Change-of-Support Models on Irregular Grids for Geostatistical Simulation

Abstract

In many domains, numerical models are initialized with inputs defined on irregular grids. In petroleum reservoir engineering, they consist of a great variety of grid cells of different size and shape to enable fine-scale modeling in the vicinity of the wells and coarse modeling in less important regions. Geostatistical simulation algorithms, which are used to populate the cells of unstructured grids, often have to address the problem of transition from the small-scale statistical data stemming from laboratory cores analysis and seismic processing to the multiple larger scale geological supports. The reasonable generalization of the above-mentioned problem is integrating the point-support data to simulations on irregular supports. Classical geostatistical simulation methods for generating realizations of a stationary Gaussian random function cannot be applied to unstructured grids directly, because of the uneven supports. This article provides a critical review of existing geostatistical simulation methodologies for unstructured grids, including fine-scale simulations with upscaling and direct sequential simulation algorithms, and presents two different generalizations of the discrete Gaussian model for this purpose, thereby discussing the theoretical assumptions and the accuracy when implementing these models.

Appendix: Covariance Relations

This section demonstrates that simulations produced with generalized DGM 2 are biased in terms of covariance relative to the theoretically expected result. The theoretical expression of the covariance between $Z(v_p )$ and $Z(v_q )$ for two blocks $v_p$ and $v_q$can be expressed as follows

$$\begin{aligned} cov({Z( {v_p }),Z( {v_q })})&=\frac{1}{| {v_p } || {v_q} |}\mathop \int \limits _{v_p } \mathop \int \limits _{v_q }C( {x-x'})\mathrm{d}x\mathrm{d}{x}' \nonumber \\&=\frac{1}{| {v_p} || {v_q } |}\mathop \int \limits _{v_p } \mathop \int \limits _{v_q } C( {x-{x}'})\mathrm{d}x\mathrm{d}{x}' \nonumber \\&=\frac{1}{| {v_p }|\vert v_q \vert }\mathop \sum \limits _{i=1}^\infty \phi _i^2 \mathop \int \limits _{v_p } \mathop \int \limits _{v_q } \rho ^i({x-{x}'})\mathrm{d}x\mathrm{d}{x}'. \end{aligned}$$

(14)

For DGM 2, the block-to-block covariance is equal to

$$\begin{aligned} \mathrm{cov}( {Z( {v_p }),Z({v_q})})&=\mathrm{cov}({\phi _{v_p} ({Y_{v_p } }),\phi _{v_q } ( {Y_{v_q } })})\nonumber \\&=\sum \nolimits _{i=1}^\infty {\phi _i^2 r_p^i r_q^i cov( {Y_{v_p } ,Y_{v_q } })^i}\nonumber \\&= \sum \limits _{i=1}^\infty {\phi _i^2 r_p^i r_q^i \left( {\frac{1}{| {v_p } || {v_q } |r_p r_q }\mathop \int \nolimits _{v_p } \mathop \int \nolimits _{v_q } \rho ( {x-x'})\mathrm{d}x\mathrm{d}x'}\right) ^i} \nonumber \\&=\sum \limits _{i=1}^\infty {\phi _i^2 \left( {\frac{1}{| {v_p } || {v_q } |}\mathop \int \nolimits _{v_p } \mathop \int \nolimits _{v_q } \rho ( {x-x'})\mathrm{d}x\mathrm{d}x'}\right) ^i,} \end{aligned}$$

(15)

which does not coincide with the expected theoretical result. The difference $\Delta$ between the theoretical and implied by DGM 2 covariance values is

$$\begin{aligned} \Delta =\mathop \sum \limits _{i=2}^\infty \phi _i^2 \left( {\frac{1}{| {v_p } || {v_q } |}\mathop \int \nolimits _{v_p } \mathop \int \nolimits _{v_q } \rho ^i( {x-x'})\mathrm{d}x\mathrm{d}x'-\left( {\frac{1}{| {v_p } || {v_q } |}\mathop \int \nolimits _{v_p } \mathop \int \nolimits _{v_q } \rho ( {x-x'})\mathrm{d}x\mathrm{d}x'}\right) ^i}\right) , \end{aligned}$$

(16)

which finalizes the proof.