Control of Human-Induced Seismicity in Underground Reservoirs Governed by a Nonlinear 3D PDE-ODE System

Induced seismicity caused by fluid extraction or injection in underground reservoirs is a major challenge for safe energy production and storage. 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 nonlinear and robust (MIMO Super-Twisting design), producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainties 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 the 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 carbon operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation while preserving production objectives.

💡 Research Summary

This paper addresses the pressing problem of human‑induced seismicity that arises from fluid injection and extraction in subsurface reservoirs, a challenge that threatens the safety and public acceptance of many emerging energy technologies such as geothermal, carbon capture and storage (CCS), and hydrogen storage. The authors formulate a coupled three‑dimensional (3‑D) diffusion partial differential equation (PDE) for reservoir pressure together with a logistic‑type ordinary differential equation (ODE) that models the seismicity rate (SR). The PDE captures Darcy flow through a heterogeneous permeability field, while the ODE links the time derivative of pressure (the “input” to the ODE) to the evolution of SR, incorporating both a linear pressure‑driven term and a nonlinear term that drives SR toward a spatially varying intrinsic seismicity background R*. The two subsystems are strongly coupled: the control inputs are the well‑wise fluid fluxes Q(t), which appear as source terms in the PDE, and the pressure time‑derivative ut appears as the driving term in the ODE.

The control objective is to regulate two sets of spatially averaged outputs simultaneously: (i) regional pressures y_u, defined as volume averages of u over selected sub‑domains V_u_i, and (ii) regional seismicity rates y_R, defined analogously over sub‑domains V_R_j. Both output vectors must track prescribed reference trajectories r_u(t) and r_R(t) while respecting actuator saturation (bounded Q) and operating under significant parametric uncertainty (permeability k(x), viscosity η(x), coefficients γ_1, γ_2, and the intrinsic seismicity field R*). The authors assume that the number of control inputs exceeds or equals the number of regulated outputs, and that each output region contains at least one well, guaranteeing that the nominal input‑output matrix B_0 has full row rank and thus a right pseudoinverse exists.

The core contribution is a robust, continuous‑output‑feedback controller based on a Multi‑Input Multi‑Output (MIMO) Super‑Twisting algorithm. Defining the tracking errors σ_u = y_u – r_u and σ_R = (1/γ_10 R*_0)(y_R – r_R), the error dynamics can be written compactly as σ̇ = Ψ(t) + B(t)Q(t), where Ψ aggregates known reference derivatives and unmodeled terms, and B(t) =