# 多目标水平井轨道设计方法研究Study on Design Method of Multi-target Horizontal Well Track

Abstract: By designing multiple control points or target zones in multi-target horizontal wells, the well track could be extended in the reservoirs at maximum level, by which the drilling rate and single well yield were enhanced. With the rapid development of geosteering technology, the multi-target horizontal wells had application prospects in the development of small layers, fault reservoirs, thin reservoirs and remaining oil. Therefore, the designing target was no longer a single horizontal section, and it was composed of multi-control targets, so the trajectory design was greatly constrained and difficult. Based on the theory of space arc trajectory, a general mathematical model for multi-target horizontal well design was established. A vector analysis theory is used for deriving analytic expression of variables of design, and a special software for trajectory design is developed for rapid and accurate design of well tracks under multi-constraining conditions.

1. 引言

2. 设计模型

2.1. 点目标设计模型

Figure 1. The design model for multi-control point horizontal wells

$\mathrm{cos}\gamma =\frac{{T}_{S}-{L}_{1}-{L}_{m}}{{L}_{m}+{L}_{2}}$ (1)

${L}_{m}=\frac{{L}^{2}+{L}_{1}^{2}-{L}_{2}^{2}-2{L}_{1}{T}_{S}}{2\left({T}_{S}+{L}_{2}-{L}_{1}\right)}$ (2)

${L}_{m}=R\cdot \mathrm{tan}\frac{\gamma }{2}$ (3)

${T}_{S}=\left({N}_{T}-{N}_{A}\right){l}_{a}+\left({E}_{T}-{E}_{A}\right){m}_{a}+\left({H}_{T}-{H}_{A}\right){n}_{a}$

$L=\sqrt{{\left({N}_{T}-{N}_{A}\right)}^{2}+{\left({E}_{T}-{E}_{A}\right)}^{2}+{\left({H}_{T}-{H}_{A}\right)}^{2}}$

$R=\frac{{L}^{2}+{L}_{1}^{2}-{L}_{2}^{2}-2{L}_{1}{T}_{S}}{2\sqrt{{L}^{2}-{T}_{S}^{2}}}$ (4)

${L}_{1}={T}_{S}-\sqrt{{T}_{S}^{2}+{L}_{2}^{2}-{L}^{2}+2R\sqrt{{L}^{2}-{T}_{S}^{2}}}$ (5)

${L}_{2}=\sqrt{{L}^{2}+{L}_{1}^{2}-2{L}_{1}{T}_{S}-2R\sqrt{{L}^{2}-{T}_{S}^{2}}}$ (6)

2.2. 线目标设计模型

Figure 2. The design model for multi target-control points and lines

$\text{cos}{\gamma }_{1}=\frac{{T}_{0}-{L}_{\text{mm}}-{L}_{\text{nn}}\text{cos}\theta }{{L}_{\text{mm}}+{L}_{22}+{L}_{\text{nn}}}$ (7)

$\text{cos}{\gamma }_{2}=\frac{{T}_{\text{t}}-{L}_{\text{nn}}-{L}_{\text{mm}}\text{cos}\theta }{{L}_{\text{mm}}+{L}_{22}+{L}_{\text{nn}}}$ (8)

${L}_{\text{mm}}=\frac{{L}_{\text{ad}}^{\text{2}}-{L}_{\text{22}}^{\text{2}}-\text{2}{L}_{\text{nn}}\left({T}_{t}+{L}_{22}\right)}{2\left({T}_{0}+{L}_{22}+{L}_{\text{nn}}-{L}_{\text{nn}}\mathrm{cos}\theta \right)}$ (9)

${L}_{\text{nn}}=\frac{{L}_{\text{ad}}^{\text{2}}-{L}_{\text{22}}^{\text{2}}-\text{2}{L}_{\text{mm}}\left({T}_{0}+{L}_{22}\right)}{2\left({T}_{\text{t}}+{L}_{22}+{L}_{\text{mm}}-{L}_{\text{mm}}\mathrm{cos}\theta \right)}$ (10)

${L}_{\text{mm}}={R}_{\text{1}}\text{tan}\left(\frac{{\gamma }_{1}}{2}\right)$ (11)

${L}_{\text{nn}}={R}_{\text{2}}\text{tan}\left(\frac{{\gamma }_{2}}{2}\right)$ (12)

3. 设计方法

Figure 3. The schematic diagram of target trajectory design

3.1. 目标段轨道设计

1) 2个目标点间的轨道设计。当目标段只有2个目标点时，直接用直线来连接目标点，如普通的水平井。

2) 3个目标点的轨道设计。当目标段只有3个目标点且不在同一直线上时，可采用图4所示的3种方法进行设计。

Figure 4. The design method of three target point trajectory

$\mathrm{cos}\frac{{\gamma }_{1}}{2}=\frac{{L}_{2}^{2}+{L}_{3}^{2}-{L}_{1}^{2}}{2{L}_{2}{L}_{3}}$

$\mathrm{cos}\frac{{\gamma }_{2}}{2}=\frac{{L}_{1}^{2}+{L}_{3}^{2}-{L}_{2}^{2}}{2{L}_{1}{L}_{3}}$

$R=\frac{{L}_{1}}{2\mathrm{sin}\frac{{\gamma }_{1}}{2}}=\frac{{L}_{2}}{2\mathrm{sin}\frac{{\gamma }_{2}}{2}}$

3) 4个目标点以上的轨道设计。可直接应用点目标设计模型来穿越全部控制点。

3.2. 目标间轨道设计

3.3. 着陆前轨道设计

4. 应用实例

Table 1. The data of target design for Well A

Table 2. The data of profile design for Well A

Figure 5. The vertical plan of well trajectory of Well A

5. 结论

1) 提出的多目标控制点井眼轨道设计方法较好地解决了复杂地质结构和油藏条件下轨道设计难题，为井眼轨道设计和控制提供了理论依据。

2) 多目标控制点水平井设计时首先要保证入靶的井斜和方位，一般将多个靶点中的前两个定义为梯形靶，以方便采用线目标模型进行设计。

3) 若采用单圆弧连接两相邻靶点轨迹起伏较大时，可考虑将位于上一圆弧的轨道修正为空间圆弧加稳斜段(或稳斜段 + 空间圆弧)使轨迹更为平滑。

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