Modelling Asymmetrical Facies Successions Using Pluri-Gaussian Simulations

Abstract

An approach to model spatial asymmetrical relations between indicators is presented in a pluri-Gaussian framework. The underlying gaussian random functions are modelled using the linear model of co-regionalization, and a spatial shift is applied to them. Analytical relationships between the two underlying gaussian variograms and the indicator covariances are developed for a truncation rule with three facies and cut-off at 0. The application of this truncation rule demonstrates that the spatial shift on the underlying gaussian functions produces asymmetries in the modelled 1D facies sequences. For a general truncation rule, the indicator covariances can be computed numerically, and a sensitivity study shows that the spatial shift and the correlation coefficient between the gaussian functions provide flexibility to model the asymmetry between facies. Finally, a case study is presented of a Triassic vertical facies succession in the Latemar carbonate platform (Dolomites, Northern Italy) composed of shallowing-upward cycles. The model is flexible enough to capture the different transition probabilities between the environments of deposition and to generate realistic facies successions.

Appendix: Analytical Expression of the Triple Gaussian Integral

In a similar fashion as Kendall et al. (1994), we consider three correlated gaussian variates being in their respective intervals as a set of three dependent events. With the truncation rule displayed in Fig. 1 and thresholds that equal 0, facies 1 at location x and facies 2 at location x+h correspond to one variate being negative and two positive. The indicator covariance C ₁₂ (h) quantifies the probability of the intersection of these three events. The correlation matrix between the three gaussian variates is the following:

$$ {\displaystyle \sum }(h)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill {\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{h}\right)\hfill & \hfill \frac{\boldsymbol{\rho} {\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\left|\boldsymbol{h}+\boldsymbol{a}\right|\right)}{{\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{a}\right)}\hfill \\ {}\hfill {\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{h}\right)\hfill & \hfill 1\hfill & \hfill \boldsymbol{\rho} \hfill \\ {}\hfill \frac{\boldsymbol{\rho} {\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\left|\boldsymbol{h}+\boldsymbol{a}\right|\right)}{{\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{a}\right)}\hfill & \hfill \boldsymbol{\rho} \hfill & \hfill 1\hfill \end{array}\right) $$

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The probability can be written as a triple integral of the corresponding gaussian density g _Σ(h)(u,v,w):

$$ {C}_{12}(h)={\displaystyle {\int}_{-\infty}^0{\displaystyle {\int}_0^{+\infty }{\displaystyle {\int}_0^{+\infty }{g}_{\varSigma (h)}\left( u, v, w\right) dudvdw}}} $$

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Thanks to the gaussian integral symmetry property, the probability of intersection of the events is the complementary of the probability of their union (Kendall et al. 1994). Therefore, by definition of the union, the intersection of the three events can be expressed as a sum of the corresponding single and pair events and so the triple integral as a sum of the single integrals that equal to 0.5 and double integrals with their respective correlation coefficient:

$$ \begin{array}{l}{C}_{12}(h)\\ {}\kern1em =\frac{1}{2}(1-3*0.5+{\displaystyle {\int}_{-\infty}^0{\displaystyle {\int}_0^{+\infty }{g}_{{\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{h}\right)}\left(u,v\right) dudv}}+{\displaystyle {\int}_{-\infty}^0{\displaystyle {\int}_0^{+\infty }{g}_{\frac{\boldsymbol{\rho} {\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\left|\boldsymbol{h}+\boldsymbol{a}\right|\right)}{{\boldsymbol{\rho}}_{{\boldsymbol{Z}}_1}\left(\boldsymbol{a}\right)}}\left(u,v\right) dudv}}\kern2.3em +{\displaystyle {\int}_0^{+\infty }{\displaystyle {\int}_0^{+\infty }{g}_{\boldsymbol{\rho}}\left(u,v\right) dudv}})\end{array} $$

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Sheppard (1899) gives then the solution of the double integral that allows to obtain the final expression of the transition probability between facies 1 and 2 (Eq. 15):

$$ {\displaystyle {\int}_0^{+\infty }{\int}_0^{+\infty }}{g}_{\rho}\left( u, v\right) dudv=\frac{1}{2}-{\displaystyle {\int}_0^{+\infty }{\int}_0^{+\infty }}{g}_{\rho}\left( u, v\right) dudv=\frac{1}{4}+\frac{1}{2\pi} \arcsin \left(\rho \right) $$

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