Logarithmic Mean: Permeability Averaging, Antilog Calculation, and Radial Flow Interpretation
The logarithmic mean is an averaging technique in which a set of measurements is converted to logarithms, the arithmetic average of those logarithms is taken, and the antilogarithm of that average is reported as the representative value. The result is mathematically identical to the geometric mean, the nth root of the product of n values, and it is the preferred average wherever the underlying physical property spans several orders of magnitude or behaves multiplicatively rather than additively. In oil and gas work the property that most often demands this treatment is permeability, which in a Western Canadian Sedimentary Basin reservoir can range from 0.001 millidarcies (mD) in a tight Montney siltstone to several thousand mD in a clean Leduc reefal carbonate. A simple arithmetic average of such numbers is dominated by the single largest value and badly overstates the effective permeability the reservoir actually delivers, so petrophysicists and reservoir engineers fall back on the logarithmic mean to capture the central tendency of the distribution. The reason the method works is that permeability, porosity-permeability transforms, grain-size distributions, and fluid-saturation ratios tend to follow a log-normal distribution: the histogram of the raw values is heavily right-skewed, but the histogram of their logarithms is close to symmetric and bell-shaped, so the arithmetic mean of the logs is an unbiased estimate of the centre. The logarithmic mean also appears in well-test interpretation, where the average permeability sensed by a pressure transient flowing radially toward a wellbore is closer to a logarithmic or geometric blend of the near-well and far-field values than to a flow-weighted arithmetic average. It surfaces again in mud logging and core analysis, where engineers average gas-show concentrations, total organic carbon readings, or capillary-pressure thresholds that vary multiplicatively. The same antilog-of-average-log machinery underlies the log mean temperature difference used to size produced-fluid heat exchangers and the decibel and pH scales that compress wide ranges into a workable span. Because a single zero or negative value makes the logarithm undefined, the logarithmic mean is only valid for strictly positive data, and analysts typically substitute a small floor value, such as 0.0001 mD, for non-detect permeability before averaging. Understanding when to apply it, rather than reaching reflexively for the arithmetic mean, is one of the practical skills that separates a defensible permeability estimate from a misleading one.
Key Takeaways
- Identical to the geometric mean: The antilogarithm of the arithmetic average of the logarithms equals the nth root of the product of n values. For permeabilities of 1, 10, and 100 mD, the logarithmic mean is the antilog of (0 + 1 + 2) divided by 3, which is 10 mD, versus an arithmetic mean of 37 mD. The 27 mD gap shows why averaging method matters for reserve and rate forecasts.
- Best for log-normal data: Permeability, grain size, capillary entry pressure, and gas-show intensity in WCSB plays such as the Montney and Duvernay are right-skewed and roughly log-normal. The logarithmic mean gives an unbiased central estimate where the arithmetic mean is dragged upward by a few high outliers, overstating deliverability and inflating booked reserves.
- Requires strictly positive values: The logarithm of zero or a negative number is undefined, so non-detect or zero permeability intervals must be floored at a small value such as 0.0001 mD before averaging, or excluded with a documented cutoff. Analysts note the substitution method in their petrophysical workflow because it materially changes the result for tight reservoirs.
- Radial flow connection: A pressure transient expanding radially from a well in a heterogeneous reservoir senses an effective permeability that lies closer to the logarithmic or geometric blend of layer permeabilities than to a thickness-weighted arithmetic average, which is why well-test-derived kh and core-derived kh frequently disagree until the averaging basis is reconciled.
- Widely embedded in oilfield math: The same antilog-of-mean-log structure underlies the log mean temperature difference (LMTD) used to size produced-fluid and gas-plant heat exchangers, the decibel scale, pH, and the Richter scale. Recognizing the pattern helps engineers choose the correct average whenever a quantity behaves multiplicatively rather than additively.
Calculating the Logarithmic Mean of Core Permeabilities
Consider a 12 m Cardium sandstone interval cored at Pembina with plug permeabilities of 0.5, 2, 8, 15, 40, and 220 mD. Taking base-10 logarithms gives roughly -0.30, 0.30, 0.90, 1.18, 1.60, and 2.34. Their arithmetic average is about 1.00, and the antilogarithm of 1.00 is 10 mD, the logarithmic mean. The plain arithmetic mean of the same six plugs is about 48 mD, nearly five times higher, because the single 220 mD plug dominates the sum. A reservoir engineer building an inflow performance forecast for this Cardium well would use the 10 mD logarithmic mean as the layer-representative permeability, since the arithmetic figure would imply a deliverability the rock cannot sustain across the full interval.
Logarithmic Mean Versus Flow-Weighted Averages
No single average is universally correct; the right one depends on flow geometry. For beds in parallel with flow along bedding, an arithmetic average weighted by thickness is exact. For beds in series with flow across bedding, the harmonic mean governs. For areally random heterogeneity with no preferred flow path, the logarithmic or geometric mean best predicts effective permeability, a result confirmed by both stochastic simulation and field history matching. WCSB reservoir studies on Viking and Mannville channels routinely report all three averages so reviewers can see the spread. When a well test returns a permeability between the harmonic and arithmetic bounds and near the logarithmic mean, it usually signals genuine three-dimensional heterogeneity rather than a measurement error.
Fast Facts
The logarithmic mean of permeability is so reliable a predictor of effective flow in randomly heterogeneous rock that it is sometimes called the Matheron conjecture, after the French geostatistician Georges Matheron who formalized the result in the 1960s. For a two-dimensional medium with log-normally distributed permeability, the effective permeability equals the geometric mean exactly, a tidy theoretical anchor that practising reservoir engineers lean on every time they upscale a fine-grid permeability cube into a coarse simulation model of a Montney or Duvernay pad.
Related Terms
The logarithmic mean is most often applied to permeability, the rock property whose log-normal spread makes ordinary averaging unreliable, and it is computed across intervals defined by porosity cutoffs in the same petrophysical pass. The averaged kh values feed directly into well test interpretation, where transient-derived permeability is compared against the core-based logarithmic mean to diagnose heterogeneity, and into net pay calculations that decide which intervals carry the average at all. Each of these terms connects because the choice of average changes the reserve and deliverability numbers downstream.
Real-World WCSB Scenario: Reconciling Core and Well-Test Permeability at Pembina
An operator developing a Cardium oil pool near Pembina cored two wells at a cost of roughly 90,000 CAD per cored well and measured a thickness-weighted arithmetic permeability of 55 mD across the pay. A subsequent buildup test on the first horizontal producer returned an effective permeability of only 11 mD, alarming the asset team who feared a damaged completion or an over-optimistic core program. Recomputing the core data as a logarithmic mean produced 12 mD, almost exactly matching the well test and showing that the rock was simply log-normally heterogeneous, not damaged.
The reconciliation saved an unnecessary 180,000 CAD stimulation workover that had been proposed to fix the imagined skin damage. The asset team adopted the logarithmic mean as the standard reporting basis for all future Cardium core averages, and the next three development wells came in within 15 percent of their pre-drill deliverability forecasts, restoring confidence in the subsurface model and the booked reserves underpinning the project economics.