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Among the fens located in Michigan, the average inter-fen distance is just over 5 kilometers.
Almost two-thirds of the fens have at least one adjacent fen less than 5 kilometers away. This fen is located on a narrow strip of coarse-grained outwash material adjacent to the River Raisin in Lenawee County, Michigan Fig 1. The predominant glacial material in this area is a fine-grained glacial till that occurs on both sides of the River Raisin. Ives Road Fen is adjacent to two surface water bodies—the River Raisin to its east and a small pond to its northwest.
The distance between the pond and the River Raisin is about meters, but the difference in their surface water elevations is almost 5 meters. As a result, the land surface slopes away rather steeply from the pond towards the river, with Ives Road Fen located between the surface water bodies. Due to this slope, there is no ponding of water at the fen location, as any overland flow that may enter the fen runs off towards the River Raisin.
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At the regional scale, the topography slopes gently towards River Raisin from the Hillsdale mound as seen in Figs 2 and 3. Typically, in such areas where there is a general topographic slope with relatively negligible local topographic relief, regional groundwater flow systems develop [ 31 ]. Apart from River Raisin, there are no other large surface water bodies near the fen. The nearest large lakes Lake Erin, Sand Lake, and Wamplers Lake are almost 16 kilometers away to the west of the fen at the northeastern edge of the Hillsdale groundwater mound.
The shallow glacial geology in a narrow band along River Raisin consists of coarse-grained deposits created by a glacial outwash channel. Under this well-sorted material, older fine-grained clay-like materials are found. Evidence for this is seen in the well logs from the statewide water well database, Wellogic [ 32 ].
At the regional scale, the shallow outwash aquifer is bordered on its west by a series of moraines and till plains composed of medium- and fine-textured tills. Further west is Hillsdale mound, a large glacial interlobate area composed of outwash material. The borehole data indicate that there is a fairly thick more than 30 meters and extensive clay layer in the till plain, beneath which a confined aquifer is found.
From the borehole data we can also infer that there may be openings in the clay layer near the River Raisin caused by erosional processes of the river. These openings connect the lower confined aquifer to the shallow outwash aquifer adjacent to the River Raisin. The deeper bedrock unit in this area is the Coldwater Shale formation, which is a confining unit and can be assumed to be a no-flow boundary.
In order to accomplish the study objectives, a coupled geologic modeling and hierarchical, multi-scale groundwater modeling approach was used to understand the multi-scale groundwater flow system. To model the complex, 3-dimensional geology in the area of interest, a Transition Probability approach was used. In this approach, lithologic data from Wellogic , a state-wide well log database [ 32 ] was used to map the geologic variability. The general approach in hierarchical groundwater modeling is to model as large an area as necessary to capture the regional-scale dynamics, and then progressively refine the model in smaller areas of interest, utilizing a site-scale model s to resolve variability at that scale.
A hierarchical patch dynamics modeling approach developed by Li and colleagues [ 33 ]—[ 39 ] that enables multi-scale modeling in a highly flexible and efficient manner was used. The geologic model was incorporated into the groundwater model to characterize the complex 3-dimensional geology in the study area. A particle tracking approach that uses the results velocity vectors from the groundwater models was used to delineate the sources of water and the delivery mechanisms to the fen. More details on the coupled approaches used in this study are provided next. The conventional methodology in groundwater modeling is to divide the vertical extent of the model into geologic layers, such as glacial and bedrock aquifers.
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Lithologic data from Wellogic [ 32 ] indicates that using this quasi-3D approach of geologic layers cannot accurately model 3-dimensional variability in aquifer materials. In order to account for this kind of variability, a fully 3-dimensional geologic model was created using a Transition Probability approach. This geologic model was then used to inform a conceptual hydro-geologic model to predict the hydrologic connections that support the fen.
The first step in this approach was to classify the different geologic materials using the lithologic descriptions from the borehole data. In principle, the borehole data may be classified into any number of such materials. Using the above-mentioned classification scheme for the lithologic data, the geologic model was constructed using more than well logs.
Using the borehole data, the transition probability matrix of auto- and cross-correlations between the different aquifer materials as a function of vertical lag spacing was created.
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Thus, a matrix of graphs was created, denoting the spatial variability of each aquifer material in the vertical direction, which was then fit to a geo-statistical model using a Markov chain analysis. In order to convert this vertical model into a 3-dimensional model, an anisotropy ratio, i. More details on the transition probability approach are available in [ 40 ]. The geologic model was manually calibrated by adjusting the values of the anisotropy ratio. The vertical extent of each aquifer material was reflected in its average vertical thickness, which was inferred directly from the data.
For instance, if the data showed that the average thickness of a piece of clay was 5 meters, and if the anisotropy ratio was set to 10, then the horizontal extent of the clay would be 50 meters. The geologic model was calibrated by visually comparing the simulation results to the borehole logs and to the known large-scale geologic structure. The selection of an appropriate anisotropy ratio is a rather subjective decision, which can, however, be guided by larger scale understanding of the geologic depositional processes that create spatial continuity of deposits. Carle et al.
Details of the calibrated geologic model are provided in Table 1. The calibrated values of vertical anisotropy are 15 and 8. Therefore, their average horizontal lengths are about the same, which is consistent with the conceptual understanding of a continuous confined aquifer below a continuous clay layer. Therefore, they are likely to have lesser spatial continuity in the lateral directions. Since each realization of the geologic model represents only one likely geologic scenario, realizations of the geologic model were simulated.
Ideally, each realization of the geologic model can be incorporated into the groundwater model, thus creating realizations of the groundwater model. A more pragmatic approach was to incorporate the ensemble mean of the realizations of the geologic model, which is an unbiased statistical representation, into the groundwater model. Importantly, this averaging procedure also made it possible to evaluate the likelihood of the openings in the clay layer discussed previously.
This averaging procedure can have a significant impact on the particle tracking used to delineate the sources of water. Obviously, the particle paths predicted by the average of realizations will be less dispersed than those predicted by averaging the particle paths predicted by the realizations. A 3-dimensional view of the ensemble mean of the realizations from the geologic model is presented in Fig 4.
The presence of borehole data means that this opening in the clay layer is not an artefact of the model as each realization honors the data, which also implies that the averaging procedure would have no impact on the prediction at this location. Indeed, Fogg et al. These inter-connections may be critical in creating upwelling of water from the regional system towards the River Raisin, and potentially to Ives Road Fen.
Based on the understanding of the local and regional hydrogeologic and topographic features, it is clear that the regional discharge area is River Raisin, which must receive water from Hillsdale groundwater mound, the regional recharge area. From the 3-dimensional geologic model it is clear that water from the mound reaches River Raisin through the confined aquifer beneath the clay layer. It is important to note that this pipeline is not literally a uniform, 1-dimensional conduit but rather a tortuous, 3-dimensional preferential path created by the presence of high hydraulic conductivity materials with embedded confining materials as predicted by the geologic model.
The upstream and downstream end of this pipeline are connected to the regional mound and the River Raisin respectively. This hydrologic connection to River Raisin is created by several openings in the clay layer, possibly created by erosional processes along the river. Ives Road Fen benefits from this complex geologic process. Fig 8 presents a schematic of the conceptual model for this regional connection between the Hillsdale mound and Ives Road Fen. At the local scale, the fen likely receives water from the small pond and local recharge area to the west that provide water to the shallow outwash aquifer.
The hierarchical groundwater models were developed using Interactive Groundwater [ 34 ]—[ 37 ], an interactive modeling environment that is live-linked to GIS-enabled groundwater databases. In the hierarchical groundwater modeling framework, for a generic model, M p , l , which refers to model patch p in level l , the governing equations for steady-state groundwater flow can be given as: 1. Each child or patch model in the hierarchical model derived its boundary conditions from its parent model in the form of a prescribed head boundary; this is called down-scaling.
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Each parent model, in turn, incorporated the finer details from its child models through up-scaling. A hierarchy of steady state groundwater flow models was developed for Ives Road Fen using the data-enabled, multi-scale modeling framework, such that the multi-scale hydrologic processes were adequately resolved. Since many studies have noted that fens are characterized by saturated conditions throughout the year without being inundated for any significant length of time i.
Groundwater models were created at watershed, regional and local scales, which were linked to each other through an iterative two-way head coupling mechanism see Eq 1. This mechanism consists of down-scaling in which the child models derive boundary conditions from its parent model, and up-scaling in which the parent models aggregate information from the child model to reflect local conditions. Details of the hierarchical model discretization grid sizes and resolutions are provided in Table 2.
One of the approaches to avoid this issue of dry cells is to use a coarse vertical discretization [ 45 ]. However, the complex geology in the study area necessitated the use of a fine vertical grid to resolve the geologic model. The issue of dry cells is also encountered when adjacent model cells have hydraulic conductivities varying over several orders of magnitude [ 46 ]. To overcome this issue, we used an iterative vertical discretization scheme.
In this approach, the groundwater model is first discretized using one vertical layer i. In the next iteration, the water table from the previous iteration is used to sub-divide the saturated thickness of the aquifer at a finer resolution, say NZ of 2 or 4, and then solved.
This process is repeated until the desired vertical resolution is achieved. As a result, the vertical grid spacing is not uniform in the model and depends on the saturated aquifer thickness at each location. This iterative process ensures that the occurrence of dry cells in the model is minimized, if not eliminated. Topography and aquifer geometry, which form the hydro-geologic framework, are critical to the formation of local, intermediate, and regional systems of groundwater flow [ 31 ], [ 47 ]. The bottom of the glacial aquifer was simulated as a no-flow boundary, since the bedrock unit in this area is a confining unit Coldwater Shale formation.
The area with the extensive till plain was simulated using a calibrated lower rate of recharge, which was appropriate for an area with significant clay thickness. The two-way head dependent boundary condition allows water flux to enter or leave the aquifer depending on the head gradient between the aquifer and the boundary as shown in Eq 2.
Lake leakances ranged from 0. Since streams are 1-dimensional features that were converted to equivalent 2-dimensional grid cells in the groundwater model, their leakance values were converted as shown in Eq 3. Stream leakances ranged from 1 md -1 for 1 st order stream to 50 md -1 for 6 th order streams. The land surface was simulated as a one-way drain with a leakance, whose drain elevation was set to the land surface elevation.
This accounted for groundwater seepage in areas where the water table elevation exceeded the land surface elevation as given in Eq 4. The drain leakance was set to 1 d -1 , which was later calibrated. The groundwater models were calibrated manually using static water levels from water well records in the statewide database, water level measurements at 3 locations within the fen, and base-flow estimates at USGS stream-flow gaging stations.
The hydraulic conductivity values were calibrated to match the hydraulic heads at the wells from the statewide database. The calibrated hierarchical groundwater model was used to perform 3-dimensional, reverse particle tracking to identify the sources of water to the fen. The particle tracking algorithm was implemented within the Interactive Groundwater modeling environment [ 34 — 37 ], and used an Eulerian scheme to move the particles using advection only.
About particles were uniformly distributed in the fen area and tracked backwards through the hierarchical groundwater models in a seamless manner. This process was repeated until the particle came to a stop upon reaching a source of water i. One of the limitations of this approach is that the particle tracking was performed in a domain in which the geologic variability was smoothed by averaging among many realizations, which can potentially under-represent the complexity of the system.
However, critical aspects of the geologic model such as the openings in the clay layer, which can have a significant impact on the particle tracking, have a much greater statistical likelihood see Fig 6. In other words, while the lateral extent of the source area delineations may be under-estimated by the smoothed geologic realization, the connectivity between the Hillsdale mound and the fen has a significantly higher likelihood of occurrence.
Also, the particle tracking approach was used only to indicate the sources of water to the fen, and not to precisely predict the flux that enters the fen or the relative contributions from the different sources delineated. The final calibrated groundwater model parameters are presented in Table 3. The base-flow into the River Raisin predicted from the groundwater model was , m 3 d -1 , which was in agreement with the observed base-flow for the River Raisin at Adrian USGS Site of , m 3 d Fig 9 shows the comparison of observed water levels with the predicted groundwater model heads for the regional and local models.
At the regional scale, there were data points to compare with the model, which decreased to 37 at the local scale, including 3 data points within the fen boundary. From the calibration plots we see that the regional scale model had an R 2 of 0. There are, however, a number of potential factors that lead to weaker performance of the groundwater model at the local scale, including: i poor data quality due to measurement error, temporal bias or geo-spatial inaccuracies, ii inability to resolve local geologic features due to lack of input data, iii inaccurate groundwater model conceptualization and iv ineffective calibration.
While the latter two factors are, in general, applicable to any model, the first two factors may have a greater impact in this case, because the data used for calibration were from the statewide database, which included data collected at various points in time across decades and during various seasons. Given that the range of head values is smaller at the local scale and the lower sample size, measurement errors can have a greater impact on the relationship with predicted values.
The advantage of using these data is that the data quantity is significantly larger, especially at regional scales, than typically used in groundwater studies. However, in the approach used in this study, the hydraulic conductivity varied in every model cell. Relative to the simplest parameterization, i. However, a limitation of this approach is that it assumed that the hydraulic conductivity for each aquifer material was the same throughout the model.