Cuttings transport problem has long been recognized as one of the key difficulties in drilling horizontal wells, and the models in cuttings transport research are usually formulated with highly nonlinear equations set. When using Newton methods to solve real engineering problems with nonlinear equations set, the problems of result dependence on initial values, Jacobian matrix singularity, and variable outflow of its definition domain in iterations are three of the oftenencountered difficulties. In this paper, the ant colony algorithm is applied to solve the twolayer cuttings transport model with highly nonlinear equations set. The solutionsearching process of solving nonlinear equations set is transformed into an optimization process of searching the minimum value of an objective function by applying ant colony algorithm. Analyzing the results of the example, it can be concluded that ant colony algorithm can be used to solve the highly nonlinear cuttings transport model with good solution accuracy; transforming the solutionsearching process of solving nonlinear equations set into an optimization process of searching the minimum value of the objective function is necessary; the real engineering problem should be simplified as much as possible to decrease the number of unknown variables and facilitate the use of ant colony algorithm.
Cuttings transport problem has long been recognized as one of the key difficulties in drilling horizontal wells. Over the past 30 years, considerable effort has been expended on solving cuttings transport problem in drilling horizontal wells. Many researchers developed various models [
Nevertheless, solving the complicated highly nonlinear model to get a reasonable and stable solution has long been a challenge to researchers. Usually, the Newton methods, including the Newton iteration method, Discrete Newton method, and Newton Downhill method, are used in solving nonlinear equations set. However, the result solved by using the Newton methods is highly dependent on the initial values, and finding proper initial values for nonlinear equations set is not an easy job. Meanwhile, since the gradient or the Jacobian matrix has to be calculated and updated in the iteration, singularity problem of Jacobian matrix often occurs in the computation, and this problem will probably make the iteration prematurely terminated. In addition, when these Newton methods are applied to solve real engineering problems in which the variables usually have to fall within their specific definition domain, the solutionsearching process often causes the variable outflow of its definition domain, which often leads to failure of getting reasonable results. Obviously, the result dependence on initial values, Jacobian matrix singularity, and variable outflow of its definition domain in iterations are three of the oftenencountered difficulties when using Newton methods to solve real engineering problems.
Recently, some researchers [
The objective of this paper is to apply the ant colony algorithm to solve the cuttings transport problem with highly nonlinear equations set so as to simplify the process of solving cutting transport model and provide a new way to solve nonlinear engineering problems.
In order to formulate the model of cuttings transport problem, material and momentum balance analysis are needed. In the formulation of material and momentum balance equations,
Under steady flow conditions, assuming no slip between the liquid and solid phases, the material balances can be expressed as follows.
For solid phase [
For liquid phase,
Schematic of cuttings condition under twolayer model [
Under steady flow conditions, the forces acting on the cuttings bed and suspension layer must equal zero. Therefore, the momentum balances can be written as follows.
For cuttings bed,
For suspension layer,
The cuttings concentration in the suspension layer is assumed to submit to the diffusion law, which can be expressed as follows:
In (
From Figure
The analysis above shows that (
When solving the cuttings transport problem in horizontal section (i.e.,
Adding (
Comparing (
The pressure loss per unit length
Through the analysis above, it can be seen that solving the five nonlinear equations set can be simplified into two steps.
First, solve the equations set of (
In order to apply the ant colony algorithm to the cuttings transport problem, an objective function has to be formulated before the calculation. Transforming (
Ant colony algorithm (ACA) is a heuristic algorithm initially proposed by Marco Dorigo in 1992 and has been widely used in many areas [
The initial design of ant colony algorithm is only applicable to discrete domains, such as in TSP problems. When the algorithm is applied in continuous domains, it should be modified. For example, the selection probability is calculated by the fitness value which is related to the objective function value rather than the distance between two discrete cities in the TSP problem.
In this paper,
For the formulation of the fitness value of each ant, choose arbitrary values within their definition domain (e.g., 1.53 m/s, 0.059 m) and then calculate the value of
Objective function:
Fitness value:
Variables:
Definition domain:
The solutionsearching procedure with ant colony algorithm is shown in Figure
Flow loop of equation solving with ant colony algorithm.
(
In the ant colony algorithm, the ants will search solutions within the preset generations. If the ants could not find solution with the preset generations, probably, more ants and generations are necessary. The quantity of ants and iteration generations can be adjusted by doing tests for different problems. Generally, 6–10 ants and 300–700 generations are sufficient for a common optimization.
(
When using the ant colony algorithm to solve the cuttings transport model, it only needs setting the scopes of optimization parameters, which are the cuttings bed height and the suspension layer velocity, rather than providing accuracysensitive initial values. The two parameters will change within the preset scope, and the ant will find solution within their preset scope. This successfully avoids the difficulty of providing the resultsensitive initial value selection problem. The scopes of optimization parameters can be set according to common drilling experiences.
(
The detailed solutionsearching process with ant colony algorithm is as follows.
First, the ant colony algorithm randomly assigns positions (i.e., cuttings bed height and suspension layer velocity) within the two preset optimization scopes for each ant to initiate the ant positions. After doing this, each ant has an initial position with two parameters (i.e., cuttings bed height and suspension layer velocity). Then, these chosen parameters for each ant are sent to evaluate the objective function (i.e., (
The parameters used in the cuttings transport model are shown in Table
Parameters used in the model.
Parameters  Values 

Hole size, m  0.127 
Drill pipe size, m  0.04826 
Consistency coefficient, Pa. 
0.295 
Flow index  0.698 
Drilling fluid density, kg/m^{3}  1.102 × 10^{3} 
Well inclination angle  90° 
The ant population size, maximum generations, and the pheromones evaporation coefficient are taken as 10, 700, and 0.8, respectively.
Figures
Initial distribution of ants.
Distribution of ants at generation = 55.
Final distribution of ants at generation = 550.
The relationship between the average objective function value and iteration generations is shown in Figure
Relationship between the average objective function values and generations.
The cuttings transport problem is also solved with Discrete Newton method. When using Discrete Newton method, one should be very careful with dealing with the singularity problem of the Jacobian matrix and try to keep the variation of variables in the reasonable scope in the iteration. If the values of variables go beyond the reasonable scope, it can lead to obtain unreasonable results or possibly failure to converge. However, when using ant colony algorithm, there is no need to calculate the gradient and Jacobian matrix, so the singularity problem of the Jacobian matrix is avoided. Moreover, the scope of variable variation can be artificially set within the reasonable scope. For instance, the cuttings bed height should vary within the limit of hole diameter, so the variation range of cuttings bed height can be artificially set within the scope (0, 0.127 m), so the variable outflow of its definition domain problem is avoided. Meanwhile, since the ant colony algorithm initiates values of variables randomly in the optimization domain, the optimization result has no dependence on the initial values. Therefore, using ant colony algorithm can avoid the result dependence on initial values problem, Jacobian matrix singularity problem, and variable outflow of its definition domain problem. The solution result comparison between the Discrete Newton method and ant colony algorithm is presented in Table
Comparison of results calculated by Discrete Newton method and ant colony algorithm.
Flow rate, m^{3}/s  Variables  Discrete Newton method  Ant colony algorithm  Error 



1.0282  1.0270  0.12% 

0.0178  0.0179  0.56%  


1.1688  1.1663  0.21% 

0.0111  0.0110  0.90%  


1.3285  1.3303  0.14% 

0.0070  0.0070  0.00% 
The values of pressure loss per unit length
The values of all the variables at flow rate 0.014 m^{3}/s.
Flow rates, m^{3}/s  Suspension layer velocity, m/s  Cuttings bed height, m  Pressure loss per unit length, Pa/m  Cuttings concentration in suspension layer  Velocity of cuttings bed, m/s 

0.014  1.3303  0.007  756.23  0.0025  0 
The cuttings transport model is formulated with nonlinear equations set, and traditionally it is solved by the Newton methods. When the Newton methods are used to solve the nonlinear equations set, the result dependence on initial values, Jacobian matrix singularity, and variable outflow of its definition domain are three of the oftenencountered difficulties. These problems will lead to the failure of getting reasonable results when Newton methods are used to solve real engineering problems.
Compared to the Newton methods, the ant colony algorithm method does not need the selection of resultsensitive initial values but only needs a comparatively large solutionincluded scope, which greatly decreases the difficulty of providing resultsensitive initial values. Since there is no Jacobian matrix in the ant colony algorithm, the new method avoids the Jacobian matrix singularity problem which often occurs when using Newton methods. The ant colony algorithm searches solution in the preset scope, so there is no problem such as variable outflow of its definition domain. Therefore, compared to Newton methods, using ant colony algorithm makes solving the nonlinear cuttings transport model easier and more stable and provides a new way of solving cuttings transport problem.
The present study mainly focuses on proposing a new method of using ant colony algorithm to solve the cuttings transport model. The new method effectively avoids the initial values selection, the singularity problem of Jacobian matrix, and the variable outflow of its definition domain problem in solving the model and meanwhile does not decrease the accuracy, which greatly simplifies the process of solving the nonlinear equations model. The application of using the ant colony algorithm to solve the cuttings transport problem in the field will be further explored in the future research.
The ant colony algorithm can be used to solve cuttings transport model with highly nonlinear equations set, and the solutions solved by ant colony algorithm and Discrete Newton method show good agreement with each other.
Transforming the solutionsearching process of solving nonlinear equations set into an optimization process of searching the minimum value of the objective function is necessary in applying the ant colony algorithm to real nonlinear engineering problems.
The real engineering problem should be simplified according to its physical characteristics as much as possible in order to decrease the number of unknown variables and facilitate the use of ant colony algorithm.
The authors declare that they have no conflicts of interest.
This work is supported by the Qingdao Independent Innovation Project (no. 165123jch); the Fundamental Research Funds for the Central Universities (no. 16CX02025A); Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (no. IRT_14R58).