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Robust Output Tracking for Induced Seismicity Mitigation in Underground Reservoirs Governed by a Nonlinear 3D PDE-ODE System

Diego Gutiérrez-Oribio, Ioannis Stefanou

TL;DR

The paper addresses preventing induced seismicity in underground reservoirs governed by a nonlinear 3D PDE-ODE cascade. It proposes a robust, output-feedback controller based on a MIMO Super-Twisting algorithm to track region-averaged pressure and seismicity rate, achieving finite-time or exponential convergence despite parameter uncertainty. By proving eISS/ISS properties for the diffusion and SR subsystems and deriving a closed-form controller, the approach provides a practical, continuous-input solution requiring minimal model information. The Groningen reservoir case study demonstrates safe production with potential CO$_2$-neutral operation, illustrating significant real-world impact for energy storage, geothermal, and carbon-management applications, while highlighting avenues for incorporating richer physics and stochastic seismicity in future work.

Abstract

This paper presents a robust output-feedback controller for induced seismicity mitigation in geological reservoirs described by a coupled 3D PDE-ODE model. The controller is a MIMO Super-Twisting design, producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainty and spatial heterogeneity. Two operational outputs are regulated simultaneously: regional pressures and seismicity rates computed over reservoir sub-regions. Closed-loop properties are established via explicit bounds on the solution and its time derivative for both the infinite-dimensional dynamics and the nonlinear ODE system, yielding finite-time or exponential convergence of the tracking errors. The method is evaluated on a Groningen gas-field case study in two scenarios: gas production while not exceeding the intrinsic seismicity of the region, and combined production with CO$_2$ injection toward net-zero operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation without sacrificing production objectives.

Robust Output Tracking for Induced Seismicity Mitigation in Underground Reservoirs Governed by a Nonlinear 3D PDE-ODE System

TL;DR

The paper addresses preventing induced seismicity in underground reservoirs governed by a nonlinear 3D PDE-ODE cascade. It proposes a robust, output-feedback controller based on a MIMO Super-Twisting algorithm to track region-averaged pressure and seismicity rate, achieving finite-time or exponential convergence despite parameter uncertainty. By proving eISS/ISS properties for the diffusion and SR subsystems and deriving a closed-form controller, the approach provides a practical, continuous-input solution requiring minimal model information. The Groningen reservoir case study demonstrates safe production with potential CO-neutral operation, illustrating significant real-world impact for energy storage, geothermal, and carbon-management applications, while highlighting avenues for incorporating richer physics and stochastic seismicity in future work.

Abstract

This paper presents a robust output-feedback controller for induced seismicity mitigation in geological reservoirs described by a coupled 3D PDE-ODE model. The controller is a MIMO Super-Twisting design, producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainty and spatial heterogeneity. Two operational outputs are regulated simultaneously: regional pressures and seismicity rates computed over reservoir sub-regions. Closed-loop properties are established via explicit bounds on the solution and its time derivative for both the infinite-dimensional dynamics and the nonlinear ODE system, yielding finite-time or exponential convergence of the tracking errors. The method is evaluated on a Groningen gas-field case study in two scenarios: gas production while not exceeding the intrinsic seismicity of the region, and combined production with CO injection toward net-zero operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation without sacrificing production objectives.

Paper Structure

This paper contains 12 sections, 1 theorem, 42 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Output Feedback Tracking Control Design
  4. Prevention of Induced Seismicity in Groningen reservoir
  5. Model setup and validation
  6. Scenario 1: Gas extraction
  7. Scenario 2: Gas extraction and CO$_2$ injection
  8. Comparison and limitations
  9. Conclusion
  10. Controllability of the diffusion process
  11. Properties and bounds of control \ref{['eq:Q']}
  12. Model implementation

Key Result

Theorem 1

Let system eq:errordyn under Assumptions A1--A4 be driven by the control eq:Q with. Suppose that the uncertain control coefficient $\Delta B(t)$ in eq:Bbounds and the perturbation term $\Psi(t)$ in eq:errordyn are assumed to fulfil for almost all $t \in T$. Then, there exist matrices $K_1,K_2$ defined as where $L>0$ is sufficiently large, and $\bar{K}_1,\bar{K}_2$ are arbitrary positive diagonal

Figures (12)

  • Figure 1: Groningen gas reservoir with regions for controlling pressure and SR based on the locations of major towns in the area. Background image was obtained from the public image https://zoek.officielebekendmakingen.nl/stcrt-2017-28922.html.
  • Figure 2: Monthly and cumulative extraction of gas in Groningen.
  • Figure 3: Normalized spatial density map of the SR in Groningen representing the 712 events that occurred between 12-1991 and 01-2023. Top image shows the magnitude and location of the real seismic events, from which their spatial density is determined. The bottom image depicts the simulated spatial density of the events. The normalization was made with the maximum value of the spatial density of $R(x,t)$ over the reservoir.
  • Figure 4: Average SR over the reservoir and cumulative number of seismic events in Groningen. Real data (blue line) and simulated data (orange line).
  • Figure 5: Average pressure over the reservoir (see also b:doi.org/10.1029/2023GL105455).
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Remark 5