Henk M. Haitjema

On the residence time distribution in idealized groundwatersheds

The relative cumulative frequency distribution of residence times F(T) is calculated for an entire groundwatershed under steady-state conditions and assuming Dupuit-Forchheimer flow. It appears that F(T) is always the same: F(T) = 1 -exp(−TT), provided that the aquifer recharge rate and T are constant over the groundwatershed. T is the weighted mean residence time for the groundwatershed and is defined at T = nHN, where n is the aquifer porosity, H is the saturated aquifer thickness, and N the areal recharge rate owing to precipitation. In such an idealized groundwatershed the function F(T) appears independent of the groundwatershed size, shape, and nature of the stream network. It is also independent of regional variations in the hydraulic conductivity, provided the aquifer is locally homogeneous. Under unconfined flow conditions, where H varies, the relative frequency distribution of the residence time does depend on all of these parameters, but may be approximated by F(T), as demonstrated for the case of a one-dimensional groundwatershed. The frequency distribution of the residence times, dFdT, is the response curve for an instantaneous unit pulse of a conservative tracer, applied over the entire groundwatershed. The findings are of significance for studying the effects of non-point source pollution on the scale of one or more watersheds.

Comparing a three-dimensional and a Dupuit-Forchheimer solution for a circular recharge area in a confined aquifer

Abstract An exact, closed-form analytic solution is presented for three-dimensional steady-state flow due to a source disc. The solution is used to model a circular recharge area at the upper boundary of a confined aquifer. Streamlines and contour plots of equipotentials are compared with those obtained from a Dupuit-Forchheimer solution. The latter solution is approximate in that resistance to vertical flow is ignored. As a further illustration a groundwater contamination problem is solved; that of a leaching circular pond in a uniform flow field. Some streamlines and the shape of the downstream plume are plotted for both the three-dimensional and the Dupuit-Forchheimer solution. It appears that when the recharge area is sufficiently large the Dupuit-Forchheimer solution is quite adequate, particularly when tracing streamlines. The results, therefore, confirm the validity of the Dupuit-Forchheimer assumption when modeling regional flow, even under circumstances of areal recharge. However, when the size of the recharge area is in the order of the aquifer thickness a truly three-dimensional solution is needed.

Approximate analytic solutions to 3D unconfined groundwater flow within regional 2D models

Abstract We present methods for finding approximate analytic solutions to three-dimensional (3D) unconfined steady state groundwater flow near partially penetrating and horizontal wells, and for combining those solutions with regional two-dimensional (2D) models. The 3D solutions use distributed singularities (analytic elements) to enforce boundary conditions on the phreatic surface and seepage faces at vertical wells, and to maintain fixed-head boundary conditions, obtained from the 2D model, at the perimeter of the 3D model. The approximate 3D solutions are analytic (continuous and differentiable) everywhere, including on the phreatic surface itself. While continuity of flow is satisfied exactly in the infinite 3D flow domain, water balance errors can occur across the phreatic surface.

An analytic element solution to unconfined flow near partially penetrating wells

An analytic element solution for steady-state unconfined flow near one or more partially penetrating wells in an ambient flow field has been developed. The phreatic surface is modeled using a three-dimensional version of the Zhukovski function. Boundary conditions at control points on the phreatic surface and seepage face at the well are maintained by use of distributed singularities outside the flow domain. The phreatic surface and seepage face for a single well is compared with two different numerical solutions and a laboratory experiment found in the literature. A sample case showing flow near two wells in an ambient flow field is presented.

An analytic element model for transient axi-symmetric interface flow

Abstract The problem of axi-symmetric flow of two immiscible fluids is solved by use of the analytic element method. A sharp interface divides the flow into two domains with different but homogeneous fluids. The sharp interface is realistic for truly immiscible fluids, and is an approximation for miscible fluids such as fresh and salt water in coastal aquifers. A specific discharge potential is introduced which is discontinuous across the interface. The jump in the specific discharge potential depends both on the difference in fluid viscosity and fluid density. The discontinuous specific discharge potential is realized by use of a singularity distribution along the interface. For the case of axi-symmetric flow, the singularity distribution may be approximated by a set of either sink rings or vortex rings. The approach is implemented in a computer program, which is validated by simulating the movement of a spherical inclusion of one fluid in an infinite domain of another fluid, and comparing the results with an exact solution. Finally, the practical use of the program is demonstrated by simulating a saltwater upconing problem underneath a partially penetrating well. The critical pumping rate for which the interface remains stable is determined and compared with both a laboratory experiment and numerical calculations reported in the literature.

Use of Nested Flow Models and Interpolation Techniques for Science-Based Management of the Sheyenne National Grassland, North Dakota, USA

Noxious weeds threaten the Sheyenne National Grassland (SNG) ecosystem and therefore herbicides have been used for control. To protect groundwater quality, the herbicide application is restricted to areas where the water table is less than 10 feet (3.05 m) below the ground surface in highly permeable soils, or less than 6 feet (1.83 m) below the ground surface in low permeable soils. A local MODFLOW model was extracted from a regional GFLOW analytic element model and used to develop depth-to-groundwater maps in the SNG that are representative for the particular time frame of herbicide applications. These maps are based on a modeled groundwater table and a digital elevation model (DEM). The accuracy of these depth-to-groundwater maps is enhanced by an artificial neural networks (ANNs) interpolation scheme that reduces residuals at 48 monitoring wells. The combination of groundwater modeling and ANN improved depth-to-groundwater maps, which in turn provided more informed decisions about where herbicides can or cannot be safely applied.

Using the stream function for flow governed by Poisson's equation

Abstract For two-dimensional groundwater flow governed by Laplaces equation there exists both a discharge potential function and a stream function, which are each others conjugate harmonics. The stream function is constant along a streamline, and the difference in its value for two streamlines represents the total flow between these streamlines. Most regional groundwater flow problems, however, exhibit areal recharge and leakage from adjacent aquifers. These problems are governed by Poissons equation; no stream function exists for that case. In the context of the analytic element method (AEM), a technique has been developed which allows the stream function to be used in models which include areal recharge or leakage. The AEM uses superposition of analytic functions; the majority satisfy Laplaces equation, whereas only a few functions satisfy Poissons equation. By treating these solutions to Poissons equation separately, the stream function can be utilized in a manner similar to that for flow without areal recharge or leakage. The approach is applied to streamline tracing and, more importantly, to the modeling of leaky slurry walls and no-flow boundaries, both open and closed.

How Aquifer Characteristics of a Watershed Affect Transit Time Distributions of Groundwater: D. Abrams and H. Haitjema Groundwater x, no. x: x-xx

Transit time distributions (TTDs) have a number of applications in the hydrologic sciences. Our work aligns with the group of researchers who use groundwater flow models to assess the transit time distribution of groundwater, not considering the other components of flow that comprise stream water (Kauffman et al., 2008; Sanford et al., 2012; Engdahl, 2017). If the input of a solute is ubiquitous over the watershed and known through time, then a convolution of the TTD and solute input yields the output concentration of groundwater discharging to a well or stream in the watershed over time (Małoszewski and Zuber, 1982).