A mass-conservative method for the integration of the two-dimensional groundwater (Boussinesq) equation


  • E. Cordano,

    Corresponding author
    • Dynamics in the Agro-Ecosystems Technological Platform Sustainable Agro-Ecosystems and Bioresources Department, IASMA Research and Innovation Centre, Fondazione Edmund Mach, San Michele all'Adige, Trento, Italy
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  • R. Rigon

    1. Il Dipartimento e' ora: ingegneria Civile, Ambientale e Meccanica, Università di Trento, Trento, Italy
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Corresponding author: E. Cordano, Dynamics in the Agro-Ecosystems Technological Platform Sustainable Agro-Ecosystems and Bioresources Department, IASMA Research and Innovation Centre – Fondazione Edmund Mach, Via E. Mach 1, San Michele all'Adige, Trento I-38010, Italy. (emanuele.cordano@gmail.com)


[1] This work presents a new conservative finite-volume numerical solution for the two-dimensional groundwater flow (Boussinesq) equation, which can be used for investigations of hillslope subsurface flow processes and simulations of catchment hydrology. The Boussinesq equation, which is integrated for each grid element, can take account of the local variations of topography and soil properties within the individual elements. The numerical method allows for wetting and drying of the water table, without “ad hoc assumptions.” The stability and convergence of the method is shown to be guaranteed a priori by the properties of the solver itself, even with respect to the boundary conditions, an aspect that has been neglected in previous literature. The numeric solutions are validated against some approximate analytical solutions and compared to those of another (1-D) numerical solver of the Boussinesq equation. The solver capabilities are further explored with simulations of the Panola experimental hillslope where the bedrock topography, which is accurately known, causes complex wetting and drying patterns; in this situation, the importance of a two-dimensional description of subsurface flows to obtain properly simulated discharges becomes clear. Finally, a comparison is made between the results of the presented algorithm and the output of the GEOtop hydrological distributed model, which simulates variably saturated soils; the findings of the comparison are discussed.

1. Introduction

[2] Subsurface water flows play a fundamental role in determining the partition between surface and subsurface water flows. In the past, parameterizations of groundwater flows in hillslope and catchment hydrology were often developed on the basis of conceptualized approaches [Beven and Kirkby, 1979; Barling et al., 1994; Troch et al., 2003, 2004; Akylas and Koussis, 2007; Harman and Sivapalan, 2009, and references therein] which were mainly aimed at the identification of runoff source areas and the estimation of discharge at the hillslope toe. In these studies, steady-state approaches were often used, but they have proven to be insufficient for many purposes, especially revealing the real patterns of soil moisture distribution [Western and Grayson, 1998]. Therefore, in subsequent researches, these have been modified to introduce more dynamic conditions [e.g., Barling et al., 1994; Chirico et al., 2003] and/or effects due to the downslope topography [e.g., Hjerdt et al., 2004; Lanni et al., 2011]. Eventually, these latter studies also proved to be of interest in determining hillslope stability [e.g., Dietrich and Montgomery, 1994; Iverson, 2000; D'Odorico et al., 2005] by coupling the simplified hydrology with geomechanics.

[3] In the more traditional contexts of groundwater analysis, analytical solutions for unconfined groundwater flow have been produced for suitably simplified conditions and simple geometries [Hantush, 1967; Barenblatt et al., 1990; Fan and Bras, 1998; Lockington et al., 2000; Parlange et al., 2000; Kim and Ann, 2001; Rai et al., 2006; Song et al., 2007; Telyakovskiy et al., 2010]. These studies have been a valuable reference in understanding the processes dynamics; however, they have been less useful in exploiting the information generated by the growing knowledge of catchment topography.

[4] Very rarely do hillslope hydrologists use complete three-dimensional (3-D) numerical models [e.g., as those in Harbaugh et al., 2000; Panday and Huyakorn, 2004], developed in mainstream groundwater studies, and they have always looked for the greatest algorithmic simplicity. In hillslope literature, the search for conceptualized or analytical methods has been justified by the avoiding the computational burden of solving the complete 3-D form of the groundwater equations, which seemed too great, but also by the perception that hillslopes water table dynamics has different characteristics than in aquifers, and, therefore, requires a special care. In fact, soils can more frequently dry than large-scale aquifers, and the presence of rugged bedrocks and shallow perched water tables can bring to disconnected patterns of soil moisture [Tromp-van Meerveld and McDonnell, 2006]. This is called wetting-and-drying phenomenon and has been studied thoroughly in surface water literature [Stelling and Duynmeyer, 2003]. However, it has been less investigated in groundwater literature, and, for instance MODFLOW [Harbaugh et al., 2000] has been shown to fail to in modeling it [Doherty, 2001; Werner et al., 2006; Sokrut et al., 2007; Painter et al., 2008].

[5] Overall, considering the various approaches, it can be verified that there is a gap in literature: there are many 3-D solvers for the scopes of groundwater analysis, which are possibly unsuitable to model hillslope problems; there are contributions that investigate the one-dimensional (1-D) version of the groundwater equation, in which the lateral width of a hillslope is appropriately parameterized [e.g., Troch et al., 2003, 2004, and references therein]. But much fewer, and much more recent, are the two-dimensional (2-D) solvers [Rocha et al., 2007; Harman and Sivapalan, 2009; Cayar and Kavvas, 2009; Dehotin et al., 2011].

[6] This paper tries to support the statement that a 2-D simplification of groundwater flow, based on a new numerical method, can answer the needs of spatial information for hillslope hydrology (in relation to hillslope-toe discharge and soil saturation patterns—therefore contributing also to some hillslope stability problems) and for modern catchment hydrology.

[7] To obtain this result, the paper implements a new and clean numerical method that merges the achievements of Brugnano and Casulli [2008] and Casulli [2009], and, in particular, extends the latter (which deals with free surface waters) to the case of Darcian flow.

[8] The paper is organized as follows: in section 2, the form of the Boussinesq equation and its terms are discussed; in section 3, the equation is discretized according to the new conservative scheme; in section 4, the effect of boundary conditions on the structure of the solver is presented; in section 5, the model is validated against appropriate analytical and numerical solutions, and the model is applied to the Panola experimental site to confirm its behavior at the presence of rugged terrain. In the same section the model's results are also compared to those of GEOtop.

2. Boussinesq's Groundwater Equation

[9] The governing equation for groundwater flow used in this paper has been derived from the hydraulic theory of unconfined, saturated, groundwater flow in a sloping aquifer. It is based on Darcy's law and the continuity equation (soil water and groundwater budget). It was first introduced by Boussinesq [1877] and it will be referred throughout the paper as BEq. It has been applied by many authors in studies of free aquifers and pumping [e.g., Manglik and Rai, 2000; Don et al., 2005; Rai et al., 2006] and in the study of recession curves of a hillslope draining into a surface water body [e.g., Childs, 1971; Brutsaert, 1994; Troch et al., 2003]. The integration of the BEq is finally gaining increased interest in relation to coupled surface/subsurface models [e.g., Pruess et al., 1999; Panday and Huyakorn, 2004; Kollet and Maxwell, 2006; Rigon et al., 2006].

[10] Recently, Hilberts et al. [2005, 2007] generalized the BEq, including unsaturated groundwater flow, using the concept of drainable porosity, which can be applied to studies of hillslope hydrology, runoff production, and landsliding; Cordano and Rigon [2008] were able to derive it from simplifications of comprehensive 3-D Richards' equation.

[11] The BEq applied to a generic hillslope has the following 2-D form:

display math(1)

where η [L] is the unknown piezometric head (water-table elevation); x, y are planimetric Cartesian coordinates; t is time; inline image is the total water volume stored in a soil column per planimetric unit area; inline image is the thickness of the aquifer, which is a function of η and space since it is defined as inline image where inline image is the bedrock elevation; inline image[L−1] is the divergence operator; inline image [L−1] is the gradient operator; Q [L T−1] is a source term, which also accounts for boundary conditions (as explained in the next sections); and KS [L T−1] is the saturated hydraulic conductivity. Multiplying inline image [L] by the domain area gives the total volume of water stored. As shown in Cordano and Rigon [2008], inline image can be calculated as the integral of volumetric water content over the vertical depth, assuming a vertical hydrostatic distribution of soil water pressure:

display math(2)

where the z axis is positive upward, zs is the elevation of the terrain surface, and inline image is the volumetric soil water content, variable in space. Then, the drainable porosity [dimensionless] is calculated by deriving inline image with respect to η:

display math(3)

[12] Remarkably, while inline image and inline image are always positive quantities, inline image can be lower than bedrock elevation and even assume negative values, when a water table does not exist over the bedrock. This fact is crucial in modeling the wetting-and-drying phenomenon, and, when not recognized, implies the introduction of tricks to adjust the numerical solution [for instance fixing a minimum low water level, as in Painter et al., 2008].

[13] Equation (1) is written in conservative form, i.e., the time derivative is with respect to the volume of total stored water per unit area, which is the conserved quantity, and the second addendum of the equation contains the divergence of mass fluxes. By applying definition (3) and the derivation chain rule to the left-hand side of equation (1), BEq can be rewritten in the form usually found in papers and manuals [e.g., Bear, 1972; Harbaugh et al., 2000; Painter et al., 2008]:

display math(4)

[14] However, writing of the equation in either form (4) or (1) is not the same thing in terms of numerical resolution [Celia et al., 1990]. In fact, at the edge of the wet zones, the water level inline image is not a smooth function and the two equations are, therefore, not equivalent [e.g., Leveque, 2002]. In practice, in order to conserve mass in the presence of wetting and drying, other authors usually introduce “ad hoc” iteration procedures for control [called outer iteration procedure, a “type of damping or under-relaxation,” in Painter et al., 2008] which are unnecessary. Actually, other authors [Dehotin et al., 2011, and literature cited therein] even use a different “Boussinesq” equation, where the coefficient within square brackets, i.e., inline image, the water-table transmissivity, is set not dependent on z, i.e., on the water table level η, and no mention of the concept of “drainable porosity” or “specific yield” is made. The latter case, even if it seems to reproduce standard groundwater case studies in a fairly reasonable manner, cannot account properly for wetting and drying because of nonvanishing transmissivity in dry cells. To solve a generic free-surface hydrodynamic problem, i.e., the shallow water equation (SWEq) or the BEq, Brugnano and Casulli [2008] found a rigorous numerical method. Such a method is conservative, can simulate wetting or drying fronts, and is the one used in the following sections. This paper, in fact, also extends Casulli's [2009] method to the case of using equation (1), in order to account for subgrid heterogeneity of topography and hydraulic conductivity.

3. Application of a Mass-Conservative Scheme to the Boussinesq Equation

[15] In this section, we show that equation (1) can be cast as a system of the form:

display math(5)

where inline image is the array of the unknown quantities (one for each of the Np grid cells), inline image is the array of known parameters, and inline image are the values of the field conserved quantity, i.e., the water volume stored in the ith cell, a nonlinear function of η. All these “vectors” are denoted by the symbol inline image, or harpoon, to distinguish them from space vectors, denoted by inline image. In this section, we also show that the discretizations adopted endow the matrix inline image the properties, further specified in Appendix The Structure of the Numerical Solver Utilized, that make the system numerically solvable with the use of a Newton method [e.g., Kelley, 2003] and the conjugate gradient algorithm [e.g., Schewchuk, 1994] having convergence to solutions guaranteed a priori [Brugnano and Casulli, 2008].

[16] The discretization of equation (1) requires three steps: (1) definition of the grid, (2) integration of equation (1) over the ith cell (in space), and (3) choosing a time marching technique (discretization in time).

3.1. Properties of the Grid

[17] The integration domain of equation (1) needs to be divided into Np convex, nonoverlapping polygons (grid elements), delimited by a total number of Nl sides (or edges), joining in Nv vertices (such that inline image). In the example in Figure 1, there are four polygons, 12 sides, and nine vertices.

Figure 1.

Graphical representation of the one-dimensional problem used for the comparison between numerical and linearized analytical solutions of BEq. The soil is bounded by two reservoirs where water surface elevations are inline image and inline image at x=0 and x=L, respectively. An intermediate dynamical state is represented.

[18] In order to simplify the estimation of fluxes between cells, the polygons are organized in an orthogonal grid [Casulli and Walters, 2000]. Such a grid is drawn so that the segment joining the centers of two adjacent polygons and the edge common to them both have a nonempty intersection and are orthogonal to each other.

[19] The topological connection of these geometric elements can be described through the use of an adjacency matrix of dimensions inline image, composed by as many rows as there are polygons in the grid and as many columns as there are sides [e.g., Cormen et al., 2001, section 22.1, pp. 527–531], such that:

display math(6)

[20] Aij is a sparse matrix and it is usually implemented as a data structure in the form of an adjacency list that can be represented in a very compact way, only occupying the necessary contiguous space in computer memory [e.g., Davis, 2006, and references therein]. The sum of each row of the adjacency matrix gives the number of sides of each polygon. The sum of each column gives at most two, since an edge is shared by two polygons at most. The edges of the ith polygon can then be grouped into the following subset:

display math(7)

[21] In practice, all the edges and all the polygons are progressively numbered and ordered into two arrays with Nl and Np elements, respectively. Each polygon has, as an attribute, a vector containing the indexes of its edges, i.e., the elements of inline image.

[22] It is also necessary to assess the correspondences between two adjacent polygons and their shared edge, i.e., given the polygon i and the edge inline image, what is the index r so that inline image when inline image. Since A has at most two nonzero entries per column, the polygon r, if it exists, is unique. Thus, a function between inline image and r, such that inline image, can be found by inspecting the following inline image matrix:

display math(8)

[23] Since inline image is a sparse matrix with few nonzero entries, only the non-null elements of inline image are saved in the actual implementations. Each polygon contains, as an attribute, an array with the values of inline image related to each inline image. If inline image and inline image, the jth edge belongs to the boundary. For the next computations, it is useful for each ith polygon to define the subset inline image as the subset of the edges which are not part of the boundary:

display math(9)

[24] The subset inline image will be used later for the discretization of the divergence term in equation (1).

3.2. Spatial Discretization

[25] Integrating equation (1) over the ith polygon, one obtains:

display math(10)

where pi is the planimetric area of the cell. Therefore, observing that the volume of stored water in the ith cell is:

display math(11)

given that hw is the water volume per unit area between the bedrock and the free surface level (η) as defined by equation (2), and then applying the divergence theorem, the following integrated form of the BEq is obtained:

display math(12)

where inline image is the length of jth edge, inline image is the differential along the jth edge, inline image is the outcoming versor orthogonal to the jth edge, and inline image is the derivative (gradient) of η, estimated at the jth edge and orthogonal to it. Remarkably, it can be noticed from equation (11) that the total stored volume is dependent on the spatial variability of hw within the cell and, therefore, on the subgrid variability of the drainable porosity, the bedrock and surface elevation. Due to the orthogonality of the grid, the spatial gradient component can be approximated with the finite difference:

display math(13)

where inline image [L] is the distance between the centers of the ith and inline imageth polygons. Thus, equation (12) is transformed into:

display math(14)

where inline image is a transport coefficient [L 3T−1] along the jth edge estimated with an upstream weighting scheme as follows [e.g., Painter et al., 2008]:

display math(15)

[26] The main advantage of this upstream weighting estimator, inline image, is that it prevents water from leaving a nearly dry cell and allows water to flow to initially dry cells during a flooding [e.g., Painter et al., 2008]. However, it can also be easily modified on the basis of knowledge of the local variability of hydraulic transmissivity (the integrand) in the volumes under consideration. It is remarkable that inline image is a property of the jth edge and is symmetric with respect to the level of η in adjacent cells, i.e., inline image. If the variations of hydraulic conductivity and porosity in space are known, for instance with a stochastic theory of the medium [e.g., Dagan, 1989], the above integral can be estimated when the cells are big enough to ensure ergodicity with a Monte-Carlo method.

3.2.1. Time Discretization

[27] Each term in equation (14) depends on time (which has been left implicit so far), and it can be discretized in a semi-implicit way as follows:

display math(16)

where inline image is the time step and all the superscripts indicate the time instant. The transport coefficient inline image is estimated at time n (and is therefore known), as is inline image [L2T−1], the water-table recharge (or a sink) averaged over the whole ith polygon. The gradient is treated implicitly and the solution of equation (16) must be sought by solving an algebraic nonlinear system:

display math(17)

for inline image to Np. The system in equation (17) is a particular case of equation (5) where inline image is replaced with inline image and the following equalities are taken:

display math(18)


display math(19)

[28] From equation (18), inline image results to be a symmetric and positive semidefinite matrix, which satisfies the T2 property, defined in the Appendix, with diagonal entries:

display math(20)

and off-diagonal entries:

display math(21)

where inline image is the Kronecker delta symbol. It can be shown that, in each row of inline image, the sum of the entries is zero:

display math(22)

[29] Finally, equation (17) is rewritten with the index notation as follows:

display math(23)

[30] From equation (18), it follows that the unit vector, inline image, is the eigenvector of inline image associated to the null eigenvalue. This happens when the hydraulic head is uniformly distributed and, according to Darcy's law, there is no flux.

[31] The stability and convergence to the exact solution is verified according to the conditions (A9) or (A10) (see Appendix The Structure of the Numerical Solver Utilized).

[32] However, it can happen that inline image is reducible (and then inline image for a certain value of i and inline image for certain adjacent cells i and l). In this case, no water flux occurs through the sides of some cells and there is a disconnected “wet” domain. Consequently, the matrix inline image has as many eigenvectors inline image corresponding to the zero eigenvalue as there are wet domains, i.e., groups of connected cells. The condition for each group of connected cells (to be compared with equation (A9)) becomes:

display math(24)

where d is the index of the group of connected cells inline image.

4. Boundary Conditions

[33] Equation (17), which represents the water balance of a generic cell, is solved by coupling with boundary conditions which can be either flux boundary conditions (Neumann type) or head boundary conditions (Dirichlet type). In the following, we analyze how each choice of boundary condition modifies the integration properties of the system and, in particular, requires the modification of equation (17).

4.1. Flux-Based Boundary Condition (Neumann)

[34] Neumann boundary conditions assess a positive or negative water flux along a certain part of the boundary. It consists in the introduction of a further source/sink term for the cells that are adjacent to the boundary of the integration domain and can be accounted for with the following reformulation of the known term bi:

display math(25)

where inline image is the subset of the edges of the ith polygon belonging to the boundary and inline image is the outgoing water flux (water discharge per unit vertical area) [L T−1] normal to the boundary. The new source/sink term affects the numerical stability of the method; therefore, equation (24) needs to be modified to account for the outgoing water flux as follows:

display math(26)

4.2. Head-Based Boundary Conditions (Dirichlet)

[35] Dirichlet boundary conditions assign the time-variable value for η at some boundary or internal cells according to a known function. Therefore, for any i cell belonging to the boundary:

display math(27)

where inline image is an external forcing known a priori. Such i cells are called “Dirichlet Cells” (DC) in this paper. In this case, the nonlinear system to be solved is formed by equation (27) for the “DC” and by equation (23) for the other cells. Therefore, such a system has a nonlinear part similar to equation (5), i.e., one resulting in a monotonic function of η but also having a linear part which is not symmetric. In fact, for a cell adjacent to a DC, the left-hand side of equation (23) depends on values from the DC. On the contrary, equation (27) does not contain any unknown values from neighboring cells. Because this asymmetry makes the new algebraic equation system different from equation (5), the procedure described in Appendix The Structure of the Numerical Solver Utilized to solve it may not work correctly.

[36] It is possible to avoid this problem by splitting the matrix inline image, defined by equation (18), into two components:

display math(28)

where inline image and inline image are both symmetric matrices constructed as follows:

display math(29)


display math(30)

[37] The matrix inline image contains information about the connection between cells not belonging to the boundary, whereas inline image assumes nonzero values only in correspondence of DC. Then, equation (23) becomes:

display math(31)

[38] Equation (31) is applied to non-Dirichlet cell i; however, inline image can assume non-null values only if j refers to a DC, therefore equation (31) can be rearranged, using equation (27), as:

display math(32)

where inline image, related to DC domain, moves to the right-hand side because it is known.

[39] The final solver of the Boussinesq equation is represented by the system (32) for all the non-DC. The matrix inline image satisfies the appropriate T1 or T2 properties defined in Appendix The Structure of the Numerical Solver Utilized and then equation (32) is solved with the procedure described in Appendix The Structure of the Numerical Solver Utilized.

5. Test Cases

[40] Analytical solutions of the general nonlinear Boussinesq equation do not exist. Existing solutions are either approximations, usually obtained through a linearization of the flux term [e.g., Brutsaert, 1994; Troch et al., 2004], which corresponds to assuming a suitably constant value of the hydraulic transmissivity inline image, or they are related to particular cases. For instance, in the case of 1-D wetting processes, nonlinear analytic solutions exist based on similarity analysis [Barenblatt et al., 1990; Lockington et al., 2000] and power series expansion [Song et al., 2007]. An implementation of these analytical solutions is available as an R package [Cordano, 2011]. There are also 2-D analytical solutions for the linearized Boussinesq equation, obtained according to Hantush's approximation, which considers water table dynamics in a 2-D rectangular domain with horizontal bedrock and pumping and/or bottom leakage processes [Manglik and Rai, 2000; Rai et al., 2006].

[41] However, in order to assess the performance of our solver, we have chosen: (i) a 1-D time variable analytical solution of the linearized equation on planar horizontal topography with a sudden increase of water head at the boundary together with the exact steady-state solution relative to the same problem [e.g., Bear, 1972, exercise 7.27]; (ii) a 1-D solution on planar topography developed by Song et al. [2007] that derives from a similarity assumption and that refers to the fully nonlinear wetting process; (iii) the 1-D width-averaged numerical solution proposed by Troch et al. [2004] known as the “Hillslope storage Boussinesq” model (HsB) [Troch et al., 2003], applied to a simple planar hillslope (the HsB itself was, in turn, tested against solutions provided by Richards' equations [Paniconi et al., 2003]). Then, an application of the general nonlinear Boussinesq equation to the Panola hillslope [Lanni et al., 2011; Hopp and McDonnell, 2009; Tromp-van Meerveld and McDonnell, 2006] in which wetting and drying patterns over an irregular topography bedrock are performed and compared with the hydrological response of a planar hillslope. A further comparison between the results of the Boussinesq equation with those of a 3-D variably saturated water flow model, e.g., the GEOtop hydrological distributed model [Rigon et al., 2006], is finally discussed.

5.1. Comparison With an Analytical Linearized Solution With Planar Topography

[42] In this example, a horizontal 1-D aquifer is analyzed with two head (Dirichlet) boundary conditions applied to the two borders in contact with two reservoirs (Figure 1). At one border, the pressure is assumed to increase suddenly to a constant value inline image, generating a wave which moves to the opposite border where the water surface remains at height inline image. If the physical parameters allow the linearization of equation (4), the solution can be found analytically by following the examples illustrated in Rozier-Cannon [1984]. The mathematical problem is then reduced to solving:

display math(33)

with the following boundary conditions:

display math(34)
display math(35)

and the initial condition:

display math(36)

[43] As anticipated, assuming KS and s to be spatially uniform (i.e., constant), the analytical solution can be either (a) exact at steady state, obtaining an asymptotic solution, or (b) approximate using a linearization during the transient regime as shown in Figure 2.

Figure 2.

Water surface profiles, BEq solution compared to the linearized (p=0, p=0.5, and p=1, see equation (42)) and analytic solutions after 1, 5, 10, and 20 days from instant t=0. The porosity, s, is 0.4 and the saturated hydraulic conductivity KS is 10−4 m/s in blue, 10−3 m/s in green, 10−2 m/s in red, and 10−1 m/s in black. Solid lines refer to the numerical solution; dashed lines to the linearized solution with p=0; dotted lines to the solution with p=0.5; dot-dashed lines to the solution with p=1. The furthest right black dashed line refers to the analytical asymptotic (steady-state) solution.

[44] The asymptotic solution is easily found. In fact, the asymptotic equation, for t → ∞, is:

display math(37)

which can be easily integrated, resulting in:

display math(38)

where inline image and inline image are coefficients determined by satisfying the boundary conditions, resulting in:

display math(39)
display math(40)

[45] During the transient time, equation (33) can be linearized as follows:

display math(41)

where inline image is a constant value of the height of the water surface, which must be contained between inline image and inline image and is therefore estimated with a weighted average:

display math(42)

where p is normally estimated empirically. Equation (41) is formally equivalent to the 1-D heat equation [Rozier-Cannon, 1984] and is analytically solved using a Fourier series expansion. The detailed mathematical derivation and the explicit form of the solution are reported in Appendix The Analytical 1-D Linearized Solution of the Boussinesq Equation.

[46] The comparison of the above solutions with BEq is shown, at different times and for different values of the hydraulic conductivity, in Figure 2. In these experiments, we set the boundary conditions η1=2 m, η2=1 m; the length of the domain, L=1000 m; and the cell size of side 1 m. The porosity was set to inline image, whereas four different values of saturated hydraulic conductivity were tested: KS=10−41 m/s; KS=10−3 m/s; KS=10−2 m/s; and KS=10−1 m/s. This last value has little real physical relevance, but it was used to allow the system to reach asymptotic conditions in a reasonable time. The numerical time step used was 3600 s (1 h). The water surface profiles are illustrated at 1, 5, 10, and 20 days. Figure 2 also shows the analytic solution (of the linearized equation) obtained with different estimates of p (p=0, p=0.5, and p=1). The steady-state solution (38) is always the furthest right in the panels of Figure 2. For the lowest hydraulic conductivities, stationarity is not reached and the right boundary condition (at x=L) has no effect on the solution. As expected, the numerical solution provided by the BEq is located between the analytical solutions of the linearized equation for p=0 and p=1. In fact, if p=0, the hydraulic transmissivity is underestimated being evaluated according to inline image and corresponding to the transmissivity value of the undisturbed zone of the water table profile. The water table level obtained by our BEq solver converges to the one obtained with the linearized solution for p=0 and then tends to inline image when xL, as expected.

[47] On the other hand, for p=1 the numerical solution fits well with the linearized solution near the left boundary (at x=0) where the transmissivity can be approximated by considering inline image as water table level, despite the numerical solution presenting nonlinear effects due to strong gradients of the water table surface. The same behavior can be observed more clearly in the cases of KS=10−2 m/s (represented with red lines) and KS=10−1 m/s (black lines) 1 day after the beginning of the simulation (t=0), as illustrated at the top left of Figure 2. In the case of KS=10−1 m/s, the hydraulic transmissivity is large enough for the solution to reach stationarity after 20 days, but the linearized solution does not fit the numerical solution even partially, tending toward a linear profile. However, the numerical solution is coincident with the exact steady solution given by equation (38).

5.2. Comparison With an Analytical Nonlinear Solution With Planar Topography

[48] The second test is a limit case of the previous one, defined by equation (33), including its boundary conditions, equations (34) and (35), and the initial condition, equation (36). In this test, the water surface level at the left boundary (x=0) suddenly increases to inline image value, whereas the other boundary is initially dry and domain is indefinitely long, i.e., inline image and L → ∞ (see Figure 3).

Figure 3.

Graphical representation of the one-dimensional problem used for the comparison between numerical and Song et al.'s analytical solutions of BEq. The problem is a wetting process on an initially dry bedrock ( inline image), which is a substantially different from the case depicted in Figure 1. At x=0, the water surface level suddenly increases to inline image at the beginning of the simulation. To obtain the analytical solution, L is taken →∞.

[49] In this case, equation (33) cannot be solved approximately with a linearization because of the coexistence of “wet” and “dry” zones. In order to obtain the solution, Song et al. [2007] and Lockington et al. [2000] defined a new spatial-temporal coordinate:

display math(43)

[50] With this coordinate, the analytic solution obtained with equation (10) of Song et al.'s paper becomes:

display math(44)

where an is an infinite sequence of real numbers with the following properties:

display math(45)


display math(46)

and the values of an are calculated with the following recursive form:

display math(47)

and inline image is a constant, related to the displacement of the wetting front, which is estimated such that equation (46) is verified. Equation (47) corresponds to equation (16) of Song et al.'s paper. Song et al. [2007] gave an explicit form of equation (47) and fully described how to get the numerical values of the inline image and an terms.

[51] Figure 4 illustrates the fit between Song et al.'s analytical solution and the numerical solution obtained with the BEq code. As in the previous case, the saturated hydraulic conductivities were set to KS=10−4 m/s, KS=10−3 m/s, KS=10−2 m/s, and KS=10−1 m/s; the porosity was set to inline image in each of the four subcases; the water surface elevation, inline image, at the left boundary (x=0) was set at 1 m; and analogously to the previous experiment, results were plotted at times of 1, 5, 10, and 20 days since the initial instant. The cell size used is 1 m, the numerical time step used is 3600 s for the lower conductivities (blue and red), whereas in the other two cases (red and black) the time step used is 10 times smaller, i.e., 360 s. In fact, according to our numerical scheme, the wetting front can advance into one initially dry cell no more than once per time step and, therefore, if the time step is relatively long, the displacement of the wetting front could be underestimated.

Figure 4.

Water surface profiles, the BEq solution against Song et al.'s analytical solution after 1, 5, 10, and 20 days from the instant t=0. The porosity s is 0.4 and the saturated hydraulic conductivity KS is 10−4 m/s in blue, 10−3 m/s in green, 10−2 m/s in red, and 10−1 m/s in black. Solid lines refer to the numerical solution. Dashed lines to the analytical lines and when they are invisible, it is because they are almost perfectly superimposed to the solid ones.

[52] For this reason, disagreements between numerical and exact analytical solution may occur in earlier time steps. If the time step is not sufficiently short, the analytical wetting front anticipates the numerical one; however, for long times, the lag between the two solutions is reduced and tends to be zero. In Figure 4 this lag is significant, about 20 m, only in the case of KS=10−1 m/s (an unrealistically high conductivity chosen to test a limit case) after 1 day (86,400 s). In all the other cases, the numerical and analytical solutions fit quite well after 5 and 10 days (432,000 s and 864,000 s, respectively). After 20 days (1,728,000 s), there is also a quasi-perfect agreement except for KS=10−1 m/s when the wetting front reaches the right-hand side end of the numerical domain: in fact, while the numerical solution has the zero Dirichlet boundary condition (35) at x=L, the Song et al.'s analytical solution represents the movement of a wetting front over a semi-infinite planar dry bedrock and does not have any right-hand boundary condition. Therefore, the small disagreement is due to lateral right-hand boundary effects.

5.3. Comparison With HsB Numerical Solution

[53] The BEq code was compared with an existing 1-D code, produced by Troch et al. [2004]. The test was conducted on virtual hillslopes of variable sizes, similarly to what was presented in Troch et al. [2003]. Three artificial, 100 m long, planar hillslopes were setup with different shapes: (1) hillslope A, with a uniform width of 7 m, (2) hillslope B, (convergent) with an exponentially variable width from 50 m (at the top) to 7 m (at the toe), and (3) hillslope C, (divergent) with an exponentially variable width from 7 m (at the top) to 50 m (at the toe).

[54] In A, B, and C the slope is 0.05, the drainable porosity is uniformly distributed and equal to inline image, the saturated hydraulic conductivity is uniformly distributed and equal to inline image × 10−4 m/s (1 m/h), and the initial soil water-table thickness is equal to inline image m [Troch et al., 2003].

[55] A water-table recharge of 10 mm/day is applied to each of the three hillslopes. The following boundary conditions are applied (Experiment 1): (1) a Dirichlet condition ( inline image) at the toe according to Troch et al. [2003], (2) a no-flux condition at the top of the hillslope and at the lateral boundary.

[56] In this experiment, the time step used is 3600 s for our model and 180 s for the HsB model. These were chosen in order to have comparable computational times for both solvers, and we sacrificed time accuracy for increased spatial information. Both models work with a resolution of 1 m (but while the response of the BEq code is given pixel by pixel, the HsB model returns the mean level of the water table at each given distance along the transects, drawn in red in Figure 5).

Figure 5.

Map of the bedrock elevation of the analyzed hillslopes: (a) uniform, (b) convergent, and (c) divergent.

[57] Figure 6 shows the water-table thickness along the principal direction of the hillslopes at different times.

[58] In hillslope A, the water-table thickness tends to be uniform in the middle part of the hillslope and have relevant gradients at the top and at the toe. After 5 and 20 days, the thickness h increases with a peak at about 20 m, as expected.

[59] In hillslope B, the behavior is initially similar to hillslope A. The peak of h after 20 days is greater than in hillslope A exceeding it by 1 m.

[60] In the hillslope C, after 1 day, h has a profile similar to that of hillslope A, then decreases in the upper part of the hillslope and increases a little in the lower part.

Figure 6.

Water-table thickness, h, evaluated along the principal direction of the hillslopes, simulated for hillslopes A, B, and C with our BEq model and the HsB model (dotted lines) after 1, 5, and 20 days from the beginning of the simulation.

[61] The results of our model confirm the results of the HsB model. The fit of the two solvers is relatively good. Few differences can be seen for convergent and divergent hillslopes. The HsB model is 1-D and assumes a uniform profile of h along the width. In the case of the convergent hillslope, our model underestimates the values predicted by HsB. On the contrary, in divergent hillslopes it overestimates them. These differences are relatively small and due to a nonuniform profile of the water table along the hillslopes' cross sections. In our case, the water table depth in the middle part of the hillslope is greater for the convergent case or lower for the divergent case than close to the lateral boundary, a situation that obviously cannot be reproduced by a 1-D model.

5.4. Application to the Panola Hillslope With Wetting and Drying Patterns

[62] The capabilities of the model go beyond the description of hydrological responses of simple virtual hillslopes. Indeed, it was designed to be able to deal with complex topographies, as already shown in Lanni et al. [2011]. The Panola site was chosen to perform analyses with the proposed BEq solver. It lies within the Panola Mountain Research Watershed (PMRW), located about 25 km southeast of Atlanta, GA, USA, in the southern Piedmont. The site is of particular interest because a detailed survey was carried out to determine the topography of the bedrock, as distinguished from the topography of terrain surface, shown in Figure 7. The surface topography of the study hillslope is approximately planar while the bedrock topography is very irregular, resulting in highly variable soil depth ranging from 0 to 1.86 m with an average value of 0.63 m. This was shown by Tromp-van Meerveld and McDonnell [2006] to be the cause of the complexity of the subsurface storm flow discharge generated.

Figure 7.

(left) Map of the Panola bedrock elevation expressed in meters. The horizontal line marks the outlet where the water recharge is plotted in Figure 10. (right) Map of the 13° planar slope elevation used as comparison (expressed in meters).

[63] The site has a slope angle of 13°, is 28 m wide, and is 48 m long. Soil depths have been measured on a regular 2 m × 2 m grid and linearly interpolated to a 1 m × 1 m digital elevation model for a total of about 1815 cells. The downslope boundary of the Panola hillslope is formed by a 20 m wide trench. The upper boundary of the study hillslope is formed by a small bedrock outcrop. The soil is a sandy loam, without discernible structure and overlain by a 0.15 m deep organic-rich horizon [Hopp and McDonnell, 2009]. All other details are described by Tromp-van Meerveld and McDonnell [2006], Meerveld and Weiler [2008] and we refer the reader to those papers for further information.

[64] Overland flow is uncommon at the Panola site and is observed only during very intense thunderstorms after extended dry periods. Even during these storms, overland flow was restricted to small areas and reinfiltrated within several meters. A constant water-table recharge rate, Q (see equation (1)), of two millimeters per hour (2 mm/h) for 48 h was assumed on the basin. This is coherent with the magnitude to the event of 6–7 March 1996, reported in Hopp and McDonnell [2009]. Simulations were performed assuming hydraulic conductivity, KS=0.1 m/h (2.78 × 10−5 m/s) and completely dry initial conditions (null water thickness h=0).

[65] Due to the irregular bedrock, very soon after the beginning of the rainfall event, flow creates irregular water table levels along the hillslope which cannot be simulated with 1-D models of flow; these are clearly evident after 24 h, as shown in Figure 8. Variable zones of null water table depth also appear and, after the end of the rainfall event, small, disconnected, perched water tables remain in the domain. For comparison, Figure 9 reports the behavior of a purely planar hillslope of 13° slope under the same conditions. It is easily understood that the flow is absolutely symmetric and trivial.

Figure 8.

Figure map of Panola water thickness (expressed in meters) after a rainfall event with 2 mm/h water-table recharge rate and duration of 48 h. The maps are referred to 24, 48, 120, 240, 360, and 480 h since the beginning of the rainfall event. Soil drainable porosity is inline image and saturated hydraulic conductivity is KS=0.1 m/h (2.78 × 10−5 m/s).

Figure 9.

Figure map of water thickness (expressed in meters) over the 13° steep planar slope after a rainfall event with 2 mm/h water-table recharge rate and duration of 48 h. The maps are referred to 24, 48, 120, 240, 360, and 480 h since the beginning of the rainfall event. Soil drainable porosity is inline image and saturated hydraulic conductivity is KS=0.1 m/h (2.78 × 10−5 m/s).

[66] The irregularities of the bedrock also affect the discharge at the toe of hillslope, as collected at the trench position, and shown in Figure 10. The most evident feature is an increase of the mean transit time of water, with a consequent decrease of the discharge peak by approximately one fourth.

Figure 10.

Hydrograph (water discharge versus time) at the bottom of the Panola hillslope (solid line) and 13° planar slope (dotted line) after an event with recharge rate of 2 mm/h and duration of 48 h.

[67] However, since the recharge is constant and synchronous everywhere on the bedrock, filling and spilling phenomena (i.e., the production of multiple peaks) [Tromp-van Meerveld and McDonnell, 2006] are not present. This supports the idea that filling and spilling are due to the irregular arrival time of water infiltrating to the different depths implied by the irregular distance between terrain surface and bedrock.

5.5. Comparison With the GEOtop-Distributed Hydrological Model

[68] In order to better comprehend the capabilities of the proposed numerical method for solving the Boussinesq equation with irregular topography, a preliminary comparison with the distributed hydrological model GEOtop is presented here. The GEOtop model solves the 3-D Richards' equation and extends it to the case of saturated soils [Zanotti et al., 2004; Rigon et al., 2006; Bertoldi et al., 2006, 2010; Dall'Amico et al., 2011; Endrizzi and Gruber, 2012]. GEOtop is an open-source free software available on www.geotop.org. However, GEOtop is not completely compatible with BEq: by construction GEOtop resolves vertical infiltration, while BEq does not. A proper comparison would require coupling BEq with an infiltration module, which is beyond the scope of the present paper. Nevertheless, we effected the comparison as we believe it to be instructive to the reader. Regarding computational time, given equivalent grid sizes, BEq is much faster than GEOtop. This is because GEOtop is 3-D, rather than 2-D, and it includes many other processes, such as surface runoff and runoff reinfiltration that BEq does not consider. The comparison is made with the application of the two models to the Panola hillslope, already used and described above.

[69] The hillslope terrain is planar whereas the bedrock topography is irregular; average soil thickness is reduced to 0.5 m to minimize the effects of infiltration. The GEOtop simulations are performed with the build 1.225-9. During the simulations, a constant precipitation rate of 10 mm/h is applied uniformly over the whole hillslope for 48 h. Analogously with the case described in the previous section, the simulations were performed assuming hydraulic conductivity, KS=0.1 m/h (2.78 × 10−5 m/s), and completely dry initial conditions (corresponding to null water thickness). This means that soil water pressure head is initially uniformly distributed and equal to inline image (in practice, it is equal to a relatively high negative value). In the presented simulations with the Boussinesq equation, the soil water retention curve is modeled as a step function (i.e., it is equal to the soil porosity in saturated zones and 0 in unsaturated zones). On the other hand, GEOtop models the soil water retention curve according to VanGenuchten's [1980] formula with the following parameters: αVG=10 m−1, n=3, and m=0.667, typical values for a sandy soil. However, VanGenuchten's parameters are set such that the retention curves for the two models are very similar and GEOtop's numerical convergence is guaranteed.

[70] In GEOtop, the water-table thickness can be obtained by observing the growth of saturated soil layers over the bedrock for each point of the basin. However, for the purposes of this paper, we are not interested in the 3-D water profile in the vadose zone. Therefore, it is simply estimated from the values of pressure head just above the bedrock, assuming a slope-normal hydrostatic equilibrium profile [e.g., Cordano and Rigon, 2008, equation (10)]:

display math(48)

where ψ is the soil water pressure head obtained by GEOtop, α is the slope degree of the Panola hillslope terrain which is about 13°, z is here a slope-normal downward vertical coordinate, and d is the depth of water table.

[71] When the pressure head over the bedrock is positive, a perched water table occurs. In this case d is calculated by inverting equation (48) and the water thickness is obtained from the difference with the local soil thickness.

[72] Figure 11 illustrates the dynamic behaviors of the vertical water-table thickness over the Panola topography, simulated with the Boussinesq equation solver on the left and GEOtop on the right.

Figure 11.

Figure map of (left) BEq-generated and (right) GEOtop-generated Panola water thickness (expressed in meters) after a rainfall event with 10 mm/h water-table recharge rate/rainfall and duration of 48 h. The maps are referred to 48, 240, and 480 h since the beginning of the rainfall event. Soil drainable porosity is inline image and saturated hydraulic conductivity is KS=0.1 m/h (2.78 × 10−5 m/s).

[73] The pattern needs to be read after having weighed the effects of infiltration. While in BEq, all the precipitation is immediately provided as “recharge” over the bedrock, in GEOtop, water needs to “infiltrate” downward (sometimes “precipitation” and “water-table recharge” are used as synonyms in groundwater literature but they are different). Therefore, in BEq simulations, a perched water table is formed from the first time instant of simulation and water begins to flow downhill immediately. In GEOtop, on the other hand, water first needs to infiltrate across the vadose zone and form a water table before moving downhill. This depends on the water retention curve chosen for the simulation, but it can typically take up to 48 h. Therefore, the soil water thickness patterns from the two simulations are very similar after 48, 240, and 480 h (corresponding to 1, 10, and 20 days) from the beginning of the precipitation; during the initial 48 h, however, the two simulations are not similar. Also as a consequence of missing infiltration, the water thickness values obtained with the BEq are on average greater than the ones obtained by GEOtop, which, however, stores water in the vadose zone. Some exceptions are clarified below. Unlike Figures 8 and 9, Figure 11 illustrates water thickness stored within the soil thickness (i.e., surface water is not taken into account in these simulations).

[74] Besides the visual comparison shown in Figure 11, a scatterplot of BEq versus GEOtop water-table depth is reported in Figure 12. During the initial time steps, there is not a good correlation between the two models. Surprisingly, there is a considerable number of points where there is a higher water table in GEOtop than in BEq. This is not completely intuitive but can be explained as follows: inspection of the points on the upper left quadrant (of the top left plot) shows that these are all located close to the divide where water in BEq flows away proportionally to saturated hydraulic conductivity and the local gradient of bedrock topography, while in GEOtop it is still infiltrating. This is confirmed in the top left scatterplot of Figure 13, where the total volume of water, including unsaturated water volume per unit area, is estimated for each pixel. In particular, going back to the points close to the divides, the water simulated in the BEq experiment has already flowed away, whereas the water simulated by GEOtop has just reached the bedrock and forms a perched water table.

Figure 12.

Comparison between GEOtop-generated and BEq-generated Panola water-table thicknesses. See the corresponding spatial patterns shown in Figure 11 for the time instant: 48, 240, and 480 h.

Figure 13.

Comparison between GEOtop-generated and BEq-generated Panola water volume in a soil column per unit area. In the case of GEOtop, these plots also include the water stored in the vadose zone.

[75] As the time increases, the number of points where GEOtop water storage is larger than BEq decreases and, as expected, most of the points have a lower water-table thickness in GEOtop than in BEq. However, there is a good correlation between data from the two models, with a coefficient greater than 0.95. It can also be observed that many of the points accumulate on the x-y plane bisector: they mostly represent the parts of the basin in which the soil is completely saturated.

[76] Figure 13 shows the comparison of water volume per unit area for each pixels in GEOtop and BEq. The scatterplots of Figure 13 show a better general agreement between the two models than those based on the water-table thickness. GEOtop gives larger values of stored water volumes in the lower range of volumes, for the same phenomenology described above to explain the water table dynamics of points close to the divides. The differences between the results of the two models tend to decrease, exponentially after 10 and 20 days from the beginning, even if the delay in the GEOtop simulation caused by infiltration is still evident.

6. Conclusion

[77] A new 2-D numerical solver of the Boussinesq equation has been presented. The solver derives from a finite-volume mass budget for each element of the domain. The Boussinesq equation is reduced to a nonlinear system composed of three terms, where the known term corresponds to the initial condition and the external forcing. The unknowns appear in a nonlinear term, which is the water volume stored in each cell, and a linear term deriving from the fluxes of water between cells.

[78] The method is innovative with respect to other 2-D solvers [e.g., Harman and Sivapalan, 2009] and other widely used models [e.g., Harbaugh et al., 2000; Painter et al., 2008], in that it intrinsically preserves mass (even for large integration time steps, where accuracy is lost) and properly accounts for wetting and drying of the water table without the introduction of ad hoc internal iteration schemes. The solver can properly reproduce simple drainage problems that have analytical solutions, and it performs competitively with nonlinear analytic solutions [Song et al., 2007]. The solver also performs well with the HsB model [Troch et al., 2003], at least for the simplified setting proposed in the literature, despite that it works with 2-D water table patterns as opposed to the 1-D patterns of the HsB model.

[79] Simulations in complex topography, as those for the Panola site, show the capabilities of the BEq solver to describe the complex water flow generated in simulations of water table above rugged bedrock.

[80] A comparison with GEOtop [Rigon et al., 2006; Endrizzi and Gruber, 2012], a model solving the 3-D Richards' equations, was carried out, despite some limitations. The main obstacles to a proper simulation comparison are: (i) the absence of an integrated vertical infiltration module within the Boussinesq equation; and (ii) because of the GEOtop code, the impossibility of introducing a direct water-table recharge source in each part of the domain into the GEOtop model itself. In fact, solving either problem would require a big effort on the source codes of both models, which goes beyond the scope of this paper. Nevertheless, the results show consistent behavior and encourage one to think that, once the appropriate infiltration module is created, the results given by BEq would be credible and fit well to real case studies.

[81] Therefore, it is clear that the BEq algorithm can be successfully integrated into rainfall-runoff models of large areas if a detailed, 2-D, forecast of the water table position is necessary, especially if the runoff model is integrated with an appropriate (even simplified) infiltration model. It is also clear that the BEq algorithm could exploit the information contained in soil depth distribution maps, when available, to simulate complex patterns of soil moisture and runoff. Obviously, the model can be successfully used in all the traditional applications related to 2-D unconfined aquifers without any modification.

Appendix A: The Structure of the Numerical Solver Utilized

[82] As derived in section 3, the spatial and temporal discretization method consists in reducing the free-surface hydrodynamic problem to an algebraic system of the following form:

display math(A1)

where inline image is the array of unknown quantities (one for each of the Np grid cells), inline image is the array of known parameters, and inline image are the values of the field conserved quantity, i.e., the water volume stored in the ith cell. All of these “vectors” are denoted by the symbol inline image, or harpoon, to distinguish them from space vectors, denoted by inline image. inline image is a inline image matrix that satisfies one of the following properties [Brugnano and Casulli, 2008]

[83] • T1: a Stieltjes matrix, i.e., a symmetric M-matrix [e.g., Horn and Johnson, 1991];

[84] • T2: a symmetric positive semidefinite matrix such that inline image for each inline image and inline image [Casulli, 2009].

[85] System (A1) can also be written with the following index notation:

display math(A2)

for every ith cell of the domain. This notation highlights that the ith component, Vi, of the conserved quantity vector, inline image, is solely a function of the component inline image and, in the case analyzed in this paper, related to the ith cell itself.

[86] If the following residual function is defined:

display math(A3)

solving equation (A1) corresponds to seeking the zeros of the residual function inline image. Furthermore, when the Jacobian matrix of the residual inline image is locally a continuous, nonsingular, Lipschitz matrix around the solution inline image [e.g., Kelley, 2003], then the zeros of the residual function (i.e., the solution of equation (A1)) can be found with a Newton-like iterative method:

display math(A4)

where inline image is the Jacobian matrix of the residual function, defined as:

display math(A5)

where inline image, and m (preceded by a semicolon) is the iteration level of Newton's method. An explicit calculation shows that inline image is the sum of inline image, which is symmetric and positive by definition, and the Jacobian matrix of inline image, i.e.,

display math(A6)


display math(A7)

where “:=” indicates a definition (rather than an equality).

[87] If the volume inline image (the conserved quantity) of a generic ith grid element is a nondecreasing function of the unknown variable inline image and does not depend on the values of ζ in neighboring cells, then it can be shown that inline image is a diagonal matrix with non-null entries that are all positive, which correspond to the horizontal wet area of each cell, as will be shown in the next section. In practical applications, it is not computationally convenient to use equation (A4) and then find the inverse of the Jacobian matrix, whose storage in memory can be overwhelming and time expensive. Thus, equation (A4) is usually solved by finding the solution for the associated linear system:

display math(A8)

where the sparse Jacobian matrix, inline image, can be used in a, so-called, matrix-free implementation that is much less demanding, both computationally and on storage. Given the symmetry of inline image, the linear system (A8) can be efficiently solved with a preconditioned conjugate gradient method [e.g., Schewchuk, 1994]. Under the conditions described above, the solution of equation (A1) is unique and it can be proved that equation (A4) converges to such a solution with few iterations. In particular, when T is irreducible and satisfies the T2 property, the following condition needs to be satisfied [Casulli, 2009]:

display math(A9)

which means that no more water mass can be abstracted through sinks than is present in the domain [Brugnano and Casulli, 2008]. If T is reducible (i.e., not irreducible, the “wet” domain is decomposed into disconnected parts) and satisfies the T2 property, then equation (A9) must be changed into the following scalar product [Brugnano and Casulli, 2008]:

display math(A10)

for each inline image, eigenvector of T, giving a null eigenvalue:

display math(A11)

Appendix B: The Analytical 1-D Linearized Solution of the Boussinesq Equation

[88] From section 4.2, equation (33) is rewritten as equation (41):

display math(B1)

where KS and inline image are two suitable constants, and the following boundary conditions must be imposed:

display math(B2)
display math(B3)
display math(B4)

[89] To standardize the solution of the problem, a nondimensional form of equation (B1) is usually applied. Consequently, a new variable w is introduced:

display math(B5)

so that equation (41) (or equation (B1)) is modified and further simplified to a constant-coefficient linear equation, which is the classical form of heat equation:

display math(B6)


display math(B7)

[90] The boundary and initial conditions (B2), (B3), and (B4) are rewritten with the new variable w to obtain respectively:

display math(B8)
display math(B9)
display math(B10)

[91] Finally, the exact solution is [e.g., Rozier-Cannon, 1984]:

display math(B11)

where wn is defined for each n (from 1 to inline image) so that:

display math(B12)

and then, by Fourier series decomposition, it is:

display math(B13)

[92] In conclusion, by assembling equations (B5), (B11), and (B13), the explicit form of the solution to equation (B1) is:

display math(B14)


[93] This work is partially sponsored by the projects MORFEO and HydroAlp (University of Trento), ace-sap, ENVIROCHANGE, and IDROCLIMA (Fondazione Edmund Mach). The presented numerical algorithm for a numerical solution of BEq is coded in C (the BEq code) and freely available on request to the authors. The analytical solutions used for the comparisons of the BEq code are implemented as an R package called “boussinesq” and freely available under GPL license on Comprehensive R Archive Network (CRAN) at http://cran.r-project.org/web/packages/boussinesq/index.html. GEOtop is coded in C/C++ and freely available under GPL3 license at http://www.geotop.org. The elaborations of GEOtop results are performed with the R package “geotopbricks” freely available under GPL on CRAN website http://cran.r-project.org/web/packages/geotopbricks/. The authors thank Joseph Tomasi and Cristiano Lanni for helpful suggestions, Peter Troch for making the HsB code available, the Associate Editor and the anonymous reviewers for helpful comments and suggestions, and Stefano Endrizzi for the help and the suggestions on the GEOtop simulations.