Table 1 Thermodynamic model for the thermal water cycle

ZP1 → ZP2 (well inflow)

Δp

Friction: \(p\left(r,t\right)-{p}_{i}=-\frac{\dot{m}\cdot g}{4\cdot \pi {\cdot T}_{\mathrm{GW}}}\cdot \left(-\mathrm{0,577216}-\mathrm{ln}\left(u\right)+u-\frac{{u}^{2}}{2\cdot 2!}+\frac{{u}^{3}}{3\cdot 3!}-\frac{{u}^{4}}{4\cdot 4!}+\dots \right)\cdot {10}^{-6}\)

with \({T}_{\mathrm{GW}}=\frac{\rho \cdot g}{\mu }\cdot \mathrm{k}\cdot \mathrm{h}\) and \(u=\frac{{r}^{2}\cdot S}{4{\cdot T}_{\mathrm{GW}}\cdot t}=\frac{{\mu \cdot r}^{2}}{4\cdot k\cdot t}\cdot c\cdot \varphi\)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation \({T}_{\mathrm{ZP}1}-{T}_{\mathrm{ZP}2}=\frac{\left({h}_{\mathrm{ZP}1}-{h}_{\mathrm{ZP}2}^{*}\right)}{{{c}_{pm}}^{*}}\) with \({h}_{\mathrm{ZP}2}^{*}=h({{p}_{\mathrm{ZP}2},T}_{\mathrm{ZP}1})\) and \({c}_{pm}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}2}+{p}_{\mathrm{ZP}1}}{2},{T}_{\mathrm{ZP}1})\)

ZP2 → ZP3 (rise to pump)

Δp

Potential change and friction: \(\Delta p=\Delta {p}_{\mathrm{pot}}+\Delta {p}_{\mathrm{fric}}=\left( L\cdot g\cdot {\rho }_{f}+\frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}2}-{T}_{\mathrm{ZP}3}=\frac{\left({h}_{\mathrm{ZP}2}-{h}_{\mathrm{ZP}3}^{*}\right)}{{{c}_{pm}}^{*}}\) with \({h}_{\mathrm{ZP}3}^{*}=h({{p}_{ZP3},T}_{ZP2})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}3}+{p}_{\mathrm{ZP}2}}{2},{T}_{\mathrm{ZP}2})\)

Heat losses into the formation: \({T}_{e}={T}_{0}+({T}_{i}-{T}_{0})\cdot {e}^{\left(\frac{-k\cdot U\cdot z}{\rho \cdot v\cdot A\cdot {c}_{p}}\right)}\), A = area perpendicular to flow

with \(k=\frac{1}{R\cdot A}\), \(R=\frac{1}{2\cdot \pi \cdot L}\cdot \left(\frac{1}{{\alpha }_{\mathrm{Brine}}{\cdot r}_{1}}+\frac{ln\left({r}_{2}/{r}_{1}\right)}{{\lambda }_{\mathrm{Casing}}}+\frac{ln\left({r}_{3}/{r}_{2}\right)}{{\lambda }_{\mathrm{Cement}}}+\frac{ln\left({r}_{4}/{r}_{3}\right)}{{\uplambda }_{\mathrm{Formation}}}\right)\), and \({\alpha }_{\mathrm{Brine}}=\frac{Nu\cdot {\lambda }_{\mathrm{Brine}}}{2\cdot r}\)

ZP3 → ZP4 (pump)

Δp

Production pump pressure increase: iterative determination to match the surface system pressure

ΔT

Pump compression: \({T}_{\mathrm{ZP}4,\mathrm{compr}}-{T}_{\mathrm{ZP}3}=\frac{\left({h}_{\mathrm{ZP}4}-{h}_{\mathrm{ZP}3}\right)}{{c}_{\mathrm{pm}}}=\frac{\left({h}_{\mathrm{ZP}4,isentrop}-{h}_{\mathrm{ZP}3}\right)/{\eta }_{isentrop}}{{c}_{\mathrm{pm}}}\)

with \({c}_{\mathrm{pm}}=\frac{{c}_{p}\left({p}_{\mathrm{ZP}4},{T}_{\mathrm{isentrop}}\right)+{c}_{p}\left({p}_{\mathrm{ZP}3},{T}_{\mathrm{ZP}3}\right)}{2}\)

Pump motor cooling: \({T}_{\mathrm{ZP}4}-{T}_{\mathrm{ZP}4,\mathrm{compr}}=\frac{{P}_{\mathrm{hydr}}\cdot (1-{\eta }_{\mathrm{Motor}})}{\dot{m}\cdot {c}_{p}}\)

with \({P}_{\mathrm{Hydr}}=\dot{V}\cdot \Delta p\cdot {10}^{-3}\) and \(\dot{V}=\frac{\dot{m}}{\left({\rho }_{\mathrm{ZP}3}+{\rho }_{\mathrm{ZP}4}\right)/2}\)

ZP4 → ZP5 (rise to surface)

Δp

Potential change and friction: \(\Delta p=\Delta {p}_{\mathrm{pot}}+\Delta {p}_{\mathrm{fric}}=\left( L\cdot g\cdot {\rho }_{f}+\frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}4}-{T}_{\mathrm{ZP}5}=\frac{\left({h}_{\mathrm{ZP}4}-{h}_{\mathrm{ZP}5}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}5}^{*}=h({{p}_{\mathrm{ZP}5},T}_{\mathrm{ZP}4})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}5}+{p}_{\mathrm{ZP}4}}{2},{T}_{\mathrm{ZP}4}\))

Heat losses to the (gas filled) annulus: \({T}_{e}={T}_{0}+({T}_{i}-{T}_{0})\cdot {e}^{\left(\frac{-k\cdot U\cdot z}{\rho \cdot v\cdot A\cdot {c}_{p}}\right)}\)

with \(k=\frac{1}{R\cdot A}\) and \(R=\frac{1}{2\cdot \pi \cdot L}\cdot \left(\frac{1}{{\alpha }_{\mathrm{Brine}}{\cdot r}_{1}}+\frac{ln\left({r}_{2}/{r}_{1}\right)}{{\lambda }_{\mathrm{Pipe}}}+\frac{ln\left({r}_{3}/{r}_{2}\right)}{{\lambda }_{\mathrm{Annulus}}}\right)\), and \({\alpha }_{\mathrm{Brine}}=\frac{Nu\cdot {\lambda }_{\mathrm{Brine}}}{2\cdot r}\)

ZP5 → ZP6 (flow to heat exchanger)

Δp

Friction: \(\Delta {p}_{\mathrm{fric}}=\left(\frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}5}-{T}_{\mathrm{ZP}6}=\frac{\left({h}_{\mathrm{ZP}5}-{h}_{\mathrm{ZP}6}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}6}^{*}=h({{p}_{\mathrm{ZP}6},T}_{\mathrm{ZP}5})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}6}+{p}_{\mathrm{ZP}5}}{2},{T}_{\mathrm{ZP}5}\))

Heat losses to the environment: \({T}_{e}={T}_{0}+({T}_{i}-{T}_{0})\cdot {e}^{\left(\frac{-k\cdot U\cdot z}{\rho \cdot v\cdot A\cdot {c}_{p}}\right)}\)

with \(k=\frac{1}{R\cdot A}\) and \(R=\frac{1}{2\cdot \pi \cdot L}\cdot \left(\frac{1}{{\alpha }_{\mathrm{Brine}}{\cdot r}_{1}}+\frac{ln\left({r}_{2}/{r}_{1}\right)}{{\lambda }_{\mathrm{Pipe}}}+\frac{ln\left({r}_{3}/{r}_{2}\right)}{{\lambda }_{\mathrm{Insulation}}}\right)\), and \({\alpha }_{\mathrm{Brine}}=\frac{\mathrm{Nu}\cdot {\lambda }_{\mathrm{Brine}}}{2\cdot r}\)

ZP6 → ZP7 (heat exchanger)

Δp

2% (input value)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}6}-{T}_{\mathrm{ZP}7}=\frac{\left({h}_{\mathrm{ZP}6}-{h}_{\mathrm{ZP}7}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}7}^{*}=h({{p}_{\mathrm{ZP}7},T}_{\mathrm{ZP}6})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}7}+{p}_{\mathrm{ZP}6}}{2},{T}_{\mathrm{ZP}6})\)

Heat extraction: \({P}_{\mathrm{gt}}=\dot{m}\cdot {c}_{p}\cdot \left({T}_{i}-{T}_{o}\right)\)

ZP7 → ZP8 (flow to Injection pump)

Δp

Friction: \(\Delta {p}_{\mathrm{fric}}=\left(\frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic expansion: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}7}-{T}_{\mathrm{ZP}8}=\frac{\left({h}_{\mathrm{ZP}7}-{h}_{\mathrm{ZP}8}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}8}^{*}=h({{p}_{\mathrm{ZP}8},T}_{\mathrm{ZP}7})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}8}+{p}_{\mathrm{ZP}7}}{2},{T}_{\mathrm{ZP}7}\))

Heat losses to the environment: \({T}_{e}={T}_{0}+({T}_{i}-{T}_{0}){e}^{\left(\frac{-k\cdot U\cdot z}{\rho \cdot v\cdot A\cdot {c}_{p}}\right)}\)

with \(k=\frac{1}{R\cdot A}\) and \(R=\frac{1}{2\cdot \pi \cdot L}\cdot \left(\frac{1}{{\alpha }_{\mathrm{Brine}}{\cdot r}_{1}}+\frac{ln\left({r}_{2}/{r}_{1}\right)}{{\lambda }_{\mathrm{Pipe}}}+\frac{ln\left({r}_{3}/{r}_{2}\right)}{{\lambda }_{\mathrm{Insulation}}}\right)\), and \({\alpha }_{\mathrm{Brine}}=\frac{\mathrm{Nu}\cdot {\lambda }_{\mathrm{Brine}}}{2\cdot r}\)

ZP8 → ZP9 (Pump)

Δp

Injection pump pressure increase: iterative determination against the direction of flow starting with the undisturbed reservoir pressure

ΔT

Pump compression: \({T}_{\mathrm{ZP}9}-{T}_{\mathrm{ZP}8}=\frac{\left({h}_{\mathrm{ZP}9}-{h}_{\mathrm{ZP}8}\right)}{{c}_{\mathrm{pm}}}=\frac{\left({h}_{\mathrm{ZP}9,\mathrm{isentrop}}-{h}_{\mathrm{ZP}8}\right)/{\eta }_{\mathrm{isentrop}}}{{c}_{\mathrm{pm}}}\)

with \({c}_{\mathrm{pm}}=\frac{{c}_{p}\left({p}_{\mathrm{ZP}9},{T}_{\mathrm{isentrop}}\right)+{c}_{p}\left({p}_{\mathrm{ZP}8},{T}_{ZP8}\right)}{2}\)

ZP9 → ZP10 (flow to Wellhead)

Δp

Friction: \(\Delta {p}_{\mathrm{fric}}=\left( \frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic compression: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}9}-{T}_{\mathrm{ZP}10}=\frac{\left({h}_{\mathrm{ZP}9}-{h}_{\mathrm{ZP}10}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}10}^{*}=h({{p}_{\mathrm{ZP}10},T}_{\mathrm{ZP}9})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}10}+{p}_{\mathrm{ZP}9}}{2},{T}_{\mathrm{ZP}9}\))

ZP10 → ZP11 (flow thru casing to reservoir)

Δp

Potential change and friction: \(\Delta p=\Delta {p}_{\mathrm{pot}}+\Delta {p}_{\mathrm{fric}}=\left( L\cdot g\cdot {\rho }_{f}+\frac{\lambda \cdot L}{d}\cdot \frac{{\rho }_{f}\cdot {v}^{2}}{2}\right)\cdot {10}^{-6}\)

ΔT

Isentropic compression: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}10}-{T}_{\mathrm{ZP}11}=\frac{\left({h}_{\mathrm{ZP}10}-{h}_{\mathrm{ZP}11}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}11}^{*}=h({{p}_{\mathrm{ZP}11},T}_{\mathrm{ZP}10})\) and \({c}_{pm}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}11}+{p}_{\mathrm{ZP}10}}{2},{T}_{\mathrm{ZP}10}\))

Heat losses into the formation: \({T}_{e}={T}_{0}+({T}_{i}-{T}_{0})\cdot{e}^{\left(\frac{-k\cdot U\cdot z}{\rho \cdot v\cdot A\cdot {c}_{p}}\right)}\), A = area perpendicular to flow

with \(k=\frac{1}{R\cdot A}\), \(R=\frac{1}{2\cdot \pi \cdot L}\cdot \left(\frac{1}{{\alpha }_{\mathrm{Brine}}{\cdot r}_{1}}+\frac{ln\left({r}_{2}/{r}_{1}\right)}{{\lambda }_{\mathrm{Casing}}}+\frac{ln\left({r}_{3}/{r}_{2}\right)}{{\lambda }_{\mathrm{Cement}}}+\frac{ln\left({r}_{4}/{r}_{3}\right)}{{\lambda }_{\mathrm{Formation}}}\right)\), and \({\alpha }_{\mathrm{Brine}}=\frac{\mathrm{Nu}\cdot {\lambda }_{\mathrm{Brine}}}{2\cdot r}\)

ZP11 → ZP12 (well outflow)

Δp

Friction: \(p\left(r,t\right)-{p}_{i}=+\frac{\dot{m}\cdot g}{4\cdot \pi {\cdot T}_{\mathrm{GW}}}\cdot \left(-\mathrm{0,577216}-\mathit{ln}\left(u\right)+u-\frac{{u}^{2}}{2\cdot 2!}+\frac{{u}^{3}}{3\cdot 3!}-\frac{{u}^{4}}{4\cdot 4!}+\dots \right)\cdot {10}^{-6}\)

with \({T}_{\mathrm{GW}}=\frac{\rho \cdot g}{\mu }\cdot k\cdot h\) and \(u=\frac{{r}^{2}\cdot S}{4{\cdot T}_{\mathrm{GW}}\cdot t}=\frac{{\mu \cdot r}^{2}}{4\cdot k\cdot t}\cdot c\cdot \varphi\)

ΔT

Isentropic compression: iterative determination on Δp using Refprop plus salinity correction

Dissipation: \({T}_{\mathrm{ZP}11}-{T}_{\mathrm{ZP}12}=\frac{\left({h}_{\mathrm{ZP}11}-{h}_{\mathrm{ZP}12}^{*}\right)}{{{c}_{\mathrm{pm}}}^{*}}\) with \({h}_{\mathrm{ZP}12}^{*}=h({{p}_{\mathrm{ZP}12},T}_{\mathrm{ZP}11})\) and \({c}_{\mathrm{pm}}^{*}={c}_{p}(\frac{{p}_{\mathrm{ZP}12}+{p}_{\mathrm{ZP}11}}{2},{T}_{\mathrm{ZP}11})\)