U.S. Department of the InteriorU.S. Geological Survey

Temperature: A useful tracer for separating conduit and diffuse flow in the karstic Madison aquifer

In cooperation with

West Dakota Water Development District,

City of Rapid City,

South Dakota School of Mines and Technology

Andy Long

Patrick Gilcrease

Model shown conceptually

Inflow from sinking stream

LimestoneOpen passages

Spring discharge

Conduit flow

Slow flow

0 0

qp p q p q t r q s p s

x x x

dTA x C mC T mC T h D x T T q xC T

dt

Ts

Tq

Tr

Heat transport equation for flow in pipe

-Heat accumulation

Heat flow in

Convective heat from conduit wall

Heat from slow inflow

Heat flow out= + +

Inflow from small openings

0

s

d mq

dx

0 2 2

4 4 4q qt s sr q x q s

p

T Th q x qT T T T

t D C D x D

Change in mass flow rate from slow inflow qs

Change in velocity due to change in mass flow rate

Final heat transport equation for the conduit

02

4

d D d m

dx dx

Mixing of conduit and slow flow to a well

LimestoneOpen passages

The well is a binary mixture of conduit flow and local slow flow

(1 )d q lT T T

Tl

Tq

1+ dimensional model

Black Hills

Model applicationStudy area is dominated by streamflow recharge (~90%)

Spring Creek

swallow holes

Temperature sensorOmega RTD class A

±0.03°C at 0°C

Spring Creek

Well house

Cross-section of Madison aquifer and conduit

Heat-transport equation is solved in each model cell by finite difference

0 2 2

4 4 4q qt s sr q x q s

p

T Th q x qT T T T

t D C D x D

Data and results

Summary

The model is useful for estimating the conduit flow fraction to a well or spring

Conduit flow to the well varied temporally from 2 to 31% of total flow

Conduit velocity ranged from 44 to 353 m/d

Changes in temperature for the well were influenced by conduit velocity and the changing relative pressures of the conduit and surrounding aquifer

0 2 2

4 4 4q qt s sr q x q s

p

T Th q x qT T T T

t D C D x D

(1 )d q lT T T

ht = heat transfer coefficientD = conduit diameterqs = flow rate of slow flowTs = temperature of slow flowTq = temperature of rocksTd = temperature of water at discharge point= conduit flow fraction (weighted by head)