Problem $12$: Enhanced Geothermal Systems $($EGS$)$ is a developing technology where

two wells $($drilled boreholes$)$ in a deep, hot, granitic reservoir in the Earth are

interconnected by hydraulic fracturing. Cold fluid can be pumped down one well and then

pass through the network of hydraulic fractures, heating up along the way to the second

well. The hot fluid entering the second well is produced to the surface and flashed or run

through an Organic Rankine Cycle plant, a turbine, and a generator to produce electricity.

A recent test of flow between two such connected wells at an experimental site in Utah

involved pumping down the injection well $($designated as $"1"$ in the figure$)$ at $10$ bpm $.$

This water came from a $125,000$ bbl earthen, lined pit $(125$ Kbbl pit, designated as $"2").$

The geometry of the pit is shown in the second figure. After circulating through the

hydraulic fractures deep in the reservoir, fluid is produced from a production well

$($designated as $"3")$ after circulating through the network of hydraulic fractures. ${?}^1$ The

returned fluid flowed at about $8$ bpm and was flowed through an atmospheric separator

into an open $12,000bbl$ sump $("4")$ and allowed to cool somewhat before being returned

to the $125$ K bbl pit $("1")$ at a volume of $6,300$ bpd$.$ The fraction of steam $($pressure before

the separator is about $200$ psig $)$ is about $16\%$ by mass. The steam is lost to the

atmosphere. There is also evaporation from the pits because of the temperatures and

windy conditions. Without refilling, the $125,000125K$ bb pit is going down about $6-$inches

per day and would be depleted before the planned end of the test. There is a nearby well

that can supply additional $("$make$-$up$")$ water to the $125$ K bbl pit. What flow rate from

this well would you recommend to maintain a constant level in the $125Kbb$

pit at a distance of $4$ vertical feet below the upper lip? Bonus: Can you estimate

${?}^1$ In a commercial scenario, this fluid would then go to an ORC or steam flash.

Figure $1.$ Schematic of pumping setup for the $30-$day flow test. The pumps

shown are high$-$pressure, high$-$rate quintiplex positive displacement pumps.

Figure $2.$ Schematic of earthen pit nominally holding $125,000$ bbl of water. $1$

bbl is $42$ U$.$S$.$ gallons. These ponds are excavated and then lined with a

membrane. Dimensions are in feet. At the left is a plan view, looking down on

the pit. At the right are two cross$-$sections $($elevation views$).$