J. Geomag. Geoelectr., 49., 1659-1676, 1997

1

Transcript of J. Geomag. Geoelectr., 49., 1659-1676, 1997

Page 1: J. Geomag. Geoelectr., 49., 1659-1676, 1997

J. Geomag. Geoelectr., 49., 1659-1676, 1997

Experimental Design for Surface-to-Borehole Hydrocarbon Applications

Berthold KRIEGSHAUSERr'2, Alan TRIPP', and Lev TABAROVSKYZ

'University of Utah, Department of Geology and Geophysics, Salt Lake City, UT 84112, U.S.A. 2 Western Atlas Logging Services, 10205 Westheimer, Houston, TX 77042, U.S.A.

(Received December 20, 1996; Revised August 20, 1997; Accepted September 3, 1997)

We discuss in a simplified surface-to-borehole example the resolution of the coefficient of anisotropy A for an anisotropic layer sandwiched in an isotropic half-space as a function of transmitter-receiver configuration and magnetic field components. Our analysis supposes that the horizontal resistivity of the anisotropic bed has been established by previous logging, and that only horizontal magnetic fields will be used to determine the vertical resistivity. The resolution analysis shows that for the model studied the coefficient of anisotropy A can be resolved within 10% for a broad range of transmitter-receiver configurations if 8Bz/8t and 8By/8t are used jointly in the interpretation. Using only 8B./8t or 8B,/8t individually, A can only be resolved for a limited range of transmitter-receiver configurations. A joint interpretation of both horizontal magnetic field components facilitates a better resolution of the coefficient of anisotropy compared to a single interpretation.

1. Introduction

Surface electromagnetic (EM) exploration techniques are not commonly used for hydrocarbon exploration. The reason is the relatively poor resolution of surface EM techniques for a given depth of investigation (Wilt et at, 1995). Cross-borehole and surface-to-borehole techniques should give better resolution of a hydrocarbon target since the receiver or receiver arrays are closer to the target (Wilt et al., 1995). In addition, the geomagnetic or telluric noise level is greatly reduced at depth due to the attenuation of higher frequencies through the earth (Vozoff, 1991). One factor, which has not been addressed in the literature, is the resolution of anisotropic hydrocarbon reservoirs. Macroscopic anisotropy can occur, for example, due to interbedded thin resistive, oil rich sands and thin brine saturated or conductive shale layers. These reservoirs appear to be anisotropic with the resistivity perpendicular to bedding being different from the resistivity parallel to bedding. To measure this anisotropy we need to generate vertical current flow. An inductively coupled loop source deployed on the surface only generates a horizontal current flow. However, a grounded electric bipole source (HED) on the surface generates both a horizontal and vertical current flow (e.g., Weidelt, 1987; Strack, 1992). The vertical currents are sensitive to the resistive units and by measuring both horizontal magnetic field components (8B~/&, 8By/8t) and the vertical magnetic field component (8B.18t) it is possible to determine the anisotropy tensor. We will demonstrate this using realistic noise and reservoir rock models as well as SVD based resolution analysis. We will begin by discussing the frame work for our analysis.

2. Methodology

2.1 Modeling Accurate estimation of formation and reservoir resistivities permits optimization of produc-tion of movable hydrocarbons. Formation resistivities are linked to porosity and water saturation, e.g. via Archie's law (1942), which can be of help in estimating the oil in place (OIP). Archie's

1659

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1660 B. KRIEGSRAUSER et al.

law can be used to calculate water saturation (S.) in clean sandstones (Berg, 1995). A number of empirical laws have been derived for the cases when the formation rock matrix is conductive or the electrical properties of the water-saturated rocks depend on frequency, e.g., Hanai (1960), Waxman and Smits (1968), Bussian (1983), and Berg (1995). We can use Archie's law to derive model formation resistivities as a function of water saturation and porosity.

In general, anisotropy can be described by a 3 x 3 tensor p. The resistivity tensor p relates the electric field f to the current density f via Ohm's law

E'=pf.. (1)

In a horizontally thin layered sequence of homogeneous isotropic layers, however, the resistivity

tensor is approximated by a simple form in which the resistivity differs in horizontal and vertical

directions, i.e., parallel and perpendicular to the layering. The resistivity tensor is diagonal with

l Ph 0 P = 0 Ph

0 0

Materials with resistivity tensors of th

of anisotropy A is defined as

The vertical resistivity

resistivity, hence A > 1.

0

0

Pv

e form (2) are called "

A - Pv Ph

transversely isotropic".

(2)

The coefficient

(3)

of an transversely isotropic layer is always greater than the horizontal

In analogy to circuit theory, horizontal currents `see' the resistivities of

Transient Response

10-4

- Transient Curve

o Center of Measurement Window

a

m

10'

me

10°

/

/

/

/ /

// /

/

/ /

/

Measurement Window

i

\~

\~ \ ~

\`

1 2 3 4 5 5 Time (msec)

7 8 9 10

Fig. 1. Averaging receiver time windows as a function of recording time.

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Surface-to-Borehole Hydrocarbon Applications 1661

a transversely anisotropic layer in parallel, whereas for the vertical currents the resistivities are in series. The horizontal currents will flow mainly in the conductive thin beds. The horizontal resistivity ph can be calculated as

h Ph (4)

hi 1 Pi

where h is the thickness of the anisotropic layer, h; and pi are the thickness and resistivities of

each individual thin bed, and n is the number of thin beds composing the layer. The vertical

resistivity is

1 Pv = h ~ htPc. (5)

i=1

In fractured media, e.g. in geothermal areas or fractured reservoirs, we can have vertical conduc-

tive paths which can lead to p„ < Ph.

To measure transversely isotropic structures we need a source which generates a vertical current flow. A loop source on the surface of the earth which is inductively coupled to the ground

generates only a horizontal current flow in the earth and is insensitive to p,,. These currents will flow mainly in the more conductive water saturated bed units, and will not be sensitive to the more resistive hydrocarbons bearing layers.

Cross Section

HED

Om

Sow r

X=] 10 Ohm-m

X=2 • 0 0 0• 0 0 1000 m

* P„=1 Ohm-m 1050m

R=1 10 Ohm-mReceivers

y

1000 m

1000 mx

Plan View

I Ohm-m

100 Ohm-m

+ Origin

Fig. 2. Earth model for geologic noise simulation. The source is a 1000 m long grounded bipole (HED) at the surface, while the receivers are horizontal magnetic field sensors centered in the anisotropic layer. The origin

of the coordinate system is at the center point of the transmitter.

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1662 B. KRIEGSHAUSER et a!.

A grounded bipole source on the surface of the earth generates both a horizontal and a vertical current flow. The vertical currents generate horizontal magnetic fields which are sensitive to p,,. Hence, measuring both the horizontal and vertical component of the magnetic fields should allow to determine the coefficient of anisotropy. The forward modeling was done using the program SYSEM (Xiong, 1992). SYSEM uses the frequency-domain volume integral equations method (Weidelt, 1975) for three-dimensional structures in layered anisotropic background medium. The time-domain solution is obtained by transforming the frequency-domain responses into the time-domain.

2.2 Resolution analysis concept Resolution analysis can be based on the singular value decomposition (SVD) of the Jacobian matrix (Martin, 1971; Bard, 1974; Hohmann and Raiche, 1987; Press et al., 1989). The Jacobian matrix (J) is a partial derivative matrix relating small changes in model parameters (Ap) to changes in data (Ad),

Ad =J Op. (6)

The inverse of J maps uncertainties in the data into uncertainties in parameter estimations, and thus establishes a link between the accuracy of the data and the confidence intervals of the interpreted parameters.

Near-Surface Noise Simulation

10,

10-

1010

a,

10

. 10 ,]

,a-13

1014

10-15

I

I

I

- dB,I-D/dl

aBr/al

ae,-D / al

1.5

I

4 m

s

0.5

10 4 1o' io' 10QTime (sec)

Fig. 3. Near-surface noise simulation for 8B./8t-component. The x, y, z-coordinates of the receiver are 1000 , 1000, 1025 m. The 3-D and 1-D responses pertain to the models with and without the near-surface anomaly .

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The noise in the data comes from different sources, e.g. geomagnetic noise, electronics noise, systematic noise, and geologic noise. The random data noise comprises electronics noise or "Johnson Noise" (Ott, 1976), geo-magnetic or telluric noise (sferics, etc.), and cultural or man-made noise (pipelines, powerlines, etc.).

Electronics noise includes thermal noise, shot noise, contact noise, cross-talk, etc. Thermal noise comes from the thermal agitation of electrons within a resistant material (Ott, 1976). The main thermal noise contributors are the coil sensor and the amplifier. To cover all sources of electronics noise the electronics noise can be described as the product of the spectral noise density d (nV/ Hz) multiplied by the square root of the frequency bandwith B. The value for the spectral noise density d is around 2-20 nV/ Hz for the electronics noise. For our analysis we chose d to be 10 nV/ Hz. The amplitude has a normal or Gaussian distribution with zero mean. Since the electronics noise is random it can be reduced by stacking multiple measurements. Special stacking techniques can be applied to further enhance the signal-to-noise ratio (Strack, 1992). Geomagnetic fields caused by natural sources are usually given as spectral field or noise densities, denoted as S (nT/ Hz). The RMS magnetic field amplitude in a frequency band

Near-Surface Noise Simulation

a, E

a

10_I

1010

10

10

_11

-12

10-13

10 1,

t

I

t

I V.

A

dBr/d1

dBy°/at

dB'y-D/at

1.5

4

s m0.5

Fig. 4.

1025

loa low io' to Tine (sec)

Near-surface noise simulation for 8By/8t-component. The x, y, z-coordinates of the receiver are 0, 0, m. The 3-D and 1-D responses pertain to the models with and without the near-surface anomaly.

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1664 B. KRIEGSHAUSER et al.

f15f5f2 is defined as (Weidelt, 1987)

B(fl, f2) = f2

JftS2(f) df. (7)

Documentations of natural geomagnetic fields can be found in Spies and Frischknecht (1991), Vozoff (1991), and Macnae et al. (1984), among others. The cultural or man-made noise is usually a function of frequency and can often be removed by notch filtering or local noise compensation techniques (Stephan and Strack, 1991). To obtain the noise amplitude we have to integrate the spectral noise density over the fre-quency bandwidth of the receiver. The frequency bandwidth is usually achieved using a high-pass filter at the lower end to eliminate the low-frequency noise and a Nyquist filter at the upper end to avoid aliasing. Geomagnetic noise is significantly smaller than the electronics noise at receiver depths greater than 500 m, due to the attenuation of the natural fields through the conductive earth. Electronics and telluric noise can be reduced by enlarging the averaging gates over which the signal is integrated as time increases during data acquisition (Fig. 1). However, San Filipo and Hohmann (1983) showed that signals below 1 Hz cannot be reduced due to the 1/f power spectrum of low frequency noise. Under geologic noise we sum all the responses of the host which are not associated with the target and might mask the target response. Near-surface inhomogeneities, e.g., caused by differential weathering, can severely affect and mask subtle changes in target response (Newman, 1989). Figure 2 shows the earth model used to simulate the near-surface effect. A checkerboard

Resolution Analysis

Ad= J Ap

Ap=VS-1 U

4 = Var(A d

£p = vs-1 U

T

)

T

Ad

4 US-1 V T

Inversion

\• Otion 1 Option 2

----------

Data Noise Parameter Uncertainties

Fig. 5. Resolution analysis maps data uncertainties into parameter uncertainties. are described in Appendix A.

The mathematical concepts

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Surface-to-Borehole Hydrocarbon Applications 1665

of cells with sizes of 250 m x 250 m x 50 m in x-, y-, and z-direction was placed underneath the transmitter. The resistivity of each cell was either 1 ohm-m or 100 ohm-m. We modeled 7 cells in the x-direction and 5 cells in the y-direction. Figures 3 and 4 show the near-surface effect for the 8Bz/8t and 8B./8t-component. The 8B3-'/&-components and 8B1-D/8t-components are the responses for the model simulations with and without the near-surface inhomogeneities, respectively. The left scale of the y-axis gives the field amplitudes, while the right scale depicts the field ratio between the responses with and without the inhomogeneities. The receiver locations in x, y, z-coordinates for the 8B/&t receivers are at 1000, 1000, 1025 m, while the 8By/8t receivers are at 0, 0, 1025 m. For this particular geologic noise simulation, both figures indicate that the near-surface effects can be strong and do not vanish at late times. These effects can mask the anomalous response of a target at depth depending on target size and target depth. Figure 5 illustrates the concept of using a resolution analysis to map the noise in the data into uncertainties in parameter estimations. Appendix A describes mathematically the concepts of the resolution analysis technique. Different transmitter-receiver configurations and survey designs give different uncertainty regions in parameter space (Strack, 1992). Thus, this methodology can be used to tune different transmitter-receiver configurations and survey designs. Designs with smaller uncertainty regions in parameter space are preferable.

Resolution Analysis Flow Chart

OPTIONS

lSynthetic

dataRandom

noiseSystematic

noiseTechnical

tool parameterse

I

d

modify

no?

Dataselection

Noisemapping

i

Required

resolution

7

yesl

SELECTION

Fig. 6. Resolution analysis flow chart.

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1666 B. ICRIEGSHAIJSER et at.

Since telluric or geomagnetic and thermal noise are statistically uncorrelated , they add like

Vrandom = (Vgeomagnetic)2 +(Velectronics)z (8)

The flow chart on Fig. 6 illustrates the underlying concepts of the resolution analysis. Resolution analysis can be applied to various measurement options, for different operating domains; different components to be analyzed or different combinations of receiver or logging positions. We may want to analyze data acquired at one single logging point, and compare the resolution to a data set which combines various logging points. In any case, we need to create synthetic data sets for a particular earth model. Then we need to estimate the random and systematic noise. Tool parameters include items such as achievable transmitter-receiver moments (TRM), power supply, dynamic range of the system, frequency bandwidth of the receiver unit, trans-mitter ramp time, minimum measurable signal level, etc.

In data selection we filter the data according to certain thresholds, such that the data used for the analysis are significantly above the random noise level and satisfy technical tool parameter constraints.

Cross Section

HED

4 I

1.=t 10 Ohm-m

• • • • • • • 9 • • • p'.1 Ohm-m1000 m

1050 m

F2=2

2.=IReceivers

10 ohm-m

y

1000 m

1000 mx

Plan View

• . . • .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Receivers

Fig. 7. Earth model for anisotropic modeling study. The source is a 1000 m long grounded bipole (HED) at the surface, while a grid of magnetic field induction sensors is centered in the anisotropic layer. The origin of the coordinate system is at the center point of the transmitter.

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Surface-to-Borehole Hydrocarbon Applications 1667

The noise of these selected data is then mapped into regions of uncertainties in parameter space. If the required resolution is achieved, we can select a certain tool or survey design. If not, we can go back, and reexamine the options to determine what design parameters we need to obtain the required resolution. This mathematical model allows us to determine parameters such as number of stacks, TRM, combination of logging points and field components, subject to a specific resolution criterion.

3. Resolution Analysis Results

A simple 1-D model was used to analyze the resolution of anisotropic structures using a surface-to-borehole technique. The upper part of Fig. 7 depicts the layered earth model with an anisotropic 50 m thick layer embedded in an isotropic host. The depth of the top of the layer is at 1000 m, while the resistivities of the layer and the host are 1 ohm-m and 10 ohm-m, respectively. The coefficient of anisotropy was chosen to be A = 2, thus the vertical resistivity p„ is four times the horizontal resistivity ph. Assume that the anisotropic layer is composed of an equal volume of conductive and resistive beds. Then the resistivity values of the conductive and resistive beds composing the anisotropic layer correspond to a porosity of 30% and a water saturation of approximately 100% and 25% using Archie's law with m = n = 2 and a connate water resistivity of 0.05 ohm-m.

A suite of receivers is positioned on a mesh in the center of the anisotropic layer. The plan view (Fig. 7) depicts the mesh of receivers placed in the center of the anisotropic layer with

Transmitter in X- Direction aBx at20

2500

18

E

2000

1500

1000

500

0

0

r

a

500

INE

1000 1500 2000 2500

0 0

r 0 U C

N N E m `m

0t

16

14

12

0

8

s

4

2

0

X (m)

Fig. 8. Parameter uncertainties for A using the OBx/at component.

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1668 B. KRIEGSHAUSER et at.

respect to the transmitter. The transmitter length was 1000 m pointing in the x-direction, the spacing between the receivers in the x- and y-direction was 250 m each. The receiver grid consists of 11 by 11 receivers, and all three components of 8B/8t were calculated. The parameters analysed simultaneously were the resistivity of the host (10 ohm-m), the horizontal resistivity of the layer (ph = 1 ohm-m), the coefficient of anisotropy (A = 2), and the thickness of the layer (50 m). Although this study focusses on the resolution of the coefficient of anisotropy, we also included other parameters in the resolution analysis to obtain a more realistic estimate of the resolution of each individual parameter. The more a priori information included in the interpretation scheme, the better the resolution of the individual parameters. As an example, the resolution analysis for the coefficient of anisotropy using 8B./8t, aBylat, and using &B./8t and aBylat jointly, are shown in Figs. 8 through 10. The vertical component (8B,/at) is insensitive to the vertical resistivity (p„) and was dis-carded in this analysis in order to emphasize the joint use of both horizontal magnetic field components in resolving A. However, since 8B,18t is sensitive to ph the vertical magnetic field is useful in determining the horizontal resistivity distribution of the subsurface and could be used in a joint interpretation with the horizontal components. Although the vertical magnetic field component is insensitive to p,,, its use in a joint analysis would improve the resolution of all parameters overall, because 8B,/at aids in resolving other formation parameters besides p,,. The data were modeled for a transmitter current of 100 Amperes, and an effective receiver area of 100 m2; the noise was simulated for an average of 10,000 stacks. Data was gathered in the time-domain over a time window of 1 msec to 1 sec. Only those data points were used in the

Transmitter in X-Direction , aBy / at20

2500

18

E

2000

1500

1000

500-j£f

E3~ I~aV~ MM

0 500 1000

X (m)1500 2000

~. kFdk

2500

N d

C 'C N U C

N N E

m `m a

16

14

12

10

8

f>.4

4

2

Q

Fig. 9. Parameter uncertainties for A using the 8By/8t-component.

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Surface-to-Borehole Hydrocarbon Applications 1669

resolution analysis which satisfied the noise threshold of 1%. Figure 8 shows the resolution analysis for 8B,, /8t. The parameter uncertainties are gray shaded for the coefficient of anisotropy. Light gray reflects very good resolution, while darker grays correspond to poorer resolution. To utilize better the gray spectrum, parameter uncertainties greater than 20% are set to 20%. Each little box pertains to a receiver location and gives the resolution of this receiver position to A.

Since for all receivers with x = 0 m and y = 0 m, 8Bs/8t = 0 for an x-directed transmitter, the uncertainties are infinite. The best receiver positions to resolve A measuring 8Bs/8t for an x-directed transmitter lie in a radial band with distances to the center of the transmitter from 1000 m to 1500 m. Using only 8B,/8t to resolve A gives the parameter uncertainties shown in Fig. 9. The best receiver location measuring 8By/8t is directly underneath the center of the transmitter, because the signal to noise ratio is biggest for this receiver location. Thus, big signal to noise ratios together with the sensitivity to A gives a good resolution for the coefficient of anisotropy for this receiver locations. A second area with fairly good resolution of A is at x = 2250 m and for y from 0 m to 1000 m. In this receiver locations we observe a sign reversal in the 8B,,/8t component at later times, yielding high sensitivities to A. However, since in practice it is difficult to measure

data in time windows where the data exhibit a sign change, this area of good resolution should

be interpreted with caution.

Combining the 8B,/8t and 8B./8t measurements in a joint data set and analyzing the data gives the resolution with respect to A shown in Fig. 10. Analysing both measurements in one

Transmitter in X-Direction , aBX / at and aBv / at (joint)

E

2500

2000

1500

1000

sf

{

10 1~ i

500

O

• ~ ~ P#~~R~ MEN

i

500 1000

X (m)1500

Y!e S'',t~

ME L:9..

~

2000 2500

0 N N C

N U C

Gl N E m

a

20

18

16

14

12

10

8

6

4

2

0

Fig. 10. Parameter uncertainties for A using both 8Bz/8t- and 8By/Otcomponents.

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1670 B. KRIEG5RAU5ER et a/.

interpretation effectively enhances the resolution of each individual data set. There is now a broad area, lying roughly inside a circle centered at the transmitter with a radius of 2000 m, where the uncertainties in A are smaller than 10%. This example nicely illustrates how a joint interpretation of different components can enhance the resolution of a parameter of interest. Figures 12 and 13 show the field amplitudes for 8B,/8t and 8B~/8t as a function of time and radial distance to the transmitter. The transmitter-receiver configuration is depicted in Fig. 11. At each depth level we have 11 receivers logarithmically spaced along the diagonal axis with x-coordinates ranging from 10 m to 10000 m. The horizontal projection of the location vectors for these receivers make an angle of 45° with the transmitter wire. The fields are shown at three different depth levels: the top, middle and bottom slices depicting depth levels of 150 m, 400 m and 1025 m, respectively. Figure 12 shows that the field maxima of BB,/8t moves outward in time and radial distance as a function of depth. Hence, at a depth of 1025 m (middle of the anisotropic layer), the field maxima of 8B,/8t is between 10 and 100 msec at radial distances to the HED between X500 m to 2000 m. Figure 13 shows that the field maxima of 8B,/8t moves to later times with depth, but the maximum is always underneath the center of the transmitter. At the depth panels 150 m and 400 m the 8B,/8t-component changes sign around 50 msec and 100 msec, respectively, for

Model Setup

HE

y

{ I500 m TO 0mx

I

450, _LRemven

i

.

i

.

OM

150 m

400 m

1025 m

z

Fig. 11. Transmitter-receiver configuration for calculating 8B./8t and 8B/at for 3 different depth levels as a function of time and radial distance to the HED. The 11 receivers are spaced logarithmically along the diagonal axis with x-coordinates ranging from 10 m to 10000 m.

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Surface-to-Borehole Hydrocarbon Applications 1671

150

E c 400 a m O

1025

104

Transmitter in X -Direction , aB, / at

Time (sec) 10030

radial Distance (m)

10000

E E a

Z m D J

n E n

m LL pl O

-7

4

-9

-10

-11

-12

-13

-14

Fig. 12.

HF,1).

Field amplitudes for 8B./8t for 3 different depth levels as a function of time and radial distance to

receivers within 1000 m radial distance from the transmitter. With increasing distance, the sign reversals move to later times. This sign reversal in the 8By/8t-component is due to the downward movement of the induced current filament. As soon as this current filament moves deeper than the receiver, the polarity of 8B,/8t-component reverses. Figure 14 depicts the sensitivities to the coefficient of anisotropy A for 8B./8t (top panel) and 8By/8t (bottom panel) at a depth of 1025 m as a function of time and radial distance to HED. The component parallel to the transmitter (x-component) shows much greater sensitivity to the coefficient of anisotropy A than the y-component. Moreover, the sensitivity for the x-component does not change significantly as a function of radial distance d, as long as d is within 1500 m. Hence, the best resolution for the (9B,/&-component is obtained in the area of maximum signal strength (d ~ 1000 m).

Although the sensitivity of the 8B,/8t-component to the coefficient of anisotropy A is smaller compared to that for the 8B,/8t-component (Fig. 14), we obtain good resolution in areas of high signal strength, i.e., underneath the center of the transmitter (Fig. 9). This is because the

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1672 B. KRIEGSHAUSER et al.

Transmitter in X-Direction, £B1, / at

150

400

E

L 4 0

C

1025

104

Time (sec) 10°30

radial Distance (m)

10000

E E a

a

n E Q

a m LL

0O

-7

-8

-9

-10

-11

-12

-13

-14

Fig. 13. Field amplitudes for &By/3t for 3 different depth levels as a function of time and radial distance to the HED.

resolution analysis scheme maps the noise in the data into parameter uncertainties. Thus the resolution is a function of both sensitivities of the data to a certain parameter and the statistical noise distribution of the data. Large signal-to-noise ratios together with small sensitivities can yield similar resolution as smaller signal-to-noise ratios together with high sensitivities. The apparent artifacts in the Jacobian for 8B./at receivers (top panel) within 300 m lateral

distance from the transmitter and at late times are believed to be caused by numerical instability, since the signal level for these receiver locations is very small at late times (Fig. 12). However, the features in the aB,/8t (bottom panel) component at 1000, 2000, and 3000 m and at times 20, 100 and 500 msec, respectively, are consistent with the downward movement of the current filament.

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Surface-to-Borehole Hydrocarbon Applications 1673

Transmitter in X-Direction, dB, / at Component

10'

0 N N N E

p 10I

1o°

3a 100 300 1000 3000 10000

C N

0 O U Q

0.5

0.4

0.3

0.2

0.1

0

Transmitter in X-Direction, a; / at Component

1e

U d N m E Is

1D'

C W a

U

.3!

0.5

0.4

0.3

0.2

0.1

to'

30 100 300 1000 radial Distance (m)

3000 100000

Fig. 14. Sensitivities to the coefficient of anisotropy A for aBz/at 1025 m as a function of time and radial distance to the LIED.

(top) and aBe/at (bottom) at a depth of

4. Conclusions

Resolution analysis of applications of surface-to-borehole EM to anisotropic structures has been illustrated. For this analysis a theoretical noise model comprising geomagnetic and elec-tronics noise was developed. This noise model together with a resolution analysis scheme permits a quantitative estimation of parameter uncertainties for a particular tool and survey configura-tion as a function of technical tool constraints and survey parameters. This mathematical model

permits comparison and selection of design and survey options on a quantitative basis. To illustrate the applicability of this mathematical model, we applied our resolution analysis technique to a simplified 1-D anisotropic model, with a 50 m thick anisotropic layer sandwiched in an isotropic host. It should be emphasized, that the results obtained from this study are only valid for the particular model analysed, however, the methodology used in this study is generic in such a sense, that it can be applied to any other exploration problem. For this particular 1-D anisotropic model, receivers measuring aB,xlat give best resolution of the coefficient of anisotropy if located at radial distances between 1000 m and 1500 m from the center of the transmitter (HED) inside the anisotropic layer. Receivers measuring 8B, lot should

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1674 B. KRIEGSHAHSER et al.

not be located in-line or perpendicular to the transmitter. The optimum location for receivers measuring 8By/8t is directly underneath the center of the transmitter. Locations within 500 m from the center of the transmitter give relative good resolution as well. Receivers at x = 2250 m and y < 1000 m give resolution to within 10%. A joint interpretation of 8Bz/8t and 8B/at broadens the area where receivers can be located with respect to the transmitter to give resolution within 10%.

We wish to thank K.-M. Strack for valuable discussions and Western Atlas Logging Services for per-mission to publish this work. One of the authors (AT) received financial support from the Consortium for Electromagnetic Modeling and Inversion (CEMI) at the Department of Geology and Geophysics, Uni-versity of Utah, including BHP Minerals, CRA Exploration, Kennecott Exploration, MIM Exploration, Mindeco, Naval Research Laboratory, Newmont Exploration, Schlumberger-Doll Research, Shell Interna-tional Exploration and Production, United States Geological Survey, Unocal Geothermal, Western Atlas Logging Services, Western Mining, and Zonge Engineering. The authors also acknowledge Z. Xiong and K. Spitzer for their constructive and useful comments.

Appendix A: Resolution Analysis Concepts

Resolution analysis can be based on the singular value decomposition (SVD) of the Jacobian matrix (Martin, 1971; Bard, 1974; Hohmann and Raiche, 1987; Press et al., 1989). This resolu-tion analysis technique maps uncertainties in the data into regions of uncertainties in parameter estimates, establishing a link between the accuracy of the data and the confidence intervals of the interpreted parameter. The Jacobian matrix J is a partial derivative matrix relating small changes in model param-eters (Ap) to changes in data (Ad),

Ad = J Ap. (A.1)

This equation can be solved using the singular value decomposition (SVD) of J

J = USVT, (A.2)

where U and V are orthonormal matrices and S is a diagonal matrix containing the singular values sj of J. Combining Eq. (A.1) and Eq. (A.2) gives

Ad = USVT Ap. (A.3)

Considering Ad in Eq. (A.3) to be the noise which we want to map from data space to parameter space we can obtain the least squares solution of Eq. (A.3) for Ap as

Ap=VS-1UT Ad. (A.4)

Since the noise in the measurements Ad is normally distributed, the errors in parameters, which are linear combinations of the noise components, are normally distributed as well. Applying the variance relation (Martin, 1971)

var(Ax) = A var(x) AT, (A.5)

to Eq. (A.4) yields the parameter covariance matrix

Ep =VS-1 UT Ee US-1VT. (A.6)

Equation (A.6) provides the basis for the resolution analysis.

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Surface-to-Borehole Hydrocarbon Applications 1675

Recalling that the relative variations of actual formation parameters Ap are related to the relative variations in generalized parameters AP via Ap = VT Ap we begin determining the confidence intervals of the formation parameters p by defining the variable

Q = APT EP1 Ap. (A.7)

Since the elements of Ap are normally distributed, Q is a chi-squared distributed random variable with Np degrees of freedom, where NN is the number of parameters. EP 1 from Eq. (A.7) can be diagonalized using a rotation matrix R which yields

Ey 1 = REP' RT. (A.8)

If we substitute Ap = RT 4p, together with Eq. (A.8), into Eq.

Q ApT (REP 1RT)T AP

(RT AP)T Ey 1 (RT AP) APT EP1 Ap NP

~siOPj, i=1

(A.7) we get

(A.9)

where si are the diagonal elements of EP 1. The sum in Eq. (A.9) forms a chi-squared distributed variable with Np degrees of freedom, X,2vp For a given confidence interval (usually 0.95) we can write

P[Q <_ XNp (a)] = o!, (A.10)

which indicates that the probability of having values Q lower than XNp (a) is a. Substituting Eq. (A.7) into Eq. (A.10) and solving for the upper bounds yields

Np

/J 3iAPj =XNp(°) i=1

(A.11)

or NP Q 2

b2 _1 i=1 I

with bi representing the semiaxes of the ellipsoid of uncertainty

(A.12)

bi =XNp (a)

Si(A.13)

To o

real

The ellipsoid of uncertainty defines the region of equivalence for the generalized parameters. btain the uncertainty regions for the actual parameters, we project each semiaxis onto the axis and define the largest value as the parameter uncertainty Ap = max(IR bi).

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1676 B. KRIEGSRAUSER et al.

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