Rw Determination

Summary

Formation water resistivity (Formation water resistivity) is the most important input to every saturation equation, and there are four ways to get it: a water analysis, the SP, the apparent resistivity Apparent water resistivity in a water-bearing zone, and a Pickett plot. This page gives each and the calculator converts a water salinity to Rw at formation temperature.

Inputs and outputs

Equations

From a water analysis. The salinity of a formation water sample, as sodium chloride equivalent, gives the resistivity at the 77 °F reference temperature. A common fit is:

with the salinity \(\Salw\) in ppm. A resistivity at one temperature is moved to another with the Arps relation (temperatures in °F):

The formation temperature is the surface temperature plus the geothermal gradient times depth, or a corrected bottom-hole temperature.

From the SP. In a thick, clean, water-bearing sand the static SP relates the resistivities of mud filtrate and water:

with the formation temperature in °F. The equivalent resistivities are converted to \(\Rmf\) and \(\Rw\) at the same temperature, using a chart or an approximation.

From the apparent resistivity. In a clean water-bearing zone, the Archie equation with \(\Sw = 1\) gives \(\Rw\) directly from the resistivity and porosity:

In a hydrocarbon zone the same expression gives a value above \(\Rw\), so the lowest \(R_{wa}\) in a clean zone is the Rw pick.

From a Pickett plot. On a log-log plot of resistivity against porosity, points at \(\Sw = 1\) form a straight line of slope \(-m\) with the intercept \(\aTort\,\Rw\) at a porosity of 1. Lines of constant \(\Sw\) are parallel to it:

Single-value calculator

Behavior

Resistivity falls as salinity rises and as temperature rises. At 77 °F, 10,000, 50,000 and 150,000 ppm give 0.564, 0.131 and 0.0539 ohm·m. The temperature effect is large: heating from 77 °F to 200 °F multiplies Rw by 0.405, so 50,000 ppm falls from 0.131 to 0.0531 ohm·m and 150,000 ppm from 0.0539 to 0.0218 ohm·m. A saturation computed with Rw at surface temperature in a hot reservoir is therefore too high by roughly 57% (the square root of the ratio, \(1/\sqrt{0.405}\) = 1.57, for \(n = 2\)).

Parameter guidance

Water catalog or sample. The most reliable Rw is from a produced-water analysis of the same formation, corrected to formation temperature. Watch for contamination with mud filtrate and for fresher water from other zones. SP. Use only in thick, clean, water-bearing sands with a good \(R_{mf}\) and no large invasion effect. Result is approximate in shaly sand, because shale reduces the SP deflection: see the Clay Volume from SP page for the baselines. Rwa and Pickett. Use the 100% water zone, clean and with good porosity, and check that the pick is at formation temperature. In a hydrocarbon column, take the minimum \(R_{wa}\) over the clean intervals, not the average. Rw varies between formations and fault blocks, so choose one value per zone. Temperature. Use one geothermal gradient per area, typically around 1.0 to 1.6 °F per 100 ft, and the same temperature for \(R_w\), \(R_{mf}\) and the \(B\) term of Waxman-Smits. Shared picks are covered on the step page (Water Saturation).

Worked example

Rw at formation temperature from a 50,000 ppm water at 200 °F, then the Rwa of a water-bearing and a hydrocarbon-bearing zone, and an SP example:

import math

# 1. water analysis to Rw at formation temperature
ppm, temp = 50000.0, 200.0
rw77 = 0.0123 + 3647.5 / ppm ** 0.955
rw = rw77 * (77.0 + 6.77) / (temp + 6.77)
print(f"Rw at 77 F = {rw77:.4f} ohm.m   Rw at {temp:g} F = {rw:.4f} ohm.m")

# 2. apparent Rw from two zones (a = 1, m = 2, porosity 0.22)
a, m, phie = 1.0, 2.0, 0.22
for label, rt in (("water zone", 1.1), ("hydrocarbon zone", 25.0)):
    rwa = rt * phie ** m / a
    print(f"{label:17s} Rt = {rt:5.1f}   Rwa = {rwa:.4f}   Rwa/Rw = {rwa / rw:.1f}")

# 3. SP: SSP = -60 mV at 200 F, equivalent filtrate resistivity 0.25 ohm.m
ssp, rmfe = -60.0, 0.25
k = 61.0 + 0.133 * temp
rwe = rmfe / 10 ** (-ssp / k)
print(f"K = {k:.1f}   Rwe = {rwe:.4f} ohm.m")

Output

Rw at 77 F = 0.1310 ohm.m   Rw at 200 F = 0.0531 ohm.m
water zone        Rt =   1.1   Rwa = 0.0532   Rwa/Rw = 1.0
hydrocarbon zone  Rt =  25.0   Rwa = 1.2100   Rwa/Rw = 22.8
K = 87.6   Rwe = 0.0516 ohm.m

Assumptions and limitations

• The salinity fit is for sodium chloride solutions. Waters with a lot of calcium, magnesium or sulfate have a different resistivity at the same total dissolved solids, and need an ionic-composition correction.
• The Arps temperature correction is an approximation that works over the usual reservoir range of 75 to 350 °F.
• The formation water is a single, uniform water across the zone, with no mixing with filtrate, injected or fresher waters.
• The apparent Rw and Pickett methods rely on a clean, 100% water-bearing zone, a correct porosity and correct a, m values. Wrong m or porosity moves Rw as well.
• The SP method needs a thick, clean sand, no strong invasion, and a known mud filtrate resistivity. Salinity contrast between filtrate and water must be significant.

QC checks

• Rw from different methods agrees to within 20 to 30%. If it does not, find out why before choosing one.
• Rwa in a known water leg is flat with depth and equal to the chosen Rw; Rwa in the hydrocarbon zone is higher.
• The Rw is at formation temperature, not at the surface.
• The Pickett plot has the 100% water points on the line and the hydrocarbon points above it, with the same a, m and n that are used in the saturation equation.
• Salinity implied by the chosen Rw is consistent with water analyses from the field and with the regional trend.

Going Deeper

The relation between salinity and resistivity goes back to the early log analysis charts, and the common numerical fits (such as the one used here) reproduce those charts for sodium chloride solutions. The Arps temperature relation is a straight-line rule from measurements of brines. Neither is exact, and the disagreement between methods is often useful: a Rwa much lower than the catalog Rw points to fresh filtrate invasion or a shaly interval, and the SP method is known to read fresher than the actual water in shaly rocks. The Pickett plot is the standard tool for finding Rw and m together in a wet interval, and also shows at a glance which points are hydrocarbon bearing. In practice many wells need an Rw per zone, because formation waters are not uniform across a field.

References

1. Arps, J.J., 1953. The effect of temperature on the density and electrical resistivity of sodium chloride solutions. Transactions of the AIME, 198, 327–330.
2. Pickett, G.R., 1966. A review of current techniques for determination of water saturation from logs. Journal of Petroleum Technology, 18(11), 1425–1433.
3. Bateman, R.M. and Konen, C.E., 1977. The log analyst and the programmable pocket calculator. The Log Analyst, 18(5), 3–11.

Python reference implementation

Related methods

• Archie
• Cementation and Saturation Exponents
• Rxo and Sxo
• Total vs Effective Sw