Stationarity: Variogram Inference, Reservoir Geomodelling, and Decision of Spatial Homogeneity
Stationarity is a form of homogeneity in a single statistical characteristic of a spatial variable, and it is the working assumption that makes geostatistical reservoir modelling possible. When a geomodeller maps porosity or permeability across a formation from a handful of well control points, the data are never dense enough to estimate a separate statistical distribution at every location. Stationarity resolves this by treating the property of interest as if its key statistics, the mean, the variance, and the spatial correlation structure, do not depend on absolute position within a chosen domain. Under this assumption, every measurement contributes to a single pooled description of how the property behaves, and that pooled description can then be applied everywhere inside the domain. In practical terms, stationarity is not a property of the rock; it is a decision the geoscientist makes about how to treat the data. The classic statement of this idea is that stationarity is a decision, not a hypothesis that can be proven or disproven from the data alone. Geostatisticians distinguish several grades of the assumption. Strict stationarity requires the full multivariate distribution to be invariant under translation, which is far stronger than anyone needs. Second-order stationarity, the workhorse for kriging, requires only that the expected value be constant and independent of location, and that the covariance between two points depend only on the separation vector between them, not on where the pair sits. The intrinsic hypothesis is weaker still: it requires only that the increments, the differences between paired values, have a constant mean of zero and a variance that depends only on separation. That last form is what allows an unbounded variogram to be used even when the variance of the property itself is not finite. Local stationarity, the form most relevant to heterogeneous reservoirs, occurs when two or more adjacent, locally homogeneous sub-regions yield similar values of the property of interest, so a moving search neighbourhood can be treated as stationary even when the field as a whole is not. In Western Canadian Sedimentary Basin geomodelling, the decision of stationarity governs everything downstream. A Montney siltstone interval modelled as one stationary unit will produce a smooth, single-population permeability field; the same data split at a recognised flooding surface into two stationary zones will reproduce a sharp contrast that the single-zone model erases. Getting this decision wrong is one of the most common and most expensive errors in reservoir characterisation, because the variogram, the kriging estimate, and every stochastic realisation built on top of them inherit the assumption silently.
Key Takeaways
- A decision, not a rock property: Stationarity is a modelling choice about how to pool data, not a measurable attribute of the formation. The geoscientist declares a domain over which the mean, variance, and spatial correlation are treated as position independent, then borrows statistical strength across all wells in that domain. Because it cannot be proven from sparse data, the decision must be defended geologically, by argument from depositional setting, not by a statistical test alone.
- Second-order versus intrinsic: Second-order stationarity requires a constant mean and a covariance that depends only on the lag vector between points, which yields a bounded variogram with a finite sill. The intrinsic hypothesis is weaker: it constrains only the increments, allowing an unbounded variogram and accommodating spatial trends that a strict covariance model cannot. Most WCSB porosity work assumes second-order; fracture intensity or structural depth often needs the intrinsic form.
- Local stationarity rescues heterogeneity: Real reservoirs are rarely globally stationary. Local stationarity treats a moving search neighbourhood as internally homogeneous even when the basin-wide field is not, so kriging can use nearby data without assuming the whole formation shares one mean. Adjacent, locally homogeneous samples yielding similar property values is the operational signature that the local assumption holds.
- Trends break the constant mean: A systematic drift in the property, deepening structure, a regional porosity gradient across the Pembina Cardium, or compaction with depth, violates the constant-mean requirement of second-order stationarity. The standard fix is to model the trend explicitly and apply geostatistics to the residuals, using kriging with a trend or universal kriging rather than forcing a stationary model onto non-stationary data.
- Zonation enforces the assumption: Splitting a reservoir into stratigraphic zones bounded by flooding surfaces, unconformities, or facies changes is how geomodellers make stationarity defensible. Each zone is modelled as an independent stationary population with its own variogram and histogram, preventing a high-permeability channel sand and a tight overbank shale from being averaged into one meaningless intermediate population.
Second-Order Stationarity and the Covariance Requirement
Second-order, or weak, stationarity rests on two conditions. First, the expected value of the regionalised variable is constant and independent of location, so E[Z(x)] equals a single mean for every point x in the domain. Second, the covariance between any two points exists and depends only on the separation vector h, written C(h), not on the individual positions. When both hold, the variogram is bounded and reaches a sill equal to the stationary variance, and the relationship gamma(h) equals C(0) minus C(h) links the two functions directly. This is the assumption embedded in ordinary kriging. In a Viking sandstone unit with no obvious areal trend, second-order stationarity is usually a reasonable default, letting a single variogram describe porosity correlation in every direction from every well.
The Intrinsic Hypothesis and Unbounded Variograms
The intrinsic hypothesis weakens the requirement to the increments alone. It asks that the expected difference between two values separated by h be zero, and that the variance of that difference, twice the variogram, depend only on h. Crucially it does not require the variance of Z itself to be finite, so the variogram may rise without ever reaching a sill. This matters for properties dominated by large-scale structure: top-of-formation depth across a deforming WCSB fairway, or cumulative thickness trends in the McMurray. A power-law or linear variogram model, valid under the intrinsic hypothesis but not under second-order stationarity, captures these continuously increasing dissimilarities without forcing a false sill onto the data.
Fast Facts
The phrase that stationarity "is a decision, not a hypothesis" traces to Andre Journel, who hammered the point through the 1980s at Stanford to stop engineers from running statistical tests for stationarity that cannot logically exist: with one realisation of the subsurface and no repeated sampling at a point, there is no frequentist test that can confirm the mean is constant. The honesty of geostatistics, he argued, lies in admitting the assumption is imposed by the modeller and must be justified by geology, not discovered in the numbers.
Related Terms
Stationarity is inseparable from the variogram, the function that quantifies spatial correlation and whose very inference assumes the property is stationary over the lags being measured. It underpins kriging, the estimation method that breaks down when the constant-mean assumption fails. It is measured along a Euclidian distance, the separation metric used to bin sample pairs into lags. And it interacts with anisotropy, because a field can be stationary yet still correlate more strongly along bedding than across it, requiring a directional variogram rather than abandonment of the stationary assumption.
Real-World WCSB Scenario: Zonation in a Cardium Geomodel
A Pembina Cardium operator building a static model for a CAD 4.2 million infill program initially treated the entire Cardium interval as one stationary unit. The pooled permeability histogram spanned four orders of magnitude, and the single variogram showed a nugget effect so large the kriged map was almost flat, useless for steering the 1,800 m laterals. A geomodeller re-examined the wireline logs and split the interval at the regional coarsening-upward boundary into a clean upper conglomerate and a tight lower sandstone, declaring each a separate stationary population.
With zonation, each zone produced a coherent variogram with a clear range near 1.1 km and a modest nugget. The reworked permeability field resolved the conglomerate fairway the wells had hinted at, and three of the five planned infills were repositioned. Post-drill, those three averaged 28 percent higher 90-day cumulative oil than the original locations, recovering the modelling cost many times over and confirming that the stationarity decision, not the algorithm, drove the result.