3.7. Rivers

The river boundary condition of MINEDW simulates rivers, streams, canals, and/or ditches that interact with the groundwater system. Using the river package instead of a constant-head boundary depends on the hydrogeologic setting. For example, if the river depth and head remain unchanging, users should simulate the stream as a constant-head boundary instead of using the river package.

The following terms describe the river package of MINEDW and appear in example river networks:

  • River: A surface-water flow body that interacts with the groundwater system such as a river, stream, canal, or ditch.

  • Reach: The region between user-defined surface-water nodes where river-groundwater interaction occurs and river flow properties remain relatively constant.

  • Riverbed: The area beneath a stream with physical properties that control flux between the river and groundwater system.

The river function of MINEDW provides flexibility through the following features:

  • Rivers assigned automatically from input files or manually input,

  • Slopes defined from ground-surface elevation or user-defined constant values,

  • Riverbed hydraulic conductivity specified for each reach or set constant for an entire river,

  • External flow to rivers applied at constant or time-varying rates, and

  • Multiple rivers defined as a network with flow routing between rivers.

MINEDW users define rivers using a series of connected nodes in the model. Multiple rivers form a river network within the MINEDW model. River flow routes from one river to another in the network. External water sources (pipe discharges or river diversions) apply along rivers. Networks include reaches and tributaries.

Discharge into or out of the model domain from groundwater exchange with river nodes follows the equation below.

(3.21)\[\begin{split}q_{R} = \sum_{i=1}^{n} \begin{cases} C_{Ri}\,\bigl(H_{Ri} - H_i\bigr), & \text{for } H_i > H_{BOTi},\\ C_{Ri}\,\bigl(H_{Ri} - H_{BOTi}\bigr), & \text{for } H_i \le H_{BOTi}. \end{cases}\end{split}\]

Where:

  • \(q_R\) = discharge into or out of the groundwater system related to surface water [L³T⁻¹],

  • \(C_{Ri}\) = riverbed conductance [L²T⁻¹],

  • \(H_{Ri}\) = elevation of stream water level [L],

  • \(H_i\) = Computed head at node i [L], and

  • \(H_{BOTi}\) = elevation of the bottom of the streambed at node i [L].

Riverbed conductance (\(C_{Ri}\)) depends on four properties: the hydraulic conductivity of the riverbed, the reach length, the river width, and the riverbed thickness. The equation below shows the riverbed conductance calculation, with parameters used in calculating surface water exchange with the groundwater system.

(3.22)\[C_{Ri} = \frac{K_s\,L\,W}{B}\]

Where:

  • \(K_s\) = vertical hydraulic conductivity of the streambed at node i [L],

  • L = length of reach associated with node i [L],

  • W = width of river associated with node i [L], and

  • B = thickness of streambed directly above node i [L].

Manning’s Equation relates river flow to channel geometry as follows:

(3.23)\[Q = \frac{\mathrm{conv}}{n}\,A\,R_{H}^{2/3}\,S^{1/2}\]

Where:

  • Q = river discharge [L³T⁻¹]

  • n = Manning’s roughness coefficient [TL⁻¹/³], A = cross sectional area of the river [L²],

  • \(R_H\) = hydraulic radius (defined in the equation) [L],

  • S = slope between adjacent river nodes [L⁰], and

  • conv = constant for unit consistency [L⁰].

The hydraulic radius (\(R_H\)) and cross-sectional area (A) are defined as:

(3.24)\[R_{H} = \frac{w\,d}{w + 2d}\]
(3.25)\[A = w\,d\]

Where:

  • w = width of the river [L], and

  • d = depth of the river [L].

Water conservation between surface and groundwater systems requires that the difference between upstream and downstream discharge equals \(q_R\), as shown in the water balance equation below. Here, \(Q_{\text{in}}\) equals \(Q_{\text{out}}\) of the immediately upstream reach.

(3.26)\[q_{R} = Q_{\text{out}} - Q_{\text{in}}\]

Where:

  • \(Q_{\text{out}}\) = discharge at the downstream end of the stream reach [L³T⁻¹], and

  • \(Q_{\text{in}}\) = upstream inflow to the stream reach, including tributary inflow [L³T⁻¹].

The river function is nonlinear because discharge into or out of the groundwater system from surface water interaction (\(q_{R}\)) depends on both the local groundwater head distribution and the water elevation in the stream. Using these equations, a non-linear, iterative solution finds the river depth, which is then used to calculate the discharge into or out of the groundwater system.

The river function uses Manning’s equation to relate river flow to channel geometry and maintains water balance between surface and groundwater systems through iterative solutions.