Using multicomponent induction data to identify drilling induced fractures while drilling

A method for identifying drilling induced fractures while drilling a wellbore into a formation is disclosed. The method includes: obtaining multi-component induction data collected by a drill string including a multi-component induction tool; processing the data to estimate values for principal components; and identifying drilling induced fractures from the principal components. An instrument and a computer program product are disclosed.

CROSS REFERENCE TO RELATED APPLICATION

This application is a Non-Provisional application of U.S. Ser. No. 61/078,664, filed Jul. 7, 2008, the contents of which are incorporated by reference herein in their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention is related generally to oil field exploration and in particular to use of multi-component induction measurements for identifying fractures in surrounding formations.

2. Description of the Related Art

Electromagnetic induction and wave propagation logging tools are commonly used for determination of electrical properties of formations surrounding a borehole. These logging tools give measurements of apparent resistivity (or conductivity) of the formation that, when properly interpreted, are diagnostic of the petrophysical and other properties of the formation and the fluids therein.

In thinly laminated sand-shale sequences or shaly-sand formations, the formation electrical anisotropy becomes important in determining the hydrocarbon saturation from the resistivity measurements. Due to physical complexities of the formation, determining the electrical anisotropy can be complicated.

For example, intrinsic and macroscopic properties may give rise to anisotropy (i.e., directional dependence). Fractures may also have a substantial effect. In some instances, fractures occur as a result of drilling processes. Accordingly, fracturing as a result of drilling can affect other measurements. Unfortunately, techniques presently available do not adequately provide for characterization of anisotropy while drilling.

When drilling a well it is critical to keep the pressure in the well bore resulting from the weight of the drilling mud below the formation fracture pressure. If the pressure in the well does exceed the formation fracture pressure then the well will fracture and the drilling mud will flow out of the well. Once the drilling mud flows out of the well shallower formations may flow high pressure water or hydrocarbons back to the surface and the well will blow out.

Generally, principles of geomechanics are used to determine the appropriate mud weight to prevent high pressure formation fluids from flowing to the surface (minimum mud weight) as well as a maximum mud weight to keep from fracturing the formation. Unfortunately the actual mud weight includes both static and dynamic components. The static mud weigh is controlled by weighting agents in the mud and suspended cuttings. The dynamic component is due to the movement of the drilling mud and drill pipe (both rotation and upward and downward motion) which can increase the apparent mud weight above the static mud weight.

It is the equivalent circulating density of the mud which is critical. There are well known rules for calculating the ECD, but these may not be accurate and the strength of the formation is not know precisely. Hence during drilling the formation may fracture (drilling induced fractures). These drilling induced fracture should be avoided. Knowing in real time that the formation is beginning to fracture would allow one to reduce the mud weight and thus avoid “fraccing” (i.e., factoring) the well and a subsequent potential blow out.

There is a need for advanced methods for providing determinations of formation anisotropy. Preferably, the methods should consistently provide accurate and reliable data, be computationally efficient and useful for implementation while drilling a wellbore. Ultimately, the methods provide users with a capability to one to reduce the mud weight and avoid fraccing the well and experiencing a blow out.

BRIEF SUMMARY OF THE INVENTION

The above-discussed and other features and advantages of the present invention will be appreciated and understood by those skilled in the art from the following detailed description and drawings.

In one embodiment, the invention includes a method for identifying drilling induced fractures while drilling a wellbore into a formation, the method including: obtaining multi-component induction data collected by a drill string including a multi-component induction tool; processing the data to estimate values for principal components; and identifying drilling induced fractures from the principal components.

In another embodiment, the invention includes an instrument for identifying drilling induced fractures in a formation surrounding a wellbore, the instrument including: at least one set of coils adapted for generating a field in the formation, the set further adapted for obtaining multi-component induction data from the field; and a processor equipped to process data to estimate values for principal components; and identify a drilling induced fracture from the principal components.

In a further embodiment, the invention includes a computer program product stored on computer readable media, the product including computer executable instructions for identifying a fracture in a subterranean formation, the instructions including instructions for: receiving multi-component induction data collected by a multi-component induction tool; processing the data to estimate values for principal components; and identifying drilling induced fractures from the principal components.

DETAILED DESCRIPTION OF THE INVENTION

Refer now toFIG. 1where aspects of an apparatus for drilling a wellbore1(also referred to as a “borehole”) are shown. As a matter of convention, a depth of the wellbore1is described along a Z-axis, while a cross-section is provided on a plane described by an X-axis and a Y-axis.

In this example, the wellbore1is drilled into the Earth2using a drill string11driven by a drilling rig (not shown) which, among other things, provides rotational energy and downward force. The wellbore1generally traverses sub-surface materials, which may include various formations3(shown as formations3A,3B,3C). One skilled in the art will recognize that the various geologic features as may be encountered in a subsurface environment may be referred to as “formations,” and that the array of materials down the borehole (i.e., downhole) may be referred to as “sub-surface materials.” That is, the formations3are formed of sub-surface materials. Accordingly, as used herein, it should be considered that while the term “formation” generally refers to geologic formations, and “sub-surface material,” includes any materials, and may include materials such as fluids, gases, liquids, and the like.

The drill string11includes lengths of drill pipe12which drive a drill bit14. In this example, the drill bit14also provides a flow of a drilling fluid4, such as drilling mud. The drilling fluid4is often pumped to the drill bit14through the drill pipe12, where the fluid exits into the wellbore1. This results in an upward flow of drilling fluid4within the wellbore1. The upward flow generally cools the drill string11and components thereof, carries away cuttings from the drill bit14and prevents blowout of pressurized hydrocarbons5.

The drilling fluid4(also referred to as “drilling mud”) generally includes a mixture of liquids such as water, drilling fluid, mud, oil, gases, and formation fluids as may be indigenous to the surroundings. Although drilling fluid4may be introduced for drilling operations, use or the presence of the drilling fluid4is neither required for nor necessarily excluded from well logging operations. Generally, a layer of materials will exist between an outer surface of the drill string11and a wall of the wellbore1. This layer is referred to as a “standoff layer,” and includes a thickness, referred to as “standoff, S.”

The drill string11generally includes equipment for performing “measuring while drilling” (MWD), also referred to as “logging while drilling” (LWD). Performing MWD or LWD generally calls for operation of a logging instrument20that in incorporated into the drill string11and designed for operation while drilling. Generally, the MWD logging instrument20is coupled to or includes an electronics package which is also on board the drill string11, and therefore referred to as “downhole electronics13.”

Generally, the downhole electronics13provides for at least one of operational control and data analysis. Among other things, the downhole electronics13receives input from at least one on-board sensor15. Accordingly, the downhole electronics13may include, without limitation, a power supply, a transformer, a battery, a processor, memory, storage, at least one communications interface and the like. The electronics unit may include a signal generator and power amplifiers (not shown), which cause alternating currents of selected frequencies to flow through transmitter coils in the sensor15.

In some embodiments, the electronics unit13includes receiver circuits (not shown) for detecting voltages induced in receiver coils in the sensor15, and circuits for processing these received voltages (not shown) into signals representative of the conductivities of various layers3A-3C of the formations3. As a matter of convenience, the electronics unit13may include signal telemetry to transmit conductivity related signals to the surface of the Earth2further processing. Alternatively, the electronics unit13may store data in an appropriate recording device (not shown) for processing after the instrument20is withdrawn from the wellbore1.

Often, the logging instrument20and the downhole electronics13are coupled to topside equipment7. The topside equipment7may be included to further control operations, provide greater analysis capabilities as well as data logging and the like. A communications channel (not shown) may provide for communications to the topside equipment7, and may operate via pulsed mud, wired pipe, and other technologies as are known in the art.

Generally, data from the logging instrument20provide users with enhanced capabilities. For example, data made available from MWD evolutions may be useful as inputs to geosteering of the drill string11and the like.

Referring now toFIG. 2, an exemplary well logging instrument10(also referred to as a “tool”) for wireline logging is shown disposed in a wellbore1(also referred to as a “borehole”). As a matter of convention, a depth of the wellbore1is described along a Z-axis, while a cross-section is provided on a plane described by an X-axis and a Y-axis. Prior to well logging with the logging instrument10, the wellbore1is drilled into the Earth2using a drilling rig, such as one shown inFIG. 1.

In some embodiments, the wellbore1has been filled, at least to some extent, with drilling fluid4. The drilling fluid4(also referred to as “drilling mud”) generally includes a mixture of liquids such as water, drilling fluid, mud, oil, gases, and formation fluids as may be indigenous to the surroundings. Although drilling fluid4may be introduced for drilling operations, use or the presence of the drilling fluid4is neither required for nor necessarily excluded from well logging operations. Generally, a layer of materials will exist between an outer surface of the logging instrument10and a wall of the wellbore1. This layer is referred to as a “standoff layer,” and includes a thickness, referred to as “standoff, S.”

The logging instrument10is lowered into the wellbore1using a wireline8deployed by a derrick6or similar equipment. Generally, the wireline8includes suspension apparatus, such as a load bearing cable, as well as other apparatus. The other apparatus may include a power supply, a communications link (such as wired or optical) and other such equipment. Generally, the wireline8is conveyed from a service truck9or other similar apparatus (such as a service station, a base station, etc, . . . ). Often, the wireline8is coupled to topside equipment7. The topside equipment7may provide power to the logging instrument10, as well as provide computing and processing capabilities for at least one of control of operations and analysis of data.

Generally, the logging instrument10includes apparatus for performing measurements “downhole” or in the wellbore1. Such apparatus include, for example, at least one on-board sensor15. Exemplary sensors15may include a set of coils to provide an induction logging instrument. The sensors15may communicate with the downhole electronics13. The measurements and other sequences as may be performed using the logging instrument10are generally performed to ascertain and qualify a presence of hydrocarbons5.

One skilled in the art will recognize that certain aspects of the teachings herein may be applied in either one or both of embodiments that include logging while drilling (LWD/MWD) and wireline logging. Accordingly, the invention disclosed herein is not limited to embodiments presented, and may include any type of subterranean measurement apparatus now known or later devised and as deemed practicable. In short, the well logging tool10,20may includes adaptations as may be necessary to provide for operation during various deployments, including during drilling or after a drilling process has been undertaken or completed.

Referring toFIG. 3, one embodiment of the sensor15that includes a set of coils further including transmitter coils and receiver coils as shown. Three transmitter coils101,103and105, referred to as the Tx, Tz, and Tytransmitters are substantially orthogonally oriented (the z-axis being along the longitudinal axis of the tool, as depicted in the legend). The transmitter coils101,103and105generate magnetic fields for obtaining measurements in the formations3surrounding the wellbore1.

Corresponding to the transmitter coils101,103and105are receiver coils107,109and111, referred to as the Rx, Rz, and Ryreceivers, respectively. The receiver coils107,109and111measure aspects of the corresponding magnetic fields, and are also are substantially orthogonally oriented to each other. Additional receiver coils113and115may be included and measure two cross-components Hxy, and Hxzof the magnetic field (H) produced by the x-component transmitter.

As used herein, the well logging tool10,20may also be referred to as an “instrument10,20” or a “multi-component induction tool10,20,” and by reference numbers “10”, “20” or “10,20” without implying any particular limitation of apparatus for deploying the technology.

Other embodiments of the tool10may include those, for example, with side-by-side transmitter and receiver coils and those having other arrangements of transmitters and receivers. Varying number of coils may be used. Therefore, it should be recognized thisFIG. 3depicts one embodiment of aspects of the well logging tool10,20, and is not limiting thereof. For example, in other embodiments, the additional coils113,115are not included. Further examples include embodiments having transmitter coils and receiver coils with other or varying orientations.

As discussed herein, reference to the tool10,20and aspects thereof generally refer to the exemplary and non-limiting embodiment. More information regarding the tool10,20may be had by turning to the paper “A New Multicomponent Induction Logging Tool to Resolve Anisotropic Formations” by Kriegshäuser, et al, SPWLA 41stAnnual Logging Symposium, Jun. 4-7, 2000, the disclosure of which is incorporated by reference herein in its entirety.

As discussed herein, and for convention, there are four coordinate systems involved in the processing of multi-component induction logging data. The four coordinate systems include one for each of the Earth2, the formation3, the tool10and the coils15. In typical embodiments, and by convention herein, the Earth2coordinate system has a z-axis Z in the direction of gravity, an x-axis X pointing to the magnetic North, and a y-axis Y in the direction of magnetic East. The other three coordinate systems are defined relative to the earth coordinate system28, depicted inFIG. 3A.

Referring toFIG. 3A, the Earth coordinate system28is shown, with the axes as described in the preceding paragraph. The formation coordinate system21is also depicted, and shown relative to the earth coordinate system28. An z-axis Zfof the formation coordinate system21is normal to a bedding plane, while an x-axis Xfand a y-axis Yfare on the bedding plane, each of the axes (Zf, Xf, Yf) being orthogonally oriented. Two formation angles, depicted as a formation dip angle (θ′) and a formation azimuth angle (Φ) describe a relationship between the Earth coordinate system28and the formation coordinate system21. Typically, the formation dip angle (θ′) and the formation azimuth angle (Φ) are determined by use of inversion processing. The tool coordinate system is depicted inFIG. 4B.

Referring toFIG. 3B, a z-axis Ztof the tool coordinate system22describes the trajectory of the wellbore1and is normal to a tool plane. The x-axis Xtand the y-axis Ytof the tool coordinate system22are on the tool plane, each of the axes (Zt, Xt, Yt) being orthogonally oriented. The relation of the tool coordinate system22to the Earth coordinate system28is described by a tool dip angle (Ψ) and a tool azimuth angle (Γ).

A combination of coordinate systems is depicted inFIG. 5. More specifically, the formation coordinate system21, the tool coordinate system22and a coil coordinate system23are depicted in relation to each other. Referring toFIG. 5, a dip plane is also shown. The dip plane is described in the coil coordinate system23, and includes an x-axis Xc, a y-axis Ycand a z-axis Zc. The z-axis Zcof the coil coordinate system23shares the same z-axis Ztas the tool coordinate system22, while the y-axis Ycof the coil coordinate system23is offset from the y-axis Yfof the formation coordinate system21by the formation azimuth angle (Φ). A relative tool rotation angle (φ) (not shown inFIG. 5) is described by the x-axis Xcfor the coil coordinate system23, with respect to the high side of the wellbore1.

In operation, the tool10measures co-axial and co-planer components called main components, and also measures cross-components. These measurements provide information to determine, among other things, resistivity and anisotropy of the formation3. With this information, other salient information may be obtained or derived. For example, with tool orientation data, the formation dip angle (θ′) and the formation azimuth angle (Φ) can be obtained. As disclosed herein, algorithms based on and operational within neural networks can be used advantageously to check data quality and determine the formation dip angle (θ′) and the formation azimuth angle (Φ) in real-time, or near real-time.

As used herein generation of data in “real-time” is taken to mean generation of data at a rate that is useful or adequate for making decisions during or concurrent with processes such as production, experimentation, verification, and other types of surveys or uses as may be opted for by a user. As a non-limiting example, real-time measurements and calculations may provide users with information necessary to make desired adjustments during the drilling process. In one embodiment, adjustments are enabled on a continuous basis (at the rate of drilling), while in another embodiment, adjustments may require periodic cessation of drilling for assessment of data. Accordingly, it should be recognized that “real-time” is to be taken in context, and does not necessarily indicate the instantaneous determination of data, or make any other suggestions about the temporal frequency of data collection and determination.

As further context for the teachings herein, consider aspects of formation composition and associated properties.

Classic evaluation of properties of formations3assumes that the formation3is homogenous and isotropic. With the advent of tools for measuring conductivities in orthogonal directions, more accurate evaluations may be made. For example, the tensorial nature of electrical properties can be obtained and used for improved saturations. Today, tensorial resistivities measurements are available from both wireline and logging while drilling instruments.

Generally, in sedimentary reservoir rocks, two types of anisotropy (directional dependence) are found. A first is that of intrinsic anisotropy (i.e., anisotropy originating in the mineral crystal structure or preferred orientation of rock components) or in layering at the microscopic level; another type is macroscopic anisotropy (i.e., anisotropy where layering is visible to the human eye, and may be composed of different, but homogenous layers).

Most commonly, macroscopic anisotropy is associated with thin sand-shale layers (such as laminated shaly-sands). Such anisotropic sands are common in deep water fans and turbidites.

Fractures, depending whether they are open or closed and the on the conductivity of the fracture filling material can also give rise to anisotropy. Fractures generally are oriented at high angles relative to a bedding plane and give rise to azimuthal anisotropy in the bedding plane. In thinly bedded formations3, fractures will give rise to biaxial anisotropy. With respect to the origin of fractures, consider two cases: a first case involving drilling induced fractures, with a fracture filling fluid originating from the drilling mud (oil- or water based); and, a second case where natural fractures result from tectonic stresses, with a fracture filling fluid similar to the formation fluids. Thus, a main difference for the two cases is the assumed composition and resistivity of the fracture filling fluid. SeeFIG. 6.

InFIG. 6, aspects of biaxial anisotropy are shown. InFIG. 6A, biaxial anisotropic rock (i.e., formation3) is shown. This includes superposition of transverse isotropic medium and oriented natural fractures. InFIG. 6B, biaxial anisotropy is shown resulting from a drilling induced fracture51in a transverse isotropic medium (i.e., formation3).

With simple parallel and series resistor networks, the effect of the fractures51and the fluid in the fractures and their relation to biaxial anisotropy can be understood. Starting with a laminated sands and shales (horizontal layers in the x-y-plane) with thin vertical fracture oriented in the x-z-plane, the effect of these fractures on the electrical resistivity, R, for the three principal directions Rx, Ry, Rzmay be characterized. Reference may be had toFIG. 7.

For a conductive filling fluid in the fracture51, such as water based mud, and as shown inFIG. 7A, a presence of fracturing will result in a decrease of the resistivities in all directions parallel to the fracture51. Little or no influence on Rywill be realized. Thus, the fracture effect is most pronounced on the X and Z directions. Fracturing results in an anisotropy in the horizontal plane with Rx<Ry.

For a resistive filling fluid in the fracture51, such as oil based mud, and as shown inFIG. 7B, a presence of fracturing will result in a dramatic increase of the resistivity in the direction perpendicular fracture plane (Ry) but practically no effect on the resistivity in directions parallel to the fracture. Thus, the impact of fractures is most pronounced on Y direction. In this case, fracturing results in an anisotropy in the horizontal plane with Ry>Rx.

Accordingly, it may be expected that the magnitude of the effects of biaxial anisotropy to be controlled mainly by the resistivity contrast between the unfractured formation and the fracture porosity and fluid in the open fracture.

Now, quantitatively consider the effect of fractures on a transverse isotropic medium. First, start with a transverse isotropic media with near vertical fractures making an angle, α with respect to the bedding normal. For quantitative considerations, it is assumed the fractures are normal to the bedding plane. The derivation follows from the laws for parallel and series resistor networks, as depicted inFIG. 8.

Now, also with reference toFIG. 9, to model the biaxial anisotropy attributable to fracturing, we express the resistivities in terms of the horizontal resistivities of unfractured media (Rform,h), a macroscopic manifestation of a transverse isotropic media consisting of unfractured laminated sand and shales, where Rform,hrepresents parallel to the bedding plane; vertical resistivity of an unfractured media (Rform, v) the macroscopic manifestation of a transverse isotropic media, where Rform,vrepresents the resistivity perpendicular to the bedding plane; fracture filling fluid (mud, mud filtrate, or formation fluids) resistivity (Rm, Rmf, respectively); and fracture porosity, φf. The resistivity of the unfractured laminated shaly may then be estimated by Eqs. (1) and (2):

Rform,h=(1-Vsh)·Rsd+Vsh·Rsh,h;(1)Rform,v=[(1-Vsh)Rsh+VshRsh,v]-1;(2)
where:Vshrepresents laminar shale content;Rsdrepresents resistivity of the porous sand fraction (without fracture), andRsdmay be estimated as: Rsd=φ−m·SW−n·RW;Rsh,hrepresents resistivity of the shale sand fraction (without fracture), parallel bedding;Rsh,hrepresents resistivity of the shale sand fraction (without fracture), perpendicular bedding.

Consider now the effects of water-based and oil-based mud drilling induced fractures with this model.

In a first aspect of effects of evaluation of drilling induced fractures, forward modeling drilling induced fractures are considered. An exemplary model has two pore systems with different petrophysical properties. For these properties, and with regard to the intergranular pore system of the sand fraction, it is assumed, that the pore system with “sand” porosity, φsd, can be characterized by Archie's equation with the parameters, m=n=2. With regard to the fracture system, a description of electrical properties of fractured rocks using Archie's equation leads to small exponents m ranging from about 1 to about 1.2. In this study, ideal fracture values were found to be about where m=n=1.

In order first calculations for the three principal components of the resistivity tensor, Rx, Ry, and Rz, a forward calculation with the following input parameters were applied. For the sand fraction: φsd=0.25; m=n=2; Rw=0.035 Ohm-m; and Sw=1.0, 0.5, 0.1. For the shale fraction: Rsh,h=1 Ohm-m; and Rsh,v=3 Ohm-m.

For the fracture filling fluid (whole mud or possibly mud filtrate), we use a relatively fresh mud resistivity of 1 Ohm-m for the water-based mud and 200 Ohm-m for the oil-based mud. In addition, the water saturation in the drilling induced fractures may be varied. For both water-based and oil-based muds, the impact of water saturation for three water saturations are evaluated, where Sw=1.0, 0.5, 0.2. Shown inFIG. 10are the results for water-based mud. Resistivities Rx, Ry, Rz, are plotted as function of the laminar shale volume fraction for unfractured and fractured (φf=0.02) formation.

More specifically,FIG. 10shows resistivities as function of the laminated shale content of the formation for water based mud. Curves are calculated for unfractured and fractured formation (φf=0.02). Corresponding results for an oil-based mud are shown inFIG. 11. InFIG. 11, resistivity as function of the laminated shale content of the formation for oil-based mud drilling induced fractures is shown. Curves are calculated for unfractured and fractured formation (φf=0.02).

It is seen that the results depend strongly on fracture filling fluid type and saturation. Accordingly, it is found that for the resistivities parallel to the fracture plane Rxand Rz, that in the water-based mud case, resistivities are influenced only slightly by the drilling induced fracture. The small effect results from the “parallel conductor effect” which is characterized by the influence of the conductive component (i.e., conductive fluid in the fracture). Note that the small effect on Rxand Rzis easily masked by the presence of the highly conductive shale laminations. In the oil-based mud case, there is no noticeable effect on Rxand Rzbecause the resistive fracture is masked by the “parallel conductor” effect of the more conductive formation.

Accordingly, for the resistivity perpendicular to the fracture plane, Ry, in case of a water-based mud, there is no noticeable effect because the effect of the conductive fracture (low resistivity) is small relative to the resistive formation (i.e., the “series resistor” effect is dominated by the high formation resistivity. In case of an oil-based mud, even for small fracture porosity is strong influenced by the resistive fracture. The effect is most pronounced as the laminated shale content increases.

For a transversely anisotropic formation, a ratio of the vertical to horizontal resistivity (Rv:Rz) is generally presented, but when there are two sources of directional dependence for the anisotropy it is less clear which ratio should be used. Shown inFIG. 12are the vertical (Rz) to horizontal resistivity perpendicular to the fracture plane (Ry). In general, for the laminated and fractured rock:the ratio Rz:Rxreflects mainly anisotropy resulting from lamination, because fracturing has a similar contribution on both resistivities, whereas the ratio Ry:Rxreflects mainly anisotropy resulting from fracturing, because fracturing has different contribution on both resistivities, but lamination has a similar influence.

FIG. 12depicts resistivity ratios Rz:Ryand Ry:Rxas a function of the shale content for water-based mud and oil-based mud. Calculations for a fractured formation (φf=0.02) and different formation water saturations.

InFIG. 12, the ratios Rz:Ryand Ry:Rxfor a biaxial anisotropic medium clearly show that the vertical anisotropy, Rz:Ry, still dominates anisotropy in a sand-shale layered system, but the maximum no longer occurs at 50% Vshalelaminar, but moves to higher or lower values depending on whether the drilling induced fracture occurs when drilling with a water or oil-based mud, respectively. The anisotropy in the bedding plane5, (e.g., horizontal anisotropy ratio (Ry:Rx)) shows that this ratio increase significantly for oil-based mud filled drilling induced fractures51. However, for water-based mud there is very little to no effect except when the Vshalelaminar content is very small (<10%).

With respect to fracturing perpendicular to bedding the ratio Ry:Rxcarries the information on the fracture anisotropy. Combining Eqs. (1) and (2) results in Eq. (3):

RyRx=(1-ϕf)·{1+ϕf·(RmfRform,h+Rform,hRmf)}+ϕf2(3)
which confirms that the anisotropy effect is controlled by fracture porosity, φf, and the resistivity contrast of formation and fracture filling material, Rmf/Rformation. The dependence of the azimuthal resistivity anisotropy on fracture porosity is shown inFIG. 13.

InFIG. 13, the ratio Ry:Rxis shown as a function of fracture porosity, φf, for water-based mud (left) and oil-based mud (right). Indicated in the figures is the parameter Rmf:Rformation. Note the different scales for the two cross plots.

Consider now a resistivity model for naturally fractured rock Sedimentary rocks that are subject to external stresses can fracture. Refer again to the simplified model, with vertical fractures orthogonal to the sedimentary laminations and parallel to the xz-plane. Applying the same model as for the drilling induced fractures, it is further assumed that the individual fractures are connected.

However, in contra-distinction to the drilling induced fracture, it may be assumed that natural fractures contain the same fluid components as the sand pore system. However, in these examples, different water saturations in the fractures51have not been accounted for. Therefore equations (1) to (3) must be modified by introducing a saturation term for the fracture filling fluids, Sw,f. For hydrocarbon bearing intervals, it may be assumed that the water saturation in the fractures51is lower than the water saturation in the fine pores of the laminated sands. Accordingly, resistivities in x, y and z directions may be estimated by Eqs. (4), (5) and (6), respectively.
Rx=((1−φf)·Rform,h−1+φf·Sw,f·Rw−1)−1(4);
Ry=((1−φf)·Rform,h+φf·Sw,f−1·Rw(5);
Rz=((1−φf)·Rform,v−1+φf·Sw,f·Rw−1)−1(6).

FIG. 14shows the calculated resistivities for Rx, Ryand Rzagain versus shale content Vsh. For computations herein, the input parameters are now provided. For the sand fraction, inputs include: φ=0.25, m=n=2, Rw=0.035 Ohm-m; sand water saturation, Sw=0.4, 0.2, 0.1. For the shale fraction, inputs include: Rsh,h=1 Ohm-m, Rsh,h=3 Ohm-m. Regarding the fracture porosity, inputs include: φf=0.02, the fracture water saturation, Sw,f=0.05. InFIG. 14, resistivity as function of the laminated shale content for naturally fractured formation is shown. Curves are calculated for unfractured and fractured formation (φf=0.02, fracture water saturation Sw,f=0.05). The sand of the laminated formation has a porosity of φsd=0.25 and water saturation is as indicated.

FIG. 15illustrates the effect of fractures on the ratio of the two horizontal resistivities, Ry:Rx. The ratio, Ry:Rx, decreases rapidly as the shale content increases. The low horizontal resistivity of the shale masks the effect of the fractures. Similarly increasing sand water content decreases the importance of the conductivity of the water in the fractures and the resulting azimuthal anisotropy decreases.

InFIG. 15, the resistivity ratio Ry:Rx, is shown versus shale content. For non fractured formation and fractured formation (φf=0.02, fracture water saturation, Sw,f,=0.05). The sand of the laminated formation has a porosity of φsd=0.25 and water saturation is as indicated.

Therefore, the azimuthal resistivity anisotropy, Ry:Rx, is considered indicative of fracturing in a hydrocarbon bearing, low laminated-shale content, high horizontal formation resistivity (i.e., resistivity high compared with the fracture resistivity) which is proportional RW/Sw(indicating low shale content, low water saturation, and low porosity). In the limit of no laminated shale, the biaxial anisotropy reduces to a simple uni-axial azimuthal anisotropy.

Shown inFIG. 16are results for a laminated, tight sand (perhaps an interbedded shale-silt formation) with a porosity of φsd=0.1. The azimuthal resistivity anisotropy, particularly for low shale content increases very strong. For the case of a clean fractured sand (Vsh=0),FIG. 17shows the increase of the effect with decreasing formation conductivity originated by further decreasing porosity.

FIG. 16shows the resistivity ratio, Ry:Rx, versus laminated shale content for unfractured and fractured formation (φf=0.02, fracture water saturation Sw,f=0.05). The assumed porosity of the hydrocarbon bearing sands is 10% and the water saturations are as indicated.FIG. 17, shows the resistivity ratio, Ry:Rx, versus sand porosity for a fractured formation (φf=0.02, fracture water saturation Sw,f=0.05). The laminated formation has a shale content Vsh=0 and Vsh=0.05 (right). The water saturation of the sand is indicated.

Turn now to the multi-component induction data and the processing and interpretation to find the best fitting biaxial resistivity model. A schematic configuration of the multi-component induction tool10is shown inFIG. 18. In this embodiment, the tool10,20includes three orthogonally-oriented transmitter coils (Tx, Ty, Tz) and three also orthogonally-oriented receiver coils (Rx, Ry, Rz). At every logging depth this tool measures all nine components of a magnetic field generated by the tool10, typically using multiple frequencies in the low frequency range.

In a transverse isotropic (TI) formation or in a formation with plane boundaries, particularly in a thinly laminated formation, the electromagnetic field of an arbitrary source may be described as a combination of two modes: transverse electric, TE, and transverse magnetic, TM. The TE mode has a planar electric field parallel to the boundary planes. In the TM mode, the magnetic field is planar.

Traditional induction transmitters (oriented normally to the boundaries) excite only the TE mode. The resulting electromagnetic field is sensitive only to horizontal resistivity, Rh. Transverse transmitters (parallel to the boundaries) excite two modes, TE and TM. Since the electric field contains both components, parallel and perpendicular to the boundaries, the transversal measurements Hxxand Hyydepend on both vertical and horizontal resistivity, Rvand Rh. If there is no relative dip between the tool10and formations3, all cross-components are equal to zero as there is no coupling between a transmitter and an orthogonal receiver. If the relative dip is not zero, all transmitters excite both TE and TM modes and all nine components depend on Rhand Rv.

In a bi-axially anisotropic formation3, even with one of the principal axis of resistivity tensor oriented perpendicular to plane boundaries, the introduction of two modes is practically meaningless because a solution does not exist for TE nor TM. This means that all components of the measured magnetic tensor depend on all three principal formation resistivities, Rx, Ryand Rz.

The current pattern and the behavior of corresponding magnetic fields excited by transversal coils are very complex, even in a uniform TI formation. Shoulder beds, bed boundary interfaces, the borehole (filled with either resistive or conductive drilling mud), as well as invasion cause additional significant distortions. Accordingly, some of these effects are now briefly discussed and illustrate challenges associated with understanding physics of the multi-component tool10and developing reliable processing software.

Current patterns. In an anisotropic formation3, currents are not parallel to the electric field. This leads to a concentration of electric charges in the formation volume. As a result, the currents, which are planar and circular in the isotropic formation, follow a 3D flow pattern in the case of anisotropic medium.

Skin-effect. The transversal Hxxand Hyycomponents exhibit much stronger skin-effect than the conventional Hzzcomponent. To compensate for that, one needs to measure responses at lower frequencies (consequently, move to lower signal levels) or acquire data at multiple frequencies and use sophisticated multi-frequency skin-effect correction techniques.

Bed boundary effects. Even in isotropic formations, electrical charges appear at interfaces of beds if the electric field has a component perpendicular to bed boundaries. The electrical charges affect the direction of original currents causing sharp turns in the current lines. If a resistivity contrast is high enough, the resulting magnetic field may even change sign near bed interfaces causing the appearance of infamous “horns” on the logs. As a result, transversal induction logs do not follow the resistivity distribution and are harder to understand and interpret.

Borehole and invasion effects. Transversal induction measurements are affected by borehole and invasion much more than conventional co-axial measurements. Even in an oil-based mud, where conventional deep induction measurements do not require corrections, the effect on Hxxand Hyycomponents with similar spacing may increase up to 50% depending on the measurement frequency. The eccentricity effect on the transversal components is very strong as well, especially in conductive mud and high formation resistivity. It is very difficult to correct for such an effect because relative dip, in an anisotropic formation, may affect the measurements in a similar way making it very difficult to separate the two effects.

Azimuthal sensitivity. Traditional deep resistivity measurements with axially symmetric design (both galvanic and induction) do not have azimuthal sensitivity. On the contrary, the transversal coils provide strong azimuthal sensitivity. As a result, multi-component induction measurements can be used for a number of applications.

Accordingly, a solution to these challenges provided herein includes a new processing technology. That of multi-frequency focusing (MFF). MFF provides for reducing environmental effects and simplifying data dependence on the formation parameters. MFF processing is based on an expansion of the measured field into a frequency series. In MFF processing, one of the series terms does not depend on the borehole and invasion properties. MFF is a method of deriving this term from the multi-frequency measurements. After MFF, a simple 1-D vertical layered model can be used for the data interpretation because near-zone effects are effectively removed by MFF. MFF responses are very deep and have simplified dependence on the formation parameters including very small bed boundary effects. As discussed herein, the MFF processing algorithm is presented in terms of bi-axially anisotropic formations in more detail.

As discussed herein, “multi-component” generally refers to magnetic field components sufficient to obtain axial component (conventionally Hzzcomponent, and the cross components Hxxand Hyy) MFF provides a technique for correcting the complicated data. In MFF processing, it is critical to have at least one of cross component data, fully orthogonal coils or tilted coils.

Consider MFF processing with regard to processing multi-component induction data for biaxial anisotropy. In the following discussion a bi-axially anisotropic formation is considered where, in a principal coordinate system (x, y, z), a conductivity tensor is described by Eq. (8)

It may also be assumed that a multi-component induction logging tool acquires at every depth a magnetic matrix, which is recorded in the tool coordinate system22(x′,y′,z′), as provided in Eq. (9):

H^′=(hx′⁢x′hx′⁢y′hx′⁢z′hy′⁢x′hy′⁢y′hy′⁢z′hz′⁢x′hz′⁢y′hz′⁢z′)(9)
where the first index indicates an orientation of a transmitter and the second index specifies the orientation of a receiver. For clarity, a thick anisotropic layer is considered. In this case an arbitrary position of the tool10can be fully described with the three Euler angles: tool azimuth, φ, relative dip, θ, and tool rotation, ψ.

It can be shown that matrix (9) cannot be diagonalized if the relative dip is not zero. In special cases, when the tool azimuth angle φ is 0, 90, 180, or 270 degrees in the principal coordinate system the matrix (9) has five non-zero components. In all other situations when the relative dip is not zero even in the earth coordinate system20, the matrix (9) is full.

The situation becomes different when a multi-frequency focusing is applied to the magnetic matrix to extract terms proportional to ω3/2. In a deviated well (i.e., non-zero relative dip) the matrix of MFF components looks similar to the magnetic matrix, as provided in Eq. (10):

It can be shown that in deviated wells in the earth coordinate system20, the components of the multi-frequency focused magnetic field become diagonilized (contrary to the original magnetic matrix), as provided by Eq. (11):

Accordingly, the MFF matrix may be expressed in the tool coordinate system22(see Eq. (10)) through the MFF matrix in the earth coordinate system20(see Eq. (11)) using a tensor rotation rule, as provided by Eq. (12):
Ĥ′MFF={circumflex over (R)}ĤMFF{circumflex over (R)}T(12)

The rotation matrix, {circumflex over (R)}, can be obtained from the Euler's rotation theorem, provided in Eq. (16):
{circumflex over (R)}={circumflex over (R)}z(ψ)·{circumflex over (R)}y(θ)·{circumflex over (R)}z(φ)  (13)
where {circumflex over (R)}z(ψ), {circumflex over (R)}y(θ), {circumflex over (R)}z(φ) are standard Euler rotation matrices expressed through sin and cos of corresponding angles.

Considering Eqs. (10), (11) and (13), it may be observed that in the right-hand side of the Eq. (12), there are six (6) unknowns. The unknowns include three principal components {tilde over (h)}xx, {tilde over (h)}yy, {tilde over (h)}zzand three angles φ, θ, ψ. Also, it may be observed that the left-hand side of the Eq. (12) includes nine (9) measurements. As Eq. (12) is correct at every logging depth, an algorithm has been derived and yields maximum stability and robustness in recovering the unknowns.

The algorithm for obtaining the three (3) resistivities (Rx, Ry, Rz) generally includes two sequential steps: 1). Processing for three (3) unknown angles and three (3) multi-frequency focused principal components; 2). Inverting the three (3) principal multi-frequency focused magnetic field components for three (3) principal resistivities.

At the first step at every logging depth, the azimuth angle, φ is incrementally changed from 0 to 180 degrees, the relative dip, θ, is changed from 0 to 90 degrees, and the rotation angle, ψ, is changed from 0 to 360 degrees. For every set of φ, θ, and ψ, the system of equations leading to Eq. (12) may be solved for the unknown principal components {tilde over (h)}xx, {tilde over (h)}yy, {tilde over (h)}zzusing the least-squares technique. The values of φ, θ, ψ, {tilde over (h)}xx, {tilde over (h)}yy, {tilde over (h)}zzobtained may then be placed back into Eq. (12) and calculate the synthetic values of multi-frequency focused responses, {tilde over (h)}. Using some norm, a misfit value of the measured and synthetic MFF components may be calculated for every combination of angles. The angle with the minimum misfit value is then selected. The principal components from the optimal set of angles are used in the following interpretation for σx, σy, and σz.

After the three angles and three principle components are determined, the second processing step is to recover the true formation resistivities, (Rx, Ry, Rz). It should be noticed that in the case of the bi-axial anisotropy, each principal component depends on all three (3) formation resistivities (Rx, Ry, Rz), which makes the sequential processing for the three (3) resistivities impossible. In the thick layer, an inversion algorithm was developed that uses a look-up table that contains pre-calculated values of the principal components for all possible combinations of the principal conductivities. To reduce the size of the look-up table, three formation conductivities are considered to be in decreasing order (σxis the largest). Also, the principal components are converted to apparent conductivities and normalized by the largest value (σapp({tilde over (h)}zz)). These measures and the smoothness of the function allow construction of a very compact two-dimensional look-up table calculated for σx=1. This yields ten (10) different values of σx/σy(from 1 to 22.63), and ten (10) similar values of σx/σz(actually, due to the decreasing order of conductivities, only the upper triangle of this table is calculated and used). The calculations for the look-up table and for the following numerical tests were conducted using the newly developed forward modeling code for bi-axially anisotropic formations.

To validate the theory and algorithms described above, measurements of the tool10have been simulated for a thick bi-axially anisotropic layer for various sets of parameters (three (3) angles and three (3) conductivities). The results for two models are presented below.

To demonstrate the single frequency response, a skin-effect correction was applied using ten (10) frequencies of the tool10. The measurements were converted to apparent conductivities (S/m). The results in the tool coordinate system22and in formation coordinate systems21are presented below, as Eqs. (14a) and (14b), respectively:

The tool responses after multi-frequency focusing and normalizing to the apparent conductivity in the tool coordinate system22and in formation coordinate systems21are presented below, as Eqs. (15a) and (15b), respectively:

It may be seen that the numerical results (Eqs. (15)) agree with the theoretical Eq. (11) within the numerical accuracy of the calculation of the MFF components (all cross-components in the Earth coordinate system28are close to zero). Moreover, it may be noted that the increasing order of the principal diagonal components in matrix (15b) correctly reflects the decreasing order of the true formation conductivities in this model, which is not the case for the single frequency components.

The multi-frequency focused responses (18a) were used as an input in the least-square optimization algorithm described above. The output from this code provided the following values: fi=15.0; theta=45.0; psi=30.0; hh(1)=304.3; hh(2)=469.8; and hh(3)=716.4.

It may be seen that all the angles and the principal components were recovered with high accuracy (consider hh(i), and comparison with the calculated principal components (15b)). The obtained principal components were then used to estimate true formation resistivities, (Rx, Ry, Rz), as described above. The results obtained for this model were: Rxx=0.999; Ryy=1.999; and Rzz=7.999. Now consider the second test model.

The single frequency responses in the tool coordinate system22and in formation coordinate systems21corrected for the skin-effect and converted the results to apparent conductivities (S/m) are presented in Eqs (16a) and (16b):

The tool responses after multi-frequency focusing and normalizing to apparent conductivity in the tool coordinate system22are provided by Eqs. (17a) and (17b):

It can be seen that the numerical results of Eqs. (17) agree with the theoretical Eq. (11) within the numerical accuracy of the calculation of the MFF components (all cross-components in matrix 17b are close to zero). Again, it may be noticed that the increasing order of the principal diagonal components in (17b) correctly reflects the decreasing order of the true formation conductivities in this model, which is not the case for the single frequency components.

The multi-frequency focused responses (17a) were used as an input in the least-square optimization algorithm described above. The output from this code provided the following values: fi=33.0; theta=66.0; psi=71.0; hh(1)=133.4; hh(2)=195.4; and hh(3)=222.1.

It may be seen that all the angles and the principal components were recovered with high accuracy (consider hh(i), and comparison with the calculated principal components (17b)). The obtained principal components were then used to estimate true formation resistivities, (Rx, Ry, Rz), as described above. The results obtained for this model were:Rxx=3.29; Ryy=6.36; and Rzz=9.31.

In both Model 1 and Model 2, the recovered resistivities (Rx, Ry, Rz) for the formation4agree very well with the original resistivities (Rx, Ry, Rz) for the formation4.

A field trial was completed where the processing described herein was applied for evaluations of bi-axially anisotropic formations using data from a well having drilling induced fractures. The well included a 12.25″ vertical well and was drilled overbalanced with a 13.2 ppg synthetic oil-base mud. A wide spectrum of logging while drilling (LWD) and wireline logging data (data acquired after drilling) was acquired, including LWD propagation resistivity, wireline array induction data, multi-component induction data, and cross-dipole acoustic measurements data.

In a review of the data, the presence of drilling induced fractures in some intervals was immediately apparent based on the large difference between LWD resistivity and array induction resistivity data and separation of the shallow and deep array induction resistivity curves. The induction measurements were strongly affected by fractures as well. Hxxand Hyycomponents showed strong azimuthal sensitivity oscillating around each other in perfect correlation with the rotation angle. A typical manifestation of vertical fractures was high formation dips recovered from the induction data. The dip angles were up to about 90 degrees. In this case, the induction data proved to respond more to vertical fractures rather than to orientation of horizontal and vertical resistivities, meaning that TI model was not appropriate for interpretation any longer.

To test the teachings provided herein, four (4) depths were selected where the fractures were most pronounced. The results of the bi-axial anisotropy processing at these depths are presented in Table 1. For comparison, results from enhanced TI processing for fractured formations are included. The obtained resistivities in the Earth coordinate system28are presented in columns 2-4. It can be observed that in all cases ρxand ρzare slightly higher then ρhand ρv(columns 7, 8) correspondingly. The resistivity anisotropy in horizontal plane (ρy/ρx) is consistently between 8 and 10. The orientation of fractures (or the orientation of principal y-axis) obtained with algorithm provided herein (column 5) in all cases agrees within two (2) degrees with the rotation technique used in the enhanced TI processing (column nine (9)). The relative misfit of the measured and synthetic data presented in column six (6) indicates the good quality of inversion and consistency of the fractured formation with the bi-axial anisotropic model.

The model derived equations allow an inversion of processed data. Table 1 shows a realistic data set from a shale section. Results are based on the biaxial anisotropy model with fracture porosity, φf, and formation resistivities, Rx, Ry, Rz) and mud filtrate resistivity, Rmf=200 ohm-m:

TABLE 2Input data set (left) and result of inversion (right)for natural fracturesInputResultRyRzRform,hDepthRx(ohm(ohm(ohmRform,v(m)(ohm m)m)m)m)(ohm m)φfD135.60.756.501.400.731.360.029D213.00.604.751.500.591.470.021D252.30.676.751.450.651.270.027D260.00.685.951.300.661.380.027

In summary, presented herein are resistivity models for laminated sand-sale formations in the presence of drilling-induced and natural fractures. The principal resistivities Rx, Ry, Rzare expressed as a function of horizontal and vertical resistivities of unfractured formation, fracture porosity and resistivity of fracture which is the resistivity of mud filtrate in the case of drilling induced fractures or the connate water resistivity and saturation in the case of natural fractures. These models allowed the effect of fractures on the three principal resistivities to be studied as a function of laminated shale content and as a function of fracture porosity for both drilling induced and natural fractures. The combined three-step analysis of measured principal resistivities based on the modular concept allows a separation and quantification of the two anisotropy sources: fracturing and lamination. The analysis based on the three principal resistivities provides an estimate of fracture porosity, resistivity of the porous sand fraction (without fractures) and water saturations.

In addition, an inversion algorithm for the biaxially anisotropic formations has been presented and may be used to reliably determine three principal resistivities from multi-component induction measurements. First, in the least-square inversion the multi-frequency focused magnetic field tensor is rotated to the principal coordinate system. This rotation is characterized by three angles: φ, θ, and ψ; and three principal, diagonal magnetic field components Hxx, Hyy, and Hzz. After the three angles and three principle components are determined, principal formation resistivities are estimated using a fast inversion algorithm based on a look-up table approach. To validate the theory and algorithms described above, the multi-component induction measurements in thick biaxially anisotropic layers have been simulated for various sets of parameters and used the software to process the data. The new approach was also applied to the real measurements acquired in the well with drilling induced fractures and as a result principal formation resistivities, fracture orientation, and fracture porosity were reliably determined.

Accordingly, with a simple petrophysical model, the teachings herein have investigated the effects of fractures on the tensor resistivity properties of laminated rocks and demonstrated that fractures in such a systems give rise to biaxial anisotropy. In particular, this has focused on laminated sand-shale formations and investigated both natural and drilling induced fractures. For the case of natural fractures the effect of hydrocarbon saturation have been investigated. For drilling induced fractures both fractures flushed with water-based and oil-based muds have been considered.

The forward calculations provided demonstrate both subtle and dramatic changes in directional resistivities. The magnitude of the effects depends on whether the measurement is in or across the fracture plane and whether the fracture is filled with conductive or non-conductive fluid.

For convenience, it is considered herein that laminations lie parallel to the x-y plane and fractures are oriented in the x-z plane. Based on simple parallel and series resistor analogues and simple forward calculations, it is found that: the ratio Rz:Rxreflects mainly anisotropy resulting from lamination, because the additional conductivity of the fractures has the same contribution to both resistivities, and that the ratio Ry:Rxreflects mainly anisotropy resulting from fracturing, because fracturing effects the resistivities normal (y) and perpendicular (x) to the fracture plane differently while lamination effect is similar.

In detail the effects, particularly the fracture effects, are controlled by the contrast between the formation (unfractured) resistivity and the “fracture resistivity” (which is controlled by the fracture porosity and resistivity of fracture filling fluid). Therefore, results depend strongly on fracture filling fluid type and saturation. In particular, drilling induced fractures (oil-based mud) have practically no influence on the resistivities parallel to the fracture plane Rxand Rz(“parallel conductor effect”), but increase very markedly the resistivity perpendicular to the fracture plane, Ryeven for small fracture porosity. Therefore, there is a noticeable increase in the azimuthal resistivity ratio, RyRx, associated with the change in Ry. There is also a dependence on the formation conductivity, which in turn is controlled by shale content, porosity, water saturation. Further, drilling induced fractures (water-based mud) give rise to very small resistivities decreases parallel to the fracture plane Rxand Rz, and have practical no influence on Ry. Hence, the fracture effect manifests itself as an increase of the ratio Ry:Rxassociated with the change of Rx. This effect is relatively small and decreases with increasing formation conductivity, which depends on shale content, porosity, and water saturation. In addition, natural fractures have an influence on the resistivities which is controlled again by the contrast of formation resistivity and fracture resistivity. The latter is a function of fracture porosity, water resistivity and fracture water saturation. Model calculations demonstrate that a clear fracture signature is to be expected in the azimuthal resistivity ratio Ry:Rx, especially for relatively resistive formations. As the laminar shale content increases the resistivities of both Rxand Rydecrease and mask the relatively small fracture effect. However, it may be expected that for a relatively clean, hydrocarbon bearing fractured tight-gas sandstone, that there might be a measureable biaxial resistivity effect.

Lastly, processing algorithms have been developed for multi-component induction data that allow us to extract best fit biaxial anisotropy data. In addition to the three principal resistivity components (Rx, Ry, and Rz), the three Euler angles that describe the orientation of the laminations and fractures (φ, θ, and ψ) may also be extracte{tilde over (d)}. The model concept provided allows solving for the properties of the fractures (e.g., fracture porosity) as well as the properties of the sand and shale laminations.

Embodiments of the invention include various methods, systems, instruments and computer program products.

For convenience of referencing and to establish clarity, certain nomenclature as used herein is now presented. As used herein, the following variables generally are applied, and defined as follows:x,y,z axis directions of the orthogonal Cartesian coordinate systemx,y axis in the plane of bedding/lamination; horizontal axisz axis perpendicular to the plane of bedding/lamination, vertical axis)Rx, Ry, Rz true specific electrical resistivities of the formation (including fractures) in the three axis directionsRform,hresistivity of unfractured formation in horizontal directions (x,y)Rform,vresistivity of unfractured formation in vertical direction (z)Rsh,hresistivity of the shale sand fraction (unfractured) in horizontal directions (x,y)Rsh,vresistivity of the shale sand fraction (unfractured) in vertical direction (z)Rsdresistivity of the porous sand fraction (unfractured)Rwresistivity of the formation waterRmfresistivity of the mud filtrateφsdporosity of the sand fraction (unfractured),φffracture porositySW,sdwater saturation of the sand pores fraction,SW,fwater saturation of the natural fractures,m,n Archie parametersVsh,lamlaminar shale content{circumflex over (σ)} conductivity tensorσx,y,zconductivity tensor components in the principal coordinate systemĤ′ measured magnetic field tensor in tool coordinate systemhx′y′components of the measured magnetic field tensor in tool coordinate systemĤ′MFFmeasured magnetic field tensor after MFF in tool coordinate system{tilde over (h)}x′y′components of the measured magnetic field tensor after MFF in tool coordinate systemĤMFFmeasured magnetic field tensor after MFF in formation coordinate system{tilde over (h)}xycomponents of the measured magnetic field tensor after MFF in formation coordinate systemφ, θ, ψ Euler's angles{circumflex over (R)}z(ψ),{circumflex over (R)}y(θ),{circumflex over (R)}z(φ) Euler's rotation matrices.
It should be noted that with regard to any errors or inconsistencies between this reference information and other information provided herein, that common sense and logic should govern. Accordingly, in some instances, this reference information may be inconsistent with other portions of this disclosure.