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Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Define the gas volume
Chapters 1–2. Identify volume, porosity, saturation and physical support.
Derive spatial covariance
Chapters 3–5. Derive the mean-product correction and calculate paired cells.
Change the arrangement
Chapters 6–7. Reverse correlation and adjust spatial weighting.
Audit evidence and errors
Chapters 8–10. Build uncertainty, evidence and a reproducible cell table.
Practise and continue
Chapters 11–12. Work a changed grid and follow specialist guides.
Full chapter index · Practice and worked checks · Petroleum Geology subject root
A gas-in-place estimate can be wrong even when its average porosity and average gas saturation are individually correct. Gas occupies the pore space in each part of the reservoir, so a cell-by-cell estimate multiplies local rock volume, porosity and gas saturation before adding the results. Multiplying separate field averages can miss the way these properties vary together.
For two fictional equal-volume cells, with porosities 0.30 and 0.10 and gas saturations 0.80 and 0.20 respectively, the true gas-filled pore volume is 260,000 m³. Using average porosity 0.20 and average gas saturation 0.50 gives only 200,000 m³. The 60,000 m³ difference is the consequence of positive spatial covariance, not a unit conversion. This article derives the difference, builds a reproducible grid calculation and separates gas-filled pore space from reporting gas volume under standard conditions. It extends the Gold Standard of Petroleum Geology.
Chapter index
Define the gas volume · Chapters 1–2
1. Identify what a gas-in-place input represents
2. Write a cell-by-cell expression before averaging
Derive spatial covariance · Chapters 3–5
3. Calculate two cells with positive covariance
4. Reverse saturation and watch the bias change sign
5. Use volume weighting when cells are unequal
Change the arrangement · Chapters 6–7
6. Distinguish gas-filled rock volume from standard gas volume
7. Understand what uncertainty does and does not reveal
Audit evidence and errors · Chapters 8–10
8. Build a practical cell-table and evidence check
9. Investigate how facies sampling changes a field-wide average
10. Questions that improve an averaged reservoir estimate
Practise and continue · Chapters 11–12
11. Practise a new cell layout and test the shortcut
12. Continue from averages toward better geological inference
CHAPTER 1 OF 12 · Define the gas volume
1. Identify what a gas-in-place input represents
A simple volumetric description starts with net reservoir bulk volume, porosity and gas saturation. Net rock volume is not necessarily the whole gross mapped formation. Porosity describes how much of that net volume is pore space. Gas saturation describes the fraction of pore space occupied by gas under the model’s stated conditions. A water-saturation estimate may be used through Sg = 1 − Sw when only gas and water occupy the relevant pores.
The product Vnet × φ × Sg gives gas-filled pore volume at reservoir conditions for a uniformly described region. It does not by itself give a standard-state gas volume, recoverable gas, reserves or a forecast of production. Those require separate fluid and commercial definitions, as explored in Gas Formation Volume Factor and the Volume Basis in a Gas-in-Place Estimate.
The first audit question is which average is being used, over which spatial support and with which weighting. A porosity averaged across core plugs, a saturation averaged over log samples and a bulk volume estimated from seismic may refer to different supports. Multiplying such numbers without reconciliation may create a precise-looking quantity that corresponds to no single consistent rock volume.
CHAPTER 2 OF 12 · Define the gas volume
2. Write a cell-by-cell expression before averaging
Partition a fictional reservoir into cells indexed i. Let Vi be the net bulk rock volume of a cell, φi its fraction of connected pore space in the stated model and Sgi the fraction of that pore volume occupied by gas. Then gas-filled pore volume is Vgp = Σ Vi φi Sgi. Every product is a volume and the final sum has units of cubic metres.
A shortcut sometimes replaces the sum by total net volume multiplied by two averages: Vshortcut = Vtotal × mean(φ) × mean(Sg). That identity is valid only under appropriate definitions and relationship between the properties. Even with correctly volume-weighted means it generally fails if porosity and saturation are correlated across cells.
For equal-volume cells, the exact identity is mean(φ Sg) = mean(φ) mean(Sg) + Cov(φ,Sg), where covariance uses the matching population averaging convention. Multiply by Vtotal to obtain Vgp = Vtotal[mean(φ) mean(Sg) + Cov(φ,Sg)]. The overlooked term can be positive, negative or zero.
The result does not require a probabilistic reservoir interpretation. Covariance here is simply a compact way to describe how two measured or modeled quantities co-vary across the same cells. It is not evidence that porosity causes saturation, nor can it tell us why those variables share a spatial pattern.
CHAPTER 3 OF 12 · Derive spatial covariance
3. Calculate two cells with positive covariance
Imagine two independent map cells that each contain 1,000,000 m³ of net bulk rock. Cell A has porosity 0.30 and gas saturation 0.80. Cell B has porosity 0.10 and gas saturation 0.20. Both values are fractions rather than percentages on the multiplication line.
Cell A contains 1,000,000 × 0.30 × 0.80 = 240,000 m³ gas-filled pore volume. Cell B contains 1,000,000 × 0.10 × 0.20 = 20,000 m³. Summing first gives 260,000 m³. The separate means are 0.20 and 0.50, so the shortcut gives 2,000,000 × 0.20 × 0.50 = 200,000 m³.
The mean product is (0.24 + 0.02)/2 = 0.13. The product of means is 0.20 × 0.50 = 0.10. Their difference, Cov = +0.03, accounts for 2,000,000 × 0.03 = 60,000 m³. The shortcut underestimates the cell-sum reference by 23.1% when the error is expressed relative to the 260,000 m³ cell-sum result.
| Cell | Net bulk volume | Porosity φ | Gas saturation Sg | Gas-filled pore volume |
|---|---|---|---|---|
| A | 1,000,000 m³ | 0.30 | 0.80 | 240,000 m³ |
| B | 1,000,000 m³ | 0.10 | 0.20 | 20,000 m³ |
| Correct cell sum | 2,000,000 m³ | Volume-weighted φ = 0.20 | Volume-weighted Sg = 0.50 | 260,000 m³ |
| Product-of-means shortcut | 2,000,000 m³ | 0.20 | 0.50 | 200,000 m³ |
The table is a constructed mathematical example. It is not a reserve statement or a claim about typical geological covariance. Its purpose is to make the order of multiplication visible.
CHAPTER 4 OF 12 · Derive spatial covariance
4. Reverse saturation and watch the bias change sign
Keep the same two porosities and volumes, but exchange the gas saturations: cell A now has Sg = 0.20, and B has Sg = 0.80. The separate averages remain φ = 0.20 and Sg = 0.50. The shortcut still reports 200,000 m³.
The cell sum changes to 1,000,000 × 0.30 × 0.20 + 1,000,000 × 0.10 × 0.80 = 60,000 + 80,000 = 140,000 m³. Now mean(φSg) = 0.07 while product of means is 0.10, so covariance is −0.03. The shortcut overestimates the reference by 60,000 m³.
These two reservoirs have identical totals, mean porosity and mean gas saturation but different spatial alignment. The first places higher gas saturation in the better-porosity rock. The second places higher gas saturation in the poorer-porosity rock. Averages alone discard the co-location information that controls the gas-filled pore-space total.
This is the conceptual heart of the problem: a correct marginal summary of each variable does not determine the mean of their product. A geologist or reservoir engineer needs joined observations or a defensible joint model of the variables, not just an impressive spreadsheet of separate statistics.
CHAPTER 5 OF 12 · Derive spatial covariance
5. Use volume weighting when cells are unequal
Most reservoir cells are not equal in bulk rock volume. Some cover larger map areas, some have thicker net reservoir and some belong to a different net-to-gross interval. The appropriate weights for the volume total are wi = Vi/ΣVi. Then weighted means are φbar = Σwiφi and Sbar = ΣwiSgi, and the weighted covariance is Covw = Σwi(φi−φbar)(Sgi−Sbar).
The exact two-variable identity becomes Vgp = Vtotal[φbar Sbar + Covw]. If one calculates a simple mean of two log-derived porosities from intervals of very different rock volume, the resulting φbar is not the correct volume-weighted average even before the covariance problem begins.
Net-to-gross and effective thickness can introduce further relationships. A map region with greater net volume may also have different porosity or saturation distributions. The safest primary computation is often ΣViφiSgi rather than trying to repair each of several averaged inputs after the fact.
Do not infer geological stationarity from a large number of measurements concentrated in one well. Spatial support and sample placement determine the representativeness of each estimated variable. An apparently large dataset may still miss the gas-poor edge or the high-porosity reservoir core.
CHAPTER 6 OF 12 · Change the arrangement
6. Distinguish gas-filled rock volume from standard gas volume
If all cells share one applicable gas formation volume factor Bg expressed as reservoir volume per standard reporting volume, standard-state gas volume in the simplified model is Gstd = Vgp/Bg. A large result in standard cubic metres is not the size of a giant underground cavern; it expresses the volume the modeled amount of gas would occupy under a named reference state.
When pressure, temperature, composition or compressibility vary across cells, Bg may vary too. Then the appropriate cellwise expression is Gstd = Σ[Vi φi Sgi/Bgi] under a consistent gas formulation and reporting convention. A single averaged Bg inserted after spatial averaging may not preserve the correct sum.
Our example is intentionally about covariance of φ and Sg, not a new derivation of gas state conversion. Mixing those two issues can mask errors. First establish the gas-filled pore volume at each cell. Then apply a physically appropriate fluid-state conversion. Finally distinguish initially in-place amount from recoverable quantities, which require further flow, technical and economic reasoning.
If gas and condensate phases coexist, or if fluid composition and pore conditions change substantially, a simple dry-gas cellwise multiplication is not sufficient. State the fluid model rather than stretch a teaching equation beyond its validity.
CHAPTER 7 OF 12 · Change the arrangement
7. Understand what uncertainty does and does not reveal
The two-cell calculation shows a deterministic bias from the missing covariance term given its fictional inputs. In field work, each input is itself uncertain. Porosity, saturation, net thickness, facies boundaries and gas properties may have measurement error and geological uncertainty, sometimes correlated with one another.
The average covariance across a spatial model is not the same as the covariance of uncertain estimates generated by a statistical ensemble. One describes patterns across locations within a scenario; the other may describe how scenario-level quantities co-vary across plausible subsurface models. Both matter, but they answer different questions and use different weights.
A Monte Carlo workflow that samples φ and Sg independently by default can erase plausible dependencies and distort the range of gas-in-place volumes. Conversely, forcing a strong correlation without geological evidence is another form of unwarranted certainty. Pair the dependency model with measured co-located data, rock types, capillary behavior and independent constraints.
A transparent report shows the baseline cell sum, the shortcut if it is used for comparison, and the sensitivity to plausible spatial arrangements. Readers can then see whether uncertainty comes mainly from field extent, net rock volume, pore properties, gas distribution or fluid-state conversion.
CHAPTER 8 OF 12 · Audit evidence and errors
8. Build a practical cell-table and evidence check
A useful spreadsheet or reservoir-model export carries one row per consistent spatial unit, with columns for net bulk volume, porosity, gas saturation, fluid-state parameters and their observation or model sources. Add a product column ViφiSgi and a separate standard-state column where appropriate. Totals then sum the physical products rather than the columns’ independent averages.
| Input or check | What it contributes | Limitation to document |
|---|---|---|
| Structure and net-volume map | Spatial weights and extent | Faults, net-pay thresholds, map uncertainty |
| Core/log porosity | Local pore-space estimate | Scale, bias and rock-quality dependence |
| Water/gas saturation interpretation | Gas fraction of pore space | Contact, transition zone, fluid and tool assumptions |
| Spatially paired facies model | Co-location and covariance | Whether correlations survive between wells |
Check volume units, percentage conversions, net versus gross volume, water versus gas saturation, and the alignment of sampling supports. The cell-table result should be reproducible by another person from the stated rows and formulas. A single unexplained net/gross factor or percent conversion can outweigh the subtler covariance term.
Reserve terminology is a separate layer. Volumetric initially in-place estimates are not automatically economically producible quantities. A resource classification depends on definitions and evidence beyond the arithmetic here.
CHAPTER 9 OF 12 · Audit evidence and errors
9. Investigate how facies sampling changes a field-wide average
A reservoir can contain channel sand, levee silt and tight beds with different combinations of porosity, gas saturation and connectivity. Sampling only clean sandstone may describe that facies well but misrepresent the net bulk rock volume. A facies-weighted estimate needs a spatial model that places facies and petrophysical properties in the same cells.
Hold total net rock volume fixed and redistribute the same porosity and saturation values among cells. If gas-filled pore volume changes substantially, the estimate depends on their co-location. Paired core and logs, structure maps and a geological model can test that dependence; separate histograms alone cannot.
A covariance value summarizes a pattern, not a cause. High gas saturation co-located with high porosity may reflect facies, fluid contacts or modeling assumptions. A useful report compares positive, neutral and negative co-location cases and identifies the independent observations needed to distinguish them.
CHAPTER 10 OF 12 · Audit evidence and errors
10. Questions that improve an averaged reservoir estimate
Is the product of averages ever valid?
Yes, if the variables are uncorrelated under the same physical weighting, or if one variable is effectively constant, the covariance term vanishes. But those conditions should be supported by the chosen cells and evidence, not presumed because the shortcut is convenient.
Does positive covariance always increase reserves?
Positive covariance increases the modeled gas-filled pore volume relative to the two-factor product of means in this ideal setup. It does not establish recoverable reserves. Flow, pressure decline, trapping, technical feasibility and commercial conditions remain separate considerations.
Can I use average Sw instead of average Sg?
If gas and water are the only relevant pore-filling phases, Sg = 1 − Sw cell by cell. The covariance with Sg is the negative of the covariance with Sw. Simply applying one average Sw does not recover the missing joint spatial information.
Does a finer grid automatically solve the problem?
No. More cells help only if the cell properties and their dependencies are meaningful at that scale. Splitting poorly known rock into many tiny cells can create numerical detail without new geological information.
CHAPTER 11 OF 12 · Practise and continue
11. Practise a new cell layout and test the shortcut
Try two equal-volume cells of 500,000 m³. Cell C has φ = 0.25 and Sg = 0.70. Cell D has φ = 0.15 and Sg = 0.30. Correct gas-filled pore volume is 500,000×0.25×0.70 + 500,000×0.15×0.30 = 87,500 + 22,500 = 110,000 m³.
Separate averages are φbar = 0.20 and Sbar = 0.50, so the product-of-means result is 1,000,000×0.10 = 100,000 m³. Their covariance is (0.175+0.045)/2 − 0.20×0.50 = 0.01, which accounts for the missing 10,000 m³.
Swap the two saturations. The correct cell sum becomes 500,000×0.25×0.30 + 500,000×0.15×0.70 = 37,500 + 52,500 = 90,000 m³ while the averages remain unchanged. Explain why a reserve summary that includes only φbar and Sbar could not tell these scenarios apart.
For a harder transfer, give cell C twice the net volume of D. Compute wi explicitly before taking weighted means. State which additional field data would help establish whether the high-porosity and high-saturation regions genuinely coincide.
CHAPTER 12 OF 12 · Practise and continue
12. Continue from averages toward better geological inference
The gold standard is to define the physical target and preserve co-location. Multiply the quantities that describe the same cell before summing; state when and why averaged quantities can replace a detailed calculation; and distinguish spatial covariance from uncertainty across geological scenarios.
Read the Gold Standard of Petroleum Geology to connect volumetrics with trap, seal, facies and subsurface evidence. Read Gas Formation Volume Factor for the separate reservoir-to-reference conversion, and Permeability Across Layers for how physical connectivity changes directional flow even when rock averages are known.
Sources and further reading
U.S. Energy Information Administration: Natural gas explained provides accessible context for gas resources and reporting. SPE Petroleum Resources Management System distinguishes in-place estimates from reserve/resource classifications. USGS: Petroleum assessment framework provides examples of geological evidence at assessment scale. The numerical cells, correlations and sensitivity tests are original educational examples rather than a measured accumulation.
How X Works and The Gold Standard of Education connect the same reasoning to a general lesson: averages often conceal the relationships that determine how a complex system works.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.