Using GIS for hierarchial refinement of MODFLOW models

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1 HydroGÏS 96: Application of Geographic Information Systems in Hydrology and Water Resources Management (Proceedings of the Vienna Conference, April 1996). IAHS Publ. no. 235, Using GIS for hierarchial refinement of MODFLOW models PIERRE T. W. J. KAMPS & THEO N. OLTSHOORN Amsterdam Water Supply, Vogelenzangseweg 21, 2114 BA Vogelenzang, The Netherlands Abstract After working out a way to generate MODFLOW models directly from GIS (ARC/INFO) (Olsthoorn et al, 1993), automated model refinement to obtain GIS-generated child models properly embedded within coarser parent models was sought. The main issue then becomes the boundary of the embedded model. The method chosen is a general head boundary condition with its conductance equal to equivalent model cells. Though this yields exact results for models with smooth geology and hydrological units not crossing the boundary of the embedded model, errors persist in the case of boundary intersecting rivers, drains, etc. At these boundary sections the flow should be specified. The method is illustrated by embedding a 135 row by 75 column model with 20 by 20 m cells into the 128 row by 64 column model with 100 by 100 m cells of the complicated dune area used by the Amsterdam Water Supply Company. INTRODUCTION The Amsterdam Water Supply Company uses a 36 km 2 dune area along the North Sea as a groundwater resource. Artificial recharge by purified river water attributes to 85 % of the water recovered. The area is modelled using MODFLOW and ARC/INFO as a GIS, from which the model is generated automatically (Olsthoorn et al., 1993). The six-aquifer model reaches to a depth of 160 m and measures 6.4 by 12.8 km, i.e. 64 columns by 128 rows with cells of 100 by 100 m. It is bordered in the west by the North Sea and in the east by a polder area. It contains abstraction canals, 40 basins for artificial recharge, 12 drains of about 1 km in length and 238 abstraction wells (Olsthoorn et al., 1993). Though the mesh with the 100 by 100 m cells is sufficient to answer most of the hydrological questions, it is too coarse for many of them. To solve this problem we worked out a method to generate automatically a hierarchy of refined child models, each of which is embedded within its coarser parent. These models have to be connected such that proper results are obtained in the detail requested. Since the automated generation of an MODFLOW model directly from the GIS has been our subject in 1993, we now focus on the connections between the embedded models. Approach Out of several ways to obtain local mesh refinement we chose embedding finer models

536 Pierre T. W. J. Kamps & Théo N. Oltshoorn (child models) within coarser ones (parent models). Detailing is thus limited to the area of focus, leaving the rest of the model as it is.

2 536 Pierre T. W. J. Kamps & Théo N. Oltshoorn (child models) within coarser ones (parent models). Detailing is thus limited to the area of focus, leaving the rest of the model as it is. The approach allows refinement to be propagated indefinitely, by creating new child models within models that are child models themselves (currently we use three levels of models this way). Should the boundary of the topmost model interfere with the modelling objective, the top level model itself can be embedded within an even larger model, such as a national model provided by others. Embedding refined models in a coarser one Embedding a refined model in a larger and coarser one, does not cut a hole in the latter; the model at each level is a complete model in itself, covering the entire area within its extensions. Hence within the area covered by the child model, the results of the parent and the child model are available. In order to simulate the groundwater flow for a parent-child system of models, first the parent model must be solved, then the embedded child model, and so on, down to the most refined model. Before an embedded model can be solved, its boundary information has to be obtained from its parent model. Because any model is complete in itself, boundary information transfer is in one direction only (from the coarser to the finer model) and therefore, no iterations between models are necessary. This is valid for stationary and transient models. Figure 1 shows part of the modelled dune area with the MODFLOW-mesh of a refined child model within the main parent model. Canals, drains, wells and recharge ponds are shown, as well as the important peat layer that locally divides the top 20 m into two aquifers, of which the uppermost is only up to 5 m thick (Olsthoom et al., 1993). This parent-child model configuration will be the focus of our discussion. However, it is by far not the only embedded model available and used in the area. Boundary conditions of the child model To properly embed the child model within its parent, its boundary conditions have to be obtained. Several possibilities would seem to exist: (a) heads from the coarser model could be taken, interpolated at the centres of the outer cells of the finer model, and (b) flows at the boundaries of the finer model could be used, the latter taken from the coarse-model intercell flows. However each of these approaches poses problems in certain situations: (a) Using heads from the coarser model, interpolated at the centres of the outer cells of the finer model, is not without ambiguity. The method of interpolation is significant. It should at least include the transmissivity information and the fact that recharge on cells with given head is not considered by MODFLOW (McDonald & Harbaugh, 1988). More severe inconsistencies are caused by the refinement itself, which provokes a shift of the modelled rims of hydrological and geological features (such as the peat layer in Fig. 1). Locally, the flow in some of the boundary cells may become large as an artifact of the chosen fixed head boundary conditions.

meters 0 150 300 450 600 Fig. 1 Part of the main six-aquifer model of the Amsterdam Water Supply Company dune area with 100 by 100 m cells, with embedded child model with 20 by 20 m cells.

3 meters Fig. 1 Part of the main six-aquifer model of the Amsterdam Water Supply Company dune area with 100 by 100 m cells, with embedded child model with 20 by 20 m cells. The infrastructure includes canals, artificial recharge ponds, drains and wells. The resistance between the first two aquifers is formed by the peat layer shown. (b) Using only the intercell flows from the coarser model as the boundary condition for the child model may destroy the uniqueness of the solution of the child model, because at least one of its cells must have a given head or a general head boundary condition, which would no longer be guaranteed. A more satisfactory boundary condition is found by mixing both possibilities into a general head boundary formulation. Here the outer cells of the child models are bound

4 538 Pierre T. W. J. Kamps & Théo N. Oltshoorn to the cells of the parent model surrounding it, via a conductance. Hence the outer cells of the child model are not fixed and can adapt themselves to the calculated boundary flow that equals: 4 = ^parent-^child) 0) Only the conductance remains to be specified. This is done by ensuring that the resistance between the child boundary cell and the adjacent coarse cell equals that of the intercell flow of the original coarse model. This way the boundary flow between child and parent model equals the intercell flow across the same boundary line in case the head at the child boundary equals the interpolated head of the coarse model at the same location. Given a child model boundary cell and an adjacent a parent cell, with cell widths L ch andl^ respectively, hydraulic conductivities k ch and k pa, heads 4> ch and 4> and a linear gradient within each cell, then the flow q between the two cells is known (McDonald & Harbaugh, 1988). The conductance between the centre of child and parent cell is then: C = B chi :' ch.»\ (2) where B ch is the width of the child cell. If MODFLOW would allow several general head boundaries at a single cell, all conductances could be calculated in advance. Since this is not the case we have to add both conductances at each corner of the child model, weighted by the head of the two cells bounding each child corner: c = C lvl + C 2^Q,2 (3) All other terms of the water balance such as storage, given head, precipitation and nodal flows are completely contained within the cells and do not intersect the boundary. Therefore, the above boundary conditions are the only ones to be taken into account. These are valid for both stationary and transient models, and so a transient system of parent-child models can be simulated by first running the parents and afterwards dealing with the child models recursively down to the finest one. Test case The embedded model of Fig. 1 is used here to test the method. To this end we shall compare the flow through the general head boundary conductances with the intercell flow of the parent model at the same location. Clearly within a GIS environment the coarse and the embedded models will be created directly from the same GIS data. Inevitably the overlapping part of the two models (the area occupied by the entire child model) will then differ to some extent. One of the reasons is the better fit of geological and hydrological features by the child model. If the geology is smooth and leaving out hydrological infrastructure to avoid features intersecting the boundary then the difference between the boundary flow of the child model and the intercell flow off the parent model, totalled over all aquifers, is small (0.08%)

5 Using GIS for hierarchial refinement of MODFLOW models 539 Fig. 2 Head of parent and child model in the top aquifer at 0.5 m intervals (coloured cells) with interpolated contours for the parent model at same intervals (thick lines). Complete models, with canals, recharge ponds and drains, both generated directly from GIS.

6 540 Pierre T. W. J. Kamps & Théo N. Oltshoorn E + M. 4 - It 4 ~ Afi 1 lsr^i_tv^ 1 j II ^^ Il \ r ^ 1 V^^\ -! -- : 4 V j! j ChilcM -. Chi!d2 Parentl --- Parent2 (a) ^ r! ====?= X-Axis l r i i i r, i i i t i, i ( i i i i i! i i _ z ipr - i _ I! 1 i An 1 1 B il S 1 Childl -. Child2 Parentl --- Parent2 ^VABS il n i «.»«1 (b) X-Axis Fig. 3 (a) Head (m + msl) in boundary cells of child model and adjacent parent model cells around it, for the first and second aquifer, situation of Fig. 2; (b) boundary flow (m 3 day" 1 ) into child model and intercell flow across same line of parent model shown in Fig. 2 for aquifers 1 and 2. The horizontal coordinate is measured along the child model boundary, clockwise from the top left corner in 100 m units. Corners are at x = 0, 15, 42, 57 and 84. Figure 2 shows the parent and child models with canals, recharge basins and drains, which are largely unseen by the coarse parent model due do its cell size. The head calculated by MODFLOW can be read from the jumps in cell colour at 0.5 m intervals. The contours of the coarse parent model, after interpolation onto a 50 m ARC/INFO- GRID, are shown as drawn lines at the same 0.5 m intervals and continued within the child model. Due to the inevitable differences between the refined and the coarse model, the lines do not perfectly match the colour jumps of the child model. Clearly the two models do not fit nicely at boundary sections where infrastructure intersects. This is especially visible at the north boundary of Fig. 2. Of course there is much difference in the interior of the child model. Where the parent model only produces some crude lines, the child model now reveals detailed heads due to individual canals and recharge ponds. Because in MODFLOW the effect of any object is located in the centre of the cells, the centre of gravity of many canals is shifted due to the refinement by a maximum of

Using GIS for hierarchial refinement of MODFLOW models 541 (100-20)/2 = 40 m (see for instance the most western canal in Fig. 2). All such effects cause non-fits at the boundary of the child model.

7 Using GIS for hierarchial refinement of MODFLOW models 541 (100-20)/2 = 40 m (see for instance the most western canal in Fig. 2). All such effects cause non-fits at the boundary of the child model. While these differences are perhaps of minor significance with respect to the heads along the boundary, they cause significant artificial flows. Figure 3 shows the head in the boundary cells of the child model together with the heads in the adjacent coarse model cells for aquifers 1 and 2. The horizontal coordinate (x) is in 100 m units (i.e. coarse cell units) measured clockwise along the embedded model boundary, starting in the upper left corner (Fig. 1). Hence the corners of the embedded model area atx = 0, 15, 42, 57 and 84. The head difference between the embedded and parent model is the driving force of the boundary flow given in Fig. 3 (b). The difference in head between the first and second aquifer between* = 35 and* = 50 is due to the peat layer (see Fig. 1 ). Figure 3 (b) shows the flows across the boundary of the child model together with the intercell flow of the parent model for the first two aquifers. The flow has a saw-tooth shape because each series of five 20 m wide child boundary cells is connected to one 100 m wide parent model cell, and so share the same external head. However their average value closely follows that of the coarse model intercell flow. Since the average per five child cells is correct, it has no overall effect on the internal heads and flows of the child model (Fig. 2). The saw-tooth effect is neglected further. Intensive boundary flows occur at those sites as an artefact of the refinement. This is a kind of short circuiting between the infrastructures at the boundary cells and the adjacent cells of the coarse model. Still the total water balance of the model as a whole is not much affected, but the individual water balance parts are. Differences between boundary flow of the child model and intercell flow of the parent model are concentrated in the first two aquifers as they are induced by intersecting canals, drains, etc. In order to keep the water flow across the boundary of the embedded model as equal as possible to the intercell flow of the parent model, one must specify flows at those boundary cells for which the coarse model contains any head related boundary conditions such as rivers and drains. This way the refined model stays consistent with the coarse model and recursive refinement can be performed. For the major part of the boundary the general head boundary together with the proposed conductance guarantee a proper boundary flow while providing the embedded model with enough head information. CONCLUSIONS Embedding refined models within coarser ones is a viable option to solve more detailed hydrological problems in areas for which a coarse model is already available. Both parent and child models can be generated directly from GIS data by standard procedures (Olsthoorn et al., 1993). The boundary conditions needed for the child model require care in order for the models to be consistent. The conditions are divided in (a) given flow boundaries along adjacent coarse model cells that have some head-related boundary condition such as drains or rivers, and (b) general head boundaries for all other boundary cells, with conductances that can be calculated from the transmissivities of the coarse model. This way refinement can be done in an optimal fashion using all available data within GIS without overkill. The method can furthermore be used both with stationary

8 542 Pierre T. W. J. Kamps & Théo N. Oltshoom and transient models. The refinement being recursive, can be repeated to any level of detail. REFERENCES Olsthoorn, T. N., Kamps, P. T. W. J. & Droesen, W. J. (1993) Groundwater modelling using a GIS at the Amsterdam Water Supply. In: Application of Geographic Information Systems in Hydrology and Water Resources Management (ed. by K. Kovar & H. P. Nachtnebel) (Proc. Int. Conf. HydroGIS'93, Vienna, April 1993), IAHS Publ. no McDonald, M. G. & Harbaugh, A. W. (1988) A modular three-dimensional finite difference groundwater flow model. USGS Techniques of Water Resources Investigations Report, book 6, chapter Al.

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