## 3 Optimize the Design Model
### Five-Stage  
The key points of the well track in Figure 6 are as follows: the wellhead position \(O\); start point of build-up \(A\); end point of build-up \(B\); start point of drop-off \(C\); end point of drop-off \(D\); first target \(T^{\prime}\); and second target \(T\). Let the known parameters be the wellhead center coordinates \(O\)(\(X_{O}\),\(Y_{O}\),\(Z_{O}\)); coordinates of the first target \(T^{\prime}\)(\(X_{T^{\prime}}\),\(Y_{T^{\prime}}\),\(Z_{T^{\prime}}\)); coordinates of the second target \(T\)((\(X_{T}\),\(Y_{T}\),\(Z_{T}\)); plane allowable error \(J\); vertical allowable error \(H\); vertical depth range of kickoff point \(A\)(\(Z_{Amin}\),\(Z_{Amax}\)); range of formation strength at kickoff point \(A\)(\(UCS_{min}\),\(UCS_{max}\)); range of formation strength at drop-off point \(C\)(\(UCS_{min}\),\(UCS_{max}\)); range of \(k_{1}\) build-up rate (\(k_{1 min}\),\(k_{1 max}\)); range of \(k_{2}\) drop-off rate (\(k_{2 min}\),\(k_{2 max}\)); range of first stable inclination angle \(a_{1}\)(\(a_{1 min}\),\(a_{1 max}\)); and range of second stable inclination angle \(a_{2}\) (\(a_{2 min}\),\(a_{2 max}\)). The decision variables are set as follows: the vertical depth \(Z_{A}\) of the kickoff point \(A\), m; vertical depth \(Z_{C}\) of the drop-off point \(C\), m; vertical depth \(Z\) of the second target, m; first stable inclination angle \(\alpha_{1}\), (\({}^{\circ}\)); second stable inclination angle \(\alpha_{2}\), (\({}^{\circ}\)); build-up rate \(k_{1}\), (\({}^{\circ}\))/30 m; and drop-down rate \(k_{2}\), (\({}^{\circ}\))/30 m.  
The integrated well has the shortest track length, lowest drilling cost, minimum target error in the first target, minimum drag in the up and down string, and minimum torque in rotary drilling. The multi-objective optimization function can be expressed as  
\[\left\{ \begin{array}{l}\min L_{T}=\min(Z_{A}+L_{AB}+L_{BC}+L_{CD}+L_{DT})\\ \min C=\min\left(\frac{C_{1}}{v_{i}}\sum\limits_{i=1}^{n}L_{i}+\frac{C_{2}}{ v_{j}}\sum\limits_{j=1}^{k}L_{j}\right)\\ \min\delta=\min\left\{\left[\left(\left(X_{T^{\prime}}-X_{O}\right)^{2}+\left( Y_{T^{\prime}}-Y_{O}\right)^{2}\right)^{0.5}-S_{T^{\prime}}\right]^{2}+\left(Z_{T}-Z \right)^{2}\right\}\\ \min F=\min\left[\mu q(L_{BC}\sin\alpha_{1}+L_{DT}\sin\alpha_{2})+\sum\limits _{i=1}^{500}F_{1,i}+\sum\limits_{i=1}^{500}F_{2,i}\right]\\ \min M=\min\left[M_{0}+\mu q(L_{BC}\sin\alpha_{1}+L_{DT}\sin\alpha_{2})D_{t}/2 +\sum\limits_{i=1}^{500}\Delta M_{1,i}+\sum\limits_{i=1}^{500}\Delta M_{2,i} \right]\end{array} \right. \tag{32}\]  
Figure 6: Five-section wellbore trajectory profile. (\(k_{1 min}\),\(k_{1 max}\)); range of \(k2\) drop-off rate (\(k_{2 min}\),\(k_{2 max}\)); range of first stable inclination angle \(\alpha_{1}(\alpha_{1 min}\),\(\alpha_{1 max}\)); and range of second stable inclination angle \(\alpha_{2}(\alpha_{2 min}\),\(\alpha_{2 max})\). The decision variables are set as follows: the vertical depth \(Z_{A}\) of the kickoff point \(A\), m; vertical depth \(Z_{C}\) of the second kickoff point \(C\), m; vertical depth Z of the second target, m; first stable inclination angle \(\alpha_{1}\), (\({}^{\circ}\)); second stable inclination angle \(\alpha_{2}\), (\({}^{\circ}\)); first build-up rate \(k_{1}\), (\({}^{\circ}\))/30 m; and second build-up rate \(k_{2}\), (\({}^{\circ}\))/30 m.  
The objective functions of the shortest comprehensive well track length, lowest drilling cost, minimum target error in the first target, minimum drag of the string up or down, minimum torque in rotary drilling, and maximum horizontal open-hole extension capability are as follows:  
\[\left\{