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- 1. Well Logging Course (1st Ed.)
- 2. 1. Spontaneous Potential A. membrane potential B. Application C. Log Example of The SP
- 3. 1. 2. 3. 4. 5. 6. Early Electric Log Interpretation Formation Factor Water saturation The Porosity Exponent, m The Saturation Exponent, n A Thought Experiment For A Logging Application
- 4. usefulness of measuring the resistivity of earth formations the water saturation Sw Is the desired petrophysical parameter from resistivity measurements In this lecture the empirical basis for the interpretation of resistivity measurements is reviewed. For many years, at the outset of well logging, it was not possible to address the water saturation question any more precisely than whether the resistivity of a formation was high or low. It was through the work of Leverett and Archie that it became possible to be more quantitative about the interpretation of a formation resistivity measurement and to link resistivity to formation water resistivity, porosity (ϕ), and water saturation (Sw) Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 5
- 5. a log of SP and formation resistivity made prior to 1935 The scale “Ohms m3” presumably refers to ohm-m. It seems possible, noting the higher resistivity, that zone a-A contains more oil (has a lower Sw) than zone B-b. But how can this be verified? An early resistivity-SP log. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 6
- 6. Early SP log interpretation The “standard” procedure at the time [for verification] was to take a core sample, representative of the zones in question, and to make laboratory measurements of its resistivity under different conditions of water saturation. Fall 13 H. AlamiNia Resistivity measurements of two core samples as a function of water saturation for use in electric log interpretation. Well Logging Course (1st Ed.) 7
- 7. Leverett experiments M. C. Leverett was conducting experiments with unconsolidated sands, to determine the relative permeability of oil and water (Kro & Krw) as a function of the water saturation (Sw). As a by-product of his research, he measured the conductivity of the material in a sample chamber, after a calibration of the system constant, in order to conveniently determine the fraction of kerosene and water in his permeable samples. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 9
- 8. Leverett conclusions Calibration curve of Leverett’s core holder with sand pack, showing variation of relative conductivity as a function of water saturation. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 10
- 9. Archie experiments Shortly after the publication of Leverett’s work, G. E. Archie of Shell was making electrical measurements on core samples, with the aim of relating them to permeability. His measurements consisted of completely saturating core samples with saltwater of known resistivity Rw and relating the measured resistivity Ro of the fully saturated core to the resistivity of the water. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 11
- 10. Relation between water resistivity (Rw) and formation resistivity (Ro) Archie found that, regardless of the resistivity of the saturating water, the resultant resistivity of a given core sample was always related to the water resistivity (Rw) by a constant factor F. He called this the formation factor, and his experiments are summarized by the following relation: Next slide is an example of Archie work on cores from two different locations. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 12
- 11. the formation factor F vs. k and, almost as an afterthought, porosity, ϕ Examples of Archie attempts to correlate the electrical F with K and ϕ for watersaturated rock samples from two regions. F vs. K and phi (phi on a much compressed scale) Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 13
- 12. A summary of an exhaustive set of measurements of formation factor Although Archie was searching for a correlation with K, he finally admitted that a generalized relationship between F and K did not exist, although one seemed to exist for porosity. His summary graph shows the hopelessness of a F/k correlation. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 14
- 13. formation factor calculation However it indicates that the F is a function of ϕ and can be expressed as a power law of the form: the exponent m is very nearly 2 for the data considered. This empirical observation can be used to describe the variation in F for a fixed water resistivity when the ϕ changes: the lower the porosity, the higher the resistivity will be. The exponent m was soon named the cementation exponent, as it was observed to increase with the cementation of the grains. In general, it was recognized that m increased with the tortuosity of the electric path through the pore space. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 15
- 14. A synthesis of various resistivity/saturation experiments The practical application of resistivity measurements is for the determination of water saturation (Sw). This was made possible by another observation of Archie. He noticed that the data of Leverett and others could be conveniently parameterized after having plotted the data in the form shown in Figure. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 18
- 15. water saturation vs. relative resistivity On log–log paper, the data of water saturation versus relative resistivity plotted as a straight line, suggesting a relationship of the form: and with the porosity dependence, the final form is: The exponent n, called the saturation exponent, is very nearly 2 for the data considered. From this, an approximate expression for the Sw is: which can be used for purposes of estimation. However, a more general form, is: However, the fully saturated resistivity Ro (which is not usually accessible in formation evaluation), can be related to the water resistivity. So: Fall 13 H. AlamiNia the constants a, m, and n need to be determined for the particular field or formation being evaluated. Well Logging Course (1st Ed.) 19
- 16. interpret a resistivity measurement in terms of water saturation (Sw) in order to interpret a resistivity measurement in terms of water saturation (Sw), two basic parameters need to be known: the porosity φ and the resistivity of the water in the undisturbed formation (Rw). As a starting point, the value of the water resistivity Rw needs to be estimated. • This can be done in water zones (resistivity logs). Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 20
- 17. Example of interpretation of resistivity measurement in terms of Sw the value of the water resistivity Rw needs to be estimated. From zone D’ or zone C’ (water zones) the porosity is about 28 p.u. so F is 1/(0.28)^2, or 12.8 the apparent resistivity = 0.2 ohm-m which is assumed to be the fully water-saturated resistivity Ro, water resistivity = 0.2/12.8= 0.016 ohm-m the increase in deep resistivity in zone C to about 4 ohm-m correspond to a decrease in Sw compared to zone C’ the porosity is constant at 28 p.u. The saturation in zone C: Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 21
- 18. Example of interpretation of resistivity measurement in terms of Sw (Cont.) zone of hydrocarbon (A) indicates the same resistivity as zone C. in zone A the porosity is much lower and can be estimated about 8 p.u. Thus F in zone A is 1/(0.08)2, or 156 If it were water-filled, the resistivity would be expected to be about 2.5 ohm-m compared to the 4 ohm-m observed. Thus the zone may contain hydrocarbons, but the water saturation can be expected to be higher than in zone C. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 22
- 19. Effective parameters As Archie was aware, his equations worked well in rocks that have simple, uniform pore systems filled with saline water. Rocks with heterogeneous-pore systems, multiple-conduction mechanisms, or that are oil-wet need a more complete solution. The problems can be considered with reference to Archie’s three equations: the relation to porosity (m), the relation to Sw (n), and the definition of formation factor (F) is mainly an issue of clay conductivity Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 23
- 20. The Porosity Exponent, m Although Archie could fit his data with a single parameter, m, in general a fit of F vs. φ throughout a reservoir will require two parameters, a and m. In practice the error caused by fitting with one parameter is often small. In either case it would be better if the variations [of a and m] through the reservoir could be related to some physical property, rather than relying on a general average. Early efforts focused on finding a relation with porosity, the idea being that as porosity decreased it was likely that the tortuosity, and hence m, increased. Many relations were developed but proved to be specific to particular reservoirs or areas, and not generally applicable. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 25
- 21. the total effective m in a fractured reservoir Clearer relations can be obtained if the reservoir contains vugs or fractures. Fractures offer a straight path for current, with minimum tortuosity. in a fractured reservoir, if we can measure the proportion of porosity due to fractures and if we assume that the conductive paths through the fractures and the intergranular porosity are in parallel, with no interaction between them, we can calculate the total effective m. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 26
- 22. Methods of m Calculation More generally, whatever the cause of the variations in m, m can be measured directly from resistivity and porosity in a water zone, and then assumed to be the same in the hydrocarbon zone. Alternatively, if the water saturation can be measured by another means in addition to resistivity, One such method uses dielectric measurements in the invaded zone. then either m or n can be calculated from Archie’s equation. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 27
- 23. saturation exponent measurement It takes much longer to complete the type of experiment that leads to the saturation exponent (n) than to measure m. Each core sample must be measured at several saturation states. Displacing water with oil or gas takes time, especially in low permeability samples. Unlike m, it is not possible to derive n from logs in a water zone. As a result there is much less data on n, and values other than 2 are less often used. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 31
- 24. 1st condition in which n can be significantly different than 2 However, laboratory experiments have highlighted two main conditions in which n can be significantly different than 2. The first is related to wettability. in an oil-wet cores • the oil coats the grains and starts blocking the pore throats when even small volumes are introduced. • The result is a sharp increase in resistivity and a high n • So n values much larger than 2 In a water-wet core • the water coats the grains and provides a continuous conduction path down to water saturations of 20% or less. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 32
- 25. Resistivity-saturation measurements Resistivity-saturation measurements on carbonates that have been flushed to make them water-wet or oil-wet, a large increase in n Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 33
- 26. 2nd condition in which n can be significantly different than 2 The second condition occurs in rocks in which the pore space is no longer uniform but consists of an irregular mixture of different sized pores. When oil or gas is introduced into such rocks, the water in some pore types displaced more easily For example the oil should easily displace the water in fractures, but may not do so in vugs if they are poorly connected. Carbonates are particularly heterogeneous, and also more likely to be oil-wet, so that for both reasons the relation between resistivity and Sw is likely to be complicated, with n not equal to 2 and also varying with saturation. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 34
- 27. setup for measuring the resistivity of a homogeneous formation It consists of a current source of intensity and a voltage-measurement electrode M at some distance r from the current emission at point A. The resistivity of the homogeneous medium is Rt, so its conductivity σ is given by σ = 1/Rt . (Conductivity is usually written as σ in measurement physics, and as C in log interpretation.) Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 36
- 28. determining formation resistivity the value of Rt is found to be: The setup can be considered as a rudimentary monoelectrode measurement device for determining formation resistivity. For this device the tool constant k is seen to be 4πr , where r is the spacing between the current electrode and the measurement point. Knowing the injected current and the resultant voltage, the resistivity of the homogeneous medium Rt may then be found. Fall 13 H. AlamiNia Well Logging Course (1st Ed.) 37
- 29. 1. Ellis, Darwin V., and Julian M. Singer, eds. Well logging for earth scientists. Springer, 2007. Chapter 4

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