A grid generating algorithm for simulating a
¯uctuating water table boundary in heterogeneous
uncon®ned aquifers
A.S. Crowe
a,*, S.G. Shikaze
a& F.W. Schwartz
b aNational Water Research Institute,Canada Centre for Inland Waters, P.O. Box 5050, Burlington, Ont., Canada L7R 4A6 b
Department of Geological Sciences,The Ohio State University,125 South Oval Mall,Columbus, OH 43210, USA
(Received 14 December 1997; revised 6 October 1998; accepted 12 October 1998)
An algorithm is presented for generating ®nite element grids that can be used to calculate the position of a ¯uctuating water table and the formation of seepage faces within a heterogeneous uncon®ned aquifer. Our approach overcomes limi-tations with existing techniques by allowing the water table to rise or decline through hydrostratigraphic boundaries yet maintains numerical and conceptual accuracy with respect to hydrostratigraphic geometry. The algorithm involves (1) limited stretching or shrinking of elements along the water table if the change in the position of the water table is small with respect to the vertical grid spacing, and (2) the addition or removal of nodes and elements in the ®nite element mesh along the water table as the change becomes large with respect to the vertical grid spacing. This technique is applicable to any 2-D or 3-D ®nite element code that contains an automatic ®nite-element grid generator. Ó 1999 Elsevier Science
Limited. All rights reserved
Key words: ®nite element method, free surface, grid generation, groundwater, hydrogeology, modelling, uncon®ned aquifer, water table.
1 INTRODUCTION
The objective of this paper is to present a method for generating ®nite element grids that can change in re-sponse to a water table that rises or falls through hydrostratigraphic boundaries, and still maintains nu-merical and conceptual accuracy with respect to hydrostratigraphic geometry. Speci®cally, our algorithm calculates the position of a ¯uctuating water table, the formation of seepage faces and the distribution of hy-draulic head within a heterogeneous uncon®ned aquifer. Although we make use of a two-dimensional domain discretized with triangular ®nite elements, the method proposed here can be incorporated into any 2-D or 3-D ®nite element code that contains an automatic ®nite el-ement grid generator.
This work provides an important step forward over other approaches that are limited in their capabilities.
For example, in some schemes the elevation of the nodes along the water table is ®xed (i.e., the grid does not deform) throughout the simulation but the calculated value of heads along the water table can change2,14,17,18. With the geometry of the cells remaining the same throughout the simulation, errors can result because the hydraulic head along the water table is not equal to the elevation of the water table.
Other schemes match water-table elevations with hydraulic heads by allowing the mesh to deform through time1,4±12,14,16. The elements that are deformed might include only the top row of elements or all rows in the domain. In most of these cases, the elements are stret-ched vertically, but a few of these algorithms allow the grid to deform both vertically and horizontally. Al-though this approach more accurately approximates the position of the water table in a homogeneous domain, problems can arise in heterogeneous domains. If the mesh expands or contracts through layer boundaries, the initial hydrostratigraphy is not preserved (Fig. 1b), Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter
PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 4 2 - 6
*
Corresponding author.
resulting in inaccurate calculations of both the position of the water table and the hydraulic head distribution within the saturated domain. Finally, with elements able to deform in this manner, numerical inaccuracies can exist causing problems with numerical instabilities and the aspect ratio should the elements stretch or compress excessively.
Our algorithm involves the addition or removal of nodes and elements along the water table, as the water table rises or falls, respectively (Fig. 1c). By adding and removing nodes and elements, the vertical grid spacing throughout the saturated zone remains constant. Hence,
the ®nite element grid will always conform to the hydrostratigraphic geometry.
2 METHOD
In a free-surface problem, both the hydraulic head dis-tribution and the water table con®guration are un-known. The main criterion for accurately simulating the position of the water table is that the elevation of a node
ialong the water table,d(x,t), is equal to the hydraulic
head at the water table node,h(x,t), at all times. Also, if the water table is at ground surface, a seepage face will form and the value of hydraulic head will be equal to the elevation of the ground surface.
Our approach considers only the saturated part of the ¯ow system (below the water table). Positive and nega-tive ¯uxes are used as boundary conditions along the top of the domain, causing the water table to rise or fall. Fig. 2 shows a typical cross section and boundary con-ditions that are described below. The governing equa-tion for transient groundwater ¯ow in the saturated zone (S in Fig. 2) is:
o
oxi Kij oh
oxj
Ss oh
ot; 1
whereKij is the hydraulic conductivity tensor [L/T],his hydraulic head [L], Ss is the speci®c storage coecient [Lÿ1],tis time [T],x
iis the coordinate vector [L] andi,
j1,2. The initial conditions are:
h x;z;0 ho x;z; 2
d x;z;0 do x;z; 3
wheredis the elevation of the free surface (F in Fig. 2)
above a datum [L],d0 is the initial elevation of the free
surface [L], h0 is the initial hydraulic head [L]. The boundary conditions for eqn (1) are:
h x;z;t H x;z;t; on bÿc (Fig. 1) 4
Kij@h
@xi
ni ÿQ x;z;t; 5
d x;z;t h x;z;d;t; on F Fig:2; 6
Fig. 2.Schematic cross section showing the computation do-main (S), the free surface (F), seepage face a±b, and constant
head boundary b±c. Fig. 1. Illustration of the geometry of the hydrostratigraphic
Kij
where H is the hydraulic head on a constant head boundary [L],Ris the rate of vertical recharge along the free surface [L/T],niis the unit outward normal vector,
Syis the speci®c yield andQis the ¯ux along a speci®ed-¯ux boundary [L/T].
Equations (2) and (3) are initial conditions that specify the hydraulic head values and the elevations of the water table at the start of the simulation. Equation (4) represents a ®rst-type or Dirichlet boundary condi-tion where speci®ed values of hydraulic head are as-signed along the boundary. The value of the speci®ed head at these boundaries can change in time, and con-stant head nodes can be turned o and/or on during a simulation. Equation (5) represents a second-type, or a Neumann boundary condition where a speci®ed ¯ux across a boundary is assigned. Equations (6) and (7) represent the boundary conditions along the free sur-face, depending on whether or not recharge ¯uxes are present. Equation (8) represents a seepage-face boun-dary where the hydraulic head is equal to elevation of the ground surface. The boundary value problem de-®ned by eqns (1)±(8) is described in further detail by Neuman and Witherspoon15.
The two-dimensional form of eqn (1), subject to ini-tial and boundary conditions, is solved in a vertical cross section using a standard ®nite-element technique. Al-though, we use a triangular ®nite-element mesh, the procedure would apply with most element types (e.g., quadrilateral ®nite elements, 3-D domains). The ®nite-element equations are formulated using the Galerkin method16. Our algorithm for generating the ®nite-ele-ment grid satis®es the following conditions:
· the position of the water table can rise or fall over time as a result of boundary conditions that can change in time,
· all nodes along the water table are located at
d(x,t)h(x,t),
· a single element does not cross over the interface between two hydrostratigraphic units (i.e., nodes located at the interface),
· seepage faces form where nodes are located at
z(x,t)h(x,t)zg(x,t) (where zg is the elevation of the ground surface).
Our method involves a combination of a limited stretching of elements and/or the addition or removal of nodes and elements along the water table. If the change in the position of the water table is small with respect to the vertical grid spacing, the elements along the water table are stretched or compressed. If the change in po-sition is large with respect to the vertical grid spacing, new elements and nodes are added or removed.
The ®rst step is to discretize the computational do-main into triangular ®nite elements using an automatic
mesh generator. This initial mesh depends on the ge-ometry of the domain, the boundary conditions and the initial elevation of the water table. The grid spacing (Dx,
Dz) is small relative to the scale of the problem in order to represent the hydrostratigraphic units and to position nodes along the interface between these units. The grid generator assigns an elevation to each node along the uppermost row of the mesh that is equal to the assigned value of hydraulic head of the water table at that node. The algorithm uses an iterative solution to determine the elevation of the water table and values of hydraulic head. Fig. 3 illustrates the adjustment of the ®nite-ele-ment mesh. At the beginning of a time step, the algo-rithm ®xes the elevation of the nodes along the water table and calculates the hydraulic heads within the ¯ow domain (Fig. 3(a),(g)). Next, nodes along the water ta-ble are moved to a position where the elevation of each node equals its value of hydraulic head (Fig. 3(b),(h)). Because only the nodes along the water table move, only the top row of elements stretch or compress. Changing the vertical dimension of an element produces a new vertical spacing ofDf. All remaining elements below the uppermost row of elements remain at a constant vertical spacing of Dz (Fig. 3(b),(h)). At the end of each itera-tion, numerical convergence is tested by calculating a residual de®ned as the sum of the absolute dierence between the head at the nodes along the water table (h(x,t)) and the elevation (d(x,t)) of this node:
residualRjd x;t ÿh x;tj 9
The solution has converged when the residual is less than a user-de®ned tolerance or closure criterion (we have chosen a tolerance of 0.001 which allows conver-gence within about 5 iterations). Moving to the next time step, the algorithm repeats this process with the opportunity to change the mesh again (Fig. 3 (c),(e),(i),(k)).
The procedure outlined above is used with most ®-nite-element codes that allow the grid to deform as the shape of the ¯ow domain changes. However, in our method, at the beginning of each new time step, if an element is stretched more than 1/4Dz beyond a regular grid spacing (Df> 5/4Dz) we form a new node and a new element. The new node is inserted at the regularDz
spacing, and the new element is inserted along the water table with a vertical element spacing of Dfnew Dfold )Dz (Fig. 3(e)). If an element stretches less than
1/4Dz beyond the regular grid spacing (Df <5/4Dz), only the top two elements are stretched, and a new node is not inserted (Fig. 3(b)). These two stretched elements are formed from the regular Dzspacing to the present position of the water table where DfnewDz+Dfold
(Fig. 3(f)). Similarly, if a node at the water table declines by more than 3/4Dz of the regular grid spacing (Df<
is simply lowered to the current value, thereby com-pressing the ®nite element, with no removal of nodes or elements (Fig. 3(h)).
Because all elements, except those at the water table, are maintained at the original vertical grid spacing of
Dz, unit boundaries remain unchanged. The extent to which our ®nite element grid maintains the geometry of the layers versus a grid generated by a method which only stretches elements is illustrated by Fig. 1. The only instance where the mesh may not coincide with the unit boundaries occurs when the water table passes into a new geologic unit. Initially, the changes in the water table elevation can result in water-table elements stretching less than 1/4Dz from DfDz (i.e., Df <
5/4Dz). If such a stretched element exists at the interface between two units, the stratigraphy will not be preserved and the boundary will actually be between 0 and 1/4Dz
units higher than the actual position. With time, the
water table will continue to rise to a point where a new element will form, at which time the stratigraphy will be once again be accurately represented and placed at the properDzspacing. The error resulting from this slight misrepresentation is very small with respect to the actual position of the water table at that particular time step and at that element location. Hence, as the water table continues to rise and the correct unit boundaries are represented, this error will diminish. However, there may be a case in this situation where the scheme can have diculty converging to a stable solution. This problem will be addressed in more detail below.
Our algorithm, like many codes that only stretch el-ements, also allows seepage faces to form when the water table intersects the ground surface. Nodes and elements along the seepage face are inserted as the seepage face expands or removed as the seepage face contracts according to the above criterion. Nodes are
not allowed to be repositioned above the ground sur-face. At the ground surface, nodes are redesignated as constant-head nodes along the seepage face. If the water table falls below the ground surface, the constant-head nodes along the seepage face revert to regular nodes.
2.1 Limitations
Convergence problems can develop when the water table moves through a boundary between geologic units that have a large contrast in hydraulic conductivity. For example, if the water table rises from a unit of low to one of high hydraulic conductivity, the elements in the lower K unit will stretch until the hydraulic heads in-crease beyond 1/4Dz(Fig. 3(d)). After this, new water-table elements will form, but these new elements will have the higherKvalue assigned to them, and the lowK
elements will shrink to the regular element size (Fig. 3(e)). Because these new elements have a higherK, the hydraulic head along the water table can decrease resulting in a drop in the water table. If these highK
elements shrink to below 3/4Dz, they will be removed and the low K elements will be stretched upwards (Fig. 4(c)). This can result in an increase in hydraulic head and result in the formation of new highKelements yet again (Fig. 4(b)). This entire sequence can repeat itself with the water table at this node oscillating (Fig. 4(a)) and convergence might never be achieved.
We have provided two solutions to rectify this problem. An algorithm is included within the code that
identi®es these oscillations. With the ®rst solution, the criterion for forming a new element is decreased from 1/4Dzto 1/10Dz, and the criterion for removing an ele-ment is increased from 3/4Dzto 9/10Dzfor the current time step. After the water table rises across the boun-dary between elements with a low (lower element) and high (upper element) hydraulic conductivity, it will form a new element. However, when the water table subse-quently falls due to the new element with a high hy-draulic conductivity, it will not remove this element because the water table is above 1/4Dz. This ®x provides convergence in most cases.
In some instances where the contrast in the hydraulic conductivity of the two hydrostratigraphic units is very large, the above solution may still not lead to conver-gence. In this case, it is recommended that the user change the hydraulic conductivity of the cell containing the element where the oscillation occurs. The computer code can identify the node causing the problem for the user. We recommend that the user start with a value of hydraulic conductivity that is midway between those of the two units. This second solution will always results in convergence.
3 APPLICATION
This section compares simulations using our adaptive griding algorithm with a scheme that allows various numbers of rows of elements to deform. Speci®cally, we allow the upper-most row or all of the rows of elements to deform. These latter cases are the techniques incor-porated in other codes. The comparison involves three dierent simulation problems. The ®rst is the simple case of a homogeneous, isotropic aquifer. The second case provides a layer with a contrasting hydraulic conduc-tivity through which the water table rises. The third case represents a more realistic domain containing numerous layers and lenses of contrasting hydraulic conductivity. Fig. 5 presents a schematic of the domain and the boundary conditions for the ®rst two cases. The initial water table elevation is ®ve metres, which coincides with the initial top boundary of the domain. With time, the water table will rise due to recharge and the size of the domain in the vertical direction will increase as new el-ements are formed. Recharge is applied uniformly dur-ing each time step and across the top boundary (Table 1). A constant head node is speci®ed at the top right corner of the domain with a hydraulic head value of ®ve metres. All other boundaries in the domain are no-¯ow.
3.1 Case 1 ± Homogeneous aquifer
The homogeneous case is used to compare our algo-rithm to others, including an analytical solution, to check the accuracy of our method. The physical and
Fig. 4.Example of a condition where convergence cannot be attained causing (a) oscillations in a portion of the water table, due to the successive (b) addition of an element, and (c) the
numerical parameters for Case 1 are tabulated in Ta-ble 1. Fig. 6 shows results from the three dierent so-lution methods at a time equal to 100 days and at steady-state. The hydraulic head along the water table is plotted versus the horizontal (x) distance. At steady state (approximately 3460 days), the maximum elevation of the water table is approximately 2 m above the starting water-table elevation. For this simple case, all three solution schemes produce essentially identical re-sults. This outcome is expected because although ele-ments can deform signi®cantly, hydraulic parameters in all elements remain the same and vertical stretching of the elements is minimal which precludes problems re-lated to the aspect ratio.
Fig. 6 also compares results obtained from the nu-merical methods to an analytical solution developed by Marino13. Although the match among the solutions is very good at early times (i.e., 100 days), the results of the analytical solution deviate slightly at later times as the water table rises. This deviation comes from limiting assumptions in the approach (Bear3, page 365). Because the numerical solutions are 2-D, they account for
ver-tical ¯ow. However, the analyver-tical solution presented by Marino13 is based on the Dupuit-Forcheimer assump-tion, and hence does not consider vertical ¯ow. The accuracy of the analytical solution also decreases as the height of the water table increases.
3.2 Case 2 ± Layered aquifer
Case 2 diers from Case 1 in that it includes a layer of lower hydraulic conductivity at an elevation of 4±6 m (Fig. 5). This example is a severe test designed to con-trast the performance of our algorithm with others that stretch elements and generally do not attempt to pre-serve the pattern of layering. As Case 1 showed, if the water-table ¯uctuations does not encompass dierent units, solutions will be comparable. Because the initial domain is only ®ve metres thick, the simulation tech-niques that involve only stretching of elements will not preserve the location and thickness of this low conduc-tivity layer. Our scheme, however, preserves the geom-etry of the low-Klayer. Fig. 7 shows the hydraulic head
Fig. 6.Comparison of three current numerical methods that involve only element stretching with our algorithm; Case 1 ±
homogeneous aquifer.
Fig. 7.Comparison of three current numerical methods that involve only element stretching with our algorithm; Case 2 ±
layered aquifer. Fig. 5. Conceptual model and boundary conditions for the
sensitivity analysis for Case 1 and Case 2.
Table 1. Hydrogeological and numerical parameters used in the simulations
Case 1 Case 2
Dx 4.0 m 4.0 m
Dz(initial size) 0.5 m 0.5 m
Dt; deltin 0.5 days, 1.02 0.5 days, 1.02
Dtmax 5 days 5 days
K1 10ÿ5m/s 10ÿ5 m/s
K2 10ÿ5m/s 2´10ÿ6 m/s
n 0.3 0.3
Ss 0.0005 0.0005
Sy 0.2 0.2
Recharge 18 cm/yr 51 cm/yr
at the water table versus horizontal distance for the three dierent schemes. As the simulations proceed, in-accuracies related to the treatment of the low conduc-tivity layer become apparent for schemes that only allow the elements to stretch. If one row of elements is stret-ched, the steady-state water table (6000 days) exceeds that from our method by approximately 2 m. The excess head develops because the low-Kunit is enlarged unre-alistically by element stretching. The low hydraulic conductivity layer is stretched from its original thickness of 2 m to a maximum thickness of 8.5 m. If all elements are stretched, the water table at steady state (4100
days) is lower by approximately 0.1 m, as compared to our method. The match between our algorithm with that in which all rows of elements are stretched is more fa-vorable because the thickness of the low conductivity layer remains close to the actual thickness as represented by our algorithm. However, the position of the low conductivity layer is distorted. Here the thicknesses of the lower most unit (K10ÿ5m/s) and the middle layer
(K2 ´10ÿ6 m/s) are stretched from 4 and 2 m,
re-spectively to a maximum of 8 and 3 m, rere-spectively. This case was designed to compare techniques that preserve or do not preserve patterns of layering. While
the code cross-veri®cation points to signi®cant dier-ences between our algorithm and those that simply stretch elements, it is not possible to compare the ac-curacy of the methods without an appropriate analytical solution. Solution methods that stretch elements over predict the elevation of the water table because they exaggerate the thickness of the low permeability layer.
3.3 Case 3 ± Heterogeneous aquifer
The ®nal example includes various zones of contrasting hydraulic conductivity, ranging from 1 ´10ÿ4 to 1 ´10ÿ7m/s. Fig. 8(a) shows the domain, and the initial water table (located at an elevation of 5 m). Note that the region above the initial water table also contains various lenses of dierent hydraulic conductivity. These lenses above the water table will not be considered in the methods that involve only element stretching. Fig. 8(b) shows the steady state water table and the computa-tional domain using our method. The low and high hydraulic conductivity lenses above the initial water table are accurately represented by our algorithm. Fig. 8(c) shows the domain and water table at steady state from the method that involves stretching all rows of elements, and clearly shows that the hydrostratigra-phy is not accurately maintained. Fig. 9 shows the steady state water tables generated by our algorithm, in comparison with the techniques that stretch one or all of the rows of elements. The methods that involve element stretching over-estimate the water table by as much as one metre.
The solution methods that simply stretch the ele-ments produce results that can be dramatically dierent from those produced by our method. Although there is no analytical solution with which to compare the results, this example clearly shows that our algorithm not only maintains the hydrostratigraphy better than the other methods, but also considers units which the stretching algorithms miss as the water table rises. Because our algorithm maintains the hydrostratigraphic geometry,
whereas the other algorithms do not, it is apparent that our solution is more representative.
4 CONCLUSIONS
Several models are capable of simulating both the hy-draulic head distribution and water-table con®guration within a uncon®ned aquifer. The simplest scheme in-volves calculating the position of the water table with a ®xed grid. The calculated heads along the water table can change but the position of the nodes along the water table remain ®xed. Other models calculate the position of the water table by having nodes and cells/elements stretch or compress in order to match the elevation of the water table nodes to the respective values of hy-draulic head. However, these models are accurate only in cases where the water table ¯uctuates within a single homogeneous unit. They are not appropriate for sys-tems in which a rising or falling water table passes through hydrostratigraphic unit boundaries. In such cases the stretching of elements distorts the geometry and changes the hydraulic parameters of the domain through which the water table ¯uctuates.
The algorithm presented here calculates the position of a ¯uctuating water table as it rises or falls through hydrostratigraphic unit boundaries, and still maintains the geometry of these units. More importantly, it maintains the distribution of hydraulic parameters by careful regeneration of the grid as appropriate. Our al-gorithm involves the addition or removal of nodes and elements along the water table, as the water table rises or falls, respectively. By adding and removing nodes and elements, the vertical grid spacing throughout the satu-rated zone remains constant. Hence, the ®nite element grid will always conform to the geometry of the units. An iterative technique is required to match the elevation and the hydraulic head of each node along the water table. Convergence is generally attained within ®ve it-erations. This technique is applicable to any 2-D or 3-D ®nite element code that contains an automatic ®nite-el-ement grid generator.
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