Publication History: This article is based on Chapter 8 of "The Log Analysis Handbook" by E. R. Crain, P.Eng., published by Pennwell Books 1986  Updated 2004, 2015, 2016, 2018. This webpage version is the copyrighted intellectual property of the author. Do not copy or distribute in any form without explicit permission. Formation Water Resistivity BASICS
Most methods for computing water saturation require knowledge of formation water resistivity at the formation temperature, so it is a necessary evil along our step-by-step path to find out how much oil and gas is in the ground.

Sodium chloride makes up the majority of the dissolved solids in hydrocarbon reservoirs, but numerous other compounds may be present. When salts dissolve, they break down into their ions, such as Na, Cl, Mg, SO4, K, Ca, and many others. Pure water has near infinite resistivity; it is these ions that make water conductive.

Water resistivity decreases with increased salinity, and for a given salinity, water resistivity also decreases with increased temperature. At any given temperature, there is a maximum salinity that can be achieved, above which salt crystals will begin to precipitate. In non-geothermal reservoirs, this limit is between 225,000 and 325,000 parts per million (ppm) total dissolved solids (TDS).

Produced water samples can be analyzed in the laboratory for their chemical composition and water resistivity. Less accurate water resistivity values can be measured at the well site or can sometimes be derived from well log data. In all cases, manipulation of the RW values to account for temperature will probably be needed. Conversion between salinity and resistivity and vice versa is commonly needed. And of course some method for estimating formation and surface temperatures will be required.

This page develops the math to handle all these critical factors, after first looking at the how lab data is presented to us in the water analysis reports. Laboratory Water Analysis REPORTS
Laboratories usually measure from 9 to 15 of the individual ions in a water sample, recorded in milligrams/litre (mg/l) or grams/cubic meter (g/m3). These two sets of units are equivalent: 1 mg/l = 1 g/m3.  At low to moderate concentrations, one mg/l is very close to 1 part per million (ppm) so mg/l and ppm tend to be used interchangeably.

In older reports, results were quoted in
grains per gallon (gpg)  One grain per US gallon equals 17.1 mg/l or approximately 17.1 ppm.

Drill stem test or produced water recoveries in your well, or from nearby wells, are analyzed for chemical content and water resistivity in the laboratory. A sample analysis is shown below.

The chemical analysis is recorded in milligrams per kilogram (mg/kg), or milligrams per liter (mg/l). Water analysis report from a drill stem test recovery, showing chemical analysis, calculated and measured water resistivity, and Stiff diagram of chemical analysis. Equivalent NaCl Water Salinity from Water Analysis
The resistivity of a water sample can be calculated from its chemical analysis. To do this, an equivalent NaCl concentration must be determined based on the ionic activity of each ion. Enter chart with total solids concentration of the sample in ppm (mg/kg) to find weighting factors for each ion present. The concentration of each ion is multiplied by its weighting factor, and the products for all ions are summed to obtain equivalent NaCl concentration.

The math is pretty simple:
1: TDS = SUM (IONi)
2: WSe = SUM (IONi * FACTRi)

Where:
TDS = total dissolved solids (ppm)
IONi = ion concentration of ith component (ppm)
FACTRi = multiplier factor for ith component (ppm)
WSe = equivalent NaCl concentration (ppm)

NUMERICAL EXAMPLE:
Assume formation-water sample analysis
460 ppm Ca,
1400 ppm SO4
19,000 ppm Na plus Cl.

Total dissolved solids concentration
TDS = 460 + 1400 + 19,000 = 20,860 ppm
Entering the chart with this total solids concentration
Ca multiplier = 0.81
SO4 multiplier = 0.45
Na+CL multiplier = 1.00

Equivalent NaCl concentration = 460 ´ 0.81 + 1400 ´ 0.45 + 19,000 ´ 1.0 = 20,000 ppm. Water Salinity from Chloride Content
Sometimes salinity is reported at the well site in ppm Chlorides instead of ppm NaCl equivalent.
6: WSa = Ccl * 1.645

Where:
Ccl = water salinity (ppm Cl)
WSa = water salinity (ppm NaCl)

Use this relationship when chloride content of the water sample is known. Usually Cl content is derived at the well site from a drill stem test recovery. It is useful as a first approximation until the water sample is analyzed more accurately at a laboratory. The relationship is for pure NaCl solutions and the factor may be higher or lower if other ions are present.

NUMERICAL EXAMPLE:
1. Chloride concentration to salinity.
WS = 11,600 ppm Cl * 1.645 = 19,000 ppm NaCl Water Resistivity from Salinity AT ANY TEMPERATURE
Crain's Model is used to convert a lab measured salinity to a formation water resistivity (RW) at any specific temperature (FT) in degrees Fahrenheit. The result is abbreviated as
RW@FT throughout this Handbook. You can use equation 5 to convert a salinity to any arbitrary temperature, for example 75F or 77F (roughly 25C) to find the resistivity at laboratory conditions.
1: FT = SUFT + (BHT - SUFT) / BHTDEP * DEPTH
2: IF LOGUNITS\$ = "METRIC"
3: THEN FT1 = 9 / 5 * FT + 32
4: OTHERWISE FT1 = FT
5: RW@FT = (400000 / FT1 / WS) ^ 0.88

SALINITY FROM WATER RESISTIVITY
Invert Crain's equation to solve for WS given RW at a a specific temperature FT1.
6: WS = 400000 / FT1 / ((RW@ET) ^ 1.14)

Where:
BHT = bottom hole temperature (degrees Fahrenheit or Celsius)
BHTDEP = depth at which BHT was measured (feet or meters)
DEPTH = mid-point depth of reservoir (feet or meters)
FT = formation temperature (degrees Fahrenheit or Celsius)
FT1 = formation temperature (degrees Fahrenheit)
RW@FT = water resistivity at formation temperatures (ohm-m)
SUFT = surface temperature (degrees Fahrenheit or Celsius)
WS = water salinity (ppm NcCl)

Use this relation if salinity is known from laboratory measurements to obtain RW from lab data at any temperature.

NUMERICAL EXAMPLE:
1. Salinity to water resistivity.
RW@FT = (400000 / 102'F / 20,000 ppm) ^ 0.88 = 0.238 ohm-m @ 102'F
(rounded to three significant digits)

2. Water resistivity to salinity.
WS = 400,000 / 102'F / ((0.250 ohm-m) ^ 1.14) = 19,000 ppm NaCl
(rounded to three significant digits) Water Resistivity from Salinity at Lab Temperature
These models generate RW at laboratory temperature of 75F or 25C.
Crain's Method (1974)
1: RW@75F = (400000 / 75 / WS) ^ 0.88

Bateman and Konen Method (1977)
2: RW@75F = 0.0123 + (3647.5 / WS^0.955)

Kennedy's Method (2015)
3:  RW@75F = 1 / (24.30853 - 0.0364 * (0.1 * WS - 29.46515957) - 0.02922 * (0.1 * WS - 29.46515957)^2)

SALINITY FROM WATER RESISTIVITY
Crain's Method (1974)
4: WS = 400000 / 75 / ((RW@75) ^ 1.14)

Baker Atlas Method (2002)

5: WS = 10 ^ ((3.562 - (Log (RW@75 - 0.0123))) / 0.955)

For all practical purposes, the three models give the same RW value (see Graph 1 below). There are minor differences above 150,000 ppm NaCl which can only be seen when water resistivity is converted to water conductivity (see Graph 2 below).

The effect on water saturation (SW%) is not very significant (+/- 0.5% SW at low SW, +/- 2% SW at high SW, at 200,000 ppm NaCl).

The 10 significant digits used in the Kennedy equation give a false sense of accuracy that is not warranted. META/SAL  Compare RW from NaCl Salinity (3 Methods) Graph 1: Rw Models - Red line = Crain, Black line = Bateman and Konen, Blue line = Kennedy Graph 2: Cw Models - Red line = Crain, Black line = Bateman and Konen, Blue line = Kennedy.
The differences above 150,000 ppm NaCl have little impact on water saturation.

NUMERICAL EXAMPLE:  RW from  Water Catalogs
Water catalogues published by your local well logging society or similar catalogues created by searching in-house data bases are a necessary tool for well log analysis. A sample is shown below. A sample of RW data from a water resistivity catalog, data is tabulated and also
posted on a map, and is based on a standard temperature of 25 degrees Celsius (77 degrees Fahrenheit). Water resistivity values in a catalog are recorded at a standard temperature, usually 75F or 77F (25C). Since water resistivity varies inversely with temperature, the catalog values must be transformed to a new value representing water resistivity at formation temperature (RW@FT) -- see next Section.

To use data from a water catalogue, it is usually necessary to do a little filtering. Nearly everything that can go wrong will raise the RW value recorded in the catalogue. Usually the minimum value from nearby offset wells is the best choice. It may be useful to gather all the values for the reservoir for a radius of 3 to 6 miles (5 to 10 Km) and prepare a histogram. On the histogram, find the point that represents the lower decile (10% of the data values are less than this value). Take the average of the data in this decile. You may want to eliminate obvious "impossible" values before you make the histogram.

The following relationships are needed to manipulate water resistivity data prior to calculations of water saturation.

1. Arps Method (1953)
1: FT = SUFT + (BHT - SUFT) / BHTDEP * DEPTH
2: KT1 = 6.8 for English units    KT1 = 21.5 for Metric units
3: RW@FT = RW@TRW * (TRW + KT1) / (FT + KT1)
4: RMF@FT = RMF@TRW * (TRW + KT1) / (FT + KT1)
5: RMC@FT = RMC@STRW + (TRW * KT1) / (FT + KT1)

Where:
SUFT = surface temperature for temperature gradient (degrees Fahrenheit or Celsius)
BHT = bottom hole temperature (degrees Fahrenheit or Celsius)
BHTDEP = depth at which BHT was measured (feet or meters)
DEPTH = mid-point depth of reservoir (feet or meters)
FT = formation temperature (degrees Fahrenheit or Celsius)
RMC@FT = mud cake resistivity at formation temperature (ohm-m)
RMC@TRW = mud cake resistivity at surface temperature (ohm-m)
RMF@FT = mud filtrate resistivity at formation temperature (ohm-m)
RMF@TRW = mud filtrate resistivity at surface temperature (ohm-m)
RW@FT = water resistivity at formation temperatures (ohm-m)
RW@TRW = water resistivity at surface temperature (ohm-m)
TRW = temperature at which water was measured (degrees Fahrenheit or Celsius)

2. Hilchie Model (1984) ALL temperatures in Fahrenheit.
6: KT1 = 10^ ( --0.340396 * log(RW@TRW) + 0.641427)
7: FT1 = SUFT + (BHT - SUFT) / BHTDEP * DEPTH
8: RW@FT = RW@TRW * (TRW + KT1) / (FT1 + KT1)
9: RMF@FT = RMF@TRW * (TRW + KT1) / (FT1 + KT1)
10: RMC@FT = RMC@STRW + (TRW * KT1) / (FT1 + KT1)

Where:
FT1 = formation temperature (degrees Fahrenheit ONLY)

Use these relations when RW@TRW is known from measured data. This transformation can be made on the chart below. The Hilchie model accounts for the slight curvature at low and high temperatures on the chart, but Arps model is quite sufficient for practical petrophysics.

NUMERICAL EXAMPLE:
1. Water resistivity at formation temperature.
English units example:
RW@FT = (0.32 ohm-m @ 77'F) * (77 + 6.8) / (102 + 6.8) = 0.25 ohm @ 102'F

Metric units example:
RW@FT = (0.32 ohm-m @ 25'C) * (25 + 21.5) / (39 + 21.5) = 0.25 ohm-m @ 39'C Schlumberger Chart GEN-9: Water resistivity - Temperature - Salinity relationships ESTIMATING SURFACE AND FORMATION TEMPERATURES
Temperature measurements specific to your area of interest are going to sparse and you will have to do some searching for useful data. The map below gives a good idea of what to use for surface temperature (SUFT). Temperature versus depth data from log headings can be plotted to estimate a best fit temperature gradient line. It doesn't have to be a straight line. See representative graphs below. In areas with sparse data, use the temperature gradient maps supplied below. Mean Annual Surface Temperature Map for North America - degrees F
The 40 deg F contour follows roughly along the US-Canada border except along the coast lines.
Contour interval is 5 deg F. Typical Depth - Temperature profiles. Create your own by best fit of BHT vs BHTDEP from data on log
headings or DST reports. Temperature gradient for USA - degrees Celsius per 1000 meters,
North American Heat Floe Map (3 MB)    Legible Legend for NA Map Temperature at 5000 meters for Australia RW from a Water Zone
Back calculation of RW@FT from log data in a clean (non shaly) zone - usually called the Rwa method, or the water zone (Ro or R0) method is commonly used when obvious water zones exist near the zone of interest. RW@FT_3 - Water resistivity from water zone
The following algorithm is used to back calculate water resistivity from a known water zone.
1: RW@FT = (PHIt ^ M) * RESD / A
2: RMF@FT = (PHIt ^ M) * RESS / A
3: RMC@FT = 2.0 * RMF@FT

Where:
A = tortuosity exponent (unitless)
M = cementation exponent (unitless)
PHIt = total porosity found by log analysis (fractional)
RESD = deepest resistivity log reading (ohm-m)
RESS = shallowest resistivity log reading (ohm-m)
RMC@FT = mud cake resistivity at formation temperature (ohm-m)
RMF@FT = mud filtrate resistivity at formation temperature (ohm-m)
RW@FT = water resistivity at formation temperatures (ohm-m)

Use this relationship if no measured values of RW are available and only if data from a clean water zone can be found. A nomographic solution is given below.

This method is often called the Rwa method

Porosity should be greater than 0.06.

Note that results are at the formation temperature. To compare these values to catalog values at 25 degrees Celsius, use the temperature transformation from the previous algorithm or the nomograph below.

RECOMMENDED PARAMETERS:
for carbonates A = 1.00  M = 2.00  N = 2.00  (Archie Equation as first published)
for sandstone  A = 0.62  M = 2.15  N = 2.00  (Humble Equation)
A = 0.81  M = 2.00  N = 2.00  (Tixier Equation - simplified version of Humble Equation)
NOTE: N is often lower than 2.0

For quick analysis use carbonate values. Values for local situations should be developed from special core data. Results will always be better if good local data is used instead of traditional values, such as those given above.

Asquith (1980 page 67) quoted other authors, giving values for A and M, with N = 2.0, showing the wide range of possible values:

Average sands              A = 1.45  M = 1.54
Shaly sands                  A = 1.65  M = 1.33
Calcareous sands         A = 1.45  M = 1.70
Carbonates                   A = 0.85  M = 2.14
Pliocene sands S.Cal.  A = 2.45  M = 1.08
Miocene LA/TX             A = 1.97  M = 1.29
Clean granular             A = 1.00  M = 2.05 - PHIe Water resistivity from water zone data (Rwa Method)

NUMERICAL EXAMPLE:
1. Assume data for water zone
Sand A Sand B Sand C Sand D
RESD        6 0       40        0.3         0.5
PHIt         0.33    0.14      0.30       0.11
A =  0.62
M = 2.15
RW@FT     0.89    0.94   0.036   0.007

Sample:
RW@FT = Rwa = (0.33 ^ 2.15) * 6.0 / 0.62 = 0.89

The RW@FT values represent the first approximation to a value of water resistivity for each of the four zones. The value for Sand D is not very realistic, and a better one will be found later when we look at shale corrections. RW from SP Log
Calculation from knowledge of the SP value in a clean zone has been a traditional method  for finding RW. It works best in clean water bearing zones, but the Rwa or R0 method would be better in this case. Shale content and hydrocarbon content reduce the SP value and cause RW to too high, giving very pessimistic saturation results.
1: FT = SUFT + (BHT - SUFT) / BHTDEP * DEPTH
2: IF LOGUNITS\$ = "METRIC"
3: THEN FT1 = 9 / 5 * FT + 32
4: OTHERWISE FT1 = FT
5: KSP = 60 + 0.122 * FT1
6: RSP = 10 ^ (-SSP / KSP)
7: IF RMF@FT > 0.1
8: THEN RMFE = 0.85 * RMF@FT
9: IF RMF@FT <= 0.1
10: THEN RMFE = (146 * RMF@FT - 5) / (337 * RMF@FT + 77)
11: RWE = RMFE / RSP
12: IF RWE > 0.12
13: THEN RW@FT = - (0.58 - 10 ^ (0.69 * RWE - 0.24))
14: IF RWE <= 0.12
15: THEN RW@FT = (77 * RWE + 5) / (146 - 337 * RWE)

Where:
BHT = bottom hole temperature (degrees Fahrenheit or Celsius)
BHTDEP = depth at which BHT was measured (feet or meters)
DEPTH = mid-point depth of reservoir (feet or meters)
FT1 = formation temperature (degrees Fahrenheit)
KSP = temperature factor (degrees Fahrenheit)
RSP = RWE / RMFE (fractional)
RMFE = equivalent mud filtrate resistivity (ohm-m)
RMF@FT = mud filtrate resistivity at formation temperature (ohm-m)
RWE = equivalent water resistivity (ohm-m)
RW@FT = water resistivity at formation temperatures (ohm-m)
SSP = static SP reading in clean zone (ohm-m)
SUFT = surface temperature (degrees Fahrenheit or Celsius)

This algorithm should only be used IF the SP log has sufficient character, the zone of interest is a clean water bearing sandstone, and the result is reasonable for the area. Use caution since many SP logs are not calibrated, and RMF or RW can be measured carelessly.

Solution of these formulae can be done on the two charts below. An Excel macro for RW from SP, published by SPWLA, is located HERE.  Nomographs for RW from SP

NUMERICAL EXAMPLE:
1. Data for Sand C:
SUFT = 25 degrees C
BHT = 65 degrees C
BHTDEP = 2225 m
DEPTH = 1000 m
RMF = 0.75 @ 25 degrees C
SP = -90 mv

FT = 25 + (65 - 25) / 2225 * 1000 = 43 degrees C
FT1 = 9 / 5 * 43 + 32 = 109 degrees F
RMF@FT = 0.75 * (25 + 21.5) / (43 + 21.5) = 0.54
KSP = 60 + 0.122 * 109 = 73.3
RSP = 10 ^ (90 / 73.3) = 16.9
RMFE = 0.85 * 0.54 = 0.46
RWE = 0.46 / 16.9 = 0.027
RW@FT = (77 * 0.027 + 5) / (146 - 337 * 0.027) = 0.051 Selection of Rw from Various Sources
Water resistivity data can be sparse or overwhelming, depending on where you are working at the moment. The usual sources in order of preference are:

1. Produced water from the zone being analyzed in the same well or nearby offset wells, analyzed for Rw in the lab.

2. Drill stem test or perf test water from the zone being analyzed in the same well or nearby offset wells, analyzed for Rw in the lab. The test should produce at least 1000 ft (300 m) of water before using the data, to prevent mud filtrate contamination from causing errors. The sample should be from the bottom of the test.

3. Produced or DST water from a nearby zone in the same geologic horizon (do not cross erosional boundaries), analyzed as above.

4. Water catalogues produced by local well log societies or government agencies.

5. Back calculated from log data in clean water bearing zone in the same well or nearby offset well (Rwa or Ro method).

6. Back calculated from nearby water bearing zone in same geologic horizon.

7. Calculated from SP in clean water bearing zone in same or nearby zone in same well or nearby offset well.

8. If no water has ever been produced in the area, back calculated from a laboratory measured or assumed PHIxSW product.

9. Local rule of thumb for water resistivity versus depth or versus geologic horizon.

Do not use:

1. Water from a DST or perf test that recovered mostly filtrate water (check water chemistry) or recovered only a small amount of water.

2. SP or Rwa in a shaly zone.

3. SP or Rwa in a hydrocarbon bearing zone.

4. SP in a carbonate or evaporite sequence.

5. SP in a low porosity zone. "META/RW" SPREADSHEET -- Water Resistivity Calculations
This spreadsheet calculates RW at formation temperature using 5 different methods
. Calculate RW at formation temperature. Metric and English Units Sample of "META/RW" for calculating water resistivity from various methods.

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