A Cox rate-and-state model for monitoring seismic hazard in the Groningen gas field

TL;DR

The paper tackles induced seismicity in the Groningen gas field by replacing the deterministic pore-pressure input in the rate-and-state model with a stochastic driving mechanism and by incorporating gas production as a covariate, formalizing a log-Gaussian Cox process where earthquakes have intensity $\Lambda(s,t)$ driven by production and state dynamics. It derives first- and second-moment expressions for the state variable $\Gamma$ and uses delta-method approximations to obtain moments of the rate, facilitating unbiased estimating equations for parameter inference. Parameter estimation relies on a data-driven unbiased estimating-equation approach with Monte Carlo estimates of the intractable intensity, and posterior monitoring is performed via parallel MALA across a spatial grid, applied to Groningen data. The approach yields interpretable hazard maps with quantified uncertainty, highlights production-driven increases in seismic hazard, and provides a flexible framework that can be extended with additional covariates or reservoir-model comparisons for improved risk assessment in practice.

Abstract

To monitor the seismic hazard in the Groningen gas field, we modify the rate-and-state model that relates changes in pore pressure to induced seismic hazard by allowing for noise in pore pressure measurements and by explicitly taking into account gas production volumes. We analyse the first and second-moment structure of the resulting Cox process, propose an unbiased estimating equation approach for the unknown model parameters and derive the posterior distribution of the driving random measure. We use a parallel Metropolis adjusted Langevin algorithm for sampling from the posterior and to monitor the hazard.

Figures (6)

• Figure 1: Spatial (left-most panel) and temporal (right-most panel) projections of the $332$ earthquakes of magnitude $1.5$ or larger with epicentre in the Groningen gas field that occurred in the period from January 1st, 1995, up to December 31st, 2021.
• Figure 2: $95\%$ pointwise confidence intervals for the mean (left) and variance (right) of $\Gamma(s, t_k)^{-1}$ as a function of $k$ when $m(s, t_k ) = 6 - {1}/ { (0.5 k + 1) }$, $\alpha = 0.01$, $\Delta = 0.1$, $\gamma_0 = 0.2$ and $\sigma^2 = 2.0$. The dots correspond to the approximations in (\ref{['e:mean']}) and (\ref{['e:cov']}).
• Figure 3: $95\%$ pointwise confidence intervals for the mean (left) and variance (right) of $\Gamma(s, t_k)^{-1}$ as a function of $k$ when $m(s, t_k) = 5 + {10} / {( 2 k + 1) }$, $\alpha = 0.01$, $\Delta = 0.1$, $\gamma_0 = 0.2$ and $\sigma^2 = 2.0$. The dots correspond to the approximations in (\ref{['e:mean']}) and (\ref{['e:cov']}).
• Figure 4: Average Pearson residual against average fitted value for $25$ bins for model (\ref{['e:LGCP-simple']}). The grey lines correspond to two standard deviations bounds.
• Figure 5: Top left: Smoothed gas production over 2021 (in Nbcm for each grid cell). Top right: Estimated pressure drop (in bara for each grid cell) from 1995 until 2022. Bottom left: Mean posterior number of earthquakes in 2022 (for each grid cell, sample size $I=5,000$). Bottom right: Sample standard deviation of posterior number of earthquakes in 2022 (for each grid cell, sample size $I=5,000$).

Theorems & Definitions (2)

• Example 1
• Example 2