© L. Sha and Z. Pan, published by IFP Energies nouvelles, 2018

1 Introduction

The number of unconventional oil and gas wells, including deviated well, highly-deviated well, extended reach well, horizontal well, etc., is steadily increasing over past years. Many of them are in the complex and fiercest of drilling fields, such as in deepwater, deep stratum, and polar fields (Ma et al., 2015). Though directional and horizontal drillings are more expensive than vertical drilling, they have the potential to significantly increase well productivity and to deliver a lower overall cost of supply on a cost/unit of gas and oil produced (Dalzell, 2013). Finding minimum trajectory length and more accurate well paths to the predetermined target are two key factors in reducing drilling time and overall well cost in directional and horizontal drillings, and at the same time reducing the risk of drilling (Karimpour et al., 2016). Thus, optimizing the trajectory of complex wellbores is of great interest to minimize drilling time and cost in drilling industry. One key method to increase efficiency and successful drilling is using optimum drilling technique in trajectory wellbore designing and accurate trajectory controlling.

Complex wellbore trajectories can be optimized in various ways with the common objective of minimizing the true measured depth (TMD) of multi-targets leading to a horizontal section in the reservoir. The trajectory design model for an objective function was originally proposed by Adams and Charrier (Adams and Charrier, 1985). In 2014, using the novel particle swarm algorithm (NPSO), a deviated well trajectory design illustrated that the geometrical distances and angles need to be considered in calculating the minimum well paths (Atashnezhad et al., 2014). In 2015, a 3D well path was designed using a multi-objective genetic algorithm (Mansouri et al., 2015). In 2016, David A. Wood applied Hybrid cuckoo search optimization (HCSO) algorithm to deal with the complex wellbore trajectories optimization (Wood, 2016a). In 2016, Wood proposed metaheuristic profiling to assess performance of hybrid evolutionary optimization algorithm in complex wellbore trajectories optimization problem (Wood, 2016b).

In this paper, we propose a Fibonacci sequence based self-adjustment quantum genetic algorithm (FSQGA) for 3D wellbore trajectory optimization problems with multi-variables and complex constraint conditions. Compared to the existing algorithms, this new approach has abilities on parallel process and fast global searching, thus can provide real-time optimal results with higher efficiency and better accuracy.

In FSQGA, the Fibonacci sequence is adopted to self-adjust the step of quantum rotation gate to optimal trajectory of complex 3D wellbore of multi-targets in the Bloch spherical coordinate. Fibonacci series has a negative exponent property which can be used to update quantum rotation gate. Firstly, we show that the time complexity of the algorithm based on Fibonacci series can be declined to O(1) without increasing space complexity. Secondly, we demonstrate that FSQGA enhances ergodicity of solution space through mapping each quantum bit into a point in Bloch sphere. Finally, we use the FSQGA to optimize TMD under the nine constrain conditions of each wellbore section, casing setting depths, true vertical depth, and boundaries with total 16 variables, including the wellbore length, inclination hold angles, azimuth angles, dogleg severities, and lateral length, etc. The simulation results show that this algorithm is faster and less time consuming; and has better minimum objective function values. The simulation also shows that the TMD of complex wellbore trajectory in actual gas-oil field can be reduced up to 4.65%. In summary, this new algorithm has the potential to improve drilling efficiency, and to reduce the drilling time and drilling cost in real-time wellbore trajectory control.

2 Concept of FSQGA

In 2000, Han proposed the quantum genetic algorithm (QGA) which is a probability optimal algorithm of quantum computing theory based on quantum state, quantum state interference and quantum state superposition. QGA has small population size, fast convergence speed and excellent global searching properties (Han and Kim, 2000). In 2009, a double chain quantum genetic algorithm (DCQGA) was designed to avoid binary coding randomness which was induced by quantum bit measurement, and frequent decoding in solving continuous function optimization problems. This algorithm can also enlarge the number of optimal solutions (Li and Li, 2008, 2009). In 2010, an improved DCQGA was proposed to increase the convergence speed. However, this method lacks of robustness (Xu and Xu, 2010). In 2012, a variable step double chain QGA was proposed to enhance both the convergence speed and stability (Sha and He, 2010). Since only the first order linear regulation was adopted for angle rotation, the convergence speed is still slow specially when the change rate of the objective fitness value at the searching point is extremely low or high (Sha and He, 2012). To improve the direction and speed of convergence, the FSQGA utilizes the change rates of the two adjacent generations of the objective function to direct the search towards the global optimum solution.

2.1 Q-bit solution space transformation

Using the linear transformation, each q-bit can be represented as |ϕ ⟩ = [cosϕsinθ, sinϕsinθ, cosθ] ^T . Then m q-bits can be turned into 3m solutions in the unity solution space I ⁿ  = [ − 1, 1] ⁿ . The solution space variables which are transformed from the jth quantum bit of the ith quantum chromosome in tth iteration can be defined as:(1)where , and are coordinate values of q-bits in Bloch spherical coordinate. Each chromosome has three solutions in optimization problem. b_i and a_i are maximum and minimum values of the optimization problem, respectively.

2.2 Update the quantum chromosomes

The algorithm efficiency in QGA is determined by the rotation angle of a quantum rotation gate R. In each iteration, the phase of q-bit is updated without changing its length. Quantum chromosome is updated through the phase change of quantum rotation gate R. For each ith quantum chromosome , the quantum chromosomes can be described as:(2)

The function of R is to rotate the phase of quantum bit by Δϕ in XOY plane and Δθ in Z direction. The signs of Δϕ and Δθ decide the direction of convergence and the values decide the rate of convergence.

2.3 Negative exponent characteristic of Fibonacci series

Let F_N be the Nth element of Fibonacci series, which can be defined as:(3)

In equation (3), each element equals to the sum of the first two elements from the 3rd one. It can be shown (Eq. (4)):(4)where x = 0, 1, 2, …, positive integer. N is more than 10 (Sha, 2013). The fitting relation of x and F_N /F_N+x is shown in Figure 1. The dashed line represents fitting relation curve of e ^−0.4812x .

In Figure 1, x reflects the change rate of the objective function at the search point and F_N /F_N+x reflects the step of rotation angle of quantum rotation gate in the process of searching the optimal solution of FSQGA. The step of rotation angle decreases at high gradient points to avoid crossing over the global solution and oscillation. Conversely, the step of rotation angle increases at low gradient points to accelerate the convergence rate and robustness.

Considering fitting relation precision and range of rotation angle step, the negative exponent characteristic of fitting relation between F_N /F_N+x and x can be used to adjust rotation angle when N = 15, x ∊ [0,15], x ∊ N ⁺. The fitting relation can be described as:(5)

Analyzing the fitting error of equation (5) and making E (x) = e ^−0.4812x  − F_N /F_N+x , the relation curve between E (x) and x is shown in Figure 2.

Fig. 1 Fitting relation of x and FN /FN+x.

Fig. 2 Fitting relation error between E(x) and x. Note that the error E < 9 × 10−6 when N = 15, x ∊ [0,20], x ∊ N + and the error equals almost zero when N = 15, x ∊ [15,20], x ∊ N +. It is reasonable to use the FN /FN +x instead of e −0.4812x (Sha, 2013).

2.4 Calculate steps Δϕ and Δθ based on Fibonacci series

According to difference of chromosomes in population, the change rate of objective function fitness of adjacent two generations can be used to update the steps of the rotation angles. Δϕ and Δθ can be obtained as follow:(6)where sgn(.) is symbolic function. Δφ ₀ and Δθ ₀ are constants with Δφ ₀ = Δθ ₀ = 0.05π. x reflects the change rate of objective function fitness of adjacent two generations, defined as:(7)where int(.) is a round function which returns the nearest integer. x is the step adjustment matrix which has Size rows and Coder columns. Size is the size of population, with the value of 50–80. Coder is the number of optimal variables. is the gradient of objective function at the point of . ∇f _i max and ∇f _i min is the maximum and minimum of objective function gradients of two adjacent generations.

2.5 Analyze the algorithm complexity

Comparing FSQGA to ordinary QGA and BEGA, the algorithm time complexity T(n) and space complexity S(n) are shown in Table 1.

3 Optimal complexity wellbore trajectory

Complex wellbore trajectory can be composed of n straight line segments and m curved segments. Each section must meet 3 conditions: (1) the two adjacent sections are not straight lines; (2) there is tangency at the common point of two adjacent sections; and (3) it is 3rd order differentiable at the common point of two adjacent sections. The vertical plane cross-section of a typical complex wellbore trajectory is shown in Figure 3.

Fig. 3 Vertical plane cross-section of a typical complex wellbore trajectory (from Atashnezhad et al., 2014 and modified after Shokir et al., 2004). D1: First build-up; D2: tangent section; D3: drop-off section; D4: hold section; D5: second build-up section; and HD: horizontal section.

3.1 Objective function of optimal 3D wellbore trajectory

Under multi-variables and multi-constraint conditions in complex multi-targets wellbore trajectory, FSQGA is adopted to optimize the TMD of the objective function. All sixteen variables, including the wellbore length, inclination angles, azimuth angles, and curvature of sections, are optimized under the nine constraint conditions in each wellbore section, casing setting depths and true vertical depth. The objective function is defined as: (8)where TMD is optimal objective function, ft. X  =   (HD, phi1 ∼ phi3, theta1 ∼ theta6, D _D , D _B , D _S1 ∼ D _S3, D _KOP  ) ∈ R ¹⁶. The solution space consists of 16 dimension decision vector X which is the number of optimal variables. j is the number of casing section, j = 1, 2, 3. D ₁ to D ₅ are the calculated measured depths of specific segments of the wellbore trajectory (Shokir et al., 2004; Atashnezhad et al., 2014). The boundaries of variables are defined in Table A1 in the Appendix.

3.2 FSQGA in 3D wellbore trajectory optimization

The flowchart of FSQGA-based 3D wellbore trajectory optimization is shown in Figure 4. The θ and ϕ of the first population are generated randomly. The parameter set of FSQGA are listed in Table 2. The 16 independent variables are adjusted to realize 3D wellbore trajectory control and optimization. The objective is to minimize true measured depth min{TMD} under constraint and boundary conditions.

Fig. 4 Flowchart of FSQGA-based 3D wellbore trajectory optimization.

4 Simulation results and discussions

4.1 Simulation results

Using FSQGA to optimize 3D wellbore trajectory, the optimization simulation results of TMD is shown in Figure 5.

As shown in Figure 5, the optimal solution is approaching to the global optimization as step functions. The progressive steps of the optimal solution are shown in Table 3.

From Figure 5 and Table 3, it can be find: (1) in the iteration sections of the 0–14, 15–31, 32–39, 40–53, 54–144 and 145–200, no update on the best optimal solution since there are no improvement from the current iteration; (2) D-value almost progressively reduces with increased iterations. It is difficult to determine the maximum iteration numbers due to the unknown real global optimal TMD solution. Therefore, in FSQGA, the convergence criteria are selected as Iteration > 200 or D-value < 10 ft. TMD is tending to be a global optimal solution 1.48 * 10⁴ ft when the iteration > 145.

In Figure 6, the value of TVD fluctuates with the identical frequency and width as TMD in Figure 5. As a boundary condition, TVD is always within the range of 10 850–10 900 ft. When the iteration is greater than 145 to reach the global optimal TMD solution, TVD tends to be stable at 1.0887 * 10⁴ ft.

Fig. 5 True measurement depth (TMD) vs. # of iteration using FSQGA in 3D wellbore trajectory optimization. Iteration = 200. Note that TMD is tending to be a global optimal solution 1.48 * 104 ft when the iteration > 145.

Fig. 6 The true vertical depth (TVD) vs. # of iteration using FSQGA in 3D wellbore trajectory optimization.

4.2 The algorithm complexity

To analyze the algorithm complexity of FSQGA for TMD optimization, we define t as the number of iterations, n as the population size, k as the number of independent variables, m as the number of particles, and j as the coding length of the chromosome of each independent variable.

4.2.1 Time-complexity

The time complexity of PSO algorithm equals T(n) = O(n ² + m) which can be represented as O(n ²). The time complexity of normal QGA equals T(n) = O(eⁿ ) which can be recorded as O(cⁿ ). The time complexity of FSQGA equals T(n) = O(n) which can be denoted as O(1). The time complexity of GA equals T(n) = O(n ²).

4.2.2 Space-complexity

The space complexity of PSO algorithm equals S(n) = O(n ²) + O(nm) which can be represented as O(n ²). The space complexity of normal QGA equals S(n) = O(nkj) which can be described as O(n ³). The space complexity of FSQGA equals S(n) = O(3nk) which can be recorded as O (n ²). The space complexity of GA equals S(n) = O(nkj) which can be represented as O(n ³).

As shown in the Table 4, the time complexity of FSQGA is greatly reduced by introducing the Fibonacci sequence. Meanwhile, there is no increase in the space complexity. Therefore, this algorithm can improve the operation efficiency and reduce the running time.

4.3 Discussions

Using the TMD as the optimization objective, the 16 dimensional decision vectors, X = (HD, phi1 ∼ phi3, theta1 ∼ theta6, D _D , D _B , D _S1 ∼ D _S3, D _KOP ) ∈ R ¹⁶, are optimized under constraint conditions of the variable values of each section, casing lengths and vertical target depth. The optimization results are shown in Table 5. Comparing with other intelligent algorithms, the optimal TMD by FSQGA has higher efficiency and considerably less running time. Meanwhile, it also provides the better global optimization solution, and can thus reduce the cost of drilling with improved drilling efficiency. From the data shown in Table 5, the TMD of FSQGA is shortened by 1.46% to 4.65%. Thus the drilling cost can be reduced up to 4.65%. Moreover, FSQGA algorithm can reduce the running time significantly (from >2000 s to 3.5 s).

5 Conclusion

We proposed and developed a new fast self-adjustment QGA to optimize the 3D complexity wellbore trajectory with multiple variables and complex constraint conditions. Compared to the existing algorithm, the new method has enhanced real-time processing ability, and can provide higher optimization accuracy and speed. Firstly, Fibonacci sequence has a negative exponent characteristic which can be introduced into self- adaptive updating step of quantum rotation gate to reduce the time complexity of the algorithm to O(1) without increasing the space complexity of algorithm. Secondly, there is an enhanced ergodicity of solutions space through mapping each quantum bit point into a point in Bloch spherical coordinate. The FSQGA algorithm is more rapidly and have more quantum chromosomes of solution space than five other algorithms, using similar population sizes, number of iterations run and execution on the same computer, and therefore it is a suitable and efficient to solve complex multi-variable /multi-constraint engineering problems, such as wellbore trajectory design.

Finally, the new proposed method is demonstrated that it enhanced the global search capability in the multi targets 3D complex wellbore trajectory optimization. The drilling parameters, such as inclination hold angles, azimuth angles, dogleg severity, true vertical depths, and lateral length can be processed by using the FSQGA to optimize objective function TMD under the nine constraint conditions. The simulations show that FSQGA has better minimum objective value, faster running speed. We believe that using the new algorithm in actual drilling processing for wellbore trajectory control can enhance the real-time processing ability, improve drilling efficiency, and reduce drilling cost and time.

References

Appendix

Definitions of wellbore trajectory variables: (modified after Shokir et al., 2004).

phi1, phi2, phi3: First, second and third hold angles, degrees

theta1: Azimuth angle at kick off point, degrees

theta2: Azimuth angle at end of the first build, degrees

theta3: Azimuth angle at end of the first hold section, degrees

theta4: Azimuth angle at end of the second build or drop, degrees

theta5: Azimuth angle at end of the second hold section, degrees

theta6: Azimuth angle at end of the third build portion, degrees

D_{S 1} : Dogleg severity of the first build portion, /100 feet

D_{S 2} : Dogleg severity of first hold portion, /100 feet

D_{S 3} : Dogleg severity of second build or drop portion, /100 feet

TMD : True measured depth, feet

TVD : True vertical depth, feet

D_KOP : Depth of kick off point, feet

D_B : True vertical depth of the well at the end of drop-off section (top of the third build section), feet

D_D : True vertical depth of the well at the top of drop-off section (top of second build section), feet

HD : Lateral length (horizontal length), feet

P ₁, P ₂ : Two survey points

r : Radius of the pipes, feet

ΔMD : Interval length, feet

P_m : Mutation probability

All Tables

All Figures

	Fig. 1 Fitting relation of x and FN /FN+x.
In the text

	Fig. 2 Fitting relation error between E(x) and x. Note that the error E < 9 × 10−6 when N = 15, x ∊ [0,20], x ∊ N + and the error equals almost zero when N = 15, x ∊ [15,20], x ∊ N +. It is reasonable to use the FN /FN +x instead of e −0.4812x (Sha, 2013).
In the text

	Fig. 3 Vertical plane cross-section of a typical complex wellbore trajectory (from Atashnezhad et al., 2014 and modified after Shokir et al., 2004). D1: First build-up; D2: tangent section; D3: drop-off section; D4: hold section; D5: second build-up section; and HD: horizontal section.
In the text

	Fig. 4 Flowchart of FSQGA-based 3D wellbore trajectory optimization.
In the text

	Fig. 5 True measurement depth (TMD) vs. # of iteration using FSQGA in 3D wellbore trajectory optimization. Iteration = 200. Note that TMD is tending to be a global optimal solution 1.48 * 104 ft when the iteration > 145.
In the text

	Fig. 6 The true vertical depth (TVD) vs. # of iteration using FSQGA in 3D wellbore trajectory optimization.
In the text