38 lines
1014 B
TeX
38 lines
1014 B
TeX
\section{Derivation of \cref{EQINFWELL}}
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\begin{equation*}
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\pi_{w}=\int_{t=0}^{T}\left[ \left(P_{ur}\cdot q_{i}\cdot e^{-Dt}-C_{op}\right)e^{-rt}\right] \,dt-C_{Drill}-C_{Res}\c dote^{-rT} \
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\end{equation*}
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\begin{equation*}
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\frac{d \pi_{w}}{dT}= \left(P_{ur}\cdot q_{i}-C_{op}e^{-rT}+r\cdot C_{Res}\right)e^{-rT}=0 \
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\end{equation*}
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\begin{equation*}
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P_{ur}\cdot q_{i}\cdot e^{-DT}=C_{op}-r\cdot C_{Res}
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\end{equation*}
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\begin{equation*}
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e^{-DT}=\frac{C_{op}-r\cdot C_{Res}}{ P_{ur}\cdot q_{i}}
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\end{equation*}
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\begin{equation*}
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-DT=ln(\frac{C_{op}-r\cdot C_{Res}}{ P_{ur}\cdot q_{i}})
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\ln( P_{ur}\cdot q_{i})
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\left(\ln( P_{ur})+\ln(\cdot q_{i})\right)
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\ln( P_{ur})-\ln(\cdot q_{i})
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\end{equation*}
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\begin{equation*}
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T^{\star}=\frac{\ln(P_{ur})+\ln(q_{i})-\ln(C_{op}-r C_{Res})}{D}
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\end{equation*}
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