Mathematical Problems in Engineering

Volume 2017, Article ID 7898647, 8 pages

https://doi.org/10.1155/2017/7898647

## A New Mathematical Method for Solving Cuttings Transport Problem of Horizontal Wells: Ant Colony Algorithm

^{1}School of Petroleum Engineering, China University of Petroleum, Qingdao 266580, China^{2}College of Energy Engineering, Yulin University, Yulin 719000, China^{3}Drilling Technology Research Institute, Shengli Petroleum Engineering Corporation, Sinopec, Dongying 257000, China^{4}Laojunmiao Oil Production Plant, Yumen Oilfield, Jiuquan 735000, China

Correspondence should be addressed to Liu Yongwang; moc.361@3002gnawgnoyuil

Received 19 March 2017; Revised 1 July 2017; Accepted 16 July 2017; Published 29 August 2017

Academic Editor: Jian G. Zhou

Copyright © 2017 Liu Yongwang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 often-encountered difficulties. In this paper, the ant colony algorithm is applied to solve the two-layer cuttings transport model with highly nonlinear equations set. The solution-searching 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 solution-searching 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.

#### 1. Introduction

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 [1–10] to investigate this problem, among which the two-layer model is one of the analytical research models and is formulated with highly nonlinear equations set.

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 solution-searching 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 often-encountered difficulties when using Newton methods to solve real engineering problems.

Recently, some researchers [11–19] used artificial intelligence algorithms, such as Genetic Algorithm, Simulated Annealing Algorithm, and Artificial Fish-Swarm Algorithm, to solve nonlinear equations set and obtained satisfactory results. The artificial intelligence algorithms search solutions in the whole definition domain and the result does not depend on the initial values. Moreover, the artificial intelligence algorithm does not need to calculate the Jacobian matrix and the variable definition domain can be artificially preset according to real problems requirements. Therefore, the initial values sensitivity problem, the singularity problem in calculating Jacobian matrix, and the variable outflow of its definition domain problem can be effectively avoided when using artificial intelligence algorithms to solve problems with nonlinear equations set. Ant colony algorithm is one of the artificial intelligence algorithms and has been widely used in optimizing engineering problems. Since solving real engineering problems needs much more work on model formulation, model simplification, variable definition domain determination, model solution, and so on, it is much more complicated than solving pure mathematical nonlinear equations set. Some researchers [18, 19] tried to solve pure mathematical nonlinear equations set with ant colony algorithm, but few applications of ant colony algorithm in solving real engineering problems with nonlinear equations set have been reported.

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.

#### 2. Formulation of Cuttings Transport Problem

##### 2.1. Model Formulation

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, , , , , and refer to area, cuttings concentration, velocity, wetted perimeter, and shear stress, respectively. The subscripts, , , , and , refer to suspension layer, cuttings bed, suspension-bed interface, and total quantity, respectively.

Under steady flow conditions, assuming no slip between the liquid and solid phases, the material balances can be expressed as follows.

For solid phase [1],

For liquid phase,where is the annular area. In Figure 1, is the hole diameter and is the drill pipe diameter. SI units are adopted if units are not specially indicated.