Bayesian inversion of synthetic AVO data to assess fluid and shale content in sand-shale media

Abstract

Reservoir characterization of sand-shale sequences has always challenged geoscientists due to the presence of anisotropy in the form of shale lenses or shale layers. Water saturation and volume of shale are among the fundamental reservoir properties of interest for sand-shale intervals, and relate to the amount of fluid content and accumulating potentials of such media. This paper suggests an integrated workflow using synthetic data for the characterization of shaley-sand media based on anisotropic rock physics (T-matrix approximation) and seismic reflectivity modelling. A Bayesian inversion scheme for estimating reservoir parameters from amplitude vs. offset (AVO) data was used to obtain the information about uncertainties as well as their most likely values. The results from our workflow give reliable estimates of water saturation from AVO data at small uncertainties, provided background sand porosity values and isotropic overburden properties are known. For volume of shale, the proposed workflow provides reasonable estimates even when larger uncertainties are present in AVO data.

Appendices

Appendix A

1.1 T-matrix approach for composite porous media

The effective stiffness tensor C d ∗ of a porous medium with inclusions (shale lenses) for the dry case is given by Jakobsen et al. (2003a):

$$ {\mathbf{C}}_{{{d}}}^{{\mathbf{\ast} }}{\mathbf{=}}{\mathbf{C}}^{{\mathbf{(0)}}}{\mathbf{+}}{\mathbf{C}}_{{\mathbf{1}}} {\mathbf{:(}}{\mathbf{I}}_{{\mathbf{4}}}{\mathbf{+}}{\mathbf{C}}_{{\mathbf{1}}}^{{\mathbf{-1}}}{\mathbf{:}}{\mathbf{C}}_{{\mathbf{2}}} {)}^{{\mathbf{-1}}}, $$

(12)

where

$$ {\mathbf{C}}_{{\mathbf{1}}}{\mathbf{=}}\sum\limits_{{r=1}}^{{{N}}} {v^{\left( r\right)}\mathbf{t}^{\left( r\right)}} , $$

(13)

and

$$ {\mathbf{C}}_{{\mathbf{2}}}{\mathbf{=}}\sum\limits_{{r=1}}^{{{N}}} v^{\left( r\right)}\boldsymbol{t}^{\left( r\right)}{{\mathbf{ :G}}}_{{\mathbf{d} }}{\mathbf{:}}\mathrm{t}^{\left( r\right)}v^{\left( r\right)}. $$

(14)

In equation (14A3) ‘:’ denotes the double scalar product (Auld 1990), C ⁽⁰⁾ is the stiffness tensor of the dry porous matrix, I ₄ is the (symmetric) identity matrix for second-rank tensors, v ^(r) is the volume concentration for inclusions of type r (shale lenses) and G _d tensor is given by the strain Green’s function integrated over an ellipsoid determining the symmetry of the correlation function for the spatial distribution of inclusions (Ponte and Willis 1995; Jakobsen et al. 2003a). The T-matrix for a single inclusion (shale lens) of type r is given by Jakobsen et al. (2003a):

$$ \text{t}^{\mathrm{(r)}} =\mathbf{(C}^{{{(r)}}} -\mathbf{C}^{{\mathbf{(0)}}}{) : [I}_{{\mathbf{4}}}\mathbf{-G}^{{{(r)}}}\mathbf{:(C}^{{{(r)}}} \mathbf{-C}^{{\mathbf{(0)}}}{)]}^{{\mathbf{-1}}}. $$

(15)

Here, G ^(r) is a fourth-rank tensor given by the strain Green’s function for a material with properties given by C ⁽⁰⁾ integrated over a characteristic spheroid having the same shape as inclusions of type r (Jakobsen et al. 2003a). C ^(r) is the stiffness tensor for the inclusions of type r (stiffness tensors of shale lenses) whose elastic properties for the saturated case were used from Hornby et al. (1994) given in table A1A1.

Table A1 Elastic constants of shale used in this study for modelling shale lenses (Jones and Wang 1981; Hornby et al. 1994).

Appendix B

Fluid saturation effects

In order to calculate the effect of fluid saturation on the effective properties of a shaley-sand medium, we have used the anisotropic relations of Brown and Korringa (1975), which can be written in the symbolic or matrix notation given as:

$$ {\mathbf{S}}^{{\mathbf{\ast}}}{\mathbf{=}}{\mathbf{S}}_{{\mathbf{d}}}^{{\mathbf{\ast }}}{\mathbf{+}}\frac{\left( \left( {\mathbf{S}}_{{\mathbf{d}}}^{{\mathbf{\ast} }}{\mathbf{-}}{\mathbf{S}}_{{\mathbf{m}}}\right){\mathbf{ : } }\left( {\mathbf{I}}_{{\mathbf{2}}}\otimes {\mathbf{I}}_{{\mathbf{2}}} \right){\mathbf{ : } }\left( {\mathbf{S}}_{{\mathbf{d}}}^{{\mathbf{\ast} }}{\mathbf{-}}{\mathbf{S}}_{{\mathbf{m}}}\right)\right)}{{{\varphi} }^{{\mathbf{o}}}\left( {\mathbf{I}}_{{\mathbf{2}}}{\mathbf{\cdot} }{\mathbf{S}}_{{\mathbf{m}}}{\mathbf{\cdot} }{\mathbf{I}}_{{\mathbf{2}}}{\mathbf{-}}\mathbf{1}/{K}_{{{f}}} \right){\mathbf{-}}{\mathbf{I}}_{{\mathbf{2}}}{\mathbf{\cdot} }\left( {\mathbf{S}}_{{\mathbf{d}}}^{{\mathbf{\ast} }}{\mathbf{-}}{\mathbf{S}}_{{\mathbf{m}}}\right){\mathbf{\cdot} }{\mathbf{I}}_{{\mathbf{2}}}}. $$

(16)

Here ⊗ denotes the dyadic product, S _m is the compliance tensor of the solid mineral component (properties of mineral quartz were used in the case of shaley-sand model), S d ∗ is the effective compliance tensor for the dry shaley-sand medium and S ^∗ is the effective compliance tensor for the saturated shaley-sand medium. φ ^∘ is the total porosity and I ₂ is the (symmetric) identity matrix for second-rank tensors. In the case of a composite porous medium (shaley-sand), which is partially saturated with oil, gas and water, K _f may be regarded as the bulk modulus of an effective fluid given by Wood also known as Reuss average (Wood 1955; Mavko et al. 2009):

$$ \frac{1}{K_{f}}=\frac{S_{w}}{K_{w}}+\frac{S_{o}}{K_{o}}+\frac{S_{g}}{K_{g}}, $$

(17)

where

$$ S_{w}+ S_{o}+S_{g}\mathrm{= 1}. $$

(18)

Here S _w ,S _o and S _g represent the saturation for water, oil and gas and K _w ,K _o and K _g represent the bulk modulus for water, oil and gas. Before proceeding further, it is important to clarify that effects of fluid saturation were introduced in our rock physics modelling prior to the modelling of saturated shale inclusions. This means that the fluid saturation effects are only for the background porous sand body.