A Design Model of Target-Hitting Trajectory Drilled Into the Reservoir with Elliptical Truncated Cone Surface

The choice of location and orientation of the surface target-hitting point significantly affects the well productivity and wellbore stability of the reservoir. Thus, before starting actual drilling, the optimal location of the entering point and the direction of the well trajectory are determined by the numerical simulation method. However, the coordinates of the actual landing point and the direction vector of the trajectory are calculated assuming that the target-hitting surface of the reservoir is a plane. Compared with the curved target-hitting surface model, the planar model does not consider the actual shape of the reservoir surface, which can affect the accuracy of calculations. The inaccuracy of the simulation model can cause inaccurate landing and increase the operational costs of the drilling. In this paper, we propose a mathematical background for target-hitting trajectory design in the case of a nonplanar target-hitting surface. First, the elliptical truncated cone surface is assumed as the target-hitting surface due to its complex and diverse shape. Second, the vector algebra method is applied to develop a geometric model of the wellbore trajectory and elliptical truncated cone surface. Third, the target-hitting trajectory design model is established, and the model is solved by a genetic algorithm. Finally, a case study is performed. The off-target distance calculated by the horizontal circular target-hitting surface model is compared with the distance calculated by the horizontal elliptical target-hitting surface model and the proposed model. The case study indicates that the model established in this paper is more accurate than other models in the case of the elliptical truncated cone surface of the reservoir. The established model provides theoretical guidance for the target-hitting trajectory design for nonplanar surface reservoirs.

THE COEFFICIENTS J-M IN EQ. (22)

The coefficients J-M in Eq. (22) can be calculated as

$$ J={k}^2{t}^2+{p}^2{r}^2-{r}^2{t}^2 $$

(A-1)

$$ K=2{jkt}^2+2{stk}^2+2{opr}^2+2{qrp}^2-2{qrt}^2-2{str}^2 $$

(A-2)

$$ L1={j}^2{t}^2+4 jkst+{k}^2{s}^2+{r}^2{o}^2+4 opqr+{p}^2{q}^2-{\mathrm{q}}^2{t}^2-4 qrst-{r}^2{s}^2 $$

(A-3)

$$ M=2{stj}^2+2{jks}^2+2{qro}^2+2{opq}^2-2{sto}^2-2{qrs}^2 $$

(A-4)

$$ N={\mathrm{j}}^2{s}^2+{o}^2{q}^2-{o}^2{s}^2 $$

(A-5)

where j-t can be expressed as

$$ j=\left({P}_1-{P}_0\right)\cdot {t}_a+R\cdot \tan \frac{\theta }{2}\left({t}_1\cdot {t}_a+{t}_2\cdot {t}_a\right) $$

(A-6)

$$ k={t}_1\cdot {t}_a $$

(A-7)

$$ \mathrm{o}=\left({P}_1-{P}_0\right)\cdot {t}_b+R\cdot \tan \frac{\theta }{2}\left({t}_1\cdot {t}_b+{t}_2\cdot {t}_b\right) $$

(A-8)

$$ \mathrm{p}={t}_1\cdot {t}_b $$

(A-9)

$$ \mathrm{q}={a}_1+\frac{\left({a}_2-{a}_1\right)}{L}\left({P}_1-{P}_0\right)\cdot {t}_0+\frac{\left({a}_2-{a}_1\right)}{L}\cdot R\cdot \tan \frac{\theta }{2}\left({t}_1\cdot {t}_0+{t}_2\cdot {t}_0\right) $$

(A-10)

$$ \mathrm{r}=\frac{\left({a}_2-{a}_1\right)}{L}\cdot {t}_1\cdot {t}_0 $$

(A-11)

$$ \mathrm{s}={b}_1+\frac{\left({b}_2-{b}_1\right)}{L}\left({P}_1-{P}_0\right)\cdot {t}_0+\frac{\left({b}_2-{b}_1\right)}{L}\cdot R\cdot \tan \frac{\theta }{2}\left({t}_1\cdot {t}_0+{t}_2\cdot {t}_0\right) $$

(A-12)

$$ \mathrm{t}=\frac{\left({b}_2-{b}_1\right)}{L}\cdot {t}_1\cdot {t}_0 $$

(A-13)