Estimation of Reservoir Fracture Properties from Seismic Data Using Markov Chain Monte Carlo Methods

Abstract

The knowledge of fracture properties and its geometrical patterns is often required for the analysis of mechanical and flow properties in fractured reservoirs, as fracture characterization plays a critical role in the optimization of hydrocarbon production or estimation of storage capacity of subsurface reservoirs. A stochastic method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to estimate fracture properties using a rock physics model for fractured rocks. Two implementations are presented: a Metropolis algorithm based on a Gaussian prior distribution and an extended Metropolis algorithm with an informative prior obtained from multiple-point statistics simulations. The results are compared to a Bayesian analytical approach where the solution is based on a linearized approximation of the rock physics model. The novelty of the proposed approach is the use of a training image, that is, a conceptual geological model, to account for the spatial distribution of the fractures. Two fracture properties are considered, namely fracture density and aspect ratio, and the spatial distribution and geometrical characteristics of fractures are also investigated to understand the connectivity patterns that control fluid flow. The MCMC approach with a training image is more computationally demanding but provides geometrical models of the spatial distribution of fractures. The inversion results show that the prediction accuracy of fracture density and aspect ratio obtained by the MCMC methods is similar to the one obtained with the analytical approach, and that the MCMC methods provide a reliable assessment of the posterior uncertainty as well.

Appendix A: Hudson’s Model

Hudson’s model is based on effective medium theory, in which an elastic solid is assumed to include thin, penny-shaped cracks filled with fluid (Hudson 1981; Sava 2004). Physical properties of the inclusions, namely fracture density ($e$) and aspect ratio ($\alpha$), are used to describe the fracture systems. According to the first-order Hudson’s model (Hudson 1981; Sava 2004), the elastic rock properties of the fractured medium can then be calculated from the components of the stiffness tensor as

$$ \rho = \left( {1 - \phi } \right)\rho_{h} + \phi \rho_{fl} , $$

(A.1)

$$ V_{P} = \sqrt {\frac{{C_{33} }}{\rho }} , $$

(A.2)

$$ V_{S} = \sqrt {\frac{{C_{44} }}{\rho }} , $$

(A.3)

where

$$ C_{33} = \left( {\lambda + 2\mu } \right) - \frac{{\left( {\lambda + 2\mu } \right)^{2} }}{\mu }eU_{3} , $$

(A.4)

$$ C_{44} = \mu - \mu eU_{1} , $$

(A.5)

$$ U_{3} = \frac{{4\left( {\lambda + 2\mu } \right)}}{{3\left( {\lambda + \mu } \right)\left( {1 + \kappa } \right)}}, $$

(A.6)

$$ \kappa = \frac{{K_{fl} \left( {\lambda + 2\mu } \right)}}{{\pi \alpha \mu \left( {\lambda + \mu } \right)}}, $$

(A.7)

$$ U_{1} = \frac{{16\left( {\lambda + 2\mu } \right)}}{{3\left( {3\lambda + 4\mu } \right)}}, $$

(A.8)

with $\lambda$ and $\mu$ being the Lamé parameters of the unfractured host (background) rock; $K_{fl}$ being the bulk modulus of the inclusion fluid material; and $\rho_{h}$ and $\rho_{fl}$ being the density of the background rock and inclusion fluid, respectively. The rock porosity ($\phi$) caused by the presence of cracks is given by

$$ \phi = \frac{4\pi }{3}e\alpha . $$

(A.9)

Based on Hudson’s rock physics model, the reflection coefficients for seismic modeling are then calculated using the Aki–Richards equation (Buland and Omre 2003)

$$ \begin{aligned} r_{PP} (t,\theta ) & = \frac{1}{2}(1 + {\text{tan}}^{2} \theta )\frac{\partial }{\partial t}{\text{ln}}V_{P} \left( t \right) - 4\frac{1}{{\gamma^{2} }}{\text{sin}}^{2} \theta \frac{\partial }{\partial t}{\text{ln}}V_{S} \left( t \right) \\ & \quad + \frac{1}{2}\left( {1 - 4\frac{1}{{\gamma^{2} }}{\text{sin}}^{2} \theta } \right)\frac{\partial }{\partial t}{\text{ln}}\rho \left( t \right), \\ \end{aligned} $$

(A.10)

where $\theta$ is the incidence angle; $V_{P} \left( t \right)$, $V_{S} \left( t \right)$, and $\rho \left( t \right)$ are the time-dependent rock properties; and $\gamma$ is the $V_{P} /V_{S}$ ratio. A convolution operator is then adopted to compute the seismic signal assuming a known source wavelet.