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Several two-dimensional heat transport simulations were conducted in 2006 to examine how certain parts of the aquifer system could be expected to behave in a high-conductivity aquifer like the SRPA. This section describes results from a two-dimensional cross-sectional model that was used to

illustrate the evolution of the vertical distribution of temperature along a flow line under idealized conditions.

5.2.1 Two-Dimensional Modeling and Evolution of Temperature Profiles along a Flow Line

Temperature profiles in the eastern Snake River Plain commonly display nearly isothermal conditions through what is interpreted as the aquifer. To illustrate how temperatures evolve in the hypothetical two-dimensional cross-sectional model of groundwater and heat flow and, therefore, how the nearly isothermal conditions observed in the eastern Snake River Plain might develop, the

two-dimensional (cross-sectional view) steady-state heat flow equations (Equation 5-5) were solved for an idealized groundwater flow system. The system considered includes a vadose zone, aquifer, and subaquifer. The hydraulic conductivity of the saturated subaquifer unit is assumed to be insufficient to allow significant convection. Heat transfer in the subaquifer is therefore by conduction only. The subaquifer is included primarily to illustrate the background geothermal gradient. Heatflow in the overlying aquifer, in contrast, occurs both by conduction and advection, with the latter specified by the specific discharge in the x-direction. There is no water flow in the y-direction. A vadose zone overlies the aquifer. To maintain a 1-D flow system in the aquifer, heat conduction in the vadose zone is also

simulated as conduction without convection. Numerical solution to the problem is found using the PDE Toolbox in MATLAB.

The bottom boundary condition is most easily specified as the geothermal heat flux. While the average crustal heat flux is about 60 mW m^-2, a value of 100 mW m^-2 is specified, which is closer to that observed in the eastern Snake River plain. The upper boundary is the ground temperature at the top of the vadose zone, which, in arid regions, is typically several degrees higher than mean annual air temperature.

In these simulations, in which the primary interest is the change in the temperature field along a flow line, a temperature of zero is specified along the entire upper boundary. At the downstream boundary, a zero-gradient condition is specified, so that the only heat flux is the convective heat flux. The remaining upstream boundary condition is a primary influence on the overall temperature distribution of the system.

The temperature at this upstream boundary was specified the same as that of the upper boundary, which simulates a system where cold recharge occurs. Thermal properties used in the simulation are the bulk thermal conductivity and the heat capacity of water, which were assigned values of 2 watt m^-1 K^-1 and 4.186E06 joules m^-3 K^-1, respectively.

Given sufficient distance along a flowline, when the heat flux is along the bottom of the system, the temperature gradient will evolve so that heat flux from the top of the aquifer balances that into the aquifer from below. The final temperature field within each unit will thus be such that the product of the vertical thermal conductivity and the vertical temperature gradient will equal the heat flux supplied to the bottom of the aquifer. The distance required to produce that condition is thus proportional to the temperature difference between the initial condition and the final temperature and inversely proportional to the groundwater velocity (Figure 5-1).

If the groundwater temperature at the inflow boundary is less than the mean temperature when the heat flux is entirely upward, the horizontal temperature gradient and the second derivative in the vertical direction will both be positive. Both of these gradients will decrease with distance, until the horizontal temperature gradient is virtually zero and the vertical temperature gradient is uniform. As the accuracy of the specific discharge calculated from application of Equation 5-5 depends on how accurately those values can be determined, the distance along a flowline beyond the point where the temperature profile is externally influenced clearly affects the accuracy of that approach.

2

Specific discharge = 1e-7 m/s Avglinear velocity = ~0.1 m/day Temperature (C)

Specific discharge = 1e-7 m/s Avglinear velocity = ~0.1 m/day

A

B

Figure 5-1. Evolution of temperature distribution (contour plots) along a flow line in an idealized two-dimensional system, with (A) inflow temperatures colder than the steady-state profile for conduction-dominated flow and (B) inflow temperatures warmer than the steady-state profile for conduction-dominated flow. Line plots show temperature profiles at the specified distances along the flow line (indicated by vertical dashed white lines in the contour plots). Dashed horizontal lines show top and bottom of aquifer domain.

If the inflow temperature is greater than the average groundwater temperature under conduction-dominated conditions, then the increased heat flux through the vadose zone will cool the system until the temperature distribution again approaches that necessary to conduct all the heat flux upward. The distance needed to reach the zero-horizontal gradient condition is the same as for the previous case.

Whether water temperatures are increasing or decreasing, the temperature profiles are isothermal only very close to the upstream boundary, where the heating from below has not been sufficient to significantly change the uniform temperature of the inflow. Approximately isothermal profiles in the SRPA appear to occur, however, throughout the aquifer, and are difficult to explain as a simple result of high discharge rates (Figure 5-2A), as is commonly stated (Brott, Blackwell, and Ziagos 1981). One adjustment to the flow regime that may resolve this discrepancy is substantial transverse dispersivity. The simulation depicted in Figure 5-2 reflects a horizontal seepage velocity of approximately 1 m/day-1.

Assuming a transverse dispersivity of 10 m effectively triples the thermal conductivity within the aquifer and yields temperature profiles (Figure 5-2) much more similar to those of deep wells through the aquifer.

As several studies indicated that the longitudinal dispersivity in the aquifer is on the order of 100 m (Robertson 1974; Goode and Konikow 1990), this may not be an unreasonable estimate of the vertical mixing that occurs across large distances.

2

Specific discharge = 1e-6 m/s Avglinear velocity = ~1 m/day Temperature (C)

Specific discharge = 1e-6 m/s Avglinear velocity = ~1 m/day

A

B

Figure 5.2. Effect of a 10-fold increase in velocity (A) and an increase in vertical thermal conductivity (B) on the evolution of temperature profiles in the two-dimensional system described in the text. The increase in thermal conductivity (B) is approximately what would occur with a transverse dispersivity of 10 m. Line plots show temperature profiles at the specified distances along the flow line (indicated by vertical dashed white lines in the contour plots). Dashed horizontal lines show top and bottom of aquifer domain.