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Well log interpretation
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.^{[1]}^{[2]} The equation is
R_{w} is the formation water resistivity, R_{t} 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
For S_{xo}, the water saturation in the flushed zone, a similar expression exists:
where R_{mf} is the mud filtrate resistivity and R_{xo} 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/ϕ^{2} is usually used; in sands, F=0.62/ϕ^{2}^{[3]} (Humble formula), or F=0.81/ϕ^{2} (a simpler form practically equivalent to the Humble formula). These equations are easily programmed into spreadsheets and are available in most log interpretation software.
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: R_{w}, F, and R_{t}. 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 = R_{t}." 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 R_{w} 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
If n and m are equal to 2, and a = 1, then
Eq. 7 shows that for R_{w} constant, ϕS_{w} 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:
For a 100% water-saturated formation, S_{w} = 1 and R_{t} = R_{0}. If R_{0} 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 S_{w} will also fall on a straight line, because in Eq. 7 the coefficient is constant for constant values of R_{w} and S_{w}.
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 S_{w} = 1 is drawn from the pivot point (ϕ = 0, R_{t} = ∞) through the most northwesterly plotted points. The slope of this line defines the value of R_{w} as shown on Fig. 1, for ϕ = 10%, R_{0} = 6.5 ohm•m. For this formation, the most appropriate F – ϕ relation is F = 1/ϕ^{2}. Thus, for ϕ = 10%, F = 100. Because R_{w} = R_{0}/F, R_{w} = 0.065 ohm•m, as shown.
For other S_{w} values, R_{t} and R_{0} are related by the equation R_{t} = R_{0}/S_{w}^{2}. For S_{w} = 50%, and 1/S_{w}^{2}=4 , and R_{t} = 4 R_{0}. This relation establishes the line for S_{w} = 50%. Other S_{w} 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. R_{t} with similar results.^{[4]} 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, t_{ma} In Fig. 2, t_{ma} 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 t_{ma}, a porosity scale, a scale of formation factor (e.g., from F = 1/ϕ^{2}) can be easily derived. A vertical line drawn through F = 100 (or ϕ = 10) intersects the water line at R_{0} = 5 ohm•m; accordingly, R_{w} (= R_{0}/F) is 0.05 ohm•m.
The lines for other S_{w} values are straight lines, determined as previously described, radiating out from the R_{t} =∞, t_{ma} = 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, ρ_{ma}. For neutron logs, the intersection defines the matrix-hydrogen index, or apparent matrix porosity. Knowledge of matrix density or hydrogen index permits the ρ_{B} or ϕ_{N} scale to be rescaled in ϕ and F units. With the F scale defined, R_{w} 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, ρ_{B}, or ϕ_{N}) 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 S_{w} 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 R_{xo}, then a line through points of mud-filtrate-saturated formations (S_{xo} = 1) should have a slope related to R_{mf}. R_{mf} 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 t_{ma} or ρ_{ma}), particularly in cases where the sonic-resistivity or density-resistivity plot does not give a clear answer because of hydrocarbon saturation. The F R_{mf} line should be easier to determine because S_{xo} 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 S_{w} = 1 (using deep induction) and S_{xo} = 1 (using microlaterolog data). The points not in these trends can be divided into two groups:
- Points whose microlaterolog readings fall on the S_{xo} = 1 line but whose deep induction log readings fall below the S_{w} = 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 R_{t}.
- Points whose induction log readings fall on the S_{w} = 1 line but whose microlaterolog points fall above the S_{xo} = 1 line are possibly a result of shallow invasion in which RMLL is lower than R_{xo}.
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.
R_{wa} comparison
If water saturation is assumed to be 100%, the Archie water saturation equation (Eq. 1) reduces to
The term R_{wa} is used in Eq. 9 rather than R_{w} to indicate that this is an apparent formation water resistivity. It is only equal to R_{w} in 100% water-bearing formations. In hydrocarbon-bearing formations, R_{wa} computed from Eq. 3 will be greater than R_{w}. Indeed, by combining Eqs. 10 and 5, the relationship between S_{w}, R_{wa}, and, R_{w} can be shown to be
The R_{wa} technique can, therefore, be useful for identifying potential hydrocarbon-bearing zones and for obtaining R_{w} values.
In practice, R_{wa} 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 R_{wa} 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 (R_{mf} in the invaded zone and R_{w} 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 R_{x} or R, totally independent of the other. Simply put, invasion must be deep enough to allow a shallow investigating resistivity device to measure R_{xo} but not so deep that a deep-resistivity device cannot measure R_{t}.
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,
This equation gives the ratio of S_{w} to S_{xo}, and no knowledge of formation factor or porosity is needed. R_{xo} may be found from a microresistivity log, R_{t} from an induction or laterolog, and R_{mf}/R_{w} from measured values or from the SP curve.
The ratio is valuable in itself as an index of oil movability. If S_{w}/S_{xo} = 1, then no hydrocarbons have been moved by invasion, whether or not the formation contains hydrocarbons. If S_{w}/S_{xo} is approximately 0.7 or less, movable hydrocarbons are indicated. The value of S_{w}/S_{xo}, along with ϕ and S_{wo}, is useful in evaluating reservoirs.
To determine S_{w} from Eq. 12, S_{xo} must be known. For moderate invasion and average residual oil saturation, an empirical relation between S_{w} and S_{xo} has been found useful: S_{xo} = S_{w}^{1/5}. Inserting this into Eq. 11 gives:
Service companies provide charts for graphical solution of this equation, or it can be easily programmed into a spreadsheet.
Invaded-zone method
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, R_{i}/R_{m} must be at least 10.
Archie’s equation for the invaded zone is
where R_{z} is the resistivity of the water in the invaded zone. Because of incomplete flushing, R_{z} is a mixture of mud filtrate, R_{mf}, and formation water, R_{w}.
Studies of many logs suggest that S_{i} and S_{w} are related by
Dividing the noninvaded-zone water saturation equation (Eq. 1) by Eq. 13 and using the relationship presented in Eq. 14 yields an expression for S_{w}:
To use Eq. 15, R_{t} is taken from a deep resistivity device such as a deep induction or deep laterolog (corrected for borehole effect and bed thickness). R_{i} is taken from a shallow resistivity device such as a Laterolog 8, 16-in. normal, or SFL (corrected for borehole effect and bed thickness).
R_{z} is given by the relationship
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 R_{mf}/R_{w} on the appropriate z scale and R_{i}/R_{t} (oblique lines) to determine S_{w}. When R_{i}/R_{t} 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 R_{i}/R_{t} ≈ 1.
R_{xo}/R_{t} quicklook
The R_{xo}/R_{t} quicklook method can be used to identify hydrocarbon-bearing formations and to indicate hydrocarbon movability (producibility). When S_{w}/S_{xo} is 1 in a permeable zone, the zone will produce water or be nonproductive regardless of water saturation. A value S_{w}/S_{xo} 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
which shows that an indication of S_{w}/S_{xo} can be obtained by comparing R_{xo}/R_{t} with R_{mf}/R_{w}, where the subscript SP emphasizes that R_{mf}/R_{w} is derivable from the SP. Equivalently, the comparison can be between log R_{xo}/R_{t} and the SP curve for an indication of log S_{w}/S_{xo}.
The value of log R_{xo}/R_{t} 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 R_{xo}/R_{t} curve, properly scaled to match the SP, and the SP curve provide a quick-look location of producible hydrocarbons.
Originally, log R_{xo}/R_{t} was computed from R_{LL8}/R_{ID} or R_{SFL}/R_{ID}. Use was made of the observation that over a wide range of invasion diameters (from approximately 20 to 100 in.), R_{xo}/R_{t} depends primarily on the value of R_{LL8}/R_{ID} or R_{SFL}/R_{ID}. The relationship used for the LL8 device was
For the SFL device, it was
Much more sophisticated algorithms are now used to obtain R_{xo}/R_{t}. These values are output in real time as separate logs.
To interpret the R_{xo}/R_{t} 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 R_{xo}/R_{t} (actually –K log R_{xo}/R_{t}) curves coincide in a permeable zone, the zone will most probably produce water. If, however, the R_{xo}/R_{t} curve reads appreciably lower (i.e., to the right) than the SP, the zone should produce hydrocarbons. An R_{xo}/R_{t} value less than the SP amplitude indicates movable hydrocarbons are present.
The R_{xo}/R_{t} quick-look technique is applicable to fresh mud conditions (R_{xo} > R_{t}) in formations where invasion falls within the limits demanded by the R_{xo}/R_{t} computation. For the simpler computation technique using Eq. 18 and –25, that is for d_{i} 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 R_{xo}/R_{t} technique efficiently handles variations in formation water resistivity, R_{w}, and in shaliness. Any change in R_{w} is reflected similarly into both the computed R_{xo}/R_{t} 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 R_{xo}/R_{t} value and the SP amplitude. Finally, the R_{xo}/R_{t} quick-look technique does not require porosity data, nor use of any F – ϕ 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 R_{xo}/R_{t} 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 R_{xo}/R_{t} ratio—a result of the same shale. Therefore, the comparison is not significantly disturbed by shaliness. Neither is it disturbed by variations in R_{w}.
Estimates of water saturation and saturation ratio in clean formations can be made by comparing the R_{xo}/R_{t} and SP curves. Eq. 17 permits S_{w}/S_{xo} to be estimated, and then Eq. 12 allows S_{w} to be estimated.
Nomenclature
a_{mf} | = | mud filtrate chemical activity |
a_{w} | = | formation water chemical activity |
A | = | area, m^{2} |
d_{i} | = | diameter of invasion (in., m) |
E_{k} | = | electrokinetic potential |
E_{kmc} | = | electrokinetic potential of the mudcake |
E_{ksh} | = | electrokinetic potential of shale |
F | = | formation factor relating resistivity to porosity |
g | = | induction-response function |
G | = | induction integrated radial-response function |
h_{mc} | = | mudcake thickness |
I | = | electrical current, Amperes |
L | = | length, m |
r | = | resistance, ohm |
R | = | resistivity (ohm•m) |
R_{ann} | = | resistivity of the annulus |
R_{h} | = | resistivity in the horizontal direction (ohm•m) |
R_{m} | = | resistivity of the mud column (ohm•m) |
R_{mc} | = | resistivity of the mudcake |
R_{mf} | = | resistivity of the mud filtrate |
R_{xo} | = | resistivity of the invaded zone |
R_{t} | = | resistivity of the uninvaded formation |
R_{v} | = | resistivity in the vertical direction (ohm•m) |
R_{w} | = | resistivity of the formation connate water (ohm•m) |
R_{wa} | = | apparent water resistivity from deep resistivity and porosity |
S_{xo} | = | water saturation of the invaded zone |
S_{w} | = | water saturation in the uninvaded zone |
t | = | acoustic travel time (μs/ft) |
t_{ma} | = | acoustic travel time of the rock matrix(μs/ft) |
V | = | electrical voltage, volts |
V_{sd} | = | fraction of the total formation volume that is sand |
V_{sh} | = | fraction of the total formation volume that is shale |
ρ | = | density |
ρ_{ma} | = | density of the rock matrix |
σ | = | conductivity, mS/m |
σ_{m} | = | conductivity of the mud column, mS/m |
ϕ | = | porosity |
References
- ↑ Archie, G.E. 1942. The Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics. Trans. of AIME 146 (1): 54-62. http://dx.doi.org/10.2118/942054-G
- ↑ de Witte, L. 1950. Relations Between Resistivities and Fluid Contents of Porous Rocks. Oil & Gas J. (24 August).
- ↑ Vail, W.B. 1989. Method and Apbrtus for Measurement of Resistivity of Geological Formations From Within Cased Boreholes. US Patent 4,820,989.
- ↑ Lindley, R.H. 1961. The Use of Differential Sonic- Resistivity Plots to Find Movable Oil in Permian Formations. J Pet Technol 13 (8): 749-755. SPE-49-PA. http://dx.doi.org/10.2118/49-PA
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See also
Water saturation determination
Log analysis in shaly formations
Log analyses in tight gas reservoirs
Resistivity and spontaneous (SP) logging