Introduction

With the industrialization in the nineteenth century, the desire for energy rose continuously resulting in a high emission of CO2, which is considered to be one of the driving forces of climate change. One possibility of mitigating increasing emissions is to store CO2 in the subsurface, e.g., depleted oil and gas reservoirs, coal seams or saline aquifers, similar to underground storage of natural gas or hydrogen (Bai et al. 2014c). Examples are the CO2 capture and storage project in the Weyburn oil field in Canada and the ongoing project CLEAN (CO2 Large-scale Enhanced Gas Recovery in the Altmark Natural Gas Field) in Germany (Reinicke et al. 2011; Akintunde et al. 2013). In order to increase public acceptance the injected CO2 has to stay underground for at least 1,000 years without leaking into overlying aquifers or atmosphere (Kühn and Münch 2013).

In oil and gas fields there are normally a large number of wells that pose a great risk of anthropogenic CO2 leakage after using the reservoir for CO2 storage, some of which are abandoned and plugged. One of the biggest challenges for the wells in the carbon capture and storage (CCS) area is the corrosion fatigue of metals (Zhang et al. 2011). CO2 tends to cause severe corrosion in the wellbore itself and the production facilities, in particular, if they are not designed for CO2 service (Monne 2012; Bai et al. 2014b). The presence of galvanic corrosion, pitting and trough corrosion as well as crevice corrosion will cause local damage to the metal, which can lead to leakages through it. CO2 is also known to disintegrate the traditional types of cement giving rise to concerns that plugging material and techniques currently in use will not suffice to seal off wells in the long term (Benge 2009; Fakhreldin et al. 2010). However, the field experience and lab experiments show that the chemical degradation can only affect well integrity severely when there are already existing pathways (Carey and Wigand 2007; IEA 2009; Bai et al. 2014a). Thus, the study of the mechanical integrity (e.g., well status after a sequence of mechanical and thermal loading in the whole life of wellbores) of the casing-cement-rock composite system is of significance for the safe, long-term containment of CO2. These existing wells in the oil and gas field undergo pressure and temperature changes for their whole life, which result in damage to the wellbore, e.g., formation of cracks, micro-annulus, gas channeling, etc. The injected CO2 may react with the defected cement or the casing, causing further degradation (Hou et al. 2012; De Lucia et al. 2012).

The injected CO2 may also migrate along the defected existing wells including the CO2 injection wells and abandoned wells. The aim of this paper is to evaluate the mechanical integrity of the existing abandoned wells prior to the injection process of CO2 as well as the production/injection wells during the injection process of CO2. To abandon a well, part of the casing must be cut, and then a bottom cementation is needed to shut off the formation and further plugs above it to provide additional barriers. In the near surface, the casings are cut a length of minimum 1 m for onshore wells and 5 m for offshore wells and covered by a cement plate or a steel plate. The space between the cement plugs within the casing is typically filled with drilling fluid. Before injection of CO2 many processes, e.g., drilling process, stimulation, production, etc., could lead to failure of the wellbore. Since it is economically not worthy to open an existing abandoned well, only a simulation approach can be used to evaluate the well conditions resulting from changes in the downhole conditions.

To study mechanical integrity, the first step is to evaluate the stress distribution around a wellbore and in the annulus. For this, many researchers have developed their own theories. Thiercelin in Thiercelin et al. (1998) proposed an analytical model to describe the stress state in which the casing, cement and rock are considered to be homogeneous, isotropic and linearly elastic up to a limit value. The mechanical response of set cement under different load scenarios was studied to determine the key controlling parameters that affect well integrity. Teodoriu et al. (2010) developed an analytical model to calculate wellbore stresses and studied the casing-cement-formation interactions based on wellbore parameters. Bosma et al. (1999) used the finite element method (FEM) to study the behavior of a cement sheath under the various loads occurring during the life of a well. This model is then advanced to take more mechanisms, e.g., expansion and shrinkage, into consideration (Ravi et al. 2007). A comprehensive method called System Response Curve (SRC) was introduced firstly by Saint-Marc et al. (2008). This method replaces traditional complex simulations with wellbore component simulations and cement damage is described by various indexes.

Laboratory and modeling studies show that the principal cause of damage is the stresses induced by varying downhole conditions (Goodwin and Crook 1992; Jackson and Murphey 1993; Bosma et al. 1999; Eshiet and Sheng 2014). It is therefore important to first study the stress development due to downhole condition changes. The mechanical well conditions including the micro-annulus between the casing and the cement sheath under the influence of downhole condition changes will be determined using both analytical and numerical models. Since the focus is on the effects of downhole condition changes, for both analytical and FEM models, the initial state of stress serves as the base scenario over which the temperature and pressure-induced stresses are superimposed.

Casing-cement-rock composite system failure

Casing-cement-rock composite system failure modes

Typical failure modes of casing-cement-rock composite system include debonding of interfaces between casing and cement or between cement sheath and rock, shear damage, radial cracking, and disking, which are shown in Fig. 1 (Bois et al. 2010). Debonding of interfaces is usually caused by shrinkage during cement hydration and a depressurization or temperature decrease inside the casing. The compressive shear failure results from a large deviatoric state of stress, or large differences of principal stresses. This difference can result from a decrease in the tangential compressive stress, which can occur during cement shrinkage. Radial cracking can be due to one of the following reasons: hydration volume reduction in the construction phase of the well, over-pressurization or temperature increase inside the casing during operation. Normally the tangential stresses are compressive. An increase pressure might turn the hoop stress into tensile. If it is in excess of a certain value, a radial crack appears. Disking occurs by axial contraction of the cement sheath when cement cannot slide at its inner or outer boundaries (Bois et al. 2010).

Fig. 1
figure 1

Typical cement sheath damage

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Downhole condition changes

A description of different scenarios of downhole condition changes is given below. Several definitions have to be explained in advance, as shown in Fig. 2. Hoop stress or tangential stress, is a normal stress in the tangential direction of a cylinder. Radial stress is a stress in directions coplanar with but perpendicular to the symmetry axis of the cylinder. These stresses determine the conditions of the casing, cement as well as the rock.

Fig. 2
figure 2

Geometry of casing-cement-rock composite system

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Inner pressure increase

An increase in interior pressure inside the casing due to, for instance, drilling, perforation and stimulation, can cause an increase in radial and axial stress. However, it will decrease the tangential stress and may even turn it into tensile stress. On the one hand, an increase in pressure can cause a bigger compressive stress across two interfaces (e.g., casing/cement or cement/formation) and guarantee a good well integrity. On the other hand, the tensile stress in the tangential direction can cause tensile failure if the tensile stress exceeds the tensile strength and radial cracking occurs. Therefore, the possible damage is tensile failure in the tangential direction.

Inner pressure decrease

A decrease in inner pressure due to, e.g., use of lower mud weight during drilling, can increase the tangential compressive stress. However, it will decrease the radial and axial compressive stress and even turn them into tensile stress. The interface between casing and cement is the weakest point in the composite system. So a tensile stress can possibly lead to interface failure if the tensile stress exceeds the bond strength of the two interfaces, which can be experimentally determined in the lab. In practical situations, the bond strength is very small.

Inner temperature increase

An increase in temperature (e.g., production of formation fluid) can increase the compressive stresses, e.g., σ r , σ θ , and σ z shown in Fig. 2 in cylinder coordinate system, and lead to cement failure when the yield strength is exceeded. In this simulation, von Mises criteria will be used to verify the stress state.

Inner temperature decrease

A decrease of temperature due to, e.g., the injection of cold CO2, can decrease the radial, axial and tangential stresses of the composite system. These compressive stresses may turn to tensile stresses. Because tensile strength of the interfaces is extremely low, there is potential damage to the interface or even debonding since the bond strength is small, which is assumed to be 0 in this paper.

Simulated scenarios and required data

A well in a CCS pilot area in the Altmark Natural Gas Field in North Germany, is evaluated, as a case study, using the analytical and FEM models. The model consists of three concentric cylinders representing the casing, the cement sheath and part of the formation rock as shown in Fig. 2.

Input data

The geometry data and material properties selected for the model are listed in Tables 1 and 2, respectively, in which r m represents the mean radius between inner and outer radius of the casing, and t s represents the casing thickness.

Table 1 Geometry data
Full size table
Table 2 Mechanical and thermal properties of materials used in the pilot area
Full size table

Scenarios to be simulated

For the case study, a formation pressure (P f) of 90 MPa and inner pressure (P i) of 85 MPa are assumed as the base case. An inner pressure of 30 and −30 MPa (base case) was applied to the casing to represent an increased and decreased pressure, respectively. From an initial temperature T 0, the wellbore experienced two temperatures T 1 and T 2 (T 1  ≥ T 2). A uniform temperature of 30 and −30 °C was applied to the composite system to mimic an increased and decreased temperature (ΔT = T 1−T 2), respectively. For analytical modeling, two scenarios are focused on which are temperature decrease and pressure decrease, because these two scenarios might lead to the occurrence of micro-annulus.

Since no detailed information regarding the mechanical properties of the cement sheath for most existing wellbores is available, a parametric study on linear elastic properties of cement sheath is performed. For the base case, the Young’s Modulus and Poisson’s Ratio (PR) for cement are 15 GPa and 0.2, respectively. Different scenarios are calculated and compared to the base scenario varying the cement properties according to Table 3. Firstly, fixing the Poisson ratio at 0.2, the Young’s Modulus is changed from 5 to 25 GPa to study the influences of Young’s Modulus on cement behaviors. To study the influences of Poisson ratio, the Young’s Modulus is fixed and the Poisson ratio is changed from 0.1 to 0.3.

Table 3 Variation of mechanical properties of cement
Full size table

Assumptions

For simplicity several assumptions are made:

  • The geometry of the casing-cement-rock composite system does not change along the interval of interest and there is no axial movement.

  • Horizontal field stresses are isotropic.

  • Casing, cement sheath and rock are thermo-elastic materials.

  • The interface between casing and cement sheath is unbonded, and the interface between cement and rock is fully bonded. For fully bonded interface, the coincident nodes were coupled in the FEM model, to prevent the areas from slipping.

  • Plane strain and axisymmetry (radial symmetric) are assumed. Axisymmetry assumes that the geometry and loading are symmetric in all directions about the axis on the solid of revolution. Plane strain assumes that the body thickness is so big that the lateral deformation is negligible.

  • For the analytical model, thin-walled pipe is selected for the casing and thick-walled pipe is selected for the cement sheath and formation rock.

  • For the FEM model, thick-walled pipe is selected for all the cylinders.

  • The sign convention is that tensile stress is positive and compressive stress is negative.

Analytical modeling

Governing equations

Continuity equations (Geometry equations)

$$\varepsilon_{r} = \frac{(u + du) - u}{(r + dr) - r} = \frac{du}{dr}$$
(1)
$$\varepsilon_{\theta } = \frac{(r + u)d\theta - rd\theta }{rd\theta } = \frac{u}{r}$$
(2)

Physical equations (Hook’s law)

$$\varepsilon_{r} = \frac{1}{E}\left[ {\sigma_{r} - \mu \left( {\sigma_{\theta } + \sigma_{z} } \right)} \right] + \alpha \Delta T$$
(3)
$$\varepsilon_{\theta } = \frac{1}{E}\left[ {\sigma_{\theta } - \mu \left( {\sigma_{r} + \sigma_{z} } \right)} \right] + \alpha \Delta T$$
(4)
$$\varepsilon_{z} = \frac{1}{E}\left[ {\sigma_{z} - \mu \left( {\sigma_{\theta } + \sigma_{r} } \right)} \right] + \alpha \Delta T$$
(5)

Equilibrium equation

$$\frac{{d\sigma_{r} }}{dr} + \frac{{\sigma_{r} - \sigma_{\theta } }}{r} = 0$$
(6)

where, ε r , ε θ , and ε z denote radial, tangential and vertical strain, respectively. σ r , σ θ , and σ z denote radial, tangential and vertical stress, respectively. u denotes displacement, and α coefficient of thermal expansion.

Stress distribution and radial displacement

The assumption of plane strain implies that the vertical strain shown in Eq. 5 is zero (ɛ z  = 0) which leads to an expression of σ z . Inserting the expression of σ z into Eq. 4 yields the expression ε θ in terms of σ θ and σ r . Thus the radial displacement u can be derived from Eq. 2, as expressed in Eq. 7:

$$u = \frac{r}{E}\left[ {\sigma_{\theta } \left( {1 - \mu^{2} } \right) - \left( {\mu + \mu^{2} } \right)\sigma_{r} + \left( {1 + \mu } \right)\alpha E\Delta T} \right]$$
(7)

In this equation, only the tangential and radial stress σ θ and σ r are unknown.

For the casing, the thin-walled theory is used (Teodoriu et al. 2010). At r = r 1,

$$\sigma_{r} \left| {_{{r = r_{1} }} } \right. = P\; - P_{{{\text{c}}1}} \;\;\;\;\;\;\;\;\sigma_{\theta } \left| {_{{r = r_{1} }} } \right. = \frac{{(P_{\text{i}} - P_{\text{c1}} )r_{m} }}{{t_{s} }}$$
(8)

For the cement and formation, the thick-walled theory is used. By solving Eq. 6, Lame’s equation is obtained. The radial, tangential stresses in the cement sheath are expressed as (Ugwu 2008):

$$\sigma_{r} = P_{\text{c1}} \frac{{r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}\left[ {1 - \frac{{r_{3}^{2} }}{{r^{2} }}} \right] - P_{\text{c2}} \frac{{r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}\left[ {1 - \frac{{r_{2}^{2} }}{{r^{2} }}} \right]$$
(9)
$$\sigma_{\theta } = P_{\text{c1}} \frac{{r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}\left[ {1 + \frac{{r_{3}^{2} }}{{r^{2} }}} \right] - P_{\text{c2}} \frac{{r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}\left[ {1 + \frac{{r_{2}^{2} }}{{r^{2} }}} \right]$$
(10)

where P c is the contact pressure between two cylinders in a compound cylinder. The derivation of P c has been performed in Ugwu (2008). They are expressed as:

$$P_{\text{c1}} = \frac{FB - KC}{DB - AK}$$
(11)
$$P_{\text{c2}} = \frac{{C - AP_{\text{c1}} }}{B}$$
(12)

where

$$A = \frac{{r_{2} }}{{E_{\text{c}} }}\left[ {\left( {1 - \mu_{\text{c}}^{2} } \right)\;\left( {\frac{{r_{2}^{2} + r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right) + \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)} \right] + \frac{{r_{1} }}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{{_{\text{s}} }}^{2} } \right) + \left( {\mu_{\text{s}} + \mu_{{_{\text{s}} }}^{2} } \right)} \right]$$
(13)
$$B = - \left[ {\frac{{r_{2} }}{{E_{\text{c}} }}\left( {1 - \mu_{\text{c}}^{2} } \right)\left( {\frac{{2r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right)} \right]$$
(14)
$$C = \frac{{P\;r_{1} }}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{s} }}\left( {1 - \mu_{\text{s}}^{ 2} } \right) + \left( {\mu_{\text{s}} + \mu_{\text{s}}^{ 2} } \right)} \right] + \left[ {\left( {1 + \mu_{\text{s}} } \right)r_{1} \alpha_{\text{s}} \Delta T} \right] - \left[ {\left( {1 + \mu_{\text{c}} } \right)r_{2} \alpha_{\text{c}} \Delta T} \right]$$
(15)
$$D = - \left[ {\frac{{r_{3} }}{{E_{\text{c}} }}\left( {1 - \mu_{\text{c}}^{2} } \right)\left( {\frac{{2r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right)} \right]$$
(16)
$$K = \frac{{r_{3} }}{{E_{\text{f}} }}\left[ {\left( {1 - \mu_{\text{f}}^{ 2} } \right)\left( {\frac{{r_{4}^{2} + r_{3}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}} \right) + \left( {\mu_{\text{f}} + \mu_{\text{f}}^{ 2} } \right)} \right] + \frac{{r_{3} }}{{E_{\text{c}} }}\left[ {\left( {1 - \mu_{\text{c}}^{2} } \right)\left( {\frac{{r_{2}^{2} + r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right) - \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)2} \right]$$
(17)
$$F = \left[ {\frac{{P_{\text{f}} r_{3} }}{{E_{\text{f}} }}\left( {1 - \mu_{\text{f}}^{ 2} } \right)\left( {\frac{{2r_{4}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}} \right)} \right] - \left[ {\left( {1 + \mu_{\text{f}} } \right)r_{3} \alpha_{\text{f}} \Delta T} \right] + \left[ {\left( {1 + \mu_{\text{c}} } \right)r_{3} \alpha_{\text{c}} \Delta T} \right]$$
(18)

where, P c1 and P c2 denote the contact pressure between the casing and the cement as well as the cement and formation, respectively.

After calculation of A, B, C, D, K, F using the basic data the contact pressure P c1 and P c2 can be obtained. The resulting radial and tangential stresses according to Eqs. 8, 9, and 10 are substituted into Eq. 7, and then the expressions of radial displacement for casing, cement and formation at radius r become:

$$u_{\text{s}} = \frac{{r(P_{\text{i}} - P_{\text{c1}} )}}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{\text{s}}^{2} } \right) - \left( {\mu_{\text{s}} + \mu_{\text{s}}^{2} } \right)} \right] + \left( {1 + \mu_{\text{s}} } \right)r\alpha_{\text{s}} (T_{1} - T_{0} )$$
(19)
$$u_{\text{c}} = \frac{r}{{E_{\text{c}} }}\left\{ {\left( {1 - \mu_{\text{c}}^{2} } \right)\left[ {P_{c1}\left (\frac{{r_{3}^{2} + r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}\right) - P_{c2} \left( {\frac{{2r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right)} \right] + P_{\text{c1}} \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)} \right\} + \left( {1 + \mu_{\text{c}} } \right)r\alpha_{\text{c}} (T_{1} - T_{0} )$$
(20)
$$u_{\text{f}} = \frac{r}{{E_{\text{f}} }}\left\{ {\left( {1 - \mu_{\text{f}}^{ 2} } \right)\left[ {P_{c2} \left(\frac{{r_{4}^{2} + r_{3}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}\right) - P_{\text{f}} \left( {\frac{{2r_{4}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}} \right)} \right] + P_{c2} \left( {\mu_{\text{f}} + \mu_{\text{f}}^{ 2} } \right)} \right\} + \left( {1 + \mu_{\text{f}} } \right)r\alpha_{\text{f}} (T_{1} - T_{0} )$$
(21)

where, the subscripts s, c, and f denotes steel (casing), cement, and formation, respectively. T 0 denotes the initial temperature, T 1 denotes the temperature before temperature drop, and T 2 denotes the temperature after temperature drop.

Estimation of micro-annulus between the casing and the cement sheath

In order to estimate the micro-annulus size between the casing and the cement due to downhole condition changes, the interface between the casing and the cement is not assumed to be bonded because the bonding strength (adhesion force) is practically very small.

Temperature decrease

Assuming the casing-cement-formation composite system has a universal temperature drop (ΔT = T 2 − T 1, and T 1 ≥ T 2), the micro-annulus results from the different contraction capacity, resulting from different coefficient of thermal expansion, of the casing and the cement sheath, which can be expressed as:

$$\Delta (\Delta u) = \left| {\Delta u_{\text{s}} - \Delta u_{\text{c}} } \right|$$
(22)

where Δu s and Δu c denotes the casing contraction and cement sheath contraction, respectively. The absolute value ensures a positive value of the micro-annulus size because Δu s and Δu m are both negative.

Since only temperature drop is taken into consideration, the contraction of casing due to temperature drop from T 2 to T 1 can be written according to Eq. 19 as:

$$\Delta u_{\text{s}} = u_{\text{s}} (\Delta T_{2} ) - u_{\text{s}} (\Delta T_{1} ) = \frac{{r_{1} (\Delta P_{\text{c1}} )}}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{\text{s}}^{2} } \right) - \left( {\mu_{\text{s}} + \mu_{\text{s}}^{2} } \right)} \right] + \left( {1 + \mu_{\text{s}} } \right)r_{1} \alpha_{\text{s}} \Delta T$$
(23)

According to Eq. 20, the contraction of cement can be written as:

$$\Delta u_{\text{c}} \left| {_{{r = r_{2} }} } \right. = \frac{{r_{2} }}{{E_{\text{c}} }}\left\{ {(1 - \mu_{\text{c}}^{2} )\left[ {\Delta P_{\text{c1}} (\frac{{r_{3}^{2} + r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}) - \Delta P_{c2}^{2} \left( {\frac{{2r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right)} \right] + \Delta P_{\text{c1}} \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)} \right\} + \left( {1 + \mu_{\text{c}} } \right)r_{2} \alpha_{\text{c}} \Delta T$$
(24)

where,

$$\Delta P_{\text{c1}} = P_{\text{c1}} \left( {T_{ 1} - T_{0} } \right) - P_{\text{c1}} \left( {T_{ 2} - T_{0} } \right)$$
(25)
$$\Delta P_{\text{c2}} = P_{\text{c2}} (T_{1} - T_{0} ) - P_{\text{c2}} (T_{2} - T_{0} )$$
(26)
$$\Delta T = T_{ 2} - T_{ 1}$$
(27)

Obviously, the contraction of cement results from temperature drop in the cement and additional contact pressure ΔP c2 resulting from the contraction of outer formation rock. At r = r 3, the radial deformations of cement and formation have the same values because this interface is fully bonded. Thus,

$$\Delta u_{\text{c}} \left| {_{{r = r_{3} }} } \right. = \Delta u_{\text{f}} \left| {_{{r = r_{3} }} } \right.$$
(28)

According to Eqs. 20 and 21, this gives:

$$\begin{gathered} \frac{{r{}_{3}}}{{E_{\text{c}} }}\left\{ {\left( {1 - \mu_{\text{c}}^{2} } \right)\left[ { - \Delta P_{\text{c2}} \left( {\frac{{2r_{3}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right)} \right]} \right\} + \left( {1 + \mu_{\text{c}} } \right)r_{3} \alpha_{\text{c}} \Delta T \hfill \\ = \frac{{r{}_{3}}}{{E_{\text{f}} }}\left\{ {\left( {1 - \mu_{\text{f}}^{2} } \right)\left[ {\Delta P_{\text{c2}} \left(\frac{{r_{4}^{2} + r_{3}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}\right)} \right] + P_{\text{c2}} \left( {\mu_{\text{f}} + \mu_{\text{f}}^{2} } \right)} \right\} + \left( {1 + \mu_{\text{f}} } \right)r_{3} \alpha_{\text{f}} \Delta T \hfill \\ \end{gathered}$$
(29)

Thus,

$$\begin{gathered} \Delta P_{c2} = {\raise0.7ex\hbox{${\left\{ {\left[ {\left( {1 + \mu_{\text{c}} } \right)\alpha_{\text{c}} \Delta T} \right] - \left[ {\left( {1 + \mu_{\text{f}} } \right)\alpha_{{_{\text{f}} }} \Delta T} \right]} \right\}}$} \!\mathord{\left/ {\vphantom {{\left\{ {\left[ {\left( {1 + \mu_{\text{c}} } \right)\alpha_{\text{c}} \Delta T} \right] - \left[ {\left( {1 + \mu_{\text{f}} } \right)\alpha_{{_{\text{f}} }} \Delta T} \right]} \right\}} {\frac{1}{{E_{\text{f}} }}\left\{ {\left[ {\left( {1 - \mu_{\text{f}}^{2} } \right)\left( {\frac{{r_{4}^{2} + r_{3}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}} \right) + \left( {\mu_{\text{f}} + \mu_{\text{f}}^{2} } \right)} \right] + \frac{1}{{E_{c} }}\left[ {\left( {1 - \mu_{\text{c}}^{2} } \right)\left( {\frac{{r_{3}^{2} + r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right) - \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)} \right]} \right\}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\frac{1}{{E_{\text{f}} }}\left\{ {\left[ {\left( {1 - \mu_{\text{f}}^{2} } \right)\left( {\frac{{r_{4}^{2} + r_{3}^{2} }}{{r_{4}^{2} - r_{3}^{2} }}} \right) + \left( {\mu_{\text{f}} + \mu_{\text{f}}^{2} } \right)} \right] + \frac{1}{{E_{c} }}\left[ {\left( {1 - \mu_{\text{c}}^{2} } \right)\left( {\frac{{r_{3}^{2} + r_{2}^{2} }}{{r_{3}^{2} - r_{2}^{2} }}} \right) - \left( {\mu_{\text{c}} + \mu_{\text{c}}^{2} } \right)} \right]} \right\}}$}} \hfill \\ \hfill \\ \end{gathered}$$
(30)

Substituting Eqs. 30, 24 and 23 into Eq. 22 yields an expression of the micro-annulus between the casing and the cement due to temperature drop.

Pressure decrease

After the pressure decrease the casing will shrink. In case the initial state of cement sheath behaves plastic, which means the cement sheath cannot follow the casing, a micro-annulus occurs. If the deformation of cement sheath is ignored, Eq. 33 can be used to roughly estimate the micro-annulus size between casing and cement owing to the inner pressure drop.

$$u_{\text{s}} = \varepsilon_{\theta } \times r_{2} = \frac{{r_{2} (P_{\text{i}} - P_{\text{c1}} )}}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{\text{s}}^{2} } \right) - \left( {\mu_{\text{s}} + \mu_{\text{s}}^{2} } \right)} \right] + \left( {1 + \mu_{\text{s}} } \right)r_{2} \alpha_{\text{s}} \Delta T$$
(31)
$$u_{\text{s}}^{\prime } = \varepsilon_{\theta } \times r_{2} = \frac{{r{}_{2}(P_{\text{i}} + \Delta P - P_{\text{c1}} )}}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{\text{s}}^{2} } \right) - \left( {\mu_{\text{s}} + \mu_{\text{s}}^{2} } \right)} \right] + \left( {1 + \mu_{\text{s}} } \right)r_{2} \alpha_{\text{s}} \Delta T$$
(32)
$$\Delta u_{\text{s}} = u_{\text{s}} - u_{\text{s}}^{\prime } = \frac{{r{}_{2}( - \Delta P)}}{{E_{\text{s}} }}\left[ {\frac{{r_{\text{m}} }}{{t_{\text{s}} }}\left( {1 - \mu_{\text{s}}^{2} } \right) - \left( {\mu_{\text{s}} + \mu_{\text{s}}^{2} } \right)} \right]$$
(33)

Numerical simulation using finite element method

For a detailed simulation of mechanical integrity, the finite element method (FEM) is used. FEM is applied to a system by spatially discretizing the system and solving the FEM mathematics simultaneously across the geometry. In the FEM, the casing, cement and rock are divided into a finite number of elements so that the governing equations can be solved. When analyzed, each element must satisfy the constraints set by the user.

An advantage of axisymmetry (radial symmetry) is that it allows a portion of the solid cylinder to be modeled without losing accuracy. It is therefore sufficiently accurate to model only a quarter of the geometry in this case. It is usually assumed that the radius of model is at least 10 times that of wellbore radius so that the far field stresses in the formation remain unchanged from the initial value, as shown in Fig. 3 (Ravi et al. 2007). A 2-D, eight-node, structural solid, quadrilateral, higher-order element PLANE 183 was selected. Contact elements were used to model the connection between the casing and the cement. The inner surface of the cement is defined as the target surface and outer surface of the production pipe as the contact surface.

Fig. 3
figure 3

Model geometry in the FEM simulator ANSYS

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The casing was modeled with 40 × 4 quadrilateral elements, the cement with 40 × 10 of such elements and the formation by 40 × 30 elements, because only the influence of the downhole condition changes on well integrity, or specifically, the pressure and temperature variations were considered. The initial stresses in the composite system were not considered and there was no force between casing and cement or cement and rock.

Simulation results

Inner pressure increase

The additional tangential stresses and von Mises stresses in the casing due to pressure increase are shown in Figs. 4 and 5, respectively. Both approaches show that increasing Young’s modulus will lead to increasing additional stress while increasing Poisson ratio will lead to decreasing additional stress, and increasing Young’s modulus will lead to decreasing additional von Mises stress while decreasing Poisson ratio will lead to decreasing additional stress.

Fig. 4
figure 4

Additional tangential stress in the cement sheath due to pressure increase inside the casing

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Fig. 5
figure 5

Von Mises stress in the casing due to pressure increase inside the casing

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Inner pressure decrease

Since a pressure decrease might lead to interface debonding, the point of interest is the radial stresses at the interfaces. The radial stresses and the resulting micro-annulus at the two interfaces are shown in Figs. 6 and 7, respectively. Both analytical and numerical approaches show that the tensile stress at the first interface (casing/cement) is bigger than that at the second one, and increasing Young’s modulus will increase the tensile radial stress while increasing Poisson ratio will decrease the tensile radial stress at the first interface slightly. A higher pressure drop will cause a bigger micro-annulus.

Fig. 6
figure 6

Additional radial stress at the interfaces due to pressure drop

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Fig. 7
figure 7

Micro-annulus between casing and cement sheath due to pressure drop inside the casing

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Temperature increase

Since an increased temperature might lead to failure after the yield point is exceeded, the von Mises stresses across the casing and cement sheath are calculated and verified. The results are shown in Fig. 8 for the simulated scenarios. It can be seen that increasing Young’s modulus will increase the von Mises stress in the casing and cement owing to the temperature increase, but Poisson ratio does not have much impact.

Fig. 8
figure 8

Additional von Mises stress in the casing and cement due to temperature increase inside the casing

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Temperature decrease

For temperature decrease the radial stresses and the resulting micro-annulus at the two interfaces are shown in Figs. 9 and 10, respectively. Both analytical and numerical simulation show that the tensile stress at the first interface is bigger than that at the second one, and increasing Young’s modulus will increase the tensile radial stress while increasing Poisson ratio will decrease the tensile radial stress slightly.

Fig. 9
figure 9

Additional radial stress at interfaces due to temperature drop inside the casing

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Fig. 10
figure 10

Micro-annulus size due to temperature drop inside the casing

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Comparison of analytical and numerical simulation results

Figure 11 shows the von Mises stress comparison using the analytical and FEM models. It can be seen that the two lines are very similar with an average error of 1 %. Figure 12 shows the comparison of the different micro-annulus size using analytical and FEM techniques under conditions of pressure and temperature decrease. For pressure drops, the two lines have a maximum error of 3 %, and for temperature drops, the two lines have a maximum error of 5 %.

Fig. 11
figure 11

Von Mises stress comparison using analytical and numerical models

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Fig. 12
figure 12

Micro-annulus size for a temperature drop and b pressure drop

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Discussion

Four scenarios, which are pressure increase, pressure decrease, temperature increase and temperature decrease, have been simulated.

For pressure increase inside the casing, it can be seen that the highest tangential stress is at the casing/cement interface while the lowest is at the cement/rock interface. The stress at the cement/formation interface is less tensile or even compressive, owing to the mechanical support from the formation rock. Under the same conditions, the interior of the casing has the greatest von Mises stress. If it exceeds the yield strength, the casing will fail.

For pressure decrease, the additional radial stress is the point of interest. The additional radial stress is added to the original compressive stress, and if it exceeds the bond strength of the interface, which is assumed zero in this paper, debonding occurs. Since debonding often occurs at the casing/cement interface, a micro-annulus is assumed to be created here (Nelson and Guillot 2006). Then a decreased pressure will make the casing shrink inwardly and a gap appears providing that the cement sheath behaves plastic.

For temperature increase, the von Mises stress is the point of interest. Under the same conditions, the casing has a much bigger von Mises stress than the cement and the interior of the casing has the largest one. Cement failure will occur when the yield strength is exceeded.

For temperature decrease, radial stresses at interfaces are point of interest. A micro-annulus will occur due to the different contraction capacity (coefficient of thermal expansion) of casing and cement resulting from decreased temperature. Obviously increasing Young’s modulus can lead to an increase in the micro-annulus size and Poisson ratio does not have much impact. A greater temperature drop will apparently increase the micro-annulus size.

The different results from the analytical and FEM simulation, shown in Figs. 11 and 12, are caused by many reasons in addition to the different algorithms. For instance, the thin-walled cylinder implies a single average value for the stresses across the casing diameter but the FEM would do the same calculations for different elements across the casing. Comparison between analytical and numerical models reveals that the analytical models can generate reasonable results given the uncertainty of the input parameters.

Conclusions

An analytical model and a numerical model were used to evaluate the mechanical integrity of the wellbore, e.g., stress distribution, micro-annulus development between the casing and the cement sheath and so on. Given the uncertainties of the system, the analytical model can be applied to predict the well integrity response due to the downhole condition changes during the life of the wellbore and provide a characterization of the defects, which are used for permeability estimation. The results show that the cement with low Young’s modulus and high Poisson ratio performs better, viz., leading to smaller micro-annulus, than then the cement with high Young’s modulus and low Poisson ratio. For decreases in temperature and pressure, there is a potential debonding of the casing and the cement sheath.

Comparison between analytical and numerical models reveals that the analytical models are relatively simple and they can generate reasonable results given the uncertainty of the input parameters. In case of more complicated and practical situations considering anisotropy and differential horizontal stresses, or where time can be implied, for example, dynamic and cyclic loading and stepwise temperature variation and so on, the FEM model can be used.

Debonding of interfaces and the resulting fluid pathway has been estimated through the mechanical integrity evaluation. This can be used to calculate the permeability of the pathway. With coupled chemical simulation cement degradation might be accessed, and long-term performance of the wellbore in terms of leakage rates of CO2 can be evaluated.