Numerical Study on Transient Annular Pressure Caused by Hydration Heat during Well Cementing

Abstract

Annular pressure has been recognized as one of the most challenging problems in the petroleum industry, posing a series of threats to wellbore integrity. Annular pressure caused by thermal expansion during the cement hydration process is rarely studied by researchers. In light of the hydration heat generation process, a kinetics model for cement hydration under different curing temperatures is demonstrated in this paper. Considering interactions between temperature and cement hydration, a transient temperature prediction model during well cementing is built. On the basis of these assumptions, the prediction model of annular pressure is established, considering the change in cement temperature and the change in annulus volume. Using the models illustrated in this paper, a series of numerical simulations are performed. The changing roles of transient cement hydration degree and temperature in wellbores are analyzed thoroughly. The annular pressure during well cementing shows a rapid increase and then a decrease, which is similar to that of the temperature. In addition, a sensitive analysis of annular pressure is conducted. The analysis shows that the annular pressure increases with the geothermal gradient, the cement hydration heat, and the wellbore diameter. Suggestions and conclusions can provide safety guidance for the management of annular pressure during well cementing.

1. Introduction

Annular pressure is an abnormally high level of pressure of the annulus in a wellbore, and has been recognized as one of the most challenging problems in the petroleum industry [1,2,3]. Annular pressure build-up is widely recognized in industry and threatens the wellbore integrity. According to the statistics of the U.S. Department of Mines [4], 11,498 casing strings in 8122 wells in the Gulf of Mexico have suffered from abnormal casing pressure. According to a report by the Tarim oilfield of China, casing pressure is observed in 93% of the wells [5]. As the result of annular pressure, accidents are reported, including the casing collapsing or even the well being abandoned [6,7].

Annular pressure is caused by two reasons: thermal expansion in the annulus and gas migration through the annulus. Using the statistics of the U.S. Mining Administration, Adam T. proposed that damage to the cement sheath during the subsequent operations is the main reason for annular pressure [8]. By analyzing the field data of annular pressure, Wojtanowicz [9] and Rong [10] proposed a probability model of annular pressure. Keivin [11] established a new method to reduce the annular pressure caused by gas migration. Milanovic [12] pointed out that the microstructure caused by mechanical failure is the main reason for annular pressure. On the basis of Darcy’s law, Zhu [13] proposed a two-phase seepage model and some prediction models for annular pressure. However, research into the annular pressure caused by thermal expansion is limited. Zhang [6] and Guan [7] proposed a numerical model to predict the trapped annular pressure caused by thermal expansion.

During the well cementing process, a large amount of heat is released as a result of the cement hydration reaction [14,15]. The sealed fluids in the annulus, including the drilling fluid and cement slurry, are heated up, and this leads to annular pressure [16]. Therefore, an abnormal increase in temperature and casing pressure is frequently observed during the cement hydration stage. Therefore, it is essential to study the transient annular pressure caused by hydration heat during well cementing.

Currently, studies on the annular pressure caused by hydration heat during well cementing are very limited. In order to study the transient annular pressure caused by hydration heat during well cementing, a kinetics model of cement hydration is established in this study. On the basis of cement hydration kinetics, a transient model of annulus pressure is established. A series of numerical simulations are performed using the new model established in this paper. The transient temperature in the wellbore, the cement hydration process, and the annular pressure are analyzed. Suggestions are drawn from the numerical simulations that can provide safety guidance for the management of annular pressure.

2. Transient Model of Annular Pressure Caused by Hydration Heat during Well Cementing

As illustrated in Figure 1, the cement slurry is circulated into the annulus between the casing and the formation. Then, the wellhead is shut in and the cement slurry is trapped in the annulus. At the same time, significant cement hydration starts. The heat generated by cement hydration results in transient temperatures and annular pressure.

2.1. Kinetics Model for Cement Hydration under Different Curing Temperature

• Kinetics model for cement hydration

The hydration degree of cement α is defined as the weight fraction of reacted cement among the total cement, which is commonly calculated using the ratio between transient cement hydration heat Q(t) and the total hydration heat of cement Q_max:

$$\alpha =Q(t)/Q_{\max }$$

(1)

The rate of cement hydration degree can be expressed as:

$$d\alpha /dt=\frac{dQ/dt}{Q_{\max }}$$

(2)

Previous works showed that the cement hydration reaction includes three processes, including nucleation and crystal growth (NG), interactions at the phase boundaries (I), and diffusion (D) [17,18]. The Krstulovic–Dabic model [19] is one of the most widely accepted hydration kinetics models, and it describes the development of cement hydration degree over time during different stages:

① For the nucleation and crystal growth (NG) process,

$${[-\ln (1-\alpha )]}^{1/m}=K_{\mathrm{NG}}t$$

(3)

A differential form can be written as:

$$\frac{d\alpha }{dt}=K_{\mathrm{NG}}m(1-\alpha ){[-\ln (1-\alpha )]}^{(m-1)/m}$$

(4)

② For the phase boundaries (I) process,

$$1-{(1-\alpha )}^{1/3}=K_{I}t$$

(5)

A differential form can be written as:

$$\frac{d\alpha }{dt}=3K_I{(1-\alpha )}^{2/3}$$

(6)

③ For the diffusion (D) process,

$${[1-{(1-\alpha )}^{1/3}]}^2=K_{D}t$$

(7)

A differential form can be written as:

$$\frac{d\alpha }{dt}=\frac{3}{2}{K}_D{(1-\alpha )}^{2/3}/[1-{(1-\alpha )}^{1/3}]$$

(8)

2.

Determination of kinetics parameters

According to Equations (3)–(8), the determination of the kinetics parameters is the key to solving the kinetics model. According to the concept of hydration degree in Equations (1) and (2), the hydration heat and hydration degree during the cement hydration process can be measured continuously using an isothermal cement calorimeter [20]. Therefore, the kinetic parameters can be determined graphically as the slope of a straight line using a log–log diagram according to Equations (3), (5) and (7). Taking Equation (3), for example, plot ln(−ln(1−α)) versus ln(t). The slope of the line equals m, and the intercept divided by m represents ln(K_NG).

3.

Kinetics model under different temperature environments

The temperature in the formation at the shallow part of wellbore is very low, while the temperature is much higher at the deep part of the wellbore as a result of the geothermal gradient. The rate of the chemical reaction increases with the temperature. Therefore, the hydration rate of cement varies with depth.

The Arrhenius equation [21,22] describes the relationship between temperature and the chemical reaction rate:

$$K=A{e}^{-E_a/RT}$$

(9)

The pre-exponential factor A and activation energy E_a can be determined graphically as the slope of a straight line using a log–log diagram according to Equation (9). Plot the curve of ln(K) versus (1/T). The slope of the line equals (−E_a/R), and the intercept divided by m represents ln(A). Thus, the kinetics model for cement hydration in different temperature environments can be rewritten as:

$$\frac{d\alpha }{dt}=A_{\mathrm{NG}}{e}^{-E_{a\mathrm{NG}}/RT}m(1-\alpha ){[-\ln (1-\alpha )]}^{(m-1)/m}$$

(10)

$$\frac{d\alpha }{dt}=3A_I{e}^{-E_{aI}/RT}{(1-\alpha )}^{2/3}$$

(11)

$$\frac{d\alpha }{dt}=\frac{3}{2}{A}_D{e}^{-E_{aD}/RT}{(1-\alpha )}^{2/3}/[1-{(1-\alpha )}^{1/3}]$$

(12)

2.2. Transient Temperature Prediction Model during Well Cementing

(1)

Thermal model in casing

As shown in Figure 1b, the internal energy change to the unit cell includes the heat exchange with the cement in the annulus and the adjacent cells. Given the conservation of energy, the thermal model of the unit cell in the casing can be acquired as:

$$\begin{array}{l}({T_c|}_{t+\Delta t}-{T_c|}_t)c_f{\rho }_f{A}_c\Delta z=2\pi r_{ci}\Delta z{U}_c(T_a-T_c)\Delta t\\ +A_c{k}_f\frac{{T_c|}_{z+\Delta z}-{T_c|}_z}{\Delta z}\Delta t-A_c{k}_f\frac{{T_c|}_z-{T_c|}_{z-\Delta z}}{\Delta z}\Delta t\end{array}$$

(13)

Equation (13) can be written in a differential form:

$$\frac{\partial T_c}{\partial t}=\frac{2\pi r_{ci}{U}_c}{c_f\rho A_c}(T_a-T_c)+\frac{k_f}{c_f\rho }\frac{{\partial }^2{T}_c}{\partial z^2}$$

(14)

(2)

Thermal model of cement in annulus

As shown in Figure 1a, the internal energy change to the unit cell includes the heat exchange with the fluid in the casing, the adjacent cells, the formation, and the cement hydration heat. On the basis of the conservation of energy, the thermal model of the unit cell of the cement in the annulus can be acquired:

$$\begin{array}{l}({T_a|}_{t+\Delta t}-{T_a|}_t)c_c{\rho }_c{A}_a\Delta z=({\alpha |}_{t+\Delta t}-{\alpha |}_t)Q_{\max }{\rho }_c{A}_a\Delta z-2\pi r_{ci}{U}_c(T_a-T_c)\Delta t\Delta z\\ +2\pi r_w{U}_a(T_{e,0}-T_a)\Delta t\Delta z+A_a{k}_c\frac{{T_a|}_{z+\Delta z}-{T_a|}_z}{\Delta z}\Delta t-A_a{k}_c\frac{{T_a|}_z-{T_a|}_{z-\Delta z}}{\Delta z}\Delta t\end{array}$$

(15)

Equation (15) can be written in a differential form as:

$$\frac{\partial T_a}{\partial t}=\frac{2\pi r_w{U}_a}{c_f\rho A_a}(T_{e,0}-T_a)-\frac{2\pi r_{ci}{U}_c}{c_f\rho A_p}(T_a-T_c)+\frac{Q_{\max }}{c_f}\frac{\partial \alpha }{\partial t}+\frac{k_f}{c_f\rho }\frac{{\partial }^2{T}_c}{\partial z^2}$$

(16)

(3)

Thermal model in formation

As shown in Figure 1c, the internal energy change to the unit cell equals the heat exchange with the adjacent cells. On the basis of the conservation of energy, the thermal model of the unit cell in formation can be acquired:

$$\frac{{T|}_{t+\Delta t}-{T|}_t}{\Delta t}{c}_e{\rho }_{e}x\Delta x\Delta \theta \Delta z=\frac{{T|}_{x+\Delta x}-{T|}_x}{\Delta x}{k}_e(x+\frac{\Delta x}{2})\Delta \theta \Delta z-\frac{{T|}_x-{T|}_{x-\Delta x}}{\Delta x}{k}_e(x-\frac{\Delta x}{2})\Delta \theta \Delta z$$

(17)

Equation (22) can be rewritten in a finite difference form as:

$$\frac{{\rho }_e{c}_e}{k_e}\frac{\partial T}{\partial t}=\frac{\partial T^2}{\partial x^2}+\frac{1}{x}\frac{\partial T}{\partial x}$$

(18)

2.3. Annular Pressure Build-Up Caused by Thermal Expansion

As a result of the cement hydration heat, an obvious temperature increase in the cement slurry can be observed. The cement slurry is trapped in the annulus between the casing and the wellbore, which results in thermal expansion and the annular pressure build-up. The annular pressure build-up can be affected by two factors: (1) the change in temperature in the annulus; (2) the change in volume of the annulus. Thus, the prediction model of annular pressure build-up during well cementing can be expressed as:

$$\Delta P_a=\frac{{\alpha }_l}{k_T}\Delta T-\frac{\pi }{k_T{V}_{ann}}\left[2u{r}_{co}-u_{}^2\right]L$$

(19)

where α_l is the thermal expansion coefficient of fluid, °C⁻¹; k_T is the isothermal compressibility coefficient of fluid, MPa⁻¹; V_ann is the volume of annulus, m³; u is the radial displacement of casing, m; L is the length of casing, m.

The first item in Equation (19) is the annular pressure build-up caused by fluid expansion, and the second item in Equation (19) represents the annular pressure change caused by the annulus volume change.

The pressure build-up in the annulus can result in the radial deformation of the casing, which changes the volume of the annulus. According to the thick-walled cylinder theory [17], the radial displacement under pressure can be expressed as:

$$u=\frac{1+\mu }{E}\left[\frac{\left(1-2\mu \right)r_{co}^3+r_{ci}^2{r}_{co}}{r_{co}^2-r_{ci}^2}{P}_c-\frac{\left(1-2\mu \right)r_{ci}^2{r}_{co}+r_{ci}^2{r}_{co}}{r_{co}^2-r_{ci}^2}{P}_a\right]$$

(20)

where μ is the Poisson’s ratio of the casing; E is the elastic modulus of the casing, MPa; r_co is the outer diameter of the casing, m; r_ci is the inner diameter of the casing, m; P_c is the pressure in the casing, MPa; P_a is the pressure in the annulus, MPa.

2.4. Initial and Boundary Conditions

① The significant period of cement hydration starts after it is placed in the wellbore, and the initial degree of cement hydration is regarded as zero:

$${\alpha |}_{t=0}=0$$

(21)

② The initial temperature in the formation can be described by the thermal gradient:

$$T_e(z)=T_g+{\mathit{zg}}_{\mathrm{G}}$$

(22)

③ The formation at infinity determines the initial temperature:

$$T_{i,M}=T_{e,i}$$

(23)

④ Radial heat transfer occurs on the border between the wellbore and the formation:

$${\frac{\partial T}{\partial t}|}_{x=r_w}{\rho }_e{c}_e{r}_w\Delta x\Delta \theta \Delta z=\Delta \theta r_w{U}_a(T_a-{T|}_{x=r_w})\Delta z+{\frac{\partial T}{\partial x}|}_{x=r_w}{k}_e\Delta x\Delta \theta \Delta z$$

(24)

⑤ Before the cement-setting process, the cement is pumped and circulated into the wellbore. The initial temperature of the cement can be described by the temperature prediction model during the cement circulation stage [23]:

$$\frac{1}{v_c}\frac{\partial T_c}{\partial t}+\frac{\partial T_c}{\partial z}=\frac{2\pi r_{ci}{U}_c}{c_f{w}_c}(T_a-T_c)$$

(25)

$$\frac{1}{v_a}\frac{\partial T_a}{\partial t}-\frac{\partial T_a}{\partial z}=\frac{2\pi r_w{U}_a}{c_f{w}_a}(T_{e,0}-T_a)-\frac{2\pi r_{ci}{U}_c}{c_f{w}_a}(T_a-T_c)$$

(26)

3. Model Validation

(1)

Kinetics model of cement hydration

In order to validate the kinetics model of cement hydration, we compared the calculated results of the model in this paper and the experiments conducted by Dillenbeck [24]. Using the isothermal calorimeter, Dillenbeck [24] measured the curve of cement hydration heat at 26.6 °C and 65.5 °C, respectively.

Using the curve of cement hydration heat, the kinetics parameters for Equations (4), (6) and (8) can be acquired. As shown in

Table 1. Calculation results of hydration parameters.

Temperature	m	KND	KI	KD	α1	α2
26.6 °C	2.421	0.033	0.0111	0.0027	0.163	0.388
65.5 °C	2.556	0.086	0.0358	0.0174	0.307	0.559

, α₁ is the boundary between the NG and I processes, and α2 is the boundary between the I and D processes. According to the hydration parameters at different temperatures, the activation energy of Equations (10)–(12) for the NG, I, and D processes can be acquired as 20,787.4 J/mol, 25,407.5 J/mol, and 40,436.3 J/mol.

As shown in Figure 2 and Figure 3, the comparison results show that the model in this paper can describe precisely the hydration process of cement at different temperatures.

(2)

Thermal model of the wellbore

In addition to the cement hydration experiment, Dillenbeck [24] also conducted a field test measuring the transient temperature of cement in a wellbore during the cement hydration process. The cement slurry used in the field test is the same as the cement used in the experiment, with a density of 1.68 g/cm³. The field test was conducted in a vertical well. The well had a total depth of 3932 m. A temperature recorder was set at a position close to the well bottom (3840 m) to record the transient temperature during well construction. The diameter of the wellbore hole was 215.9 mm. The casing had a diameter of 139.7 mm. As shown in Figure 4, the comparison results between the calculated results and the measured results show that the temperature prediction error is within 5.6%.

4. Numerical Simulations and Analyses

In order to study the transient annular pressure caused by hydration heat during well cementing, a series of numerical simulations are performed through a simulation well using the models established in this paper. The target well is an inclined well, with a total depth of 3817 m and a total vertical depth of 3308 m. The volume of cement slurry injected into the wellbore is 123 m³. The kinetics parameters of cement hydration are shown in Table 1. The basic data of the simulation well are shown in

Table 2. Basic data of the simulation well.

Parameter	Value	Unit
Cement density	2.03	g/cm3
Total heat released by cement	267	kJ/kg
Surface temperature	20	°C
Geothermal gradient	2.07	°C/m
Specific heat capacity of cement	2000	J/kg·K
Thermal expansion coefficient of cement	1.5 × 10−5	°C−1
Thermal conductivity of cement	0.72	W/k·m

. The structure of the simulation well is displayed in Figure 5. On the basis of the numerical simulation, suggestions are made to provide safety guidance for annular pressure management during well cementing. The mechanical parameters of the cement are based on Zhang’s research [23,25,26].

4.1. Transient Development of Cement Hydration Degree

The heat generation during the cement hydration process is the key reason for annular pressure. Using the models established above, the cement hydration degree of cement at different depths is simulated as shown in Figure 6. The degree of cement hydration shows a slow increase at the beginning and then a rapid increase. Finally, the cement hydration degree approaches a constant slowly. In addition, the cement hydration process is obviously affected by temperature. The higher temperature in deep part of wellbore leads to the faster hydration rate of cement. The cement in the well bottom takes only 5.2 h to reaching the hydration degree of 0.5. However, the cement at the depth of 500 m takes 15.7 h to reach the hydration degree of 0.5.

As shown in Figure 7, the transient distribution of cement hydration in the wellbore is also simulated. As a result of the geothermal gradient, the cement hydration degree increases with depth. Therefore, the waiting time for cement thickening is mainly dependent on the hydration process in the shallow part of the wellbore.

4.2. Transient Development of Temperature in Wellbore

As shown in Figure 8, the transient development of the cement temperature at different depths is simulated. The transient development of cement hydration includes two stages. For the first stage, the cement temperature shows a rapid increase as the result of the cement hydration generation. For the second stage, the cement hydration slows down. Therefore, the cement temperature approaches the environmental temperature slowly during the second stage. In addition, the temperature increase caused by cement hydration heat increases with depth. The temperature increase at the well bottom is 28.89 °C, while the temperature increase at 500 m is only 9.35 as a result of the lower environmental temperature.

As shown in Figure 9, the transient distribution of cement temperature in the wellbore is also calculated using the models established in this paper. In addition, the temperature increase at different depths is also calculated, as shown in Figure 10. The simulation results show that a temperature difference can be observed at the top of the cement due to the cement hydration heat. The obvious increase in temperature starts earlier for the cement in the deep part of the wellbore as a result of the faster hydration process. Similarly, the cement temperature shows a rapid increase, and then decreases to the environmental temperature.

4.3. Transient Development of Annular Pressure

As shown in Figure 11, the transient development of annular pressure is calculated. The simulation result shows that annular pressure is mainly caused by thermal expansion. Therefore, the annular pressure shows a rapid increase and then a decrease, which is similar to that of the temperature. The simulated well displays a maximum annular pressure of 7.12 MPa, which poses a great threat to the wellbore’s integrity.

4.4. Annular Pressure under Different Geothermal Gradients

The annular pressure under different geothermal gradients is simulated using the models established in this paper. As shown in Figure 12, the annular pressure increases faster when the geothermal gradient is larger. The faster increase in annular pressure is mainly caused by the faster hydration process under a higher-temperature environment. In addition, a lower formation temperature (geothermal gradient) absorbs more heat from the wellbore, which results in a lower annular pressure. Therefore, the maximum annular pressure increases with the geothermal gradient. According to the simulation results, the maximum annular pressure under 1.5 °C/100 m is 6.0 MPa, while the maximum annular pressure under 3.0 °C/100 m is 9.5 MPa.

4.5. Annular Pressure under Different Cement Hydration Heats

As shown in Figure 13, the annular pressure under different cement hydration heats is simulated using the models established in this paper. Higher cement hydration heat leads to a higher temperature in the wellbore. Therefore, the annular pressure increases with the cement hydration heat. However, the cement hydration process is not affected by the cement hydration heat. Therefore, the time of maximum annular pressure remains the same under different cement hydration heats. Cement with lower cement hydration heat is suggested to reduce the risk of high annular pressure.

4.6. Annular Pressure under Different Wellbore Diameters

The annular pressure under different wellbore diameters is simulated using the models established in this paper. As shown in Figure 14, the annular pressure increases with the wellbore diameter. The wellbore with a smaller diameter is easily affected by the environment and loses more heat to the surrounding formation. Therefore, the wellbore with a smaller diameter has a lower temperature and a lower annular pressure. In addition, the higher temperature accelerates the cement hydration rate. Thus, the annular pressure increases faster when the wellbore diameter is larger. According to the simulation results, the maximum annular pressure with a 269.9 mm wellbore is only 2.0 MPa, while the maximum annular pressure with a 320.7 mm wellbore is 7.9 MPa.

5. Conclusions

On the basis of the numerical simulations using the new model established in this paper, suggestions are drawn from the numerical simulations that can provide safety guidance for the management of annular pressure:

(1)

Cement hydration heat is the root reason for annular pressure. The cement hydration process is obviously affected by the environmental temperature in the wellbore. A higher temperature in the well bottom accelerates the cement hydration process. Thus, the waiting time for cement thickening mainly depends on the hydration process at the wellhead.

(2)

The temperature in the wellbore shows an obvious increase as a result of the cement hydration heat. The thermal expansion caused by the temperature increase is the main reason for annular pressure. The cement temperature development follows the same rules as the cement hydration degree. In general, the cement temperature increases at first, and then decreases to the environmental temperature slowly.

(3)

The annular pressure shows a rapid increase and then decreases, which is similar to that of the temperature. As a result of the cement hydration process, the annular pressure increases with the geothermal gradient, the cement hydration heat, and the wellbore diameter. A sensitive analysis can provide safety guidance for the management of annular pressure.