Reservoir model and uncertainty: closing the QI loop

The STOIIP formula, and what each factor depends on

Stock-tank oil initially in place, in convenient mixed units:

The 6.29 is the number of million barrels in a million cubic metres (one square kilometre by one metre). Each factor comes from a different upstream input:

• Area $A$ and net pay $h$: from the structure map (Parts 2 and 3) and the oil-water contact. Both are set by the same contact, so they rise and fall together: a deeper contact enlarges the closure and thickens the oil column at once.
• Porosity $\phi$: from the porosity cube (Section 7.4), tied to the wells.
• Hydrocarbon saturation $1 - S_{\mathrm{w}}$: from the wells, since seismic reads a fluid class, not a saturation (Section 7.4); often the least robust of the rock-property terms, because $S_{\mathrm{w}}$ is the hardest to measure away from the wells.
• Formation volume factor $B_o$: from the reservoir engineers’ PVT analysis, about 1.1 to 1.5 for oil, rising with the gas dissolved in it.

In a reservoir model the formula is applied cell by cell, $\mathrm{STOIIP} = \frac{1 - S_{\mathrm{w}}}{B_o}\sum_c \phi_c V_c$, with $V_c$ the rock volume of cell $c$ above the contact. So it is the porosity map that enters, not an average porosity, and every version of the map gives its own volume. Treating each factor as a single value (deterministic volumetrics) hides the uncertainty; the sections below draw it out.

From sparse wells to a model: variogram, kriging and simulation

Porosity is not random from place to place: nearby points are alike, distant ones less so. The variogram measures how quickly that likeness fades with the separation $h$:

It rises from zero and levels off at the sill, the variance $\sigma^2$, at the range $a$, the distance beyond which two values tell nothing about each other. A common model is the spherical one, $\gamma(h) = \sigma^2\bigl(\tfrac{3h}{2a} - \tfrac{h^3}{2a^3}\bigr)$ for $h < a$ and $\sigma^2$ beyond. Wells rarely sample enough short distances to measure the range, so it is often taken from seismic, outcrop analogues or the depositional setting.

• Kriging estimates porosity between the wells as the weighted average that is unbiased and has the least error variance under the variogram. Its map is smooth: it is the best single guess at each point, and it has less variability than the real reservoir.
• Sequential Gaussian simulation visits the cells in a random order; at each it kriges from the wells and the cells already simulated, then draws a value from the Gaussian with the kriging mean and variance. Each random seed gives one realisation: a map that honours every well, reproduces the variogram and is as likely as any other (Deutsch and Journel 1998).
• Seismic as secondary data: the calibrated seismic porosity $\phi_s$ gives a trend, the regression $m(u) = \mu + \rho\,(\phi_s(u) - \mu)$ for a correlation $\rho$ with the wells, and only the residual, with variance $\sigma^2(1 - \rho^2)$, is simulated (kriging with a locally varying mean; Goovaerts 1997). Collocated cokriging, which weights the seismic value at each cell against the wells, is the common alternative (Xu et al. 1992; Doyen 2007).

The volume of every realisation is one possible answer. Sorted, they give P90, P50 and P10; their spread is the uncertainty the data leave.

Exercise: read Figure 7.6

1. The figure opens with 6 wells, seismic porosity correlated at $\rho = 0.6$ and a variogram range of 1.2 km, the range the field was built with. Plate (a) is the seismic porosity map, (b) one realisation, (c) the variogram, (d) the volume of every realisation. The headline reads P90 52.3, P50 54.8 and P10 57.9 MMbbl, a P10/P90 ratio of 1.11, and the true volume, 56.7 MMbbl, lies between the P50 and the P10. In a real field the truth is never known; here it lets you test the model.
2. Change the wells from 1 to 6 to 24. The P10/P90 ratio falls from 1.20 to 1.11 to 1.04 while the P50 stays within 2 MMbbl: each well fixes porosity where it stands and, through the variogram, for about a range around it, so the realisations have less room to differ.
3. With 6 wells, raise $\rho$ from 0 to 0.9. The ratio falls from 1.13 to 1.06, and the realisation in (b) takes on the pattern of the map in (a). Read the subtitle of (a) as well: over the map the seismic correlates with the truth at exactly $\rho$, by construction, yet at $\rho = 0.6$ the six wells measure 0.77. In a real field the wells are the only calibration there is, and a correlation measured at six of them is itself uncertain.
4. Set the range to 250 m. The realisations turn patchy, their patches average out over the closure, and the ratio falls to 1.05; but the true volume now lies above the P10. A range far too long (3000 m) misses it too. Over 40 other synthetic fields, the P90 to P10 band held the true volume in 30 of 40 at the true range, within the 27 to 37 that chance allows an 80% band, and in only 5 of 40 at 250 m. A narrow spread from a wrong model is not knowledge.
5. Switch plate (d) to the tornado and go back to 1 well. Moving the contact 20 m up or down swings the volume from 25.9 to 85.5 MMbbl, against 50.3 to 60.4 for the porosity model. With 24 wells and a contact known to 2 m, the contact still leads, 53.5 to 59.3, against 55.1 to 57.5. The contact and saturation ranges are stated inputs, not measurements, but the lesson holds widely: structure often outweighs porosity.
6. Leaf through the realisations with the slider. Each honours every well and differs only where the data say nothing. No single map, and not the smooth mean of them all, is the volume to book: the spread is the answer.

From P10/P50/P90 to business decisions

The distribution of oil in place feeds the decisions downstream:

• Reserves booking: under the SPE Petroleum Resources Management System (SPE-PRMS), probabilistic reserves are reported as 1P (proved, the P90), 2P (proved plus probable, the P50) and 3P (proved, probable and possible, the P10). Reserves are recoverable volumes, the oil in place times a recovery factor within an approved development plan, so an in-place distribution is the start of that calculation, not the reserves themselves. Securities regulators such as the US SEC frame proved reserves on reasonable certainty.
• Field development decision (FID): management compares the recoverable P50 times the oil price against the capital and operating costs discounted over the field life. Approval usually needs a positive net present value at the P50 under a pessimistic oil price.
• Risk-weighted economics: the probability of a commercial outcome is the chance that the volume exceeds the minimum economic size, the size that pays for the wells and facilities. That probability times the value if successful, minus the cost if not, is the expected value used to rank a portfolio.
• Partner negotiations: farm-outs, farm-ins and asset trades all quote P90, P50 and P10. The counterparty has its own estimate; the negotiation is partly about whose governs.
• Drilling strategy: the spread of the distribution, not only its middle, decides how many wells to drill first. A wide spread argues for one appraisal well, then a decision on scale; a narrow one for committing to full development.

Why the tornado is the most cost-effective diagnostic

A tornado chart answers the most actionable question: where should the next million dollars of uncertainty reduction go? To the input with the longest bar. For example:

• Net pay is the top bar: drill more wells, rerun the inversion with more calibration wells, or acquire higher-resolution seismic to beat the tuning thickness.
• Area or the contact is the top bar: invest in structural imaging (PSTM or PSDM reprocessing, wide-azimuth acquisition), and in pressure data that pin the contact.
• Porosity is the top bar: recalibrate the rock-physics template, add core, or drill where the realisations disagree most.
• Hydrocarbon saturation is the top bar: take more saturation data at the wells (logs, cores, pressure gradients that pin the contacts), because seismic reads a fluid class, not a saturation (Section 7.4); long-offset pre-stack data help only where they constrain density.

The tornado shows not only the uncertainty but the leverage an investment has on it. In Figure 7.6 the porosity model’s bar is measured from the realisations, while the contact and saturation bars come from ranges stated for each scenario, the usual mix in practice.

The end-to-end QI value story

Zooming out to Part 7 as a whole, the value chain is:

1. Seismic acquisition: traces over a 3D volume.
2. Sections 7.3 and 7.4 inversion: a $Z_{\mathrm{P}}$ cube from post-stack inversion, or $Z_{\mathrm{P}}$, $Z_{\mathrm{S}}$ and density cubes from pre-stack inversion, the elastic properties of every voxel, with uncertainty.
3. Section 7.4 readings: lithology, fluid-class and $\phi$ cubes, rock properties per voxel, each with its error rate; $S_{\mathrm{w}}$ comes from the wells.
4. Section 7.5 probabilistic facies: a probability cube for each class, keeping the classification uncertainty.
5. Section 7.6 reservoir model: realisations of the reservoir conditioned to the wells and the seismic, the distribution of their oil in place, and the tornado.
6. Business layer (outside QI): FID, reserves booking, portfolio ranking, drilling programme.

Each stage adds value and inherits uncertainty from the stage before. The discipline of Part 7 is to carry that uncertainty all the way through rather than hide it behind deterministic shortcuts. A team that propagates it honestly delivers a more trustworthy product, even when its headline numbers are less dramatic, and trust is what lets asset teams spend capital.

What comes next: Part 8, advanced topics

With Part 7 complete, you have the core QI workflow end-to-end. Part 8 (outside the scope of this section but the logical continuation) covers advanced and specialized topics:

• Geomechanics: elastic cubes into stress and fracture prediction.
• Time-lapse (4D) QI: repeat surveys to track production-induced pressure/saturation changes.
• Anisotropy and azimuthal attributes: when the isotropic assumption breaks.
• Full-waveform inversion (FWI): combining velocity model building with QI.
• Machine-learning QI: neural-net inversion, deep-learning classifiers, feature engineering.
• CO₂ storage QI: monitoring injected CO₂ plumes through changes in $Z_{\mathrm{P}}$, $V_{\mathrm{P}}/V_{\mathrm{S}}$ and density.

All of them build on the Part 7 foundation: you now speak the language of QI, probabilistic, uncertainty-aware, rock-physics-grounded. Every advanced topic is a specialized extension, not a replacement, of what you’ve learned here.