmodels and often have to be determined heuristically [1]. Although a highly refinedmesh will certainly add to computation time, it can also lead to convergence issues andoften will result in models that are geometrically simpler. Figure 2 illustrates such aninstance where the refinement is detrimental.

4 Governing EquationsModeling groundwater flow consists, in its simplest form, in solving the followinggoverning equations:

Groundwater Flow Equation

∇ ·T ∇h(x,y,z, t) = S∂

∂ th(x,y,z, t)+Qδwell (4.1)

where h(x,y,z) is the hydraulic head, where T is the hydraulic transmissivity andwhere the water table storage S(x,y,z)≈ constant.

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4.1 Groundwater Drawdown by PumpingThe RBF-FD algorithm that we are introducing is based on a simple subsitution whichremoves the well’s singularity altogether. This removes the need for refinement near thewell, and it is adaptable to heterogeneous aquifers as well, for piecewise constant andcontinuous transmissivities. The governing equation for the drawdown in a confinedaquifer due to pumping from a well whose position and pumping rate are known, hasthe following form

∇.T ∇h = S∂h∂ t

+Qδ (−→x −−→x w) (4.2)

where 0≤ x≤ a, 0≤ y≤ b, S is the storage coefficient,T , the transmissivity and wherethe well, located at the coordinates~xw, has constant abstraction rate Q, and where δ (x)is the Dirac delta function. The boundary conditions are the following

∂h∂y

= 0, north and south borders

h = 0, east and west borders.

4.1.1 Homogeneous Confined Aquifer

We can avoid serious numerical error associated with large gradients by performing asimple substitution to remove the singular term pertaining to the well from the PDE.Since we know the position and the rate of extraction of the well, we can immedi-ately account for it in the solution. The contribution to the solution from the well is− Q

2πT log(r) where r is the distance from the well. In the vicinity of the well, we thenlet the solution have the form

h(~x, t) = v(−→x , t)+Q

2πTlog(‖~x−~xw‖) (4.3)

where v(−→x , t) solves the now singularity-free problem

∇ ·T ∇v = S∂v∂ t

(4.4)

The full problem, illustrated in Figure 3, is stated as follows

T(

∂ 2h∂x2 +

∂ 2h∂y2

)= S

∂h∂ t

+δ(ξ ,η), (x,y) ∈ Γ (4.5)

h = 0, (x,y) ∈ Γ1

∂h∂ t

= 0, (x,y) ∈ Γ2

T(

∂ 2v∂x2 +

∂ 2v∂y2

)= S

∂v∂ t

, (x,y) ∈Ω

v = h+Q

4T πlog((x−ξ )2 +(y−η)2), ∀(x,y)

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-0.5

time = 0.9999

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-0.5

0

0.5

0.6

0.5

0.4

0.3

0.2

0.1

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1

Figure 3: Graphical representation of the local substitution technique illustrated in4.1.1, applied to the groundwater drawdown problem. The approximate solution isshown at t = 1 (we show here −h for better visualization).

The analytical solution to this problem is provided in [8]. The numerical solution isobtained using the method of lines. The RBF-FD formulation detailed above withthe polynomial-augmented polyharmonic spline radial function is used to discretizethe differential operators in space and the remaining system of ordinary differentialequations is solved with the 4th order Runge-Kutta method. The solution at t = 1 isdisplayed in Figure 3.

Figure 4 displays the decay of the relative error with the node spacing h at timet = 10. The decay of the error is as expected, algebraic, and the rates depend on thedegree of the polynomial that has been added to the expansion, but also incidentally,on the number of nodes per cluster. As a rule of thumb, the number of nodes will bechosen as twice the number of terms in the added polynomial expansion [18].

4.1.2 Heterogeneous Confined Aquifer

Heterogeneity is translated in the PDE as a variable transmissivity T . The existingmethods that are capable of handling a variable T treat it as a piecewise constant func-tion. Although it will be important to handle cases for which the media have fun-damentally different properties, such as gravel and clay, a continuous transmissivitywould likely better represent what happens in nature. We described already how toremove the well singularity when the transmissivity is constant. We next propose aremoval of the well singularity when T is continuous. We consider

∇.T (x,y)∇h = S∂h∂ t

+Qδ (−→x −−→x w) (4.6)

where we assume that T (x,y) = T1 +T2x+T3y+T4x2 +T5xy+T6y2. A modified sub-stitution will now be necessary. Let

h(~x, t) = v(−→x , t)− Q2πT

p(x,y) log(‖~x−~xw‖)

12

1/30 1/40 1/50 1/60 1/70 1/80

h

10-7

10-6

10-5

10-4

Rela

tive l

Err

or

p=3

p=4

p=5

p=6

O(h2)

O(h8)

Figure 4: Relative l∞error in solving the homogeneous drawdown problem in a con-fined aquifer. Our method converges algebraically fast as expected [18]

where p(x,y) = p1+ p2x+ p3y+ p4x2+ p5xy+ p6y2, in a local coordinate system withthe well at the origin. Plugging p(x,y) log(‖~x−~xw‖) into ∇.T (x,y)∇h gives r1(x,y)

x2+y2 +

r2(x,y) log(x2 + y2), where r1(x,y) = c1 + c2x+ c3y+ c4x2 + c5xy+ c6y2 and wherer2(x,y) = d1. The whole point of this substitution is to get rid of the singularity at theorigin (the position of the well). It is therefore necessary to impose that r1(x,y) bedivisible by x2 + y2and that r2(x,y) be null. These constraints give rise to a system ofequations whose solution will give the coefficients of p(x,y). Once p(x,y) is known,the system of equations can be divided as in (4.5) into two distinct problems associatedto the inside and the outside of the disk surrounding the well.

Interpolation of the Drawdown Values Transmissivity is usually measured fromsamples taken from inactive wells spread on the site of study. These values can easilybe interpolated with RBFs, giving rise to a smoothly varying transmissivity throughoutthe region. Figure 5 shows on the left, the blocks of piecewise constant conductivity(transmissivity divided by the aquifer thickness) values from our test problem usingMODFLOW. On the right, the conductivity is interpolated using RBFs. Since thisquantity must remain positive, we interpolate log(T ) instead.

Comparison of MODFLOW and RBF-FD Results The hydraulic conductivity val-ues are significant for the structure of the drawdown. Figure 6 shows the MODFLOWdrawdown solutions associated with the piecewise constant conductivity values as in

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Figure 5: Logarithm (base e) of the hydraulic conductivity using MODFLOW (left)and using and RBF interpolant (right). For the latter case, the RBF centers (blue dots)are chosen to be the centers of each block of constant values.

Figure 5 (left) and with smooth interpolated conductivity values as in Figure 5 (right).Although the general structure of the drawdown is similar, there is a clear differencebetween the solutions.

For the sake of a fair comparison, we will use the RBF-interpolated conductivityvalues in the MODFLOW code. For the RBF-FD settings, we choose to augment thepolyharmonic spline with a 6th degree polynomial. We thus expect at least a 5th orderof convergence using the RBF-FD method and therefore a quite sparse set of nodescompared to MODFLOW. The RBF-FD solutions for the head and the drawndown areshown in Figure 7 left and right respectively. The differences in the node distributionscan be seen in Figure 8. MODFLOW uses a dense set of regularly placed nodes, theblue dots in Figure 8 (left), while the nodes can be scattered and much sparser, as seenby the red dots in Figure 8.

Computational Complexity In this simulation, MODFLOW uses 751×775 cells forthe spatial discretization. The time steps are 1.5 days. Our simulation shows similarresults using only 120×120 nodes and time steps of 1 day.

5 ConclusionThe modern history of RBFs originated in 1971 with R.L. Hardy’s discovery of MQRBFs while attempting to draw elevation contour lines on maps, based only on sparseand scattered measurements [23]. In Hardy’s summary in 1990 of the first 20 yearsof RBFs [24], he notes that RBFs already by then had become essential in numerousareas, including geodesy, geophysics, surveying, photogrammetry, remote sensing, sig-nal processing, and hydrology. Around that same time, it was discovered that RBFs,as an internal numerical engine, could greatly enhance neural network simulators, and

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Figure 6: Drawdown solutions for the same hydraulic conductivity values interpolatedin two different manners: as a set of piecewise constant values (left) and as a smoothset of values computed via an RBF interpolation.

Figure 7: RBF-FD Solution for the heterogeneous drawdown problem on a 120×120node domain. The final hydraulic head at t = 1000 days is shown on the left and thedrawdown is shown on the right. The backward difference method was used for solvingthe system of ODEs in time with a time step dt = 1day.

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-0.4

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0

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Figure 8: The left plot shows a portion of the different node distributions. The bluedots (MODFLOW cells) are densely and regularly placed while the red dots are sparseand can be scattered (RBF-FD nodes). The figure on the right shows the difference inmeters between the computed hydraulic head values using MODFLOW and RBF-FD.

also that they could be very effective for solving PDEs. These two discoveries dra-matically increased further still the range of applications for RBFs. Especially in thelast decade, publications of engineering applications of RBFs have skyrocketed, andthere is not space here for a meaningful survey (see for ex. [?]). The very recent tran-sition from global RBF discretizations to those of RBF-FD type extend again greatlythe range of applications in which the approach has become the preferred choice (oris poised to become it). Present growth areas include seismic exploration, geophysicalfluid dynamics, tsunami modeling (forecasting), and PDEs over general surfaces.Accurate modeling of groundwater flow and solute (contaminant) transport is a societalneed that will have far reaching impact for both industry and the environment. A bettergrasp of pollutant plumes from factories or industrial activities is needed in order tobetter localize clean up, for example.

In this study, we develop a Finite Difference-based Radial Basis Functions (RBF-FD) method to solve interrelated physical processes represented by partial differen-tial equations (PDEs) in highly complex heterogeneous and anisotropic systems. Thefocus is on groundwater flow, representing physicochemical work in a complex geo-logical environment. The key of this project is to combine RBFs’ strengths in repre-senting complex geometries with its ability to accurately discretize differential opera-tors. In the last decade, the RBF approach has become a method of choice for the lat-ter, competing very successfully against other techniques, such as the finite differencemethod (FDM), the finite element method (FEM) or the finite volume method (FVM),as demonstrated most convincingly for some large-scale, well-established test prob-lems in the geosciences. The most recent development, and the focus of the presentproposal, shifts from global RBF approximations to local, more cost-effective RBF-FD discretizations. The most commonly used methods for solving the groundwater

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flow and contaminant transport equations are: the classical FDMs, FEMs, FVMs, andthe analytic element method (AEM). All of these methods have severe weaknesses inrepresenting commonly occurring heterogeneities. Furthermore, current methods formany geological-process models all rely on interpolated layers of alternating higherpermeability aquifer and discontinuous lower permeability confining materials that areextremely challenging in simulating. The interpolation methods are chosen ad-hocbetween kriging, inverse distance weighted (IDW), or linear interpolation, whicheverworks best visually. This study has lead to the development of a robust modeling ap-proach that bypasses those issues and that utilizes the RBF-FD method in an integratedmanner to both represent the complex system geometries and the physical processes.

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