Critical Conditions for Coal Wellbore Failure During Primary Coalbed Methane Recovery: A Case Study from the San Juan Basin

Abstract

Wellbore failure/collapse is a common problem encountered during coalbed methane (CBM) production. The wellbore failures lead to significant production of coal fines, reductions in well productivity, and downhole accidents. These negative outcomes make it essential to predict the conditions responsible for the failure of wellbores and to know when control measures are needed. This study proposed a semi-analytical model of coal wellbore stability during primary CBM recovery. The model incorporates the effects of sorption-induced strain, horizontal stress anisotropy, reservoir depletion, and well trajectory. Plane-strain poroelasticity and the Mogi–Coulomb failure criterion were applied to estimate critical drawdown and depletion for coal wellbore failure. A real field case from the San Juan Basin was simulated to validate the applicability of the proposed model. Wellbore failure-free operating envelopes were then generated for the life span of the field. The effects of sorption-induced strain and well trajectory on wellbore stability were analyzed, and the onset conditions for local wellbore failure and reservoir-scale coal failure were compared. Results revealed that the critical reservoir pressure (CRP_1) for local wellbore failure exceeded the pressure (CRP_2) for field-scale failure, i.e., local wellbore failure occurred earlier than reservoir failure. The CRP_1 value predicted using the model aligned well with actual field observations, thus verifying the model. The desorption-induced shrinkage effect inhibited wellbore failure when radial stress was the minimum principal stress; however, the shrinkage effect enhanced wellbore failure when radial stress was the intermediate principal stress. The optimal well trajectory for the San Juan Basin occurred at the inclination angle of approximately 20° and azimuth of 90°; and the CRP_1 had its minimum value at this trajectory. This study can help optimize field production plans, well trajectories, and well completion to mitigate coal wellbore instability.

Appendix 1: Derivation of Induced Stresses Due to Gas Desorption

Gas desorption from coal matrix surfaces leads to matrix shrinkage and a corresponding change in the stress distribution around the wellbore. An axisymmetric condition is assumed to proceed as sorption-induced stresses are derived around the wellbore, i.e., a thick-walled cylinder method is applied. Similar to the Shi and Durucan model, the desorption-induced shrinkage effect of isothermal coal is treated analogously with the thermal contraction effect associated with nonisothermal media. For isotropic, homogeneous, and linearly elastic coalbed seams, the constitute equation for sorption-induced stresses can be expressed in a cylindrical coordinate as follows:

$$\left\{ \begin{aligned} \sigma_{rr} = \left( {\lambda + 2G} \right)\varepsilon_{rr} + \lambda \varepsilon_{\theta \theta } + \lambda \varepsilon_{zz} + K\varepsilon_{\text{s}} \\ \sigma_{\theta \theta } = \lambda \varepsilon_{rr} + \left( {\lambda + 2G} \right)\varepsilon_{\theta \theta } + \lambda \varepsilon_{zz} + K\varepsilon_{\text{s}} \\ \sigma_{zz} = \lambda \varepsilon_{rr} + \lambda \varepsilon_{\theta \theta } + \left( {\lambda + 2G} \right)\varepsilon_{zz} + K\varepsilon_{\text{s}} \\ \end{aligned} \right.$$

(15)

where λ, G, and K are the Lamé’s first parameter, shear modulus, and bulk modulus, respectively, MPa; and ε_rr, ε_θθ, and ε_zz are the strains in the r, θ, and z directions, respectively. The relationships between E, ν, G, λ, and K are expressed as:

$$\lambda = \frac{E\nu }{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}},G = \frac{E}{{2\left( {1 + \nu } \right)}},K = \frac{E}{{3\left( {1 - 2\nu } \right)}}$$

(16)

The relationships between strain and displacement are expressed as:

$$\varepsilon_{rr} = \frac{{{\text{d}}u}}{{{\text{d}}r}},\varepsilon_{\theta \theta } = \frac{u}{r},\varepsilon_{zz} = \frac{{{\text{d}}w}}{{{\text{d}}z}}$$

(17)

where u and w are the radial and vertical displacements, respectively, m. Under a plane-strain condition, ε_zz = 0.

For the axisymmetric problem, the differential equation of equilibrium can be expressed as follows:

$$\frac{{{\text{d}}\sigma_{rr} }}{{{\text{d}}r}} + \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{r} = 0$$

(18)

After substituting Eq. (17) into (15), Eq. (18) can be expressed as follows:

$$\frac{\text{d}}{{{\text{d}}r}}\left[ {\frac{1}{r}\frac{{{\text{d}}\left( {ru} \right)}}{{{\text{d}}r}}} \right] = - \frac{K}{\lambda + 2G}\frac{{{\text{d}}\varepsilon_{\text{s}} }}{{{\text{d}}r}}$$

(19)

This is a displacement equilibrium equation.

Integrating Eq. (19) two times yields the solution:

$$u = C_{1} r + \frac{{C_{2} }}{r} - \frac{K}{\lambda + 2G}\frac{1}{r}\int_{{r_{\text{w}} }}^{r} {\rho \varepsilon_{\text{s}} {\text{d}}\rho }$$

(20)

where C₁ and C₂ are the two constants of integration.

It is convenient to express the sorption-induced strain ε_s in terms of the strain difference Δε_s, expressed as:

$$\Delta \varepsilon_{\text{s}} = \varepsilon_{\text{L}} b_{\text{L}} \left( {\frac{p}{{1 + b_{\text{L}} p}} - \frac{{p_{\text{res}} }}{{1 + b_{\text{L}} p_{\text{res}} }}} \right)$$

(21)

Substituting Eq. (20) back into Eq. (15) and manipulating the equations yields:

$$\left\{ \begin{aligned}& \sigma_{rr} = 2\left( {\lambda + G} \right)C_{1}^{\prime } - 2G\frac{{C_{2}^{\prime } }}{{r^{2} }} + \frac{2GK}{\lambda + 2G}\frac{1}{{r^{2} }}\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } \hfill \\ &\sigma_{\theta \theta } = 2\left( {\lambda + G} \right)C_{1}^{\prime } + 2G\frac{{C_{2}^{\prime } }}{{r^{2} }} - \frac{2GK}{\lambda + 2G}\left( {\frac{1}{{r^{2} }}\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } - \Delta \varepsilon_{\text{s}} } \right) \hfill \\ &\sigma_{zz} = 2\lambda C_{1}^{\prime } + \frac{2GK}{\lambda + 2G}\Delta \varepsilon_{\text{s}} \hfill \\ \end{aligned} \right.$$

(22)

where the constants $C_{1}^{\prime }$ and $C_{2}^{\prime }$ can be determined by imposing specified boundary conditions.

Radial stresses at the inner boundary (r = r_w) and outer boundary (r = r_o) of the formation are both zero, i.e., σ_rr (r = r_w) = 0 and σ_rr (r = r_o) = 0. Applying the boundary conditions to the first equation of Eq. (22) yields:

$$\left\{ \begin{aligned}& C_{1}^{\prime } = \frac{GK}{{\left( {\lambda + G} \right)\left( {\lambda + 2G} \right)}}\frac{1}{{r_{\text{w}}^{2} - r_{\text{o}}^{ 2} }}\int_{{r_{\text{w}} }}^{{r_{\text{o}} }} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } \hfill \\ &C_{2}^{\prime } = \frac{K}{\lambda + 2G}\frac{{r_{\text{w}}^{2} }}{{r_{\text{w}}^{2} - r_{\text{o}}^{ 2} }}\int_{{r_{\text{w}} }}^{{r_{\text{o}} }} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } \hfill \\ \end{aligned} \right.$$

(23)

Consequently, the stresses induced by sorption effect around wellbores can be expressed as:

$$\left\{ \begin{aligned}& \sigma_{rr} = \frac{E}{{3\left( {1 - \nu } \right)}}\frac{1}{{r^{2} }}\left( {\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } - \frac{{r^{2} - r_{\text{w}}^{ 2} }}{{r_{\text{o}}^{2} - r_{\text{w}}^{ 2} }}\int_{{r_{\text{w}} }}^{{r_{\text{o}} }} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } } \right) \hfill \\ &\sigma_{\theta \theta } = - \frac{E}{{3\left( {1 - \nu } \right)}}\frac{1}{{r^{2} }}\left( {\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } - r^{2} \Delta \varepsilon_{\text{s}} + \frac{{r^{2} + r_{\text{w}}^{ 2} }}{{r_{\text{o}}^{2} - r_{\text{w}}^{ 2} }}\int_{{r_{\text{w}} }}^{{r_{\text{o}} }} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } } \right) \hfill \\ &\sigma_{zz} = \frac{E}{{3\left( {1 - \nu } \right)}}\left( {\Delta \varepsilon_{\text{s}} - \frac{2\nu }{{r_{\text{o}}^{2} - r_{\text{w}}^{ 2} }}\int_{{r_{\text{w}} }}^{{r_{\text{o}} }} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } } \right) \hfill \\ \end{aligned} \right.$$

(24)

Given that r_o ≫ r_w, Eq. (24) can be simplified as:

$$\left\{ \begin{aligned} &\sigma_{rr} = \frac{E}{{3\left( {1 - \nu } \right)}}\frac{1}{{r^{2} }}\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } \hfill \\& \sigma_{\theta \theta } = - \frac{E}{{3\left( {1 - \nu } \right)}}\left( {\frac{1}{{r^{2} }}\int_{{r_{\text{w}} }}^{r} {\rho \Delta \varepsilon_{\text{s}} {\text{d}}\rho } - \Delta \varepsilon_{\text{s}} } \right) \hfill \\ &\sigma_{zz} = \frac{E}{{3\left( {1 - \nu } \right)}}\Delta \varepsilon_{\text{s}} \hfill \\ \end{aligned} \right.$$

(25)