1. Field of the Disclosure
The present disclosure is related to methods for measuring a resistance of an earth formation downhole. In particular, the present disclosure is related to providing a correction for standoff in resistance measurements.
2. Description of the Related Art
In oil exploration, resistivity measurements of an earth formation are typically obtained by lowering a resistivity tool into a borehole penetrating the formation. Generally, the tool includes at least one current transmitter which introduces current into the formation and at least one return electrode at which current returns to the tool. Resistance is measured between the two electrodes. Ideally, current flows directly from the current transmitter through the earth formation and returns through the return electrode. In typical downhole situations, however, the resistivity tool is separated from the formation by a gap filled with oil-based mud that is used for the purposes of drilling. The gap may be uniform along the length of the borehole or may experience variations due to borehole rugosity. The presence of mud and borehole rugosity affects the value of resistivity measurements obtained from the formation.
Several methods are used to reduce the effect of borehole irregularities and standoff, such as obtaining dual frequency measurements and post-processing based on a corresponding calculation; deriving a mud parameter α=(ρmω∈m∈0)−1 from direct mud cell measurements; and obtaining multi-standoff measurements and post-processing based on a corresponding calculation; among others. These methods rely on obtaining an additional measurement to derive the mud parameter and can require new tool designs and/or assumptions about the dispersive characteristics of the mud. Thus, there is a need to provide a correction to resistivity measurements in the presence of the oil-based mud in a gap based on an analysis of the data and which does not require any additional measurements or prior information about the mud's electrical properties. The present disclosure satisfies this need.
The present disclosure provides a method of estimating a resistivity property of an earth formation in a borehole filled with oil-based mud, including: conveying a logging tool into a borehole; measuring a plurality of impedance values of the earth formation, each of the plurality of impedance values being obtained at one of a plurality of measure electrodes on the logging tool; determining a coefficient related to an electrical property of the mud from a plurality of pairs of impedance values; estimating the resistivity property of the formation from the determined coefficient; and recording the resistivity property on a suitable medium.
In one aspect, the coefficient may be determined using a relationship of the form
where α is the coefficient, Aj is a real part of a measured impedance value for ajth signal pair and Bj is an imaginary part of the measured impedance measured for the t signal pair. In another aspect, determining the coefficient may include averaging over the plurality of impedance values. In one aspect, the method updates the determined coefficient using
where Z is the complex impedance measure at a measure electrode and δ is a regularizing parameter. Correction may be made to include a systematic electronic phase error. The logging tool may be conveyed into the borehole on a conveyance device selected from (i) a wireline, and (ii) a drilling tubular.
The present disclosure also provides an apparatus for estimating a resistivity property of an earth formation in a borehole filled with oil-based mud, and includes: a logging tool configured to be conveyed into a borehole; a plurality of measure electrodes on the logging tool configured to measure a plurality of impedance values of the earth formation; and a processor configured to: (a) determine a coefficient related to the electrical property of the mud from each of a plurality of pairs of impedance values, and (b) estimate the resistivity property of the formation from the determined coefficient. In one aspect, the processor is configured to determine the coefficient using a relationship of the form
where α is the coefficient, Aj is a real part of a measured impedance value for a jth signal pair and Bj is an imaginary part of the measured impedance measured for the jth signal pair. In another aspect, the processor may also be configured to determine the coefficient by averaging over the plurality of pairs of impedance values. In another aspect, the processor may be configured to update the determined coefficient using
where Z is the complex impedance measure at a measure electrode and δ is a regularizing parameter. The processor may be further configured to correct for a systematic electronic phase error. The logging tool may be configured to be conveyed into the borehole on a conveyance device selected from (i) a wireline, and (ii) a drilling tubular.
Another aspect of the invention provides a computer-readable medium for use with a tool for determining a resistivity parameter of an earth formation in a borehole. The tool includes a logging tool configured to be conveyed into a borehole and a plurality of measure electrodes on the logging tool configured to measure a plurality of impedance values of the earth formation. The medium includes instructions which enable a processor to: determine a coefficient related to an electrical property of the mud from each of a plurality of pairs of impedance values, estimate the resistivity property of the formation from the determined coefficient, and record the resistivity property on a suitable medium. The medium may be at least one of (i) a ROM, (ii) and EPROM, (iii) an EEPROM, (iv) a flash memory, and (v) an optical disk.
The present disclosure is best understood with reference to the accompanying figures in which like numerals refer to like elements, and in which:
Also shown in
In normal operation, the current loop includes the earth formation as well as mud occupying the standoff gap between the imager 300 and the formation. Thus, neglecting any dielectric effect of the formation, the complex impedance Z measured by the imager 300 may be approximately represented in the form:
Z=Rf+ReZG+iImZG (1)
where Rf is an active (real-valued) formation resistance, and ZG is the complex impedance of the standoff (“gap impedance”) between the pad and the borehole wall. While the measurement of Eq. (1) includes the complex impedance of the mud, ZG, this term is considered to be a parasitic term that is to be removed. In other words, it is desirable to isolate the formation resistivity Rf in Eq. (1). The imaginary part of the gap impedance ZG can typically be filtered out by the phase-sensitive measurements. The correction for the real part of the gap impedance Re(ZG) may be determined using the exemplary method of the present disclosure.
There is a simple relationship between the real and imaginary parts of the gap impedance:
ReZG=αImZG
where the coefficient α depends on electrical parameters of the mud:
α=(ρmω∈m∈0 )−1
and ρm and ∈m are resistivity and permittivity values of a mud and ω=2πf is the angular frequency of the applied current. If a value of the coefficient α is known, then it is possible to use this coefficient to correct for the gap impedance using:
Rf=ReZ−αImZ (2)
Denoting the real and the imaginary part of the impedance Z by A and B respectively, for convenience, the gap impedance may be written as:
Z=A+iB
Taking into account Eq. (2), then
Rf+αB=A
It is useful to consider a set of two signals Z(1) and Z(2) corresponding to two vertically or azimuthally adjacent points, such as points indicated by sensors 305a, 305b of
Rf(1)+αB(1)=A(1) Rf(2)+αB(2)=A(2) (3)
It follows from Eq. (3) that the following equality holds:
Thus, it is possible to consider an ensemble of adjacent points, including an extended set of N adjacent “signal pairs”
(Z1(1),Z1(2)), (Z2(1),Z2(2)), . . . , (ZN(1),ZN(2)) (5)
where N is a large number and all the signal pairs are taken from a small portion of a log. These two requirements do not contradict each other since the vertical image sampling step is always small, and there are tens of sensor buttons placed azimuthally. (Note that the same signal Z may be used in several signal pairs.) Furthermore: 1) the coefficient α is the same for all pairs (i.e., the mud is homogeneous within the considered part of a well), and 2) variation of the formation resistance Rf is not correlated with variation of the imaginary part of the impedance B caused by the variable standoff and/or borehole rugosity.
Eq. (4) can therefore be averaged over the whole ensemble (of Eq. (5)). Due to the second statement above (e.g., the lack of correlation between Rf and B), the second term on the left-hand side of Eq. (4) is randomly distributed with zero mean (it has equal probability to be positive or negative for a given current pair). Thus, when the average of Eq. (4) is taken over N sensors, the contribution of this term to the average decreases by a factor of √{square root over (N)}. One can expect that for at least a majority of pairs of the adjacent points, the absolute values of terms in Rf(1)−Rf(2) are small.
Averaging Eq. (4) therefore gives the following estimate:
A concrete method is introduced to perform the averaging of Eq. (6). It is empirically found that the best choice of a weight of each pair j is:
which leads to Eq. (7):
A correction with this value of α is in fact the orthogonalization of a vector {Aj(1)−Aj(2)}j=1N to a vector {Bj(1)−Bj(2)}j=1N, so that the “energy” of the corrected signal is minimized:
The value of α calculated by the exemplary method may contain some statistical and systematic error. In one aspect, to reduce error amplification, one imposes the following regularizing “protective” condition:
ReZ−αImZ≧δReZ. (9)
Eq. (9) restrains the original signal from being diminished by the correction of Eq. (2) by more than by a factor of 1/δ. Thus, for each logging point:
In one aspect, to ensure enforcement of the condition of Eq. (9), the step of Eq. (10) is added before applying the correction of Eq. (2).
Although the formation resistivity is constant, the real part 402 of the raw signal varies over a range of about 9.5 dB. This variation is purely parasitic, caused by the rugosity of the borehole. The imaginary part 404 of the raw signal is also shown. Using the methods of the present disclosure, under the conditions of
The logs of
In one aspect, the method of the present disclosure filters out the part of the raw signal, ReZ, which directly correlates with the imaginary component ImZ. In another aspect of the present disclosure, this process may also be used to correct contaminations of another origin, but which have the same correlation.
In particular, similar correlation between real and imaginary components of the parasitic impedance may be observed when some phase error β is introduced by electronics. As a result, instead of a true impedance value Z, the tool reads an imperfect impedance {tilde over (Z)}=Z(1+iβ), where β is an unknown coefficient which is small enough to justify discarding of terms ˜β2 and higher. One can use ImZ>>βReZ, since otherwise the impact of the phase error is negligible and correction is not needed. Thus, Im{tilde over (Z)}≈ImZ, and
{tilde over (Z)}=(Rf+ReZG+iImZG)(1+iβ)=Rf+ReZG−βImZG+iIm{tilde over (Z)}=Rf+(α−β)ImZ+iIm{tilde over (Z)}≈Rf+(α−β)Im{tilde over (Z)}+iIm{tilde over (Z)} Eq. (11),
which gives
Rf=Re{tilde over (Z)}−(α−β)Im{tilde over (Z)} Eq. (12).
Eq. (12) coincides with Eq. (2), but instead of the α-parameter of the mud, a new parameter {tilde over (α)}=α−β describes the joint impact of both the standoff and the electronic phase error. Thus, the method of the present disclosure automatically corrects for both standoff and electronics errors simultaneously and Eq. (7) may be used to give an estimate of {tilde over (α)}.
Some observations on the electronics error can be made. First of all, the electronics phase error may be corrected if it is relatively stable within the ensemble of sensors. A similar requirement is applied for the mud parameter α. Secondly, unlike the original mud parameter α, the coefficient {tilde over (α)} can be negative. In these cases, the criterion of Eq. (8) has to be properly modified. Third, different sensor buttons may have different values of β. Thus, to correct for the phase error, the ensemble of sensors includes measurements made by one button only.
Note that for all three cases (β=0, +0.05, −0.05) the absolute value of the {tilde over (α)}− discrepancy is the same. It has the high relative value for β=+0.05 due to the low value of {tilde over (α)} (in this case, the electronic error compensates 70% of the standoff impact).
The results for these three values of β are presented in
While the embodiments described herein have been described for use with a wireline device, they or equivalent embodiments may equally be used with measurement-while-drilling (MWD) tool.
The operation of the transmitters and receivers, and the control of the drilling direction may be controlled by circuitry which may be conveyed downhole or located at a surface location. The circuitry may control and process of the data through, for example, execution of a program or set of instructions stored on a suitable machine-readable medium. The machine-readable medium may include ROMs, EPROMs, EAROMs, Flash Memories and Optical disks. Results of the processing, such as the estimated resistivity property, may be stored on a suitable medium.
The operation of the transmitters and receivers, and the control of the drilling direction may be controlled by the downhole processor and/or a surface processor. Implicit in the control and processing of the data is the use of a computer program on a suitable machine readable medium that enables the processor to perform the control and processing. The machine readable medium may include ROMs, EPROMs, EAROMs, Flash Memories and Optical disks. The term “processor” as used herein is intended to include Field Programmable Gate Arrays (FPGAs). Results of the processing may be stored on a suitable medium.
While the foregoing disclosure is directed to the exemplary embodiments of the disclosure, various modifications will be apparent to those skilled in the art. It is intended that all such variations within the scope of the appended claims be embraced by the foregoing disclosure.
Number | Name | Date | Kind |
---|---|---|---|
5703773 | Tabarovsky et al. | Dec 1997 | A |
5999883 | Gupta et al. | Dec 1999 | A |
6597993 | Strickland et al. | Jul 2003 | B2 |
6714014 | Evans et al. | Mar 2004 | B2 |
6809521 | Tabarovsky et al. | Oct 2004 | B2 |
7365545 | Itskovich et al. | Apr 2008 | B2 |
7397250 | Bespalov et al. | Jul 2008 | B2 |
7463035 | Merchant et al. | Dec 2008 | B2 |
20030173968 | Cheung et al. | Sep 2003 | A1 |
20030222651 | Tabanou | Dec 2003 | A1 |
Number | Date | Country | |
---|---|---|---|
20090292471 A1 | Nov 2009 | US |