# Geostatistical Modelling of Cyclic and Rhythmic Facies Architectures

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## Abstract

A Pluri-Gaussian method is developed for facies variables in three dimensions to model vertical cyclicity related to facies ordering and rhythmicity. Cyclicity is generally characterised by shallowing-upward or deepening-upward sequences and rhythmicity by the repetition of facies at constant intervals along sequences. Both of these aspects are commonly observed in shallow-marine carbonate successions, especially in the vertical direction. A grid-free spectral simulation approach is developed, with a separable covariance allowing a dampened hole-effect to capture rhythmicity in the vertical direction and a different covariance in the lateral plane along strata, as in space-time models. In addition, facies ordering is created by using a spatial shift between two latent Gaussian functions in the Pluri-Gaussian approach. Rapid conditioning to data is performed via Gibbs sampling and kriging using the screening properties of separable covariances. The resulting facies transiograms can show complex patterns of cyclicity and rhythmicity. Finally, a three dimensional case study of shallow-marine carbonate deposits at outcrop shows the applicability of the modelling method.

## Keywords

Pluri-Gaussian Asymmetric facies ordering Hole-effect Separable Transiogram Carbonate## 1 Introduction

Cyclicity is defined as a characteristic facies ordering in vertical successions (Wilkinson et al. 1997; Burgess 2016). The characterisation of cyclicity needs to be addressed statistically (Wilkinson et al. 1997) in order to apprehend the variability of the resulting facies patterns and to reproduce them in earth models. Facies cycles show preferential transitions between successive facies, such that one facies tends to be observed on top of another facies. This is also called asymmetry (Carle and Fogg 1996), because the transitions between facies differ between the upward and downward directions. For example, shallow-marine carbonate rocks at outcrops (Strasser 1988; Goldhammer et al. 1990) and in subsurface reservoirs (Lindsay et al. 2006) are typically characterised by facies cycles that record upward shallowing (regression) and that consist of subtidal facies overlain by intertidal facies then by supratidal facies. The facies succession that records upward deepening is commonly incomplete or absent, due to non-deposition or erosion, such that supratidal facies are directly overlain by subtidal facies, which mark the base of a new cycle. Such sequences are illustrated in Fig. 1c, d.

Classical geostatistical methods such as sequential indicator simulation (Alabert 1989) and object-based methods (Deutsch and Tran 2002) do not in their traditional form reproduce this asymmetric facies ordering. Multipoint statistics (Strebelle 2002) should be able in theory to reproduce cyclicity, but in practice, it is not easy to obtain a three-dimensional training image showing cyclicity. The representation of asymmetry and facies ordering is, however, straightforward in one dimension with Markov chains (Carle and Fogg 1996; Parks et al. 2000; Li 2007; Purkis et al. 2012) or renewal processes (Matheron 1968), which are based on probabilities of transition between facies, but are difficult to generalise to two or three dimensions.

Rhythmicity is another important aspect observed in vertical facies successions. It is characterised by the repetition of a facies at a constant interval along a sequence, a feature that has commonly been used to interpret periodic processes of deposition (e.g., via analysis of Fischer plots) (Read and Goldhammer 1988). Note that periodic processes (i.e., repetitive processes when looking at a time series) can either result in rhythmic sequences (when sedimentation rates are similar from cycle to cycle) or non-rhythmic sequences (if sedimentation rates change between cycles). Because the space domain is considered here, we use the term rhythmicity rather than periodicity. Rhythmic stacking of facies cycles has been observed in many shallow-marine carbonate successions (Goldhammer et al. 1993; Egenhoff et al. 1999; Lindsay et al. 2006). This aspect is also shown by the red facies in Fig. 1b, d, in which thicknesses between different beds of the same facies are constant.

Rhythmicity can be quantified by geostatistical tools, such as the variogram and the transiogram, which show oscillations or dampened oscillations called hole-effects (Pyrcz and Deutsch 2014). By looking at the probability density function (pdf) of facies thicknesses, Ma and Jones (2001) show that, as the coefficient of variation of this pdf decreases, the hole-effect becomes more pronounced. This observation is in agreement with the previous remark that rhythmicity is associated with low thickness variability of vertically stacked facies cycles. This also explains why Markov processes cannot create hole-effect transiogram models (Dubrule 2017), as the corresponding thickness pdf is exponential (coefficient of variation equal to one). On the other hand, some renewal processes may be able to create dampened hole-effect transiograms, because they offer the possibility to choose a thickness pdf with a lower coefficient of variation (Matheron 1968). However, the transiograms derived from renewal processes are not always known analytically, and are thus difficult to fit to the observed rhythmicity.

Truncated (Pluri-)Gaussian methods have been successfully used to create facies models (Armstrong et al. 2011), and they have been applied to shallow-marine carbonate reservoirs (Doligez et al. 2011; Amour et al. 2012; Le Blévec et al. 2017). The contacts between facies are defined by the truncation rule applied to a random Gaussian function, which provides control on facies juxtapositions. However, in its traditional form the method does not incorporate cyclicity and rhythmicity. Le Blévec et al. (2017) have extended the Pluri-Gaussian method to the modelling of facies asymmetry in vertical successions. They produce asymmetric transition probabilities between facies by introducing a shift in the correlation of two random Gaussian functions as suggested by Armstrong et al. (2011). This is similar to the approaches of Langlais et al. (2008) or Renard and Beucher (2012), but with more flexibility in the resulting facies transiograms. However, the use of this approach to model cyclicity and rhythmicity has not yet been investigated. Pluri-Gaussian Simulations (PGS) enable the use of hole-effect models (Beucher and Renard 2016) and may lead to hole-effect facies transiograms that could model rhythmicity.

Although cyclicity and rhythmicity are common features of vertical facies successions, they may have a variable expression laterally, depending on the formative depositional processes and controls. Laterally extensive facies in shallow-marine carbonate strata are generally attributed to external (allogenic) controls that operated over an entire carbonate platform or shelf, such as relative sea-level variations (Goldhammer et al. 1990). Facies of limited lateral extent may be attributed to the nucleation, vertical build-up and lateral shifting of tidal flat islands across a carbonate platform or shelf (Pratt and James 1986). This mechanism is internal to the dynamics of the carbonate platform depositional system (autogenic) and may generate both vertical and horizontal asymmetry in the stacking of facies if the tidal-flat-island deposits obey Walther’s Law (Burgess et al. 2001; Le Blévec et al. 2016). Cyclic and rhythmic facies successions can also be overprinted by diagenetic facies after deposition; for example, hydrothermal dolomite bodies associated with faults and igneous intrusions are observed to cut across shallow-marine carbonate platform deposits characterised by rhythmic facies cycles (Jacquemyn et al. 2015).

In order to model cross-sections and volumes that exhibit cyclic and rhythmic vertical facies successions but different lateral facies patterns, it is necessary to use different vertical and lateral covariance models. This is possible via the use of separable anisotropic models (Chiles and Delfiner 2012), although such models have rarely been used for facies modelling (Matheron et al. 1988).

The aim of this paper is to extend the PGS approach of Le Blévec et al. (2017) to model facies cyclicity and rhythmicity in the vertical direction, and a range of appropriate lateral facies patterns using space-time (lateral-vertical) separable covariance models. After presenting the main definitions, the three key aspects of the modelling method and their impact on the transiograms are presented: cyclicity, rhythmicity, separability. A new method for simulating the resulting complex facies architectures is then presented, firstly for unconditional simulations and then for simulations conditioned to data. Finally, the method is applied to a case study from the Triassic Latemar carbonate platform (northern Italy), which has been widely interpreted to show cyclicity and rhythmicity.

## 2 Definitions

### 2.1 Geostatistical Quantification with Transiograms

*I*(

*x*). If the facies

*i*is present at a spatial location

*x*, \(I_i(x) = 1\) and if not, \(I_i(x) = 0\). In the stationary case, the probability of having a facies at a location

*x*is equal to the first statistical moment or proportion

*i*and

*j*. With the stationary assumption, it is assumed that the covariance depends only on the vector

*h*separating two locations (Chiles and Delfiner 2012). This paper uses the non-centered indicator covariance

*i*to facies

*j*along a certain vector

*h*

*i*along this direction (Carle and Fogg 1996). The tangent at the origin (at right) of the cross-transiogram is called the transition rate

*h*. According to Carle and Fogg (1996) (under the stationary assumption) if a facies

*j*tends to follow a facies

*i*in the direction

*h*, rather than preceding it, then

*i*, the probability that it precedes

*j*in the direction

*h*)

In the next section, it is shown how a geostatistical simulation method, the truncated Gaussian simulation, relates to these quantities.

### 2.2 Truncated Gaussian Simulations

*Z*(

*x*) is a bi-Gaussian random vector, with

*Z*(

*x*) and \(Z(x+h)\) correlated to each other according to the non-centered covariance function

*x*from the value of the random variables

*Z*(

*x*). For instance, the truncation rule with the case of only two facies 1 and 2 and one Gaussian function (TGS) controls the indicator functions

*q*is the threshold of the truncation rule. It is possible to relate every moment of the facies field mathematically to those of the Gaussian function. According to Eq. (12a) the proportion of facies one (first order moment) is

*g*(

*x*) the standardised Gaussian pdf. The cross-transiogram (second order moment) between facies one and two is

Notations for the truncated Gaussian model

\(p_i\) | Facies |

\(c_{ij}(h)\) | Non-centered covariance between facies |

\(t_{ij}(h)\) | Transiogram from facies |

\(T_{ij}\) | Transition rate from facies |

\(r_{ij}\) | Embedded transition rate from facies |

\(\rho (h)\) | Covariance of the latent standardised Gaussian function |

\(q_1\), \(q_2\) | Thresholds of the Gaussian function \(Z_1(x)\) and \(Z_2(x)\) |

\(g_{\rho (h)}\) | Standardised bi-Gaussian density with correlation \(\rho (h)\) |

### 2.3 Understanding Cyclicity and Rhythmicity with Transiograms

Rhythmicity is defined by the repetition of a facies at constant intervals. This is usually observed on experimental transiograms that show dampened hole-effects (Journel and Froidevaux 1982; Johnson and Dreiss 1989; Ma et al. 2009). Rhythmicity cannot be quantified by embedded transition rates as they are independent of facies thicknesses. Rhythmicity can first be understood when studying two facies. If those facies have constant thicknesses, the auto-transiogram varies with a constant wavelength that is equal to the sum of the two facies thicknesses (Jones and Ma 2001). This is similar with more facies as we can still regard this as the succession of two facies, the one of interest and its complement. This is interesting to consider in combination with cyclicity because the resulting sequences show a constant cycle thickness (Fig. 1d). Thicknesses along sequences are not usually constant but can show low variability which results in non-perfect rhythmicity and dampened hole-effects. The method developed here models cyclic and rhythmic sequences, quantified by transiograms.

## 3 The Cyclical Pluri-Gaussian Approach

In this section, the classical PGS is extended to render cyclicity (or asymmetry) (Sect. 3.1), rhythmicity (Sect. 3.2) and separable anisotropy (Sect. 3.3). The first two sections (Sects. 3.1, 3.2) present results in one dimension and the last section (Sect. 3.3) extends them to three dimensions.

### 3.1 Modelling Asymmetrical Facies Juxtapositions in Vertical Successions

It is clear in the realisation shown in Fig. 3 that the facies are statistically organised in shallowing-upward cycles as highlighted by the low tangent at the origin of the cross transiogram from intertidal (orange) facies to subtidal (red) facies upwards (Fig. 3c). Indeed, this transition is absent along the vertical section while the opposite transition (from subtidal facies to intertidal facies) occurs six times. However, the succession is not perfectly cyclic (Fig. 3e) because of a low, but non-null probability of the subtidal (red) facies to transition upwards directly to the supratidal (white) facies. A high variation of facies thicknesses is also noted, resulting in a non-rhythmic sequence.

### 3.2 Modelling Rhythmicity in Vertical Successions

*r*and

*b*. If the scale factor

*r*tends to infinity, the model is the cosine function. According to Eq. (15) the resulting hole-effect transiogram model is

Figure 5 clearly shows the effect of rhythmicity on the transiograms and the corresponding realisation. The facies cycles are repeated in the vertical succession (Fig. 5e) with a rhythmicity controlled by the latent Gaussian functions (Fig. 5f). Asymmetry in facies stacking is also added to create a cyclical vertical succession. After developing covariance models in one dimension, it is necessary to expand these models into two and three dimensions while incorporating anisotropy.

### 3.3 Modelling Facies Distributions in Two and Three Dimensions with Separable Anisotropy

## 4 Conditional Simulation of the Cyclical Pluri-Gaussian Model

A simple method is here presented to simulate the latent Gaussian functions with covariance presented in Sect. 3 and to condition them to the facies observed along the wells.

### 4.1 Unconditional Simulation

*N*tends to infinity (central limit theorem). Based on the knowledge of

*x*, the individual value of

*Z*(

*x*) at each location

*x*can be simulated, which enables the algorithm to be coded in parallel and to be grid-free.

*r*, skewness and location 0 (Chiles and Delfiner 2012). Therefore, to simulate this covariance, one can use Algorithm 1 by replacing the Gaussian distributions with the stable distributions. These simulations are unconditional, and they must now be conditioned to facies data observed at specific locations in the simulated volume.

### 4.2 Conditioning the Gaussian Simulation to Facies Data

- 1.
Unconditional uncorrelated simulations \(Y_1^u(x)\) and \(Y_2^u(x)\) are performed over the domain as in Sect. 4.1.

- 2.
Joint local conditional simulation of \(Z_1(x_i)\) and \(Z_2(x_i)\) is carried out only at data locations \(x_i\) using Gibbs sampling such that Eq. (34) is respected (Freulon and de Fouquet 1993).

- 3.
Back-transform of correlated \(Z_1(x_i)\) and \(Z_2(x_i)\) into uncorrelated \(Y_1(x_i)\) and \(Y_2(x_i)\) at data location according to Eq. (22).

- 4.Two separate simple kriging \(R_1(x)\) and \(R_2(x)\) based on the mismatch at data locations$$\begin{aligned} R_1(x_i)= & {} Y_1(x_i)-Y_1^u(x_i), \end{aligned}$$(35a)are performed (Chiles and Delfiner 2012)$$\begin{aligned} R_2(x_i)= & {} Y_2(x_i)-Y_2^u(x_i), \end{aligned}$$(35b)$$\begin{aligned} R_1(x)= & {} \sum _i \lambda _i^1(x) \ R_1(x_i), \end{aligned}$$(36a)with \(\lambda _i^1(x)\) and \(\lambda _i^2(x)\) the kriging weights (Chiles and Delfiner 2012). The conditional uncorrelated Gaussian random functions \(Y_1\) and \(Y_2\) at every location$$\begin{aligned} R_2(x)= & {} \sum _i \lambda _i^2(x) \ R_2(x_i), \end{aligned}$$(36b)
*x*are finally obtained by$$\begin{aligned} Y_1(x)= & {} Y_1^u(x) + R_1(x), \end{aligned}$$(37a)$$\begin{aligned} Y_2(x)= & {} Y_2^u(x) + R_2(x). \end{aligned}$$(37b) - 5.
Transform conditioned \(Y_1\) and \(Y_2\) into conditioned and correlated \(Z_1\) and \(Z_2\) according to Eq. (22).

With a separable covariance model such as that of Eq. (26), the weights \(\lambda _i(x)\) associated with data located on different lateral planes from that of the estimated location *x* are equal to zero (Chiles and Delfiner 2012). This means that the estimation at a given location only depends on the data at the same horizontal level, and the number of kriging weights, therefore, equals the number of wells intersecting this horizontal level, as shown in Fig. 11. Assuming all wells are vertical and have the same length, the number of weights for every kriged point is, therefore, simply the number of wells. If this is not the case, for instance because some vertical wells do not penetrate a particular level, it is convenient to extend them artificially by an unconditional simulation with Gibbs sampling at step (ii) (Fig. 11), so that the geometrical configuration of the data points remains the same at all levels. Therefore, the weights are the same for every horizontal plane and the dual form of two-dimensional kriging may be used, in which the data covariance matrix is inverted only once (Dubrule 1983; Chiles and Delfiner 2012). This enables rapid and efficient kriging. An example of conditional simulation with this method is given in the following section.

## 5 Case Study: The Latemar Carbonate Platform, Northern Italy

### 5.1 Qualitative and Quantitative Study of the Case-Study Dataset

As illustrated by Fig. 12, the eight vertical logs show a high number of facies transitions. The asymmetry is clear as the subtidal facies (red) tends to be on top of the supratidal (white) facies. However, complete upward-shallowing facies cycles, containing subtidal, intertidal and supratidal deposits, occur only 24 times, while there are 56 incomplete cycles in the eight logs, which means that the sequences have some cyclic features. The subtidal facies appears to show regular spacing between beds within some wells (Fig. 12, in logs N8 and N16), which would suggest a rhythmicity of this facies. This is not the case for the intertidal and supratidal facies, which show very different spacings between the beds (Fig. 12).

All this information can be verified in the experimental transiograms computed on the logs (Fig. 13, grey points). A dampened hole-effect is observable on the auto-transiogram of the subtidal deposits (Fig. 13a). The tangent at the origin of the cross-transiogram of intertidal deposits overlain by subtidal deposits (Fig. 13) is low, showing that this transition is rare. More precisely, \(T_{12}=2.06\) and \(T_{21}=0.19\), which means according to Eq. (7) that the intertidal facies is four times more likely to overlie the subtidal facies than to underlie it.

### 5.2 Inference of the Co-regionalisation Model

The parameters of the Pluri-Gaussian model are determined using the procedure described in Le Blévec et al. (2017). Only the auto- and cross-transiograms of two of the three facies are necessary to determine the parameters of the model, as the transiograms for the third facies are derived from those of the two other facies. A trial and error procedure is here chosen for determining the parameters, as this gives the possibility to incorporate conceptual knowledge. For instance, as it is known that rhythmicity and asymmetry occur, it is important to incorporate these features during the model construction, which would be difficult with an automatic fitting procedure.

Parameters of the PGS for the Latemar platform

\((q_1,q_2)\) | \((-\,0.65,0.41)\) | Thresholds for the two Gaussian functions |

\(\rho \) | \(-\,0.6\) | Correlation between Gaussian functions |

\(\alpha \) | 0.1 m | Vertical shift between Gaussian functions |

\((r_{1_x},r_{1_y},r_{1_z})\) | (500 m, 500 m, 0.8 m) | Scale factors of the first Gaussian covariance |

\((r_{2_x},r_{2_y},r_{2_z})\) | (500 m, 500 m, 0.4 m) | Scale factors of the second Gaussian covariance |

\((b_{1},b_{2})\) | \((5~\hbox {m}^{-1}, 0~\hbox {m}^{-1})\) | Vertical frequencies of the two covariances |

### 5.3 Simulation Results

The cell sizes of the grid are 0.1 m vertically and 10 m laterally in both north and east directions, which gives approximately 800,000 cells. The simulation takes approximately one minute to run with a standard Intel processor i7, but it could be much faster by performing the unconditional simulation with the spectral method in parallel, which is the longest computational step. The conditioning by kriging every surface independently, as described in Sect. 4.2, is almost instantaneous. One realisation of the field is shown in Fig. 14.

The visual aspect of the simulation is granular due to the exponential covariance model used in the lateral plane. Visually, the subtidal (red) facies tends to lie on top of the supratidal (white) facies and rhythmicity is confirmed by the regular thickness between two subtidal (red) facies bodies. The transiograms of the wells and the transiograms computed on one realisation are compared in Fig. 13. The sills derived from simulation (black diamonds in Fig. 13) are accurate, which means that the facies proportions match those in the wells. The tangent at the origin of the auto-transiograms is appropriately fitted, which means the average thicknesses of different facies bodies honours the well data. The tangent at the origin of the cross-transiograms matches the transition rates of the wells (\(T_{12}=2.1\) and \(T_{21}=0.19\)), which means the asymmetry and cyclicity are respected. The hole-effect in the realisation transiogram is less pronounced than the one used for the theoretical model, but is closer to the one observed at the wells (grey circles in Fig. 13). This might be due to the conditioning of the realisation to the wells, where the experimental transiograms show less pronounced oscillations (black lines in Fig. 13).

## 6 Discussion

In order to illustrate the method, standard parameters have been chosen so far for both the synthetic examples and the case study. However, some parameters such as the truncation rule, the covariance model, or the shift can be changed to adjust to different geological environments.

### 6.1 The Truncation Rule

#### 6.1.1 Choosing the Truncation Rule

The same truncation rule has been applied through the paper (Fig. 2) with the subtidal facies defined by the first Gaussian function and the two others by both Gaussian functions. This has implications for the geometries of individual facies bodies and for facies relationships. The facies defined by the first Gaussian function erodes the two other facies (e.g., Fig. 8e), which means that bodies of these facies can have very different geometries from those of the other facies. This behavior can be reduced by increasing the correlation \(\rho \) between the two Gaussian functions \(Z_1(x)\) and \(Z_2(x)\).

The truncation rule affects not only the facies transiograms, but also higher order statistical moments, which can have an impact on connectivity (Beucher and Renard 2016). However, these moments are not known analytically and so are difficult to use for defining the truncation rule. It is recommended that the earth modeller tries different truncation rules and inspects the visual aspect of simulations so that it matches with his or her conceptual knowledge.

#### 6.1.2 Adapting the Truncation Rule for More Facies

In this paper, only three facies have been used to illustrate the method. Two methods for generalising the truncation rule to more facies are found in the literature: either the truncation is made more complex (Galli et al. 2006) or the number of Gaussian functions is increased (Maleki et al. 2016). The first method is probably too limited to represent a large number of transition rates between facies, while the second should be able to model all transition rates (but the number of parameters would be very high). The choice between the two methods should depend on the case study and further work on this topic is required. It should also be noted that some methods that create automatic truncation rules have been developed (Deutsch and Deutsch 2014; Astrakova et al. 2015). It would be interesting to generalise these methods by incorporating the shift between the Gaussian functions in order to match asymmetric transition probabilities.

### 6.2 Elaborating More Complex Hole-Effect Models

The vertical hole-effect model used in this paper is made of two parameters \(r_z\) and *b* (Eq. 24), which provides some flexibility to match observed rhythmicity. However, the case study shows that the observed transiograms can be even more complex (Fig. 13) and two parameters might not be sufficient to represent them. The covariance model could be modified to incorporate more than one structure (Chiles and Delfiner 2012). For instance a Gaussian covariance or a cosine covariance can be added to the Gaussian cosine model (Eq. 24).

### 6.3 Walther’s Law

The method developed in this paper models cyclicity only in the vertical direction, which is consistent with observations of most outcrop and subsurface data. However, Walther’s Law suggests that the transitions between facies should be equivalent laterally and vertically if no erosion is present (Middleton 1973). This means that the facies ordering could be similar and the transition rates proportional as in Markov chains methods (Doveton 1994; Purkis et al. 2012). Thus, asymmetry could also be observed laterally. Since the shift is actually a three-dimensional vector, it is possible to model such patterns with the presented method by defining a non-vertical vectorial shift between the Gaussian functions (Eq. 22), such that the asymmetry is also lateral (Le Blévec et al. 2016).

This choice will depend on the depositional environments, processes, controls, and scale to be modelled. These aspects are typically interpreted with reference to an underlying conceptual model, such as those for allocyclically and autocyclically generated facies cycles in peritidal carbonate strata (Pratt and James 1986; Goldhammer et al. 1990). The facies architectures to be modelled are also scale dependent. Environments of deposition generally have large lateral extents (1–10 km), such that few lateral transitions between them are observed at reservoir (1–10 km) and inter-well (\(<1\) km) scales (Sena and John 2013), which limits the expression of lateral ordering of depositional environments. At smaller scales, the lateral transitions between lithofacies within depositional environments (or facies associations) may be different from the vertical transitions (Hönig and John 2015) because of erosion or lateral changes in palaeotopography. The resulting lithofacies distributions may be highly variable, potentially reflecting a facies migration that is well-ordered and obeys Walther’s Law as one end member (Obermaier et al. 2015) or more complex and less ordered facies mosaics as the opposite end member (Wilkinson et al. 1997). The choice of appropriate conceptual model at the scale of depositional environments (facies association) or lithofacies must be made by the earth modeler in collaboration with the geologist, and then used to govern the selection of parameters of the model.

## 7 Conclusion

While cyclicity and rhythmicity are commonly observed in facies architectures, few existing geostatistical algorithms can model both patterns in an efficient manner. By addressing this issue, the method developed here is useful for modelling carbonate or shallow-marine clastic reservoirs that contain such cyclical facies successions. Broadly speaking, cyclicity and rhythmicity are quantified by facies transiograms that are computed from data (e.g., vertical facies successions) and fitted with an advanced truncated Pluri-Gaussian model for performing three dimensional simulations.

The model used for the latent Gaussian functions is the linear model of co-regionalisation with a spatial shift, which creates the asymmetric cycles. The covariance of the Gaussian functions presents a dampened hole-effect, which captures the rhythmicity. As this hole-effect is generally observed only in the vertical direction, a separable covariance model, which is the product of a lateral and a vertical covariance is used so that no rhythmicity is modelled laterally along the stratigraphy. The space-time separable covariance is simulated readily by the continuous spectral method. The numerical properties of separable covariance allows rapid and efficient conditioning to data via kriging of every horizontal surface independently. The procedure has been applied successfully to model a carbonate platform environment that shows cyclicity and rhythmicity in facies architecture.

## Notes

### Acknowledgements

The authors would like to thank Total for seconding Olivier Dubrule at Imperial College and the Department of Earth Science and Engineering at Imperial College for a scholarship to Thomas Le Blevec.

## References

- Alabert F (1989) Non-Gaussian data expansion in the earth sciences. Terra Nova 1(2):123–134CrossRefGoogle Scholar
- Amour F, Mutti M, Christ N, Immenhauser A, Agar SM, Benson GS, Tomás S, Alway R, Kabiri L (2012) Capturing and modelling metre-scale spatial facies heterogeneity in a Jurassic ramp setting (Central High Atlas, Morocco). Sedimentology 59(4):1158–1189CrossRefGoogle Scholar
- Armstrong M, Galli A, Beucher H, Loc’h G, Renard D, Doligez B, Eschard R, Geffroy F (2011) Plurigaussian simulations in geosciences. Springer, BerlinCrossRefGoogle Scholar
- Astrakova A, Oliver DS, Lantuéjoul C (2015) Truncation map estimation based on bivariate probabilities and validation for the truncated Pluri-Gaussian model. arXiv preprint arXiv:1508.01090
- Beucher H, Renard D (2016) Truncated gaussian and derived methods. CR Geosci 348(7):510–519CrossRefGoogle Scholar
- Burgess P, Wright V, Emery D (2001) Numerical forward modelling of peritidal carbonate parasequence development: implications for outcrop interpretation. Basin Res 13(1):1–16CrossRefGoogle Scholar
- Burgess PM (2016) Identifying ordered strata: evidence, methods, and meaning. J Sediment Res 86(3):148–167CrossRefGoogle Scholar
- Carle SF, Fogg GE (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476CrossRefGoogle Scholar
- Carle SF, Fogg GE (1997) Modeling spatial variability with one and multidimensional continuous-lag Markov chains. Math Geol 29(7):891–918CrossRefGoogle Scholar
- Chiles JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, vol 497. Wiley, HobokenCrossRefGoogle Scholar
- Deutsch C, Tran T (2002) FLUVSIM: a program for object-based stochastic modeling of fluvial depositional systems. Comput Geosci 28(4):525–535CrossRefGoogle Scholar
- Deutsch JL, Deutsch CV (2014) A multidimensional scaling approach to enforce reproduction of transition probabilities in truncated plurigaussian simulation. Stoch Env Res Risk Assess 28(3):707–716CrossRefGoogle Scholar
- Doligez B, Hamon Y, Barbier M, Nader F, Lerat O, Beucher H (2011) Advanced workflows for joint modelling of sedimentological facies and diagenetic properties. Impact on reservoir quality. In: SPE annual technical conference and exhibition. Society of Petroleum EngineersGoogle Scholar
- Doveton JH (1994) Theory and applications of vertical variability measures from Markov chain analysis. AAPG special volumesGoogle Scholar
- Dubrule O (1983) Cross validation of kriging in a unique neighborhood. J Int Assoc Math Geol 15(6):687–699CrossRefGoogle Scholar
- Dubrule O (2017) Indicator variogram models: Do we have much choice? Math Geosci 49(4):441–465CrossRefGoogle Scholar
- Egenhoff SO, Peterhänsel A, Bechstädt T, Zühlke R, Grötsch J (1999) Facies architecture of an isolated carbonate platform: tracing the cycles of the Latemar (Middle Triassic, northern Italy). Sedimentology 46(5):893–912CrossRefGoogle Scholar
- Emery X, Lantuéjoul C (2006) TBSIM: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628CrossRefGoogle Scholar
- Emery X, Arroyo D, Peláez M (2014) Simulating large Gaussian random vectors subject to inequality constraints by Gibbs sampling. Math Geosci 46(3):265–283CrossRefGoogle Scholar
- Freulon X, de Fouquet C (1993) Conditioning a Gaussian model with inequalities. In: Geostatistics Troia 92. Springer, pp 201–212Google Scholar
- Galli A, Le Loch G, Geffroy F, Eschard R (2006) An application of the truncated Pluri-Gaussian method for modeling geology. AAPG special volumesGoogle Scholar
- Genz A (1992) Numerical computation of multivariate normal probabilities. J Comput Gr Stat 1(2):141–149Google Scholar
- Goldhammer R, Dunn P, Hardie L (1990) Depositional cycles, composite sea-level changes, cycle stacking patterns, and the hierarchy of stratigraphic forcing: examples from Alpine triassic platform carbonates. Geol Soc Am Bull 102(5):535–562CrossRefGoogle Scholar
- Goldhammer R, Lehmann P, Dunn P (1993) The origin of high-frequency platform carbonate cycles and third-order sequences (lower Ordovician El Paso Gp, west Texas): constraints from outcrop data and stratigraphic modeling. J Sediment Res 63(3):318–359Google Scholar
- Hinnov LA, Goldhammer RK (1991) Spectral analysis of the Middle Triassic Latemar limestone. J Sediment Res 61(7):1173–1193Google Scholar
- Hönig MR, John CM (2015) Sedimentological and isotopic heterogeneities within a Jurassic carbonate ramp (UAE) and implications for reservoirs in the Middle East. Mar Pet Geol 68:240–257CrossRefGoogle Scholar
- Jacquemyn C, Huysmans M, Hunt D, Casini G, Swennen R (2015) Multi-scale three-dimensional distribution of fracture-and igneous intrusion-controlled hydrothermal dolomite from digital outcrop model, Latemar platform, Dolomites, northern Italy. AAPG Bull 99(5):957–984CrossRefGoogle Scholar
- Johnson NM, Dreiss SJ (1989) Hydrostratigraphic interpretation using indicator geostatistics. Water Resour Res 25(12):2501–2510CrossRefGoogle Scholar
- Jones TA, Ma YZ (2001) Teacher’s aide: geologic characteristics of hole-effect variograms calculated from lithology-indicator variables. Math Geol 33(5):615–629CrossRefGoogle Scholar
- Journel A, Froidevaux R (1982) Anisotropic hole-effect modeling. Math Geol 14(3):217–239CrossRefGoogle Scholar
- Langlais V, Beucher H, Renard D (2008) In the shade of the truncated Gaussian simulation. In: Proceedings of the eighth international geostatistics congress, Santiago, pp 799–808Google Scholar
- Lantuéjoul C (1994) Non conditional simulation of stationary isotropic multi-Gaussian random functions. In: Geostatistical simulations. Springer, pp 147–177Google Scholar
- Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, BerlinCrossRefGoogle Scholar
- Le Blévec T, Dubrule O, John CM, Hampson GJ (2016) Building more realistic 3-D facies indicator models. In: SPE (ed) International Petroleum Technology conference, BangkokGoogle Scholar
- Le Blévec T, Dubrule O, John CM, Hampson GJ (2017) Modelling asymmetrical facies successions using Pluri-Gaussian simulations. In: Geostatistics valencia 2016. Springer, pp 59–75Google Scholar
- Li W (2007) Markov chain random fields for estimation of categorical variables. Math Geol 39(3):321–335CrossRefGoogle Scholar
- Lindsay RF, Cantrell DL, Hughes GW, Keith TH, Mueller HW III, Russell SD (2006) Ghawar Arab-D reservoir: widespread porosity in shoaling-upward carbonate cycles, Saudi Arabia. AAPG special volumesGoogle Scholar
- Ma YZ, Jones TA (2001) Teacher’s aide: modeling hole-effect variograms of lithology-indicator variables. Math Geol 33(5):631–648CrossRefGoogle Scholar
- Ma YZ, Seto A, Gomez E (2009) Depositional facies analysis and modeling of the Judy Creek reef complex of the Upper Devonian Swan Hills, Alberta, Canada. AAPG Bull 93(9):1235–1256CrossRefGoogle Scholar
- Maleki M, Emery X, Cáceres A, Ribeiro D, Cunha E (2016) Quantifying the uncertainty in the spatial layout of rock type domains in an iron ore deposit. Comput Geosci 20(5):1013–1028CrossRefGoogle Scholar
- Matheron G (1968) Processus de renouvellement purs. Course document Ecole des Mines de ParisGoogle Scholar
- Matheron G, Beucher H, de Fouquet C, Galli A, Ravenne C (1988) Simulation conditionnelle à trois faciès dans une falaise de la formation du brent. Sci Terre Sér Inform Géol 28:213–249Google Scholar
- Middleton GV (1973) Johannes Walther’s law of the correlation of facies. Geol Soc Am Bull 84(3):979–988CrossRefGoogle Scholar
- Obermaier M, Ritzmann N, Aigner T (2015) Multi-level stratigraphic heterogeneities in a Triassic shoal grainstone, Oman Mountains, Sultanate of Oman: layer-cake or shingles? GeoArabia 20(2):115–142Google Scholar
- Parks KP, Bentley LR, Crowe AS (2000) Capturing geological realism in stochastic simulations of rock systems with Markov statistics and simulated annealing. J Sediment Res 70(4):803–813CrossRefGoogle Scholar
- Peterhänsel A, Egenhoff SO (2008) Lateral variabilities of cycle stacking patterns in the Latemar, Triassic, Italian Dolomites. SEPM Spec Publ 89:217–229Google Scholar
- Pratt BR, James NP (1986) The St George Group (lower Ordovician) of western Newfoundland: tidal flat island model for carbonate sedimentation in shallow epeiric seas. Sedimentology 33(3):313–343CrossRefGoogle Scholar
- Preto N, Hinnov LA, Hardie LA, De Zanche V (2001) Middle Triassic orbital signature recorded in the shallow-marine Latemar carbonate buildup (Dolomites, Italy). Geology 29(12):1123–1126CrossRefGoogle Scholar
- Purkis S, Vlaswinkel B, Gracias N (2012) Vertical-to-lateral transitions among Cretaceous carbonate facies, a means to 3-D framework construction via Markov analysis. J Sediment Res 82(4):232–243CrossRefGoogle Scholar
- Pyrcz MJ, Deutsch CV (2014) Geostatistical reservoir modeling. Oxford University Press, OxfordGoogle Scholar
- Read J, Goldhammer R (1988) Use of Fischer plots to define third-order sea-level curves in Ordovician peritidal cyclic carbonates, Appalachians. Geology 16(10):895–899CrossRefGoogle Scholar
- Renard D, Beucher H (2012) 3D representations of a uranium roll-front deposit. Appl Earth Sci 121(2):84–88CrossRefGoogle Scholar
- Sena CM, John CM (2013) Impact of dynamic sedimentation on facies heterogeneities in lower cretaceous peritidal deposits of central east Oman. Sedimentology 60(5):1156–1183CrossRefGoogle Scholar
- Shinozuka M (1971) Simulation of multivariate and multidimensional random processes. J Acoust Soc Am 49(1B):357–368CrossRefGoogle Scholar
- Strasser A (1988) Shallowing-upward sequences in Purbeckian peritidal carbonates (lowermost Cretaceous, Swiss and French Jura Mountains). Sedimentology 35(3):369–383CrossRefGoogle Scholar
- Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21CrossRefGoogle Scholar
- Wackernagel H (2003) Multivariate geostatistics: an introduction with applications. Springer, BerlinCrossRefGoogle Scholar
- Wilkinson BH, Drummond CN, Rothman ED, Diedrich NW (1997) Stratal order in peritidal carbonate sequences. J Sediment Res 67(6):1068–1078Google Scholar

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