A Brief Review of Wellbore Stability in Coal Beds: Technological Improvements and Challenges

Analytical Models

The analytical model uses simpler methods like calculation of the radial, tangential and axial stress components at any point around the wellbore. They provide approximate values of the failure stress criteria. These include calculation of the factors by various methods like the Mohr-Coulomb failure criteria and others. The Linearized Mohr-Coulomb failure criteria assumes that the intermediate principal stress has no influence on failure. The governing fundamental equations for the Linearized Mohr-Coulomb failure criteria are given by the following Equations 1-3.

$$ \begin{array}{l} \sigma_ {1} = C _ {0} + \sigma_ {3} (1) \\ \left(\mu_ {\mathrm {i}} ^ {2} + 1\right) ^ {1 / 2} + \mu_ {\mathrm {i}} ^ {2} = \tan^ {2} \left(\pi / 4 + \phi / 2\right) (2) \\ \end{array} $$ $$ \left(\mu_ {\mathrm {i}} ^ {2} + 1\right) ^ {1 / 2} + \mu_ {\mathrm {i}} ^ {2} = \tan^ {2} \left(\pi / 4 + \phi / 2\right) $$ $$ \begin{array}{l} + \mu_ {\mathrm {i}} = \tan \left(\% 4 + \% 2\right) (2) \\ \phi = \tan^ {- 1} \mu_ {\mathrm {i}} (3) \\ \end{array} $$ Where Co is solved-for as a fitting parameter, σ1, σ2, σ3 are the principal stresses, and μ is the coefficient of internal friction.

This criterion predicts failure stress values with reasonable accuracy in ideal clean sandstone formations with an error of about 10% [12], but in the coal seams where various cleat networks are present, the results are not so accurate and show large deviations.

It is a convenient method to calculate the failure criteria for preliminary calculations when other options are not available.

Thermo-mechanical Model

Thermo-mechanical models are applied when significant temperature changes occur during drilling. This phenomenon is more common during underbalanced drilling. When fluid temperatures deviate significantly from formation temperatures Thermo-mechanical models better capture the thermal expansion of the coal matrix and the stress changes induced by temperature gradients around the wellbore.

The governing fundamental equations can be divided into two thermal and mechanical constitutive equations. These are represented by Equations 4 and 5.

$$ f _ {i} + \sigma_ {i j, j} = 0 \tag {4} $$

$$ \varepsilon_ {\mathrm {i j}} = \frac {1}{2} \left(u _ {\mathrm {j . i}} + u _ {\mathrm {i . j}}\right) $$ (5) Where σij.j and fi are the components of the stress matrix and net body force, respectively; εij is strain tensor, uj.i and ui.j represent ꝺuj/ꝺxi and ꝺui/ꝺxj, respectively, where u represents the change in position in the j or i directions [13].

Numerical Simulation Techniques

Numerical models provide a more detailed and realistic representation of the actual and complex interactions between stress, strain, and pore pressure around the wellbore. There are various methods such as the Finite Element Method (FEM) and Finite Volume method which are widely used to simulate borehole stability in CBM wells. Hongyan Qu investigated the mechanism and control of the wellbore stability of coal seams in the Qinshui basin through theoretical modelling, laboratory experiments and field operation. They obtained the variation of the collapse pressure, pore pressure and breaking (fracturing) pressure with depth. The curve which they obtained was shown in Figure 1.

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It could be inferred from the graph that breaking (fracturing) pressure would be dependent on the pore pressure and when pore pressure increases the breaking (fracturing) pressure also increases.

Linear Elastic Borehole Stability Model

One of the easiest ways to evaluate the stability of wellbore is by using a linear elastic wellbore model to predict the stresses around the wellbore. These calculated stresses can then be compared to the rock’s strength to see if shear failure might occur. If the borehole is drilled in line with the main in-situ stress direction and we assume the borehole is much longer than its diameter, it is possible to calculate the stress distribution around the wellbore [14]. The critical wellbore pressure is given by Equation 6.

( ) ( ) 1 3 3 1 1

$$ = \frac {3 \sigma_ {1} - \sigma_ {3} + A _ {\mathrm {p}} \Delta \mathrm {P} - \mathrm {U C S} + \alpha (\mathrm {N} - 1) \mathrm {P} _ {\mathrm {m}}}{\mathrm {N} + 1} \tag {6} $$

Where, Pw(critical) is the critical wellbore pressure (Pcollapse <

Pw(critical) < Pfracturing); σ1 and σ2 are the maximum and minimum

principal stresses in the plane perpendicular to the wellbore

axis; Ap is the poro-elastic constant; α is Biot’s coefficient; ν

is the Poisson’s ratio (Static); ΔP = Pm – Pf is the difference

in mud pressure and formation pore pressure; Pm is the

mud pressure; Pf is the formation pore pressure; UCS is

unconfined compressive strength; N = (1+sinφ)/(1-sinφ) is

a dimensionless parameter of frictional strength ; Where φ is

the angle of friction; Pw is the wellbore pressure [14].

p m w critical A P UCS N P P N

This model, however, is not ideal for soft rocks since it does not consider more complex interactions, such as non- linear elasticity, strain softening, strain hardening or changes in permeability around the wellbore due to stress. Despite these limitations, the linear elastic model offers several advantages: it is comparatively easy to calculate, either by manual calculations or with the aid of excel programs or other modules, and it requires fewer input parameters than more complex models, making it a practical choice for preliminary assessments. Additionally, it provides quick insights into stress conditions around the borehole without needing extensive field data [14].

Furthermore, the model can be extended to handle 3D stress conditions, where the wellbore is not properly aligned with the principal in-situ stress direction. Although the 3D equations aren’t provided here, they are documented in works by Aadnoy and Fjaer [15].

This method helps assess the risks of various well trajectories and allows for quick sensitivity analyses without needing large amounts of data. It’s especially useful when the important parameters, such as rock compressive strength and Poisson’s ratio are not well defined. When field data is known to us, it can be used to calibrate the parameters obtained from the model so that the results can provide reasonably accurate predictions.

Elastoplastic Borehole Stability Models

The previously discussed model overlooks a critical aspect of borehole stability: after yielding, rock may weaken but still retain some residual strength. As the rock yields, stress redistribution takes place, and the stress carried by the affected region can significantly impact the well’s stability. In petroleum engineering, designing drilling fluid properties and densities has been effective in controlling the size of the yielded zone to ensure borehole stability [14]. Figure 2 illustrates an ideal elastic-brittle-plastic model. In this model, the key section of the stress-strain diagram is the post-peak region.

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The behaviour of material is initially considered to be linear elastic until it reaches its peak strength, then transitioning into strain weakening, later exhibiting plastic deformation at a constant residual stress. Although this model is not universally applicable, as some materials exhibit more complex behaviour, more advanced numerical models and other algorithms are available for those cases. For instance, Vaziri & Palmer [16] and Manogharan [17] account for nonlinear elastic behaviour before a perfectly plastic state, while Van den Hoek [18] and Xiao Y [19] consider linear strain hardening and softening. Customizable models, like those in the FLAC software (Itasca Consulting Group Inc.,), can incorporate a variety of complex material behaviours. However, many sedimentary rocks can be reasonably represented by the idealized model in Figure.2. Semi-analytical solutions for this model have been provided by Detournay and Fairhurst [20] and Wang & Dusseault [21], offering an advantage since they don’t require numerical modelling. This elastoplastic approach is demonstrated for a near-horizontal well in the Ardley coal zone in central Alberta, chosen for its potential in coalbed methane extraction and CO2 sequestration (Bachu & Stewart) [22]. Similar studies on the deeper Mannville Group coals were conducted by McLellan & Hawkes [23], with successful horizontal wells drilled in these coals at 1000 m depth in north-central Alberta, Canada [14].

The Figure 3 depicts geophysical log data from the Ardley coal zone in a vertical oil well in the Three Hills Creek Field, located about 20 km southeast of Red Deer, Alberta. Coal seams between 1 and 3 meters thick are visible in the logs.

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The caliper logs show the wellbore has expanded to almost 400 mm in these coal seams, i.e., about 100% larger than the original bit diameter of 200 mm. Despite these enlargements, the wellbore remained relatively stable because the coal seams were thin. Thin coal seams limit the overall breakout volume, reducing the impact on wellbore stability. If the coal seams had been thicker, the overall breakout volume would have been significantly larger, leading to severe wellbore instability issues. Also, if similar borehole expansion occurs during horizontal drilling, the larger volume of excess cavings could present significant operational challenges. Wellbore stability studies are thus pivotal to establish the minimum mud weight of drilling fluid necessary to restrict further borehole enlargement in horizontal section of boreholes within the coal seam area [14].

Thermo-hydro-mechanical Coupling Model

During underbalanced drilling in CBM formations, wellbore stability is influenced by the Thermo-Hydro- mechanical coupling process, which involves applying the conservation of momentum, mass, and energy along with relevant constitutive laws for fluid flow, heat transfer, and stress gradients and deformation around the wellbore, as illustrated in Figure 4.

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Due to the complexity of the coupling process, certain assumptions should be considered to simplify the modelling: (a) The coal is considered as a linear elastic isotropic homogeneous porous medium that meets the plane-strain hypothesis; (b) Gas flowing in coal obeys the Darcy’s law (single phase flow); (c) Effects of unstable or abruptly changing wellbore temperature and pressure are ignored [24].

The Thermo-Hydro-Mechanical coupling model of coal deformation is established on the stress equilibrium estimation of a representative elementary volume (REV). In drilling process, coal deformation is caused by the overall changes of gas pressure, external loading, desorption contents of absorbed gas, and wellbore temperature. The relationship between stress-strain for a non-isothermal gas-desorbing CBM wellbore are given below by Equations 7-12 [25], $$ \sigma_ {\mathrm {i j}} = 2 \mathrm {G} \varepsilon_ {\mathrm {i j}} + \frac {2 \mathrm {G} \nu}{1 - 2 \nu} \varepsilon_ {\mathrm {k k}} \delta_ {\mathrm {i j}} - \alpha p \delta_ {\mathrm {i j}} - K \alpha_ {\mathrm {T}} \left(\mathrm {T} - \mathrm {T} _ {0}\right) \delta_ {\mathrm {i j}} - K \varepsilon_ {\mathrm {s}} \delta_ {\mathrm {i j}} \tag {7} $$ $$ G = \frac {\mathrm {E}}{2 (1 + \nu)} \tag {8} $$ ( ) E K ν = − (9) ( ) 3 1 2 K K α = −

1 (10) s

$$\varepsilon_{kk} = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33}$$
$$\varepsilon_s = \varepsilon_l \frac{P}{P + P_L}$$

Where, $\sigma_{ij}$ is the component of the total stress tensor; $\varepsilon_{ij}$ is the component of the total strain tensor; $\varepsilon_{kk}$ is the volumetric strain; $G$ is the shear modulus of coal; $E$ is the Young’s modulus of coal; $K$ is the drained bulk modulus of coal; $K_s$ is the bulk modulus of coal skeleton; $v$ is the drained Poisson’s ratio of coal; $\alpha$ is the Biot coefficient; $\delta_{ij}$ is the Kronecker delta (1 when $i=j$ and 0 when $i \neq j$); $p$ is the pore pressure; $T$ is temperature; $T_0$ is original coal reservoir temperature; $\alpha_T$ is the coefficient of volumetric thermal expansion of the coal skeleton; $\varepsilon_s$ is the volumetric matrix shrinkage strain induced by gas desorption [26, 27], $\varepsilon_L$ is the coefficient of sorption-induced volumetric strain, $P_L$ is the Langmuir pressure constant.

**Drilling Fluids Used for Drilling Coal Beds**

As discussed above, drilling in the coal formations is a difficult task for various reasons, such as the low fracture gradient and the presence of microfractures in the seams, more specifically in the horizontal wells and formation damage due to overbalance drilling using conventional drilling mud. There are various methods to tackle these issues, such as using multiple types of lighter drilling fluids that operate under underbalanced conditions while steering through the coal seams. Also, coal seams have high water sensitivity due to the presence of clay in the coal beds, which swells in the presence of water and the solids present in the drilling mud enter into fracture network and block the channels present in coal beds hence reducing the productivity of the coal beds.

There are various methods to tackle these issues using multiple types of lighter drilling fluids that operate under underbalanced conditions while steering through the coal seams, these include:

Oil based mud (OBM): Oil based drilling fluids can be used with proper additives to drill in the coal seams. These OBMs are of relatively low density which help in reducing the formation damage. Also, due to the absence of water the problems of clay swelling are also solved using OBM. These oil based muds have relatively higher costs and due to the environmental concerns of the OBM these are used only in the case of severe water sensitivity cases where other drilling fluids fail to provide adequate performance.

Degradable polymer drilling fluid: The degradable polymer drilling fluids used for drilling coal beds are of low density, high yield point, low fluid loss, high cutting carrying capacity and minimize reservoir damage. Also due to the degradable nature, the drilling fluid which have entered the formation as filtration loss, can be decomposed using particular enzymes reducing the viscosity and particle size of the filtrate, hence reducing of formation damage [28].

Solid free polymer drilling fluid: The solid free polymer drilling fluid has low viscosity and are designed to minimize the formation damage by reducing the fluid invasion. These muds prevent clay swelling and maintain wellbore stability, fluid loss control, and enhanced cuttings suspension [29].

Foam drilling fluid system: Foam drilling fluids are lightweight and used extensively in underbalanced drilling in CBM drilling. The major advantages of foam drilling fluids are minimized formation damage, low density fluid, and high cutting carrying capacity for effective cuttings removal [30].