Additivity
Additivity is a mathematical property of geostatistical semivariogram models stating that any linear combination of valid variogram model functions with non-negative weights is itself a valid (positive semi-definite) variogram model. This property allows geostatisticians to construct complex spatial covariance structures by summing simpler, well-understood model components, a practice called nested variogram modelling. Each component of the nested model represents a distinct scale or range of spatial correlation in the variable being modelled, and the sum of all components represents the total spatial variability of the property. Additivity ensures that the resulting nested model is mathematically permissible as a variogram (meaning it can be used as the basis of a kriging or simulation system without producing impossible results such as negative kriging variances), which would not be guaranteed if an arbitrary mathematical function were chosen to fit the experimental variogram data. In reservoir modelling and formation evaluation, the additivity of semivariogram models underpins the construction of multi-scale geostatistical models of reservoir properties such as porosity, permeability, and net-to-gross ratio, where geological heterogeneity occurs simultaneously at multiple scales from millimetre laminae to kilometre-scale stratigraphic architecture.
Key Takeaways
- A semivariogram model is valid (positive semi-definite) if and only if it cannot produce a negative kriging variance for any set of data and estimation locations; mathematically, this requires that the model be a conditionally negative semi-definite function. The basic valid model types include: the nugget effect (gamma(h) = c_0 for all h greater than zero, representing uncorrelated variance at the scale of the measurement), the spherical model (linearly increasing gamma to a sill value at the range a, then flat), the exponential model (asymptotically approaching a sill at approximately three times the range parameter), the Gaussian model (parabolic near the origin, approaching a sill), and the power model (continuously increasing variance with distance, appropriate for non-stationary variables). Each of these models has been proven to be positive semi-definite; any non-negative linear combination of these (or other valid) models is therefore also positive semi-definite by the linearity of the property. A nested model written as gamma(h) = c_0 × nugget + c_1 × spherical(a_1) + c_2 × exponential(a_2) is valid as long as c_0, c_1, and c_2 are all greater than or equal to zero.
- The physical interpretation of a nested variogram is that the geological property being modelled has spatial correlation at multiple length scales simultaneously. For porosity in a fluvially deposited sandstone reservoir, a two-structure nested model might include a short-range spherical component (range 40 to 80 metres) representing within-channel bar spatial variation and a long-range exponential component (range 600 to 1,200 metres) representing the orientation and extent of channel belts. A nugget component represents measurement error and very short-scale variability below the resolution of the well spacing. The total sill (c_0 + c_1 + c_2) equals the global variance of the property (the variance you would compute from all values across the entire modelled domain). Each sill component (c_1 or c_2) represents the fraction of total variance that belongs to the corresponding scale of variability: if c_1 = 0.6 and c_2 = 0.3 with c_0 = 0.1, then 60 percent of the total porosity variance is at the channel-bar scale, 30 percent at the channel-belt scale, and 10 percent is nugget (noise or sub-well variability).
- Additivity allows the construction of anisotropic nested variograms by specifying different ranges in different directions for each component. An anisotropic variogram is one in which the range of spatial correlation is different in different directions: for example, porosity in a tabular sandstone might have a long range of 2,000 metres along depositional strike and a short range of 300 metres across strike, reflecting the elongation of channel bodies. Each structural component of the nested model can have its own anisotropy ratio and anisotropy direction, specified as a geometric anisotropy transformation (rotating and scaling the lag distance h to an equivalent isotropic lag before evaluating the model function). The additivity property is preserved under this transformation, so a nested anisotropic variogram with components at different scales and different anisotropy directions is still a valid model as long as each component is valid and all weights are non-negative.
- The linear model of coregionalization (LMC) extends additivity to multiple geological variables that are spatially correlated with each other. In a reservoir model, porosity and permeability are correlated at any given point (high porosity tends to accompany high permeability) and also have spatial correlation. The LMC asserts that the cross-variogram between two variables (measuring how their co-variation changes with distance) must also be a valid positive semi-definite matrix of functions, and that this is guaranteed if the matrix of sill values for all pairs of variables at each scale is a positive semi-definite matrix. In practice, this means that when fitting a nested variogram model to two or more correlated variables, the sill values at each scale must be chosen consistently so that the cross-variogram sills do not imply an impossible correlation structure. Geostatistical software packages (Petrel, GSLIB, SGeMS) enforce the LMC constraints during variogram model fitting to ensure that co-simulation of multiple variables produces a geologically plausible joint spatial distribution.
- Additivity in geostatistics is distinct from the simpler and more intuitive concept of volume additivity, which states that the total pore volume in a reservoir equals the sum of the pore volumes in each individual layer or zone. Volume additivity is a geometric identity and holds regardless of whether the zones are independent or correlated. The geostatistical additivity of variogram models is a mathematical property of functions, not volumes, and concerns whether a proposed spatial covariance structure is geometrically permissible for use in kriging. Confusing the two meanings can lead to misunderstanding: a non-geologist hearing that variogram components are additive might assume this means something about how reserve volumes from different layers can be summed (which is always true and is a different statement entirely). The correct interpretation of variogram additivity is that a nested model built from multiple valid component models is itself valid and can safely be used as the basis of spatial estimation and simulation.
Fitting a Nested Variogram Model to Well Data
The practical workflow for fitting a nested variogram model begins with computing the experimental (sample) variogram from available well data. For a property such as net porosity measured at 10-centimetre intervals in 20 wells spaced 500 to 1,500 metres apart, the experimental variogram is computed by averaging the squared differences in property values between all pairs of data points at each lag distance class (0 to 50 metres, 50 to 100 metres, and so on up to the maximum separation distance that can be reliably estimated given the number of pairs). The experimental variogram is plotted as gamma vs. h, and the modeller observes its shape: does it rise continuously to a sill (suggesting a stationary process) or continue rising beyond the well spacing (suggesting non-stationarity or a very long range component)? Does the curve rise steeply at short lag (suggesting a significant nugget or short-range component) or gently (suggesting smooth, long-range variation)?
The modeller then fits a nested model function to the experimental variogram by selecting the number of structures, choosing the model type for each structure (spherical, exponential, Gaussian), and adjusting the range and sill parameters until the model curve matches the shape and magnitude of the experimental variogram points. This fitting is typically done interactively in software with visual feedback, though automated fitting algorithms using weighted least squares or maximum likelihood are also available. The critical check is that all sill contributions are positive (additivity is preserved) and that the model reproduces the apparent nugget at the origin and approaches the total sill at long lags. If the experimental variogram shows a clear change of slope at an intermediate lag, this is a diagnostic signal that two or more distinct spatial scales of variability are present, and a nested model with one component per slope change is the appropriate response.
Additivity and Simulation: Why It Matters for Reservoir Models
Stochastic reservoir simulation (sequential Gaussian simulation, sequential indicator simulation, or multi-point statistics simulation) uses the fitted variogram model to reproduce the spatial structure of the simulated property across the entire reservoir grid. If the variogram model used as input is not valid (not positive semi-definite), the simulation algorithm can produce spurious artefacts: negative simulated values in areas where the property must be non-negative, or correlated patterns that are inconsistent with the geological concept. A nested variogram model that satisfies additivity is valid by construction, which means that the simulation algorithm is guaranteed to produce outputs with the correct statistical properties (mean, variance, spatial correlation structure) matching the input model. The assurance that the variogram model is valid is one of the reasons that experienced geostatisticians insist on using the standard valid model types rather than fitting arbitrary smooth curves to the experimental variogram data.
Fast Facts
The mathematical foundations of geostatistics, including the theory of valid variogram models and the additivity property, were developed by Georges Matheron at the Paris School of Mines (Ecole des Mines de Paris) in the 1960s, building on earlier work by Danie Krige in South Africa (whose name gave kriging its name) and Herbert Sichel on the statistical properties of gold mine grades. Matheron's 1971 text "The Theory of Regionalized Variables" established the theoretical framework used in reservoir geostatistics today. The key standard variogram models (spherical, exponential, Gaussian, power) were selected partly because they are among the few simple functional forms that have been proven to be positive semi-definite in three dimensions; the common linear model (gamma(h) = c × h) is valid only in one dimension and is not positive semi-definite in three dimensions, illustrating why arbitrary curve fitting to variogram data is dangerous. Commercial geostatistical reservoir modelling software (Petrel by SLB, IRAP RMS by Aspentech, Roxar STORM) provides built-in variogram model libraries containing only valid model types, enforcing additivity constraints automatically. The WCSB's characteristically layered Cretaceous clastic sequences (Viking, Cardium, Mannville) are among the most extensively studied examples of multi-scale reservoir heterogeneity in North America, with academic and industry studies documenting variogram model structures at scales from individual sandstone laminae (10 to 20 centimetre range) to stratigraphic units (10 to 50 kilometre range).