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Identification of above-zone pressure perturbations caused by leakage from those induced by deformation

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Abstract

Pressure changes in the above zone, i.e., the overlying aquifer of an injection zone separated by a sealing caprock, are usually attributed to leakage through wells. However, pressure changes can be induced geomechanically due to rock deformation without any hydraulic connection between the injection zone and the above zone where the pressure change is observed. To account for these two causes of pressure change in the above zone, we develop an analytical solution to evaluate the deformation-induced pressure changes and we derive an asymptotic analytical solution for pressure perturbations caused by leaking wells. The analytical models compare well with available numerical/analytical solutions. Using the analytical solutions for the deformation- and leakage-induced pressure changes, we propose a graphical diagnostic plot to determine the cause of pressure change. Considering that the pressure change is caused by leakage, we then use the asymptotic solution to develop an easy-to-use fully graphical methodology to characterize leaking wells. This methodology improves a previous analysis methodology that was based on an inverse modeling algorithm that can be highly instable and computationally expensive. Based on the graphical method presented here, the slopes and intercepts of the proposed line-fitted graphs are used to determine the leak location and transmissibility. We apply the graphical method to an example problem to illustrate its application procedure and effectiveness in differentiating deformation-induced pressure changes from leaking wells. Overall, the diagnostic plot proposed here proves to be useful to determine the cause of the above-zone pressure change.

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Author information

Authors and Affiliations

  1. Craft and Hawkins Department of Petroleum Engineering, Louisiana State University, Baton Rouge, LA, USA

    Mehdi Zeidouni

  2. Soil Mechanics Lab, Swiss Federal Institute of Technology, EPFL, Lausanne, Switzerland

    Victor Vilarrasa

  3. Institute of Environmental Assessment and Water Research (IDAEA-CSIC), Barcelona, Spain

    Victor Vilarrasa

Authors
  1. Mehdi Zeidouni
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  2. Victor Vilarrasa
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Corresponding author

Correspondence to Mehdi Zeidouni.

Appendix: derivation of asymptotic solution

Appendix: derivation of asymptotic solution

For late-time period: s→0, and we can make the following approximations (Abramowitz and Stegun 1970)

$$K_{0} \left( x \right) \approx - \gamma - \ln (x/2)$$
(33)

on the basis of which, Eq. (16) simplifies to

$$\bar{q}_{lD} = \frac{{ - \frac{{\gamma + \ln (\sqrt s R_{D} /2)}}{s}}}{{ - \frac{{\gamma + \ln (\sqrt s r_{lD} /2)}}{1} - \frac{{\gamma + \ln \left( {\sqrt {\frac{s}{{\eta_{D} }}} r_{lD} /2} \right)}}{{T_{D} }} + \frac{1}{\alpha }}} .$$
(34)

Further rearrangement to make the Laplace variable appear only in the denominator yields

$$\bar{q}_{lD} = \frac{{T_{D} }}{{s\left( {T_{D} + 1} \right)}}\left( {1 + \frac{{\ln \left( {\frac{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} R_{D}^{2} }}{{r_{lD}^{2} }}} \right)}}{{\ln \left( {\frac{{e^{2\gamma } r_{lD}^{2} }}{4}\frac{1}{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}s} \right)}}} \right) ,$$
(35)

or

$$\bar{q}_{lD} = \frac{1}{{\left( {1 + 1/T_{D} } \right)}}\left( {\frac{1}{s} - \ln \left( {\frac{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} R_{D}^{2} }}{{r_{lD}^{2} }}} \right)\left( {\frac{ - 1}{{s\ln \left( {\frac{{e^{2\gamma } r_{lD}^{2} }}{4}\frac{1}{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}s} \right)}}} \right)} \right)$$
(36)

The inverse of Eq. (36) is

$$q_{lD} = \frac{1}{{\left( {1 + 1/T_{D} } \right)}}\left( {1 - \ln \left( {\frac{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} R_{D}^{2} }}{{r_{lD}^{2} }}} \right)L^{ - 1} \left( {\frac{ - 1}{{s\ln \left( {\frac{{e^{2\gamma } r_{lD}^{2} }}{4}\frac{1}{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}s} \right)}}} \right)} \right)$$
(37)

The last term in the RHS of Eq. (37) should be evaluated in the real-time domain. Yeh and Wang (2007) present a late-time approximation of the Laplace inverse of 1/(s ln (s/κ)) in calculating wellbore flux for the constant pressure well test problem. The inverse is based on truncating four terms of the expansion series provided by Ritchie and Sakakura (1956) for the Laplace inverse of 1/(s ln (s/κ)). Three-term and two-term approximations of the inversion were presented by Jaeger (1943) and Carslaw and Jaeger (1959), respectively. The following is obtained for the last term in the RHS of Eq. (37), considering a three-term truncated series

$$L^{ - 1} \left\{ {\frac{ - 1}{{s\ln \left( {\frac{{e^{2\gamma } r_{lD}^{2} }}{4}\frac{1}{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}s} \right)}}} \right\} = \frac{1}{{\ln \left( {\frac{{4\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}{{e^{2\gamma } r_{lD}^{2} }}t_{D} } \right)}} - \frac{\gamma }{{\left( {\ln \left( {\frac{{4\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}{{e^{2\gamma } r_{lD}^{2} }}t_{D} } \right)} \right)^{2} }} + \frac{{\gamma^{2} - \frac{{\pi^{2} }}{6}}}{{\left( {\ln \left( {\frac{{4\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}{{e^{2\gamma } r_{lD}^{2} }}t_{D} } \right)} \right)^{3} }}$$
(38)

Using Eqs. (15), (33) and (36), we obtain

$$p_{mD} = \frac{1}{{2\left( {1 + T_{D} } \right)}}\left( \begin{aligned} \ln \left( {t_{D} } \right) - \gamma - \ln \left( {\frac{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} R_{D}^{2} \rho_{D}^{2} }}{{4\eta_{D} r_{lD}^{2} }}} \right) + \ln \left( {\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} \frac{{R_{D}^{2} }}{{r_{lD}^{2} }}} \right) \hfill \\ \ln \left( {\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} \frac{{\rho_{D}^{2} }}{{\eta_{D} r_{lD}^{2} }}} \right)L^{ - 1} \left( {\frac{ - 1}{{s\ln \left( {\frac{{e^{2\gamma } r_{lD}^{2} }}{4}\frac{1}{{\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} }}s} \right)}}} \right) \hfill \\ \end{aligned} \right) .$$
(39)

Eq. (39) can also be written as

$$p_{mD} = \frac{1}{{\left( {1 + T_{D} } \right)}}\left( {\frac{ - \gamma }{2} - \frac{1}{2}\ln \left( {\frac{{R_{D}^{2} }}{{4t_{D} }}} \right)} \right) - \frac{{q_{lD} }}{{2T_{D} }}\ln \left( {\eta_{D}^{{\frac{1}{{\left( {T_{D} + 1} \right)}}}} e^{{\frac{{2T_{D} }}{{\alpha \left( {T_{D} + 1} \right)}}}} \frac{{\rho_{D}^{2} }}{{\eta_{D} r_{lD}^{2} }}} \right) .$$
(40)

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Zeidouni, M., Vilarrasa, V. Identification of above-zone pressure perturbations caused by leakage from those induced by deformation. Environ Earth Sci 75, 1271 (2016). https://doi.org/10.1007/s12665-016-6090-7

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