We start this article with an example that many geoscientists may find uncomfortable, if not downright contentious.

A top structure depth map from a depth conversion is most commonly used for two tasks: providing a depth prognosis to Top Reservoir prior to drilling; and in the calculation of gross rock volume (GRV) for use in hydrocarbons initially in place (HIIP).

It cannot be stressed strongly enough that there is a key misunderstanding in the oil industry on the difference between the uncertainty at a point (depth prognosis) and the uncertainty of a surface (GRV estimation). The GRV estimation critically depends on spatial correlation being reproduced correctly in the mapping, whereas the depth prognosis does not. This has serious implications for estimating GRV and its uncertainty.

The conventional use of depth surfaces for depth prognosis for well targeting purposes is entirely valid. However, the use of the same surfaces for gross rock volume estimation is not generally valid. This may come as a surprise for many geoscientists, as this assumption forms a key part of their work.

We are going to illustrate the problem with an experiment. The first panel (**A**) in the image above, is a topographic elevation map obtained from satellite measurements. This surface is known. Assuming an imaginary hydrocarbon contact we can calculate a true gross rock volume and area of possible closures, shown as the green and orange areas in second panel (**B**). The true volume is 300 MMm^{3} and the true area 9.84 km^{2}. The orange area shows the connected extent above contact of the central closure. Note that the central structure is not connected to the smaller structure to the west.

Panels (**C**) and (**D**) show the effect of smoothing of the original structure using a moving average filter. In (**C**) the topographic surface was smoothed using a 3 x 3 node moving average before clipping with the contact; this is the smallest possible centred smoother we could use. The 3 x 3 moving average reduces the GRV by over 7% to 278 MMm^{3}. Panel (**D**) is with a 5 x 5 moving average, which reduces the GRV estimate to 261 MMm3, an underestimate of 13%. This is GRV bias resulting from smoothing and it is always present because our maps are uncertain estimates, not the true subsurface map. Smoothing also changes the connectivity of the predicted surface. For the 5 x 5 moving average (**D**) the main structure is erroneously shown to be connected to the western structure.

Concerned? Well, read on…

### Estimation of GRV

The estimation of the possible volume of oil or gas in place in the subsurface is a fundamental, routine and important element of the geoscientist’s role. Whether it be for a lead or prospect, post-discovery or later in field life, volumetric calculations form the basis of economic value and inform our decisions on such diverse activities as drilling, relinquishment, development, farming in or out and purchase or sale of assets.

The basic equation for calculating HIIP could be summarised as one volumetric measurement and a series of multiplication factors. The volumetric measurement is of course gross rock volume (GRV) and the multipliers the factors such as net-to-gross ratio, porosity, water saturation and formation volume factor.

It is reasonable to say that GRV is the most significant factor in estimating in-place hydrocarbon volume for the majority of prospects or fields. But compared to other parameters in the HIIP equation GRV is unique: there is no tool with which it can be directly measured. Instead, GRV is estimated indirectly, using a top structure depth map and knowledge of (or assumption about) the trapping mechanism. The GRV is calculated by integrating between the top structure depth map (referred to as Top Reservoir), a base or thickness grid and a hydrocarbon contact.

GRV estimation is clearly linked to depth map estimation. Uncertainty in depth maps is therefore an important factor in understanding GRV uncertainty. A typical workflow for estimating a depth map includes seismic time interpretation, gridding, time-to-depth conversion and residual mapping. Uncertainty is present at all steps including the time interpretation, gridding, estimation of the velocity field and the choice of the method of depth conversion and the intervals or layers to be used.

### Depth Conversion Uncertainty

The accuracy and validity of a depth conversion is primarily assessed by comparing the depth converted surfaces at the wells to the target formation tops. The difference between predicted and observed depths is usually referred to as a residual. A statistical summary of the residuals allows comparison between different depth conversion cases and is a measure of the depth uncertainty accuracy and variability. In order to tie depth converted surfaces to wells, the residuals are usually mapped and added to the depth converted surface to obtain a final depth conversion that ties the wells. The mapping of residuals will necessarily be smooth.

A depth conversion prepared in this way is typically referred to as ‘deterministic’, although geostatisticians would call this a best estimate. Of several deterministic depth conversions, a preferred one is often referred to as a ‘base case’.

If kriging is used as the method of residual mapping then, in addition to the depth conversion uncertainty summarised by the residuals, an estimate of the interpolation uncertainty is also obtained from the kriging standard deviation. A kriging approach combined with a conventional depth conversion will then include the two essential elements of uncertainty:

- The uncertainty arising at a single point through there being multiple valid models and parameters by which we can depth convert;

- The spatial uncertainty arising from lateral prediction between data points and the spatial correlation/ dependency model.

For a depth converted surface the spatial uncertainty varies laterally, being small when close to data points (usually wells) and becoming larger away from them, as illustrated below.

Many of the steps involved in producing a depth map involve spatial smoothing. Seismic volumes are laterally smooth due to the finite size of the Fresnel zone caused by limited bandwidth. A seismic interpretation is smoothed, perhaps to remove noise or improve its appearance.

Velocity functions from a few well data are simple, smooth functions. Velocity maps from well data are based on sparse control points and therefore smooth. Residuals are mapped and, again because they are sparse, the residual map is also smooth.

### Smooth Estimation and GRV

Because all estimators are smoothers and some smoothing is inherent at all stages in the process of producing a depth map, a mapped depth surface at Top Reservoir is always smoother than the true depth surface, if we were able to observe it.

Smoothness, particularly in the presence of noise, is generally helpful in depth prognosis and has no important side effects, but it has a significant and detrimental impact on connectivity and GRV. This can be demonstrated, as we did at the beginning of this article, by starting with a known surface, calculating a GRV and then applying a smoother to the surface and recalculating the GRV. The GRV from the smoothed surface will be lower than the true GRV and this error (bias) becomes more pronounced as the level of smoothing is increased.

From this simple case we can see very clearly that even minimal smoothing modifies the GRV, potentially significantly. It is probably surprising to many readers that a small moving average filter on your maps or picks could change the volume by as much as 7–13%, but this outcome is well known and understood in geostatistics.

In general, smoothing biases GRV downwards, but structures can also appear more connected than they really are, increasing the apparent connected volume of a prospect, so the final outcome is complex. Smoothing has either little or some beneficial impact on depth prediction, so when wells come in on prognosis we can easily convince ourselves that if the depth map is good, so is the GRV calculated from that map – but it is not, and the GRV estimate can be significantly in error.

The table opposite gives some general guidelines on the expected discrepancy between GRV estimated by simple deterministic map cases compared to the mean GRV estimated from analysis of geostatistical realisations. These are typical values based on practical experience from many GRV uncertainty studies over nearly 25 years.

It should be noted that this article is a simple, headline summary of some of the gross effects of smoothing on GRV estimation. GRV estimation bias is strongly related to the ratio of the column height relative to the depth uncertainty, this effect being more pronounced the lower the relief of the structure and the greater the depth uncertainty. Connectivity also plays an important role. A full technical explanation would require more space than we have here, so we will just proceed directly to a solution to the problem.

### Geostatistical Simulation for Estimating GRV and GRV Uncertainty

Geostatistics provides the necessary tools for solving the problem of GRV estimation and its uncertainty using a depth conversion and depth uncertainty. Geostatistical simulation, typically using an algorithm such as Sequential Gaussian Simulation (SGS), allows us to generate multiple realisations of the uncertain depth surface.

An example is shown in the image below, where Panel (**A**) shows a depth map over a structure. The long, narrow north-south oriented high is the structure on which we will compute GRV. Panel (**B**) shows the closure (orange) at lowest closing contour for this structure. The lowest closing contour is at 1,843m with an apparent spill to the west. The GRV of this structure is calculated as 843 MMm^{3}. Panels (**C**), (**D**) and (**E**) show three example geostatistical simulations of the depth structure. In each case the lowest closing contour connected to our target well location A has been computed automatically and the resulting closure is coloured orange. Note that, despite the result from the best case depth map, there may not be a single structure, instead it may be comprised of two or three sub-structures with independent spillpoints.

We can combine the closure results for many realisations to form an isoprobability closure map, illustrated below. The probability of a single north-south structure is 77%, indicated by the dark blue colours. By computing the connected volume to lowest closing contour for each realisation, the GRV can be calculated for each one, connected to the proposed well location. Using geostatistical realisations in this way we are converting the depth and spatial uncertainty into a GRV uncertainty. The GRV uncertainty distribution, for volume connected to Well A, is shown below right. The mean GRV from the geostatistics is only marginally higher, in this case, by about 3%. This is because (a) this is a high relief structure with relatively small depth uncertainty and (b) this is the volume connected to the well, and there may be disconnected volumes that we would have to increment in order to obtain the total possible volume.

### Summarising Uncertainty

Our simple guide to gross rock volume uncertainty can be summarised as:

- Depth maps and testing different depth conversion models build up a picture of the point uncertainty on depth;

- Interpolation of well residuals using kriging adds the spatial element to the point uncertainty from (1);

- Smooth best estimate depth maps from steps (1) and (2) are ideal for depth prognosis but should not be used for estimating GRV or connectivity;

- Only geostatistical simulations should be used for estimating GRV and GRV uncertainty;

- Geostatistical simulation allows the full analysis of GRV uncertainty, connectivity and ‘what if’ scenarios that would not be valid on deterministic maps.