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Modeling fault activation due to fluid production: Bayesian update by seismic data

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Abstract

The reactivation of faults and the generation of fractures can be caused by stress changes due to injection and/or production of fluids into and/or from the subsurface. The simulation of these processes, which could be associated with (micro-)seismicity, is affected by a high uncertainty. The aim of this work is at developing a mathematical framework to quantify and possibly reduce the prior modeling uncertainties by assimilation of seismic data. The mechanics of fault (re-)activation is simulated by a Finite Element (FE) numerical model where the discontinuous displacements between the fault surfaces are suitably considered using appropriate Interface Elements (IEs). The study is carried out by using a stochastic approach, with a global sensitivity analysis (gSA) based on Sobol’ indices initially performed to estimate the influence of the input parameters on the model solution. Then, a Markov Chain Monte Carlo (MCMC) sampling technique based on the generalized Polynomial Chaos expansion (gPC) surrogate solution is used to update the prior information conditioned on seismic observations. The methodology is tested on a 3D synthetic test case. The uncertain input is the natural stress regime and the Mohr-Coulomb parameters characterizing the fault activation criterion. A good reduction of the prior uncertainty is obtained, showing that the assimilation of seismic data can have a promising potential for improving the subsurface characterization.

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Abbreviations

x :

Cartesian coordinates of a point in the three-dimensional space

t :

time/loading step variable

y :

state vector of the system

y ∗ :

forcing term of the system

u :

vector of the model parameters

\(\mathcal {G}\) :

forward problem operator

q :

vector of the quantities of interest

\(\mathcal {M}\) :

map from y to q

\(\mathcal {S}\) :

solution operator of the forward problem

U :

random vector of the model parameters

Q :

random vector of the quantities of interest

Y :

random vector of the model output

ρ :

joint probability density function of a given random vector

Z :

random vector of the model parameters defined over a unit hypercube

V :

variance

S i :

first order Sobol’ index

S 1,…, s :

higher order Sobol’ index

D :

random vector of the noisy measurements

D t :

random vector of the true observable

𝜖 :

random vector of the observational error

\(\mathcal {H}\) :

map from the model state to the true observable

π :

conditioned probability density function

τ L :

shear stress limit

τ 0 :

fault cohesion

ϕ :

fault friction angle

σ n :

normal effective stress acting on the fault

ϕ k :

k-th univariate basis function

\(\tilde {\boldsymbol {Y}}\) :

gPC approximation of Y

c α :

vector of coefficients of the gPC expansion

Φα :

multivariate gPC basis functions

α, β :

multi-index vectors

\(\tilde {\boldsymbol {c}}_{\boldsymbol {\alpha }}\) :

approximation of the gPC expansion coefficients

w :

integration weights

σ 1, σ 2, σ 3 :

principal stresses

M1,M2 :

ratio of the principal stress σ1 and σ2 to σ3

σ :

effective normal stress vector

τ :

shear stress vector

s :

displacement field vector

A a :

activated area of the fault

δ s :

sliding of the fault

k :

hydraulic conductivity

p :

pore pressure

δ t :

slippage of the fault elements

M 0 :

seismic moment

G :

shear modulus of the rock formation

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Acknowledgments

The research has been carried out in the framework of the UNESCO IGCP 641 project “Mechanisms, Monitoring and Modeling Earth Fissure generation and Fault activation due to subsurface Fluid exploitation (M3EF3)”.

Funding

This work has been partially funded by the University of Padova project “Data Assimilation algorithms for reservoir geomechanics and induced seismicity.”

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Authors and Affiliations

  1. Department of Civil, Environmental and Architectural Engineering, University of Padova, Via Marzolo 9, 35131, Padova, Italy

    Claudia Zoccarato, Massimiliano Ferronato, Andrea Franceschini, Carlo Janna & Pietro Teatini

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  1. Claudia Zoccarato
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Zoccarato, C., Ferronato, M., Franceschini, A. et al. Modeling fault activation due to fluid production: Bayesian update by seismic data. Comput Geosci 23, 705–722 (2019). https://doi.org/10.1007/s10596-019-9815-3

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