Introduction

In the process of oil and gas resources exploration and development, changes in surge and swab pressure occur in the casing running process. Small pressure changes may cause significant changes in the formation permeability. Calculation of surge and swab pressure is a key problem in drilling fluid density design.

The inaccurate calculation of surge and swab pressure may easily cause complicated conditions such as downhole overflow and well kick. At the same time, it may cause reservoir damage and reduce the efficiency of the development of oil and gas resources. Therefore, the surge and swab pressures should be accurately calculated and analyzed. When starting and running casing, downhole surge and swab pressure fluctuation should be minimized to avoid pressure abrupt changes, thus ensuring the downhole safety [1,2,3,4,5].

1 The Calculation Model of Surge and Swab Pressure without Considering the Casing Coupling

To study the effect of coupling on the surge and swab pressure, first of all, we should analyze the surge and swab calculation process without considering the coupling effect. Figure 1 illustrates the schematic diagram of the casing rate, when the casing is running down into the hole, the casing is closed, and the pump is not working.

Fig. 1.
figure 1

The rate schematic diagram of the casing downward motion.

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When the casing is running down, the annulus flow rate Vf of mud consists of two parts: the mud upward movement rate Va caused by the displacement force of the casing and the mud upward movement rate Vb caused by the adhesive force of the casing. Therefore, the annulus flow rate Vf can be expressed as [6]

$$ {V}_f={V}_a+{V}_b $$
(1)

where Vf is the mud annulus flow rate, Va is the mud upward movement rate, and Vb is the mud upward movement rate.

Based on the conservation law, the following equation is obtained:

$$ \frac{\pi }{4}{D}_2^2{V}_c=\frac{\pi }{4}\left({D}_1^2-{D}_2^2\right){V}_a, $$
(2)

where D1 is the diameter of the borehole, D2 is the casing outer diameter, and Vc is the casing running velocity.

After the transformation, we obtain the following equation:

$$ {V}_a=\frac{D_2^2{V}_c}{D_1^2-{D}_2^2}. $$
(3)

Buvkhardt [7, 8] has put forward the relationship between Vb and Vc and the equation is as follows:

$$ {V}_b={MV}_c, $$
(4)

where M is the mud adherence constant. The constant M can be determined from Fig. 2 [6,7,8].

Fig. 2.
figure 2

Mud adhere constant.

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Thus, we obtain

$$ {V}_f={V}_a+{V}_b={V}_c\left(\frac{D_2^2}{D_1^2-{D}_2^2}+M\right). $$
(5)

When the casing is tripped out, Eq. (5) can be written as follows:

$$ {V}_f=-{V}_c\left(\frac{D_2^2}{D_1^2-{D}_2^2}+M\right). $$
(6)

The calculation formular of surge and swab pressure is different for the laminar flow and turbulent flow of the mud.

The surge and swab pressure equation for the laminar flow, without considering the coupling effect, is as follows [9]:

$$ p=\pm \frac{4 KL}{D_1-{D}_2}{\left[\frac{4\left(2n+1\right)V}{n\left({D}_1-{D}_2\right)}\right]}^n, $$
(7)

where p is the surge and swab pressure, L is the measured depth, K is consistency coefficient, K is 0.01 mPa.sn, n is the flow mode index, and V is the average drilling fluid velocity, m/s. The + sign refers to the surge pressure, and the - sign means the swab pressure.

For turbulent flow, the surge and swab pressure equation without considering the coupling effect is as follows [9]:

$$ p=\pm \frac{2 f\rho {LV}^2}{D_1-{D}_2}, $$
(8)

where ñ is the density of the drilling fluid, g/cm3, and f is the friction coefficient, dimensionless. The + sign refers to the surge pressure, and the - sign means the swab pressure.

2 The Calculation Model of Surge and Swab Pressure Considering the Casing Coupling

In Fig. 3, Vf is the mud annulus flow rate between the coupling and wellbore and Vfc is the mud annulus flow rate between the coupling and wellbore. Based on the mass conservation law, the following equation can be obtained:

$$ \frac{\pi }{4}\left({D}_1^2-{D}_2^2\right){V}_f=\frac{\pi }{4}\left({D}_1^2-{D}_3^2\right){V}_{fc}, $$
(9)
Fig. 3.
figure 3

Drilling fluid flow analysis considering the coupling, the pump is not working.

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where D1 is the diameter of the wellbore, D2 is the outer diameter of casing, and D3 is the outer diameter of coupling.

The flow rate of the drilling fluid in the annulus of casing Vf and the flow rate in the annulus of coupling Vfc are both comprised of two parts: the flow rates Val and Va2 caused by the casing/coupling displacement force, and the flow rates Vbl and Vb2 caused by the casing/coupling adhesive force.

$$ {V}_f={V}_{a1}+{V}_{b1}, $$
(10)
$$ {V}_{fc}={V}_{a2}+{V}_{b2}. $$
(11)

Based on the mass conservation law, the following equation is obtained:

$$ \frac{\pi }{4}\left({D}_2^2+{D}_3^2\right){V}_c=\frac{\pi }{4}\left({D}_1^2-{D}_2^2\right){V}_{a1}+\frac{\pi }{4}\left({D}_1^2-{D}_3^2\right){V}_{a2}. $$
(12)

From Eq. (4), we obtain the formulas to calculate the flow rates vbi and vb2 caused by the casing and coupling adhesive forces:

$$ {V}_{b1}={M}_1{V}_c, $$
(13)
$$ {V}_{b2}={M}_2{V}_c. $$
(14)

After the transformation, we obtain the following equations:

$$ {V}_f=\frac{V_c\left({D}_2^2+{D}_3^2\right)+\left({D}_1^2-{D}_2^2\right){V}_{b1}+\left({D}_1^2-{D}_3^2\right){v}_{b2}}{2\left({D}_1^2-{D}_2^2\right)}, $$
(15)
$$ {V}_{fc}=\frac{V_c\left({D}_2^2+{D}_3^2\right)+\left({D}_1^2-{D}_2^2\right){V}_{b1}+\left({D}_1^2-{D}_3^2\right){v}_{b2}}{2\left({D}_1^2-{D}_3^2\right)}. $$
(16)

Thus, the surge and swab calculation equations based on the flow patterns are shown as follows: in the case of laminar flow

$$ p=\pm {\left\{\frac{4{KL}_1}{D_1-{D}_2}\left[\frac{4\left(2n+1\right){V}_f}{n\left({D}_1-{D}_2\right)}\right]\right.}^n+\frac{4{KL}_2}{D_1-{D}_3}{\left[\frac{4\left(2n+1\right){V}_f}{n\left({D}_1-{D}_2\right)}\right]}^n, $$
(17)

and in the case of turbulent flow

$$ p=\pm \left(\frac{2 f\rho {L}_1{V}_f^2}{D_1-{D}_2}+\frac{2 f\rho {L}_2{V}_{fc}^2}{D_1-{D}_3}\right). $$
(18)

Considering the laminar flow in the casing and the turbulent flow in the coupling annulus, the resulting formula is

$$ p=\pm \left\{\frac{4{KL}_1}{D_1-{D}_2}{\left[\frac{4\left(2n+1\right){V}_f}{n\left({D}_1-{D}_2\right)}\right]}^n+\frac{2 f\rho {L}_2{V}_{fc}^2}{D_1-{D}_3}\right\}, $$
(19)

where L1 is the casing length, and L2 is the coupling length.

3. Example of Calculations

The case study was used as an example to analyze the effect of coupling on the calculation results of the surge and swab pressure. The specific well data are presented in Table 1.

Table 1
Full size table

The case study was used as an example to analyze the effect of coupling on the calculation results of the surge and swab pressure. The specific well data are presented in Table 1.

(1) The surge and swab pressure considering the coupling

Calculating, we obtain D2/D1 = 0.82 and D3/D1 = 0.9. The mud adherence constant M can be obtained from Fig. 2, M1 = 0.48, and M2 = 0.5. Based on Eqs. (4), (13), and (14), Vb1 = 0.864 m/s, Vb2= 0.9 m/s, Vf = 4.86 m/s, and Vfc 8.3l m/s. Using the Reynolds number calculation equation, the Reynolds number values in casing and coupling are 11243 and 15918, respectively. Therefore, the calculated surge and swab pressure is 132.4 kg/cm2.

(2) The surge and swab pressure without considering the coupling

Through calculating, D2/D1 = 0.82M1 = 0.48, vb = 0.864 m/s, Vf = 4.657 m/s, and the Reynolds number is 3614. Based on the calculated data, the surge and swab pressure is 106.5 kg/cm2.

The results show that the pressure calculated considering the effect of coupling is 25% higher than that ignoring the coupling.

4 Conclusions

This paper has put forward the surge and swab calculation process considering the effect of coupling, for a case of a closed pipe, when the pump is not working. The calculation results show that when the coupling is ignored, the calculation error is considerably high and may cause serious drilling accidents. Neglecting the impact of a collar may cause a large error in the calculation of surge and swab pressure. Therefore, the impact of the collar must be taken into consideration when calculating the surge and swab pressure.