Journal of Applied Mathematics

Volume 2017, Article ID 2412397, 12 pages

https://doi.org/10.1155/2017/2412397

## Simulation of Wellbore Stability during Underbalanced Drilling Operation

Chemical and Petroleum Engineering Department, American University of Ras Al Khaimah, Ras Al Khaimah, UAE

Correspondence should be addressed to Reda Abdel Azim; ea.ca.karua@mizaledba.ader

Received 13 June 2017; Accepted 2 July 2017; Published 15 August 2017

Academic Editor: Myung-Gyu Lee

Copyright © 2017 Reda Abdel Azim. 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

The wellbore stability analysis during underbalance drilling operation leads to avoiding risky problems. These problems include (1) rock failure due to stresses changes (concentration) as a result of losing the original support of removed rocks and (2) wellbore collapse due to lack of support of hydrostatic fluid column. Therefore, this paper presents an approach to simulate the wellbore stability by incorporating finite element modelling and thermoporoelastic environment to predict the instability conditions. Analytical solutions for stress distribution for isotropic and anisotropic rocks are presented to validate the presented model. Moreover, distribution of time dependent shear stresses around the wellbore is presented to be compared with rock shear strength to select appropriate weight of mud for safe underbalance drilling.

#### 1. Introduction

Very recent studies highlighted that the wellbore instability problems cost the oil and gas industry above 500$–1000$ million each year [1]. The instability conditions are related to rocks response to stress concentration around the wellbore during the drilling operation. That means the rock may sustain the induced stresses and the wellbore may remain stable without collapse or failure if rock strength is enormous [2]. Factors that lead to formation instability are coming from the temperature effect (thermal) which is thermal diffusivity and the differences in temperature between the drilling mud and formation temperature. This can be described by the fact that if the drilling mud is too cold, this leads to decreasing the hoop stress. These variations in hoop stress have the same effect of tripping while drilling which generates swab and surge and may lead to both tensile and shear failure at the bottom of the well.

The interaction between the drilling fluids with formation fluid will cause pressure variation around the wellbore, which results in time dependent stresses changes locally [3]. Therefore, in this paper the interaction between geomechanics and formation fluid [4] is taken into consideration to analyze time dependent rocks deformation around the wellbore.

Another study shows that the two main effects causing collapse failure are as follows: (1) poroelastic influence of equalized pore pressure at the wellbore wall and (2) the thermal diffusion between wellbore fluids and formation fluids [3–5].

Numerous scientists presented powerful models to simulate the effect of poroelastic, thermal, and chemical effects by varying values of formation pore pressure, rock failure situation, and critical mud weight [3, 6]. These models mentioned that controlling the component of the water present in the drilling fluid results in controlling the wellbore stability. More or less, there are many parameters that could be controlled during the drilling operation as unfavorable in situ condition [7, 8]. In addition, mud weight (MW)/equivalent circulation density (ECD), mud cake (mud filtrate), hole inclination and direction, and drilling/tripping practice are considered the main parameters that affect wellbore mechanical instability [9, 10].

The factors that affect the mechanical stability are membrane efficiency, water activity interaction between the drilling fluid and shale formation, the thermal expansion, thermal diffusivity, and the differences in temperature between the drilling mud and formation temperature [11, 12].

This paper presents a realistic model to evaluate wellbore stability and predict the optimum ECD window to prevent wellbore instability problems.

#### 2. Derivation of Governing Equation for Thermoporoelastic Model

The equations used to simulate thermoporoelastic coupling process are momentum, mass, and energy conservation. These equations are presented in detail in this section.

##### 2.1. Momentum Conservation

The linear momentum balance equation in terms of total stresses can be written as follows:where is the total stress, is the gravity constant, and is the bulk intensity of the porous media. The intensity should be written for two phases, liquid and solid, as follows:Equation (1) can be written in terms of effective stress as follows: where is the effective stress, is the pore pressure, and is the identity matrix. This equation for the stress-strain relationship does not contain thermal effects and, to include the thermoelasticity, the equation can be written as follows: where is the fourth-order stiffness tensor of material properties, is the total strain, is the thermal expansion coefficient, and is the temperature difference. The isotropic elasticity tensor is defined aswhere is the Kronecker delta and is the Lame constant. is the shear modulus of elasticity. The constitutive equation for the total strain-displacement relationship is defined as follows:where is the displacement vector and is the gradient operator.

##### 2.2. Mass Conservation

The fluid flow in deformable and saturated porous media can be described by the following equation:where is the Biots coefficient and assumed to be = 1.0 in this study, is the pore fluid pressure, is the temperature, is the thermal expansion coefficient, is the fluid flux, and is the sink/source, and is the specific storage which is defined by where is the compressibility of solid and is the compressibility of liquid. The fluid flux term in the mass balance in (7) can be described by using Darcy’s flow equation because the intensity has been assumed constant in this study:where is the permeability of the domain. The Cubic law is used in determining fracture permeability.

##### 2.3. Energy Conservation

The energy balance equation for heat transport through porous media can be described as follows:where is the heat flux, is the heat sink/source term, and is the heat storage and equalsIn this study, conduction and convection heat transfers are considered during numerical simulation. The heat flux term in (10) can be written aswhere is the velocity of the fluid. The first term on the right hand side of (12) is the conduction term and the second term is the convective heat transfer term and is the effective heat conductivity of the porous medium, which can be defined as

##### 2.4. Discretization of the Equations

First one discretizes the thermoporoelastic governing equations by using Greens’ theorem [13] to derive equations weak formulations. The weak form of mass, energy, and momentum balance in (1), (7), and (10) can be written as follows, respectively: where is the test function, is the model domain, is the domain boundary, is the traction vector, superscripts refer to the value of the corresponding parameters on opposite sides of the fracture surfaces, respectively, is the specific storage, is the porosity, is the volumetric Darcy flux, is the thermal expansion coefficient, is the fluid sink/source term between the fractures, is the heat flux, is the specific heat capacity, and are mechanical and hydraulic fracture apertures, is the heat sink/source term, is the thermal expansion coefficient, is the thermal conductivity, and refers to the fracture plane.

Then the Galerkin method is used to spatially discretize the weak forms of (14) to (18). The primary variables of the field problem are pressure , temperature , and displacement vector . All of these variables are approximated by using the interpolation function in finite element space as follows:where is the corresponding shape function and , , and are the nodal unknowns values.

#### 3. Validation of Poroelastic Numerical Model

The verification of poroelastic numerical model against analytical solutions (see Appendix) is presented in this section. A two-dimensional model of circular shaped reservoir with an intact wellbore of 1000 m drainage radius and 0.1 m wellbore radius is used (see Figure 1). The reservoir input data used are presented in Table 1. The numerical model is initiated with drained condition obtained by using Kirsch’s problem [14]. These conditions with the analytical solution equations for drained condition for the given pore pressure, displacement, and stresses [15, 16] are presented in the Appendix. Flow chart describes the solution process for pressure and displacement for poroelastic model and also for temperature for thermoporoelastic frameworks is presented in Figure 2. The numerical results obtained are plotted against the analytical solutions in Figures 3–6.