Categorical facies modeling

Categorical facies modelling is the spatial assignment of discrete labels, shale/sand/conglomerate, channel/levee/floodplain, ore/waste/overburden, to every cell of a 2D or 3D grid, conditioned on hard data (wells, outcrops). Three families of methods dominate: multi-facies SISIM (extending §8.3), TRUNCATED GAUSSIAN SIMULATION (a transform-based alternative), and OBJECT-BASED modelling (placement of geological objects). Each has a sweet spot.

Multi-facies SISIM, generalised

For $K$ facies, define indicator fields $I_k(x) = \mathbb{1}[\text{facies}(x) = k]$, $k = 1, \dots, K$. At each unsampled cell $x_v$:

• Run simple indicator kriging on each facies indicator using the current conditioning set: $\hat{P}_k(x_v)$ for $k = 1, \dots, K$.
• Apply ORDER-RELATION CORRECTION: clip each $\hat{P}_k$ to $[0,1]$; renormalise so $\sum_k \hat{P}_k = 1$. The result is a valid PMF on the K facies.
• Draw a category from the cumulative-sum: pick $u \sim \text{Uniform}(0,1)$, select the $k$ such that $\sum_{j<k} \hat{P}$j \le u < \sum{j \le k} \hat{P}_j.
• Update the conditioning set with the drawn facies (one new indicator value per cell, propagating through ALL K indicator-kriging systems on downstream cells).

For each cell SISIM solves $K$ kriging systems, $K$ times the cost of single-indicator SISIM. Production runs limit the search neighbourhood (16-32 conditioning points within ~1.5 ranges) and use random multiple-grid paths.

Proportion trends: how geology enters cell-based modelling

Geological knowledge often imposes a non-stationary prior: shale-dominated near the top of a sequence, sand in the middle, conglomerate at the base; or proximal facies near a sediment source and distal facies further away. Encode this with a depth/lateral-dependent prior $\mathbf{p}(x) = (p_1(x), \dots, p_K(x))$. Then simple indicator kriging is applied to residuals:

so the local mean is the trend value, and kriging only models residuals from it. Trend specification is itself a modelling decision, from data (well-log proportions plotted vs depth and smoothed) or from geological interpretation (sequence-stratigraphic framework).

Truncated Gaussian Simulation (TGSIM)

An alternative cell-based method: simulate a SINGLE underlying multiGaussian field $Y(x)$ (e.g. via SGS, §7.3), then threshold it at $K-1$ cutoffs $t_1 < t_2 < \dots < t_{K-1}$ to produce facies:

Pros: ONE Gaussian field, ONE variogram model, automatic order-relation consistency, easy proportion control (set $t_k$ from desired global proportions via the standard-normal quantile). Cons: the cutoff scheme imposes an ARTIFICIAL ORDERING on the facies (facies $k$ is always "between" $k-1$ and $k+1$ in Y-space). Adjacent facies in the truncation order MUST be spatially adjacent, impossible for, say, sand bracketed by shale on both sides without going through an intermediate "transition" facies.

PLURIGAUSSIAN extends TGSIM to two correlated Gaussian fields with a 2D facies-region map, breaking the strict ordering constraint at the cost of more modelling decisions.

Object-based methods

For complex geological shapes (channels, levees, lobes, deltas) that two-point statistics cannot capture, OBJECT-BASED methods drop pixel-by-pixel simulation entirely. Geological objects with parameterised shapes (sinuous channel + levee, ellipsoidal sand body) are placed in the grid via Poisson processes with intensities controlled by facies proportions. Examples: fluvsim (channelised reservoirs), ellipsim, FLUMY.

Pros: geologically realistic shapes; explicit object hierarchies (channel-belt → channel → levee). Cons: HARD TO CONDITION to dense well data, placing objects to honour every well's facies often requires iterative retraction-replacement schemes that scale poorly. For dense data, cell-based methods (SISIM, TGSIM, or MPS in §9) are usually preferred.

Choosing a method

Try it

• Defaults: trend strength = 0.50, range = 5.0, 8 realisations. The TOP canvas shows 8 SISIM realisations as vertical columns (depth top-to-bottom). Three facies are colour-coded: gray = shale, gold = sand, dark-brown = conglomerate. Dotted lines mark the 6 hard-data wells. Note all realisations honour the wells.
• Increase trend strength to 1.00. Strong vertical gradient: shale dominates the top, sand the middle, conglomerate the bottom, even in cells far from hard data. The BOTTOM canvas shows P_sim(facies) at each depth as horizontal bars per facies; the dashed white line is the per-cell prior P_k(z). With strong trend, P_sim hugs the trend prior away from hard data.
• Set trend strength to 0.00 (stationary). The trend prior becomes a flat line at the global proportions (1/6 shale, 3/6 sand, 2/6 conglomerate from the 6 hard data). Now realisations are dominated by local conditioning + spatial continuity, not by a geological trend.
• Vary the range from 1.5 to 10.0. Short range → realisations look noisy with rapid facies switching; long range → thick, continuous facies bodies. Long range also makes hard-data influence reach further (smoother probability-bar shapes).
• Click Resample. All realisations change while hard data and probability bars remain consistent, the per-cell probabilities are model-determined, only the draws are stochastic.
• Compare globally: with N = 8 realisations the global facies proportions in the readout will be noisier than with N = 16 (or 32 if implemented), but should converge to the trend-weighted prior as N → ∞.

A reservoir team has 12 wells in a clastic sequence with vertical proportion curves showing shale 0.6 / sand 0.3 / conglomerate 0.1 at the top grading to 0.1 / 0.4 / 0.5 at the base, and they need 100 realisations for flow simulation. Which method, multi-facies SISIM with vertical trend, TGSIM, or object-based, would you propose and why? What would change your answer if there were only 4 wells?

What you now know

Categorical facies modelling has three main approaches. Multi-facies SISIM (cell-based, two-point) is the workhorse for moderate-density data and simple connectivity, easily extended with proportion trends. TGSIM (and plurigaussian) trade flexibility for algorithmic simplicity and automatic consistency, at the cost of artificial facies orderings. Object-based methods give realistic geological shapes but struggle with dense conditioning. Method choice is dictated by data density, geological complexity, and downstream use, flow simulation, mining selectivity, contaminant transport.

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References

• Matheron, G., Beucher, H., de Fouquet, C., Galli, A., Guerillot, D., Ravenne, C. (1987). "Conditional simulation of the geometry of fluvio-deltaic reservoirs." SPE 16753. (Original truncated-Gaussian framework.)
• Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford. (SISIM with proportion trends.)
• Deutsch, C.V., Journel, A.G. (1998). GSLIB, 2nd ed. Oxford. (sisim, tgsim, fluvsim.)
• Armstrong, M. et al. (2011). Plurigaussian Simulations in Geosciences, 2nd ed. Springer.
• Pyrcz, M.J., Deutsch, C.V. (2014). Geostatistical Reservoir Modeling, 2nd ed. Oxford. (Practitioner's comparison of facies-modelling methods.)