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Well logs provide insight into the formations and conditions in the subsurface, aimed primarily at detection and evaluation of possibly productive horizons.

## Determination of saturation

Water saturation is the fraction of the pore volume of the reservoir rock that is filled with water. It is generally assumed, unless otherwise known, that the pore volume not filled with water is filled with hydrocarbons. Determining water and hydrocarbon saturation is one of the basic objectives of well logging.

### Clean formations

All water saturation determinations from resistivity logs in clean (nonshaly) formations with homogeneous intergranular porosity are based on Archie’s water saturation equation, or variations thereof. The equation is ....................(1)

Rw is the formation water resistivity, Rt is the true formation resistivity, and F is the formation resistivity factor. F is usually obtained from the measured porosity of the formation through the relationship ....................(2)

For Sxo, the water saturation in the flushed zone, a similar expression exists: ....................(3)

where Rmf is the mud filtrate resistivity and Rxo is the flushed zone resistivity.

For simplicity, the saturation exponent n is usually taken as 2. Laboratory experiments have shown that this is a reasonable value for average cases. For more exacting work, electrical measurements on cores will produce better numbers for n, a, and m. When core measured values are unavailable, the values of a and m in Eq. 4 can be estimated as follows: in carbonates, F=1/ϕ2 is usually used; in sands, F=0.62/ϕ2 (Humble formula), or F=0.81/ϕ2 (a simpler form practically equivalent to the Humble formula). These equations are easily programmed into spreadsheets and are available in most log interpretation software. ....................(4) ....................(5)

The accuracy of the Archie equation, Eq. 1 and its derivatives, depends in large measure, of course, on the accuracy of the fundamental input parameters: Rw, F, and Rt. The deep resistivity measurement (induction or laterolog) must be corrected, therefore, for borehole, bed thickness, and invasion (see the page Formation resistivity determination for more details). It is almost never safe to make the assumption "deep = Rt." The most appropriate porosity log (sonic, neutron, density, magnetic resonance, or other) or combination of porosity and lithology measurements must be used to obtain porosity, and the proper porosity-to-formation factor relationship must be used. Finally, the Rw value should be verified in as many ways as possible: calculation from the SP curve, water catalog, calculation from nearby water-bearing formation, and/or water sample measurement.

Alternate methods for determining water saturation include analysis of cores cut with low-invasion oil-based muds (OBMs) and single well chemical tracer (SWCT) tests. These independent methods can be used to calibrate log analyses.

### Resistivity vs. porosity crossplots

Combining Eqs. 1 and 5, the Archie saturation equation may be written ....................(6)

If n and m are equal to 2, and a = 1, then ....................(7)

Eq. 7 shows that for Rw constant, ϕSw is proportional to is the quantity of water per unit volume of formation. To emphasize the proportionality between ϕ and , Eq. 7 may be rewritten: ....................(8)

For a 100% water-saturated formation, Sw = 1 and Rt = R0. If R0 for water-saturated formations is plotted on an inverse square-root scale vs. ϕ, all points should fall on a straight line given by .

Furthermore, the points corresponding to any other constant value of Sw will also fall on a straight line, because in Eq. 7 the coefficient is constant for constant values of Rw and Sw.

Fig. 1 shows several points plotted over an interval in which formation-water resistivity is constant (as indicated by constant SP deflections opposite the thick, clean permeable beds). Assuming that at least some of the points are from 100% water-bearing formations, the line for Sw = 1 is drawn from the pivot point (ϕ = 0, Rt = ∞) through the most northwesterly plotted points. The slope of this line defines the value of Rw as shown on Fig. 1, for ϕ = 10%, R0 = 6.5 ohm•m. For this formation, the most appropriate Fϕ relation is F = 1/ϕ2. Thus, for ϕ = 10%, F = 100. Because Rw = R0/F, Rw = 0.065 ohm•m, as shown.

For other Sw values, Rt and R0 are related by the equation Rt = R0/Sw2. For Sw = 50%, and 1/Sw2=4 , and Rt = 4 R0. This relation establishes the line for Sw = 50%. Other Sw lines may be defined in a similar manner.

If the matrix composition remains constant over the formations under investigation, the basic measurement from the sonic, density, or neutron logs can be plotted directly vs. Rt with similar results. This is possible because of the linear relationship between porosity and bulk density, sonic transit time, or neutron-hydrogen index response. An example of a sonic-induction crossplot is shown in Fig. 2. The transit time has been plotted against the induction resistivity for several levels. The northwesterly points define the 100% water saturation line. The transit-time value at the point where this line intersects the horizontal line of infinite resistivity is the matrix-transit time, tma In Fig. 2, tma is found to be approximately 47.5 μs/ft (156 μs/m). This corresponds to a matrix velocity of 21,000 ft/sec (6,400 m/s).

By knowing tma, a porosity scale, a scale of formation factor (e.g., from F = 1/ϕ2) can be easily derived. A vertical line drawn through F = 100 (or ϕ = 10) intersects the water line at R0 = 5 ohm•m; accordingly, Rw (= R0/F) is 0.05 ohm•m.

The lines for other Sw values are straight lines, determined as previously described, radiating out from the Rt =∞, tma = 47.5 pivot point.

Density and neutron logs can be crossplotted against resistivity in a manner identical to the sonic logs. For density logs, the intersection of the 100% water line with the infinite-resistivity line yields the matrix-density value, ρma. For neutron logs, the intersection defines the matrix-hydrogen index, or apparent matrix porosity. Knowledge of matrix density or hydrogen index permits the ρB or ϕN scale to be rescaled in ϕ and F units. With the F scale defined, Rw can be calculated as for the sonic-resistivity crossplot, and lines of constant water saturation can be constructed in a similar manner.

These resistivity-vs.-porosity crossplots require that formation water resistivity be constant over the interval plotted, that lithology be constant, that invasion not be deep, and that the measured-porosity log parameter (i.e., t, ρB, or ϕN) can be linearly related to porosity. This last condition implies that the time-average transform for the conversion of t into porosity is appropriate.

The neutron-resistivity crossplot is not as satisfactory in gas-bearing formations as are the sonic- or density-resistivity crossplots. The apparent porosity measured by the neutron log in gas zones is often much too low. This results in overstated Sw values in gas zones. Indeed, in a gas zone, the neutron resistivity may indicate a porous gas-bearing zone to be near zero porosity and 100% water bearing. In contrast, the sonic- or density-resistivity tends to be slightly optimistic in gas zones (i.e., porosities may be slightly high and water saturations slightly low).

### Microresistivity vs. porosity crossplots

This method is particularly useful for older logs or cases in which the analyst has only a paper copy of the log. A resistivity-porosity plot can also be made using the values from a shallow-investigation resistivity log such as the microlaterolog, MSFL, or MCFL log. If the microresistivity log reads approximately Rxo, then a line through points of mud-filtrate-saturated formations (Sxo = 1) should have a slope related to Rmf. Rmf is an important parameter, and this check of its value by means of a sonic-microresistivity or density-microresistivity crossplot is often useful.

These plots are also valuable for improved determinations of matrix parameters (either tma or ρma), particularly in cases where the sonic-resistivity or density-resistivity plot does not give a clear answer because of hydrocarbon saturation. The F Rmf line should be easier to determine because Sxo is usually fairly high even in hydrocarbon-bearing formations.

Fig. 3 shows a resistivity-porosity plot in which both the deep induction reading and the microlaterolog at the same levels are plotted in a series of water-bearing formations. The porosity values were derived in this case from a neutron-density crossplot. The plots from the two logs define two trends corresponding respectively to Sw = 1 (using deep induction) and Sxo = 1 (using microlaterolog data). The points not in these trends can be divided into two groups:

1. Points whose microlaterolog readings fall on the Sxo = 1 line but whose deep induction log readings fall below the Sw = 1 line (Points 2, 9, and 10) are probably the result of either deep invasion or adjacent-bed effect in which deep resistivity is greater than Rt.
2. Points whose induction log readings fall on the Sw = 1 line but whose microlaterolog points fall above the Sxo = 1 line are possibly a result of shallow invasion in which RMLL is lower than Rxo.

Resistivity-porosity plots are thus often more informative if the short-spaced resistivity or medium-induction values are also plotted. Not only does this permit an appreciation of invasion effects, but it may also indicate moved oil.

## Rwa comparison

If water saturation is assumed to be 100%, the Archie water saturation equation (Eq. 1) reduces to ....................(9)

The term Rwa is used in Eq. 9 rather than Rw to indicate that this is an apparent formation water resistivity. It is only equal to Rw in 100% water-bearing formations. In hydrocarbon-bearing formations, Rwa computed from Eq. 3 will be greater than Rw. Indeed, by combining Eqs. 10 and 5, the relationship between Sw, Rwa, and, Rw can be shown to be ....................(10)

The Rwa technique can, therefore, be useful for identifying potential hydrocarbon-bearing zones and for obtaining Rw values.

In practice, Rwa is obtained by simply dividing the deep induction resistivity (or deep laterolog resistivity) by the formation factor obtained from a porosity log or a combination of porosity logs. Today, a continuous Rwa computation is made over a long interval of the borehole in real time. If one has only paper logs, many individual manual computations are made so as to approximate a continuous computation.

### Resistivity-ratio methods

In resistivity-ratio methods, it is assumed that a formation is divided into two distinct regions—a flushed zone and a noninvaded zone. Both zones have the same F, but each contains water of a distinct resistivity (Rmf in the invaded zone and Rw in the noninvaded zone). The resistivities of the two zones must be measurable or derivable from logs, and methods for determining the resistivity of the water in each zone must be available.

Because of the necessary assumptions, the resistivity-ratio methods have limitations, but when no porosity or formation factor data are available, they are sometimes the only choice. The principal limitation arises from the inability of any resistivity device to measure either Rx or R, totally independent of the other. Simply put, invasion must be deep enough to allow a shallow investigating resistivity device to measure Rxo but not so deep that a deep-resistivity device cannot measure Rt.

Another difficulty appears when hydrocarbons are present. In this case, some knowledge or assumption of the value of the flushed or invaded zone saturation is necessary.

### Flushed-zone method

If n = 2 is assumed and Eq. 1 is divided by Eq. 3, ....................(11)

This equation gives the ratio of Sw to Sxo, and no knowledge of formation factor or porosity is needed. Rxo may be found from a microresistivity log, Rt from an induction or laterolog, and Rmf/Rw from measured values or from the SP curve.

The ratio is valuable in itself as an index of oil movability. If Sw/Sxo = 1, then no hydrocarbons have been moved by invasion, whether or not the formation contains hydrocarbons. If Sw/Sxo is approximately 0.7 or less, movable hydrocarbons are indicated. The value of Sw/Sxo, along with ϕ and Swo, is useful in evaluating reservoirs.

To determine Sw from Eq. 12, Sxo must be known. For moderate invasion and average residual oil saturation, an empirical relation between Sw and Sxo has been found useful: Sxo = Sw1/5. Inserting this into Eq. 11 gives: ....................(12)

Service companies provide charts for graphical solution of this equation, or it can be easily programmed into a spreadsheet.

The invaded-zone method is useful for water saturation determination when only an ES, IES, or other early-resistivity log is available and no porosity-log or formation-factor data exist. (This section also uses some early nomenclature.) For application of the method, Ri/Rm must be at least 10.

Archie’s equation for the invaded zone is ....................(13)

where Rz is the resistivity of the water in the invaded zone. Because of incomplete flushing, Rz is a mixture of mud filtrate, Rmf, and formation water, Rw.

Studies of many logs suggest that Si and Sw are related by ....................(14)

Dividing the noninvaded-zone water saturation equation (Eq. 1) by Eq. 13 and using the relationship presented in Eq. 14 yields an expression for Sw: ....................(15)

To use Eq. 15, Rt is taken from a deep resistivity device such as a deep induction or deep laterolog (corrected for borehole effect and bed thickness). Ri is taken from a shallow resistivity device such as a Laterolog 8, 16-in. normal, or SFL (corrected for borehole effect and bed thickness).

Rz is given by the relationship ....................(16)

where z is the fraction of the invaded zone pore water, which is formation water, and 1 – z is the fraction that is mud filtrate. Experience has indicated that z varies from 0.075 in cases of normal invasion to 0.035 in cases of deep invasion or vuggy formations.

Fig. 4 solves Eq. 15. It is entered with Rmf/Rw on the appropriate z scale and Ri/Rt (oblique lines) to determine Sw. When Ri/Rt is close to unity, some caution is required. The formation may be extremely invaded or there may be little invasion, or it may be dense and impermeable. On the other hand, many good hydrocarbon-bearing reservoirs will have Ri/Rt ≈ 1.

## Rxo/Rt quicklook

The Rxo/Rt quicklook method can be used to identify hydrocarbon-bearing formations and to indicate hydrocarbon movability (producibility). When Sw/Sxo is 1 in a permeable zone, the zone will produce water or be nonproductive regardless of water saturation. A value Sw/Sxo significantly less than 1 indicates that the zone is permeable and contains some hydrocarbons, and that the hydrocarbons have been flushed (moved) by invasion. Thus, the zone contains producible hydrocarbon.

Eq. 11 can be written as ....................(17)

which shows that an indication of Sw/Sxo can be obtained by comparing Rxo/Rt with Rmf/Rw, where the subscript SP emphasizes that Rmf/Rw is derivable from the SP. Equivalently, the comparison can be between log Rxo/Rt and the SP curve for an indication of log Sw/Sxo.

The value of log Rxo/Rt is computed from solving the three or more resistivity logs for invasion parameters. It is used as an overlay comparison curve with the SP. Separations between the log Rxo/Rt curve, properly scaled to match the SP, and the SP curve provide a quick-look location of producible hydrocarbons.

Originally, log Rxo/Rt was computed from RLL8/RID or RSFL/RID. Use was made of the observation that over a wide range of invasion diameters (from approximately 20 to 100 in.), Rxo/Rt depends primarily on the value of RLL8/RID or RSFL/RID. The relationship used for the LL8 device was ....................(18)

For the SFL device, it was ....................(19)

Much more sophisticated algorithms are now used to obtain Rxo/Rt. These values are output in real time as separate logs.

To interpret the Rxo/Rt quick-look curve, the impermeable zones must be eliminated by reference to the SP, GR, or microlog curves or by resistivity ratios. Then, if the SP and Rxo/Rt (actually –K log Rxo/Rt) curves coincide in a permeable zone, the zone will most probably produce water. If, however, the Rxo/Rt curve reads appreciably lower (i.e., to the right) than the SP, the zone should produce hydrocarbons. An Rxo/Rt value less than the SP amplitude indicates movable hydrocarbons are present.

The Rxo/Rt quick-look technique is applicable to fresh mud conditions (Rxo > Rt) in formations where invasion falls within the limits demanded by the Rxo/Rt computation. For the simpler computation technique using Eq. 18 and –25, that is for di 30 to 70 in.; for the more sophisticated techniques, that is, between 20 and 120 in. Even in the more restrictive case, however, any errors are optimistic. In other words, water zones may appear to be hydrocarbon-productive. This constitutes a safeguard against overlooking pay zones, and it is considered a desirable feature in any quick-look approach.

The Rxo/Rt technique efficiently handles variations in formation water resistivity, Rw, and in shaliness. Any change in Rw is reflected similarly into both the computed Rxo/Rt and the SP amplitude. Thus, comparing the two curves still permits formation-fluid identification. Shaliness also affects the two curves in a similar manner. All other things remaining constant, shaliness reduces the Rxo/Rt value and the SP amplitude. Finally, the Rxo/Rt quick-look technique does not require porosity data, nor use of any Fϕ relationships.

Fig. 5 is an example of a shaly gas sand at 3,760 through 3,788 ft and several water-productive sands with varying amounts of shaliness. The productive-gas sand is identified by the separation between the Rxo/Rt and SP curves. Water-productive zones are shown by lack of separation. In shaly water zones, the variation in the SP curve is essentially the same as the variation in the Rxo/Rt ratio—a result of the same shale. Therefore, the comparison is not significantly disturbed by shaliness. Neither is it disturbed by variations in Rw.

Estimates of water saturation and saturation ratio in clean formations can be made by comparing the Rxo/Rt and SP curves. Eq. 17 permits Sw/Sxo to be estimated, and then Eq. 12 allows Sw to be estimated.

## Nomenclature

 amf = mud filtrate chemical activity aw = formation water chemical activity A = area, m2 di = diameter of invasion (in., m) Ek = electrokinetic potential Ekmc = electrokinetic potential of the mudcake Eksh = electrokinetic potential of shale F = formation factor relating resistivity to porosity g = induction-response function G = induction integrated radial-response function hmc = mudcake thickness I = electrical current, Amperes L = length, m r = resistance, ohm R = resistivity (ohm•m) Rann = resistivity of the annulus Rh = resistivity in the horizontal direction (ohm•m) Rm = resistivity of the mud column (ohm•m) Rmc = resistivity of the mudcake Rmf = resistivity of the mud filtrate Rxo = resistivity of the invaded zone Rt = resistivity of the uninvaded formation Rv = resistivity in the vertical direction (ohm•m) Rw = resistivity of the formation connate water (ohm•m) Rwa = apparent water resistivity from deep resistivity and porosity Sxo = water saturation of the invaded zone Sw = water saturation in the uninvaded zone t = acoustic travel time (μs/ft) tma = acoustic travel time of the rock matrix(μs/ft) V = electrical voltage, volts Vsd = fraction of the total formation volume that is sand Vsh = fraction of the total formation volume that is shale ρ = density ρma = density of the rock matrix σ = conductivity, mS/m σm = conductivity of the mud column, mS/m ϕ = porosity