Indian Geotechnical Journal

, Volume 44, Issue 1, pp 101–106

Critical Drawdown Pressure of Depleted Reservoir

Authors

    • State Key Lab of Petroleum Resources and ProspectingChina University of Petroleum
    • Faculty of Petroleum EngineeringChina University of Petroleum
  • Jingen Deng
    • State Key Lab of Petroleum Resources and ProspectingChina University of Petroleum
  • Xiangdong Lai
    • State Key Lab of Petroleum Resources and ProspectingChina University of Petroleum
  • Lianbo Hu
    • State Key Lab of Petroleum Resources and ProspectingChina University of Petroleum
  • Zijian Chen
    • State Key Lab of Petroleum Resources and ProspectingChina University of Petroleum
Technical Note

DOI: 10.1007/s40098-013-0071-5

Cite this article as:
Yan, C., Deng, J., Lai, X. et al. Indian Geotech J (2014) 44: 101. doi:10.1007/s40098-013-0071-5
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Abstract

For long time production, most oilfields have entered the later development stage, and the pore pressure is seriously depleted. The pressure depletion will affect in situ stress, and change the stress state around the borehole, and then the affect the critical drawdown pressure to cause sand production. The theoretical formula of two horizontal in situ stress changes with pore pressure is obtained based on generalized Hoek’s law, and then the stress distribution formula around the borehole in pressure depleted reservoir is established. The calculation model of sand production critical bottom hole flowing pressure in depleted reservoir is established using the Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively. The influence of pressure depletion on critical drawdown pressure is analyzed. The results show that: with the pore pressure decreasing, the horizontal in situ stress and critical drawdown pressure become smaller; the error of prediction model based on Mogi–Coulomb criterion is the minimum, and it is more in line with the actual field data; the prediction model based on Mohr–Coulomb criterion is conservative but most safe. The establishment of prediction model provides a guidance for actual production decisions in pressure depleted reservoir.

Keywords

Pressure depletionIn-situ stressSand productionCritical drawdown pressureRock strength criterion

Introduction

Sand production, which is the failure of formation due to in situ stress and fluid flow, is widespread during production of oil and gas sandstone reservoirs [20]. A great deal of work has been down in the general area of sand production (e.g. [13, 17, 18, 23]). Some models view sand production as a mixed hydro mechanical process [5, 15]. Some others base their sanding model solely on mechanical stability [3, 21]. For brittle rock with high strength, sand production occur only mechanical failure happens [24]. With the development of oil and gas, if the formation energy can not effectively supply, the reservoir pressure will deplete, which will increase the risk of sand production [12]. Sand production of depleted reservoir has been researched [1, 16, 22, 24], but hadn’t considered the influence of pressure depletion on in situ stress. Mohr–Coulomb criterion and Drucker–Prager criterion are the mostly used failure criterions for sanding prediction [4, 11, 14]. But the Mohr–Coulomb criterion is too conservative while the Drucker–Prager criterion tends to be unsafe [6, 25]. Mogi–Coulomb criterion has been proved more suitable for evaluate borehole breakouts [2, 26]. This paper analyzes the effect of pressure depletion on in situ stress, and critical bottom hole flowing pressure model in pressure depleted reservoir is established based on Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively.

The Effect of Pressure Depletion on In-situ Stress

Because the overburden pressure is generated by the weight of the formation, the reservoir pressure depletion has almost no effect on the overburden pressure. Under these conditions, if the oil and gas reservoir is with flat geologic structure, thin formation, and there is little difference in porous elastic properties from surrounding rock, due to the deformation in horizontal plane caused by pressure depletion almost negligible, the oil and gas reservoir is approximately with no horizontal deformation, i.e.:
$$ \Updelta \varepsilon_{h} = \Updelta \varepsilon_{H} = 0 $$
(1)
where \( \Updelta \varepsilon_{H} \), \( \Updelta \varepsilon_{h} \) are the strains in the maximum horizontal in situ stress direction and minimum horizontal in situ stress direction caused by pressure depletion, respectively.
According to the generalized Hoek’s law, the reservoir constitutive relationship before the oilfield is developed is as follows:
$$ \left\{ \begin{gathered} \varepsilon_{v} = \frac{1}{E}\left[ {\sigma_{v} - \alpha P_{p} - \mu \left( {\sigma_{h} - \alpha P_{p} + \sigma_{H} - \alpha P_{p} } \right)} \right] \hfill \\ \varepsilon_{H} = \frac{1}{E}\left[ {\sigma_{H} - \alpha P_{p} - \mu \left( {\sigma_{v} - \alpha P_{p} + \sigma_{h} - \alpha P_{p} } \right)} \right] \hfill \\ \varepsilon_{h} = \frac{1}{E}\left[ {\sigma_{h} - \alpha P_{p} - \mu \left( {\sigma_{v} - \alpha P_{p} + \sigma_{H} - \alpha P_{p} } \right)} \right] \hfill \\ \end{gathered} \right. $$
(2)
where E is the Young’s modulus; μ is the Passion’s ratio; \( \sigma_{H} \) and \( \sigma_{h} \) are the maximum and minimum horizontal in situ stress, respectively; \( \sigma_{v} \) is the overburden pressure; Pp is the original pore pressure; \( \alpha \) is effective stress coefficient.
When the pore pressure depletes to Pp1, keep the overburden pressure unchanged, and then the constitutive relationship in horizontal direction satisfies the following equation:
$$ \left\{ \begin{gathered} \varepsilon_{H1} = \frac{1}{E}\left[ {\sigma_{H1} - \alpha P_{p1} - \mu \left( {\sigma_{v} - \alpha P_{p1} + \sigma_{h} - \alpha P_{p1} } \right)} \right] \hfill \\ \varepsilon_{h1} = \frac{1}{E}\left[ {\sigma_{h1} - \alpha P_{p1} - \mu \left( {\sigma_{v} - \alpha P_{p1} + \sigma_{H} - \alpha P_{p1} } \right)} \right] \hfill \\ \end{gathered} \right. $$
(3)
According to the hypothesis of Eq. (1), there is:
$$ \left\{ \begin{gathered} \varepsilon_{H1} = \varepsilon_{H} \hfill \\ \varepsilon_{h1} = \varepsilon_{h} \hfill \\ \end{gathered} \right. $$
(4)
Insert Eqs. (2) and (3) into Eq. (4), calculate and rearrange:
$$ \left\{ \begin{gathered} \sigma_{H1} = \sigma_{H} + \frac{1 - 2\mu }{1 - \mu }\alpha (P_{p1} - P_{P} ) \hfill \\ \sigma_{h1} = \sigma_{h} + \frac{1 - 2\mu }{1 - \mu }\alpha (P_{p1} - P_{P} ) \hfill \\ \end{gathered} \right. $$
(5)
where \( \sigma_{H1} \) and \( \sigma_{h1} \) are the maximum horizontal stress and minimum horizontal stress after pore pressure depleted, respectively.

Borehole Stress Analysis

The in situ stress existed in the formation before a well is drilled. After drilling, borehole fluid pressure replaces the rock and provides support, and the stress around the borehole will be redistributed [9]. Assuming the formation around the borehole is porous elastic medium, so the stress distribution can be obtained using the following mechanical model. In an infinite plane, a circular hole with uniform internal pressure is under the effect of two horizontal stresses in the plane of the infinity, and under the effect of overburden pressure in the vertical direction [7], as shown in Fig. 1. The maximum tangential stress appears on the borehole wall. When the pore pressure depletes to \( P_{P1} \), the borehole effective stress for a vertical well is as follows:
$$ \left\{ \begin{gathered} \sigma^{\prime}_{r} = P_{wf} - \alpha P_{wf} \\ \sigma^{\prime}_{\theta } = (1 - 2\cos 2\theta )\left[ {\sigma_{H} + \frac{1 - 2\mu }{1 - \mu }\alpha (P_{p1} - P_{P} )} \right] + (1 + 2\cos 2\theta )\left[ {\sigma_{h} + \frac{1 - 2\mu }{1 - \mu }\alpha (P_{p1} - P_{P} )} \right] - P_{wf} - \alpha P_{wf} \\ \sigma^{\prime}_{z} = \sigma_{V} - 2\mu (\sigma_{H} - \sigma_{h} )\cos 2\theta {\kern 1pt} - \alpha P_{wf} \\ \end{gathered} \right. $$
(6)
where \( \sigma_{r}^{'} \), \( \sigma_{\theta }^{'} \), \( \sigma_{z}^{'} \) are the radial, tangential and axial effective stress, respectively; \( P_{wf} \) is the borehole fluid pressure; \( \theta \) is borehole circumferential angle.
https://static-content.springer.com/image/art%3A10.1007%2Fs40098-013-0071-5/MediaObjects/40098_2013_71_Fig1_HTML.gif
Fig. 1

Mechanical model of a vertical well

In the process of oil and gas production, formation fluid will flow into the borehole, and the additional stress induced by seepage on the borehole wall is as follows:
$$ \left\{ \begin{gathered} \sigma_{r1} = - f(P_{wf} - P_{P1} ) \hfill \\ \sigma_{\theta 1} = \left[ {\frac{\alpha (1 - 2\mu )}{1 - \mu } - f} \right](P_{wf} - P_{P1} ) \hfill \\ \sigma_{z1} = \left[ {\frac{\alpha (1 - 2\mu )}{1 - \mu } - f} \right](P_{wf} - P_{P1} ) \hfill \\ \end{gathered} \right. $$
(7)
where \( \sigma_{r1} \), \( \sigma_{\theta 1} \), \( \sigma_{z1} \) are radial, tangential, axial stress caused by seepage; \( f \) is reservoir porosity.
Take Eqs. (6) and (7) superposed, when the reservoir pressure depletes to \( P_{P1} \), the effective stress on the borehole wall for a vertical well is as follows:
$$ \left\{ \begin{aligned} \sigma^{'}_{r} & = P_{wf} - \alpha P_{wf} - f(P_{wf} - P_{P1} ) \\ \sigma^{'}_{\theta } & = \left( {1 - 2\cos 2\theta } \right)\left[ {\sigma_{H} + \frac{1 - 2\mu }{1 - \mu }\alpha \left( {P_{p1} - P_{p} } \right)} \right] + \left( {1 + 2\cos 2\theta } \right)\left[ {\sigma_{h} + \frac{1 - 2\mu }{1 - \mu }\alpha \left( {P_{p1} - P_{p} } \right)} \right] \\ \quad - P_{wf} - \alpha P_{p} + \left[ {\frac{\alpha (1 - 2\mu )}{1 - \mu } - f} \right](P_{wf} - P_{P1} ) \\ \sigma^{'}_{z} & = \sigma_{V} - 2\mu \left( {\sigma_{H} - \sigma_{h} } \right)\cos 2\theta - \alpha P_{wf} + \left[ {\frac{\alpha (1 - 2\mu )}{1 - \mu } - f} \right](P_{wf} - P_{P1} ) \\ \end{aligned} \right. $$
(8)

Critical Bottom Hole Flowing Pressure Prediction Model

In the open-hole wells, the formation generally have high strength, only the formation is failure, it is likely to cause sand production [8]. Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion are selected in this paper for researching on critical bottom hole flowing pressure in pressure depleted reservoir.

Based on the Mohr–Coulomb Criterion

The Mohr–Coulumb criterion can be expressed as following [7]:
$$ \sigma_{1} - \alpha P = (\sigma_{3} - \alpha P)\tan^{2} \left( {\frac{\pi }{4} + \frac{\phi }{2}} \right) + 2C\tan \left( {\frac{\pi }{4} + \frac{\phi }{2}} \right) $$
(9)
where \( \sigma_{1} \) and \( \sigma_{3} \) are the maximum and minimum principal stress, respectively; \( \phi \) is the internal friction angle; \( C \) is the cohesion force; \( P \) is the pore pressure.
When the Mohr circle constructed by radial and tangential effective stress reaches the rock strength, shear failure occurs on the borehole wall, and sand production occurs [11]. Due to pressure depletion, the stress state changes. If the pressure drawdown remains unchanged, sand production may occur. In the depletion process, the critical bottom hole flowing pressure that keeps formation from sand production is constantly changing with the pore pressure. Insert Eq. (8) into Eq. (9), when the reservoir pressure depletes from \( P_{P} \) to \( P_{P1} \), the critical bottom hole flowing pressure can be expressed as:
$$ P_{wf} = \frac{{3\sigma_{H} - \sigma_{h} + (3\zeta - f)P_{P1} - 2\zeta P_{P} - 2CK - fK^{2} }}{{(1 - f - \alpha )K^{2} - \zeta + f + \alpha + 1}} $$
(10)
where,
$$ \zeta = {{\alpha (1 - 2\mu )} \mathord{\left/ {\vphantom {{\alpha (1 - 2\mu )} {1 - \mu }}} \right. \kern-0pt} {1 - \mu }} $$
(11)
$$ K = \tan \left( {45^{ \circ } + \frac{\phi }{2}} \right) $$
(12)
where \( P_{wf} \) is the critical bottom hole flowing pressure.

When \( P_{P1} = P_{P} \), Eq. (9) will be simplified to the critical bottom hole flowing pressure prediction model at the initial development stage of the oilfield without depletion.

Based on the Drucker–Prager Criterion

The Drucker–Prager criterion considers the effect of intermediate principle stress, and its expression is as follows [2]:
$$ \sqrt {J_{2} } = K_{f} + RI_{1} $$
(13)
where:
$$ J_{2} = \frac{1}{6}[(\sigma_{1} - \sigma_{2} )^{2} + (\sigma_{2} - \sigma_{3} )^{2} + (\sigma_{3} - \sigma_{1} )^{2} ] $$
(14)
$$ I_{1} = \sigma_{1} + \sigma_{2} + \sigma_{3} - 3\alpha P $$
(15)
where \( I_{1} \) is the first stress tensor invariant; \( J_{2} \) is the second stress tensor invariant; \( K_{f} \) and \( R \) are both rock strength parameters, their relationship with parameters of the Mohr–Coulomb strength criterion is as follows:
$$ K_{f} = \frac{3C}{{\sqrt {9 + 12\tan^{2} \phi } }} $$
(16)
$$ R = \frac{2\tan \phi }{{\sqrt {9 + 12\tan^{2} \phi } }} $$
(17)
Take Eq. (8) into Eq. (13), the critical bottom hole flowing pressure in pressure depleted reservoir determined by Drucker–Prager criterion is as follows:
$$ p_{wf} = \frac{{ - M - \sqrt {M^{2} - 4FN} }}{2F} $$
(18)
where:
$$ \left\{ \begin{gathered} F = \zeta^{2} - \zeta + 3 - 3R^{2} (2\zeta - 3\alpha )^{2} \hfill \\ M = \zeta (A + B) - 3A - 6R(K_{f} + RA + RB)(2\zeta - 3\alpha ) \hfill \\ N = A^{2} + B^{2} - AB - 3(K_{f} + RA + RB)^{2} \hfill \\ A = 3\sigma_{H} - \sigma_{h} - \zeta P_{P1} \hfill \\ B = \sigma_{V} + 2\mu (\sigma_{H} - \sigma_{h} ) - \zeta P_{P1} \hfill \\ \end{gathered} \right. $$
(19)

Based on the Mogi–Coulumb Criterion

The Mohr–Coulomb strength criterion does not consider the effect of intermediate principal stress on rock strength, so the rock strength calculation results are often conservative [27]. The octahedral shear stress is introduced by Mogi, and then the formula is improved as follows [2]:
$$ \tau_{oct} = f(\sigma_{m,2} ) $$
(20)
where \( \tau_{oct} \) and \( \sigma_{m,2} \) are octahedral shear stress and effective intermediate principal stress, respectively, their expressions are as follows:
$$ \tau_{oct} = \frac{1}{3}\sqrt {(\sigma_{1} - \sigma_{2} )^{2} + (\sigma_{2} - \sigma_{3} )^{2} + (\sigma_{3} - \sigma_{1} )^{2} } $$
(21)
$$ \sigma_{m,2} = \frac{{\sigma_{1} + \sigma_{3} }}{2} - \alpha P $$
(22)
The linear form of Mogi–Coulomb strength criterion is:
$$ \tau_{oct} = m + q\sigma_{m,2} $$
(23)
where m and q are rock strength parameters, their relationship with the parameters of Mohr–Coulomb strength criterion is as follows:
$$ m = \frac{2\sqrt 2 }{3}C\cos \phi $$
(24)
$$ q = \frac{2\sqrt 2 }{3}\sin \phi $$
(25)
Take Eq. (8) into Eq. (23), so the critical bottom hole flowing pressure in depleted reservoir determined by Mogi–Coulomb criterion is:
$$ p_{wf} = \frac{{ - D - \sqrt {D^{2} - 4CE} }}{2C} $$
(26)
where:
$$ \left\{ \begin{gathered} C = 2\zeta^{2} - 6\zeta + 6 - {{9q^{2} (\zeta - 2\alpha )^{2} } \mathord{\left/ {\vphantom {{9q^{2} (\zeta - 2\alpha )^{2} } 4}} \right. \kern-0pt} 4} \hfill \\ D = 2\zeta (A + B) - 6A - {{9(\zeta - 2\alpha )(2mq + q^{2} A)} \mathord{\left/ {\vphantom {{9(\zeta - 2\alpha )(2mq + q^{2} A)} 2}} \right. \kern-0pt} 2} \hfill \\ E = 2(A^{2} + B^{2} - AB) - {{9(2m + qA)^{2} } \mathord{\left/ {\vphantom {{9(2m + qA)^{2} } 4}} \right. \kern-0pt} 4} \hfill \\ A = 3\sigma_{H} - \sigma_{h} - \zeta P_{P1} \hfill \\ B = \sigma_{V} + 2\mu (\sigma_{H} - \sigma_{h} ) - \zeta P_{P1} \hfill \\ \end{gathered} \right. $$
(27)

Analysis

When the bottom hole pressure is less than the critical bottom hole flowing pressure, sand production will occur. The critical drawdown pressure in pressure depleted reservoir is:
$$ \Updelta p = p_{p1} - p_{wf} $$
(28)

When \( P_{P1} = P_{P} \), Eq. (28) will be simplified to the model at the initial development stage of the oilfield.

Figure 2 gives the change law of critical drawdown pressure calculated by three models along with the reservoir pressure changes (counting parameters are shown in Table 1). As can be seen from the figure, the critical drawdown pressure reduces as the reservoir pressure decreases gradually, the risk of sand production increases with pressure depletion [12]. In the whole production process, the critical drawdown pressure based on Drucker–Prager criterion is always the maximum, and that based on Mogi–Coulomb criterion is less, and that base on the Mohr–Coulomb criterion is the minimum. No matter which criterion is used, the critical drawdown pressure decreasing rate are lower than the pore pressure decreasing rate; that is to say in the process of depletion, the critical bottom hole flowing pressure is gradually decreased. Therefore, in the production process of depleted oilfield, the practice of keeping the production drawdown pressure or bottom hole flowing pressure constant is not desirable. If the drawdown pressure is kept constant, it may lead to sand production in later oilfield development, and affect the subsequent production. If the critical bottom hole flowing pressure is kept constant, it will make the drawdown pressure too conservative, and oil production will be restricted.
https://static-content.springer.com/image/art%3A10.1007%2Fs40098-013-0071-5/MediaObjects/40098_2013_71_Fig2_HTML.gif
Fig. 2

Variation of critical drawdown pressure with reservoir pressure

Table 1

Counting parameters

\( \sigma_{H} \) (MPa)

\( \sigma_{h} \) (MPa)

\( \sigma_{v} \) (MPa)

μ

\( \alpha \)

\( f \)

\( \phi \) (°)

\( C \) (MPa)

54

48

42

0.25

0.7

0.2

37

12

When the reservoir pressure decreases to 5.5 MPa, the critical drawdown pressure based on Mohr–Coulomb criterion is reduced to zero. When the reservoir pressure continues to decrease, the critical drawdown pressure based on Mogi–Coulomb criterion and Drucker–Prager criterion also gradually decreases to zero. At this time, no matter how much the drawdown pressure is, sand production will occur. Even the strength of consolidated sandstone is high, when the reservoir pressure decreases to a certain extent, sand production problems may be faced. If production continues, sand control measures must be taken.

Case Study

According to the above method, the critical value of four wells in Yacheng 13-1 gas field in South China Sea after reservoir pressure depletion (pore pressure coefficient has depleted from 1.03 to 0.52) are calculated, and the results is compared with the tested value, the results are shown in Table 2.
Table 2

Comparison of calculation results of critical drawdown pressure

Well No.

Critical drawdown pressure (MPa)

Critical drawdown pressure by testing (MPa)

Mohr–C

D–P

Mogi–C

A

6.7

9.4

8.0

8.7

B

5.7

8.2

6.8

6.2

C

6.2

8.9

7.6

7.9

D

5.3

7.4

6.4

5.7

Average relative error (%)

14.7

20.7

8.6

 

It can be seen from the comparison, the critical drawdown pressure based on Mogi–C has the minimum error with the actual value; it is followed by that based on Mohr–C; the critical drawdown pressure based on D–P has the maximum error. But the calculated results based on Mohr–C are lower than the actual values, and it indicates that the critical drawdown pressure based on Mohr–C is the safest.

The effect of intermediate principal stress is important in rock strength [19], so Mohr–C is too conservative, due to ignoring the strengthening effect of the intermediate principal stress. The influence of intermediate principal stress is considered in D–P, but the strengthening effects of the intermediate principal stress and minimum principle stress are the same in this criterion. The D–P has been proved to overestimate the intermediate principal stress effect and predict a higher strength than reality [10, 6]. The Mogi–C neither ignores the strengthening effect of intermediate principal stress, as is done by Mohr–C, nor does it predict a strength as unrealistically high as does the D–P. The Mogi–C is shown to accurately model laboratory failure data on a range of different rocks types [2, 26], so the critical drawdown pressure based on Mogi–C is the closest to the actual value.

Conclusions

  1. (1)

    The in situ stress change law along with the reservoir pressure depletion is derived, and based on this the dynamic stress distribution around borehole is obtained for the pressure depleted reservoir in the development process.

     
  2. (2)

    The critical bottom hole flowing pressure calculation models in pressure depleted reservoir are established based on Mohr–Coulomb criterion, Drucker–Prager criterion and Mogi–Coulomb criterion, respectively.

     
  3. (3)

    The critical drawdown pressure based on Drucker–Prager criterion is the maximum, and that based on Mogi–Coulomb criterion is less, and that based on Mohr–Coulomb criterion is the minimum. The reservoir pressure depletion will lead to critical drawdown pressure decrease, but the decreasing rate is less than the pore pressure.

     
  4. (4)

    The critical drawdown pressure based on Mogi–Coulomb criterion is the closest to the actual value, but that based on Mohr–Coulomb criterion is the safest.

     

Acknowledgments

The authors gratefully acknowledge the support of Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51221003), National Natural Science Foundation Project of China (Grant No. 51174219) and National Oil and Gas Major Project (Grant No. 2011ZX05009-005).

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© Indian Geotechnical Society 2013