Determining anisotropic subsurface properties using electromagnetic measurements

Abstract

The properties of geological formations such as vertical resistivity values, horizontal resistivity values, dip angles, and azimuth angles can be determined by inverting electromagnetic (EM) well log data, based at least in part on anisotropic formation models and cost functions. The cost function can include a data mismatch term, a smoothness term, and an entropy term. In some embodiments, one or more of the data mismatch term, the smoothness term, and the entropy term can be expressed as a function of vertical conductivity and horizontal conductivity. The cost function can include one or more regularization parameters based at least in part on the data mismatch term. In addition, the cost function can include one or more relaxation factors based at least in part on the ratio of the Hessian of the smoothness term and the data mismatch term.

Description

Technical Field

[0001] The present disclosure relates to identifying properties of geological formations using downhole electromagnetic measurements. More specifically, the present disclosure relates to identifying horizontal resistivity, vertical resistivity, dip and azimuth of geological formations. Background Art

[0002] This section is intended to introduce the reader to various aspects of the art that may be relevant to various aspects of the present technology, which are described and / or claimed below. It is believed that this discussion helps to provide the reader with background information to help better understand the various aspects of the present disclosure. Therefore, it should be understood that these statements should be read from this perspective and not as any form of admission.

[0003] The extraction of hydrocarbons from a wellbore drilled into a geological formation is a very complex endeavor. In many cases, decisions related to hydrocarbon exploration and extraction can be informed by measurements of a downhole logging tool transmitted deep into the wellbore. The measurements can be used to infer the properties or characteristics of the geological formations surrounding the wellbore. An example of such a downhole logging tool is an electromagnetic downhole logging tool (e.g., an induction logging tool and a propagation induction logging tool). Summary of the invention

[0004] The following describes an overview of certain embodiments disclosed herein. It should be understood that these aspects are presented only to provide the reader with a brief overview of these certain embodiments, and these aspects are not intended to limit the scope of the present disclosure. In fact, the present disclosure may include multiple aspects that may not be described below.

[0005] One embodiment of the present disclosure relates to a method. The method includes obtaining multi-axis electromagnetic (EM) measurements in a borehole through a geological formation using one or more multi-axis EM downhole logging tools via a processor. The method also includes inverting the multi-axis EM measurements via the processor based at least in part on a formation model to determine the horizontal resistivity, vertical resistivity, dip, and azimuth of the formation, wherein inverting the multi-axis EM measurements based at least in part on the formation model includes minimizing a cost function having a data mismatch term, an entropy term, and a smoothness term, wherein the smoothness term includes a horizontal smoothness term and a vertical smoothness term. In addition, the method includes generating a horizontal conductivity log or a vertical conductivity log, or both, of the geological formation via the processor based at least in part on the output of the inversion of the multi-axis EM measurements.

[0006] Another embodiment of the present disclosure relates to an article of manufacture comprising a tangible, non-transitory, machine-readable medium comprising instructions that, when executed by a processor, cause the processor to receive multi-axis electromagnetic (EM) measurements associated with a geological formation obtained by one or more multi-axis EM logging tools. The instructions also cause the processor to invert the multi-axis EM measurements based at least in part on a formation model to determine horizontal resistivity, vertical resistivity, dip, and azimuth of the formation, wherein inverting the multi-axis EM measurements based at least in part on the formation model includes minimizing a cost function having a data mismatch term, an entropy term, and a smoothness term. The smoothness term includes a horizontal smoothness term based at least in part on a horizontal relaxation term. The smoothness term also includes a vertical smoothness term based at least in part on a vertical relaxation term, wherein the vertical relaxation term, the horizontal relaxation term, or both are based at least in part on a ratio of the smoothness term and the data mismatch term. In addition, the instructions cause the processor to generate a horizontal conductivity log curve, a vertical conductivity log curve, or both based at least in part on the output of the inversion of the multi-axis EM measurements.

[0007] Another embodiment of the present disclosure relates to a system. The system includes one or more multi-axis electromagnetic (EM) logging tools configured to obtain one or more multi-axis EM measurements from a geological formation. The system also includes a processor and a memory storing instructions configured to be executed by the processor. The instructions cause the processor to receive multi-axis EM measurements from the one or more multi-axis EM logging tools. The instructions also cause the processor to invert the multi-axis EM measurements based at least in part on a formation model, wherein the inversion includes minimizing a cost function having a data mismatch term, a smoothness term, and an entropy term to determine the horizontal resistivity, vertical resistivity, dip, and azimuth of the formation. Inverting the multi-axis EM measurements includes dynamically adjusting one or more regularization terms during inversion based at least in part on the data mismatch term, wherein the dynamic adjustment of the one or more regularization terms modifies the weights of the smoothness term, the entropy term, or both. In addition, the instructions cause the processor to generate a plurality of horizontal resistivity values ​​and a plurality of vertical resistivity values ​​associated with the geological formation based at least in part on the output of the inversion of the multi-axis EM measurements.

[0008] Various improvements may be made to the above features relative to various aspects of the present disclosure. Other features may also be incorporated into these various aspects. These improvements and additional features may exist alone or in any combination. For example, the various features discussed below with respect to one or more illustrated embodiments may be incorporated into any of the above aspects of the present disclosure alone or in any combination. The brief overview presented above is intended to familiarize the reader with certain aspects and contexts of the embodiments of the present disclosure, without being limited to the claimed subject matter. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Various aspects of the present disclosure may be better understood by reading the following detailed description and referring to the accompanying drawings, in which:

[0010] Figure 1 is a schematic diagram of a well logging system that can obtain electromagnetic (EM) measurements that can be used to identify horizontal resistivity, vertical resistivity, dip, and azimuth of a formation according to aspects of the present disclosure;

[0011] Figure 2 Flowcharts showing various processes that may be performed based on analysis of EM logging data in accordance with aspects of the present disclosure;

[0012] Figure 3 is a schematic diagram of a downhole multi-axis array that can be used to obtain EM measurements according to aspects of the present disclosure;

[0013] Figure 4 is a schematic diagram of a layered formation model that can be used to determine horizontal resistivity values, vertical resistivity values, dip and azimuth according to aspects of the present disclosure;

[0014] Figure 5 is a flow chart of a process for determining horizontal resistivity values ​​and vertical resistivity values ​​using one or more electromagnetic logging tools in accordance with aspects of the present disclosure;

[0015] Figure 6 is a flow chart of an inversion for determining horizontal resistivity values, vertical resistivity values, dip values, and azimuth values ​​according to aspects of the present disclosure;

[0016] Fig. 7A are examples of horizontal resistivity logs and vertical resistivity logs determined by inversion in a chirp model according to aspects of the present disclosure;

[0017] Figure 7B are examples of dip logs and azimuth logs determined by inversion in a chirp model according to aspects of the present disclosure;

[0018] Fig. 8A is an example of a horizontal dip log and a vertical resistivity log determined by inversion in an anisotropic Oklahoma model with a relative dip of 30 degrees in accordance with aspects of the present disclosure;

[0019] Figure 8B is an example of a dip log and an azimuth log determined by inversion in an anisotropic Oklahoma model with a relative dip of 30 degrees according to aspects of the present disclosure;

[0020] Fig. 9Aare examples of horizontal dip logs and vertical resistivity logs determined by inversion in an anisotropic Oklahoma model with a relative dip of 60 degrees in accordance with aspects of the present disclosure; and

[0021] Fig. 9B is an example of a dip log and an azimuth log determined by inversion in an anisotropic Oklahoma model with a relative dip of 60 degrees in accordance with aspects of the present disclosure. DETAILED DESCRIPTION

[0022] One or more specific embodiments of the present disclosure will be described below. These described embodiments are examples of the currently disclosed technology. In addition, in order to provide a brief description of these embodiments, certain features of the actual implementation may not be described in the specification. It should be understood that in the development of any such actual implementation, such as in any engineering or design project, many implementation-specific decisions can be made to achieve the developer's specific goals, such as complying with system-related and business-related constraints, which may vary from implementation to implementation. In addition, it should be understood that such development work may be complex and time-consuming, but for ordinary technicians who benefit from this disclosure, it will still be a routine task of design, fabrication and manufacturing.

[0023] When introducing elements of various embodiments of the present disclosure, the articles "a," "an," and "the" are intended to indicate that there are one or more elements. The terms "comprising," "including," and "having" are intended to be inclusive and mean that there may be additional elements in addition to the listed elements. Additionally, it should be understood that references to "one embodiment" or "an embodiment" of the present disclosure are not intended to be interpreted as excluding the existence of other embodiments that also incorporate the recited features.

[0024] As described above, oil and gas exploration organizations can make certain oil and gas production decisions based on well logging data, such as determining where to drill. More specifically, well logging downhole tools obtain well logging measurements, which can be processed by suitable computing equipment (e.g., normalized, denoised, provided as input to a model, etc.) to generate well logging data. As referred to herein, a "well log" is a measurement or characteristic derived from a measurement and depth or time or both of one or more characteristics (e.g., resistivity, conductivity, inclination, and azimuth, etc.) inside or around a wellbore, and can therefore be used to identify a location within the wellbore corresponding to an area of ​​interest (e.g., a "bed" or layer or formation of hydrocarbons, organic sediments, sedimentary rocks, etc.). In at least some cases, the well logging data can be converted into one or more visual representations (e.g., well logging curves) that are displayed in hard copy form or on an electronic display, wherein each of the one or more visual representations can describe the well logging data obtained from the well logging measurements.

[0025] One type of logging measurement that can be used to inform oil and gas production decisions is an electromagnetic (EM) logging measurement. Typically, electromagnetic logging measurements can be obtained using one or more electromagnetic logging tools, each of which includes a pair of transmitter coils and receiver coils. Conventional electromagnetic logging tools (e.g., electromagnetic logging tools that use only coaxial transmitter coils and coaxial receiver coils) can obtain electromagnetic logging measurements (e.g., induction logging measurements or propagation logging measurements) that are processed to generate resistivity or conductivity log curves, but lack the sensitivity to generate anisotropic resistivity or conductivity log curves (e.g., when the horizontal resistivity or conductivity is different from the vertical resistivity or conductivity). Existing electromagnetic logging measurement processing methods include inverting the EM logging measurements using a parameterized formation model, where a cost function quantifies the error between simulated and in-situ EM logging measurements, and the formation model can be modified (e.g., iteratively) based on the error. As used herein, "horizontal resistivity" is generally the resistivity in a direction parallel to the bedding plane or interface, and "vertical resistivity" is generally the resistivity in a direction perpendicular to the bedding plane or interface. Existing processing methods do not have sufficient resolution to identify certain geological formations, such as thin layers (e.g., 2 feet, 1 foot, or less than 1 foot).

[0026] Thus, the present disclosure relates to techniques for generating anisotropic resistivity (e.g., or conductivity) logs and dip and azimuth logs by processing anisotropic resistivity log measurements. Typically, anisotropic resistivity log measurements may be acquired by a multi-axis EM logging tool (e.g., having a multi-axis transmitter coil and / or a multi-axis receiver coil). For example, the multi-axis EM logging tool may be a tri-axis logging tool. It should be noted that the multi-axis transmitter coil and the multi-axis receiver coil may both be transverse, both may be tilted, both may be axial, one may be axial and the other transverse, or tilted, or one may be transverse and the other may be tilted. As used herein, "transverse," "axial," and "tilted" refer to the relative orientation of the dipole moments of the transmitter coil and the receiver coil relative to the longitudinal axis of the tool.

[0027] In some embodiments, the resistivity log measurements are inverted based on a cost function that includes multiple terms related to horizontal resistivity and vertical resistivity. More specifically, the cost function can include a data mismatch term, an entropy term, and a smoothness term. As discussed in more detail below with respect to equations (1)-(4), each of the latter two terms (e.g., the entropy term and the smoothness term) can include a horizontal term (e.g., a horizontal entropy term and a horizontal smoothness term) and a vertical term (e.g., a vertical entropy term and a smoothness term). In some embodiments, the horizontal smoothness term and the vertical smoothness term can each include a respective relaxation factor that typically accounts for a factor in EM electrical logging that is a sensitivity difference between vertical conductivity and horizontal conductivity. In some embodiments, the inversion can include determining two regularization terms that can be expressed as being proportional to the data mismatch term to avoid potential bias that may be caused by the regularization terms. In this way, the techniques of the present disclosure improve methods for determining physical properties of geological formations in which anisotropy of conductivity and / or resistivity may exist by including vertical and horizontal terms in the cost function so that the resolution of anisotropy (e.g., changes in resistivity) is not suppressed during inversion.

[0028] With this in mind, Figure 1 A logging system 10 is shown in which the systems and methods of the present disclosure may be employed. The logging system 10 may be used to convey an electromagnetic (EM) logging tool 12 through a geological formation 14 via a wellbore 16. The EM logging tool 12 may be conveyed on a cable 18 via a logging winch system 20. Figure 1 The logging winch system 20 is schematically shown as a mobile logging winch system carried by a truck, but the logging winch system 20 can be substantially fixed (e.g., substantially permanent or modular long-term installation). Any suitable cable 18 for logging can be used. The cable 18 can be wound or unwound on a drum 22, and an auxiliary power supply 24 can provide energy to the logging winch system 20 and / or the EM logging tool 12.

[0029] Furthermore, although the EM logging tool 12 is described as a wireline downhole tool, it should be understood that any suitable delivery means may be used. For example, the EM logging tool 12 may alternatively be delivered as a logging while drilling (LWD) tool as part of a bottom hole assembly (BHA) of a drill string, on a slickline or through coiled tubing, etc. For purposes of the present disclosure, the EM logging tool 12 may be any suitable measurement tool for obtaining NMR logging measurements through the depth of the wellbore 16.

[0030] Many types of EM logging tools 12 can obtain EM logging measurements in the wellbore 16. These include, for example, Schlumberger Technology Corporation's Rt scanner, LWD periscope, and Geosphere tools, but electromagnetic logging measurements from other downhole tools from other manufacturers can also be used. The EM logging tool 12 can provide the EM logging measurements 26 to a data processing system 28 via any suitable telemetry (e.g., via an electrical signal pulsed through the geological formation 14 or via mud pulse telemetry). The data processing system 28 can process the EM logging measurements 26 to identify horizontal conductivity and / or horizontal resistivity, vertical conductivity and / or vertical resistivity, dip, and azimuth at various depths of the geological formation 14 in the wellbore 16.

[0031] To this end, the data processing system 28 can therefore be any electronic data processing system that can be used to perform the systems and methods of the present disclosure. For example, the data processing system 28 can include a processor 30 that can execute instructions stored in a memory 32 and / or a storage device 34. As such, the memory 32 and / or the storage device 34 of the data processing system 28 can be any suitable article that can store instructions. To name just a few examples, the memory 32 and / or the storage device 34 can be a ROM memory, a random access memory (RAM), a flash memory, an optical storage medium, or a hard drive. The display 36, which can be any suitable electronic display, can use the EM logging measurements 26 to provide visualization, logging curves, or other indications of properties in the geological formation 14 or the wellbore 16.

[0032] Figure 2 A method 40 is shown of various processes that may be performed based on analysis of well logs according to aspects of the present invention. The location of hydrocarbon deposits within a geological formation may be identified based on well log measurements (process block 42). In some embodiments, the well log measurements may be analyzed to generate a map or outline showing an area of ​​interest having geological formations.

[0033] Based on the location and properties of the identified hydrocarbon deposits, certain downhole operations may be performed on the location or portion of the geological formation 14 (process box 44). That is, the hydrocarbon exploration organization may use the location of the hydrocarbon deposits to determine locations in the wellbore to isolate them for extraction of fluids, fracturing, and / or drilling into land. Thus, the hydrocarbon exploration organization may use the location and properties of the hydrocarbon deposits and associated overburden to determine a path to drill into land, how to drill into land, etc.

[0034] After the exploration equipment is placed in the geological formation 14, the hydrocarbons stored in the hydrocarbon deposits can be extracted through natural flow wells, artificial lift wells, etc. (box 46). In addition, the extracted hydrocarbons can be transported (box 48) to a refinery, etc. through transportation vehicles, pipelines, etc. Further, the extracted hydrocarbons can be processed according to various refining procedures (box 50) to develop different products using the hydrocarbons.

[0035] It should be noted that the processes discussed with respect to method 40 may include other suitable processes that may be based on the location and properties of hydrocarbon deposits indicated in seismic data obtained via one or more seismic surveys. As such, it should be understood that the above processes are not intended to depict an exhaustive list of processes that may be performed after determining the location and properties of hydrocarbon deposits within a geological formation.

[0036] Considering the foregoing, Figure 3 An embodiment of an EM logging tool 12 is shown, which is a multi-axis EM tool (e.g., an Rt Scanner tool from Schlumberger Technology Corporation) having mutually orthogonal and juxtaposed transmitter and receiver coils. As shown in the illustrated embodiment, the EM logging tool 12 includes three transmitters 52, three first receivers 54 (e.g., balanced receivers), and three second receivers 56 (e.g., primary receivers). In general, the three transmitters 52 induce eddy currents in the formation that flow parallel to orthogonal planes oriented with their normals in the X (e.g., along axis 58), Y (e.g., along axis 60), and Z directions (e.g., along axis 62), which are defined by the direction of the magnetic dipole moment of each of the three transmitter coils. In this way, Figure 3 The illustrated EM logging tool 12 can measure all nine orthogonal couplings to determine formation resistivity and resistivity anisotropy as well as formation dip. Although the illustrated embodiment of the EM logging tool 12 is a tri-axial EM tool (e.g., Figure 3 Each of the receivers 56 shown in FIG. 5 is along axis 58 , axis 60 , and axis 62 ), but the number of axes including receivers is not limited to three, but may be two or more.

[0037] The illustrated example of the EM logging tool 12 is shown as being communicatively coupled to a data processing system 28. As discussed herein, the EM logging tool 12 (e.g., a multi-axis logging tool) can obtain measurements within a borehole 16 of a geological formation 14. A processor 30 of the data processing system 28 can receive these measurements. The memory 32 can store information such as control software, lookup tables, configuration data, and the like. In addition, the memory 32 can store information such as anisotropic formation models and cost functions described in more detail herein. The memory 32 can include volatile memory, such as random access memory (RAM), and / or non-volatile memory, such as read-only memory (ROM). The memory 32 can store a variety of information and can be used for a variety of purposes. For example, the memory 32 can store processor executable instructions, including firmware or software to be executed by the processor 30. In some embodiments, the memory 32 is a tangible, non-transitory machine-readable medium that can store machine-readable instructions for execution by the processor 30. The memory 32 can include ROM, flash memory, a hard drive, or any other suitable optical, magnetic, or solid-state storage medium, or a combination thereof. Memory 32 may store data, instructions, and any other suitable information.

[0038] As discussed herein, one existing formation model used to determine resistivity and resistivity anisotropy is a one-dimensional model in which the geological formations are assumed to be inclined and planarly stratified. Figure 4 A diagram of a layered formation model 64 is shown, which may represent a one-dimensional formation model employed in the inversion. Typically, the layered formation model 64 assumes that the horizontal resistivity R h and vertical resistivity R v The horizontal resistivity R is constant in the y direction (e.g., along axis 66) and the x direction (e.g., along axis 68), but varies in the z direction (e.g., along axis 70). The angles θ and φ are the relative inclination and azimuth of the well path. The varying horizontal resistivity R along axis 66 h and vertical resistivity R v is shown as a plurality of planes 71. In some embodiments, each of the plurality of planes 71 may be a pixel or the top or bottom interface of a layer of a stratigraphic model. Figure 4 In the illustrated embodiment of the layered stratigraphic model 64, plane 71a is located at a first position along axis 70, and plane 71b is located at a second position along axis 70. Furthermore, it is assumed that the pixels between plane 71a and plane 71b have a constant vertical resistivity and a constant horizontal resistivity. This is also true for all other pixels in the model. It is noteworthy that the horizontal resistivity R h and vertical resistivity R v May vary from pixel to pixel.

[0039] In some embodiments, the formation is subdivided into multiple layers or multiple pixels, each layer or pixel has an equal thickness (e.g., 1 inch, 2 inches, 3 inches, 6 inches, 12 inches, etc.). In one embodiment, the subdivision of multiple layers is along the well path, and the layer thickness along the well path can be referred to as the apparent thickness. Although shown as an infinitely extended slab, each of these layers can be represented as a pixel. It is worth noting that the formation is described as a collection of equal-thickness slabs, regardless of the actual layer boundaries in the inversion. It is a collective behavior, or an image created by all layers or pixels, which reveals where the layer boundaries are and what their resistivity is. The pixel inversion (inversion, inversion, inversion, inversion) is opposite to the aforementioned parameter inversion, in which not only physical properties but also geometric properties must be used to describe the structure. The mixed use of physical and geometric terms in parameter inversion makes the inversion vulnerable to data noise. The premise of pixel inversion is that, in fact, there is no such obvious and sharp jump in resistivity, and this jump can be accurately described by a square wave-like function without causing any error.

[0040] To help illustrate the above discussion, Figure 5 An example process 72 for determining physical properties associated with a geological formation according to the present disclosure is described in FIG. 7A . Generally, process 72 includes obtaining (process block 74) data associated with one or more multi-axis EM logging tools (e.g., Figure 3 In some embodiments, process block 74 may include positioning one or more multi-axis EM logging tools into one or more wellbores within a geological formation. In some embodiments, multi-axis EM measurements may be performed in real time, for example, by a data processing system 28 communicatively coupled to the EM logging tool 12 to acquire the multi-axis EM measurements.

[0041] Process 72 also includes inverting (process block 76) the multi-axis EM measurements based on the anisotropic resistivity formation model by minimizing the cost function. In some embodiments, as described above, the anisotropic resistivity model may assume that the horizontal resistivity R h and vertical resistivity R v The cost function may include a data mismatch term, an entropy term, and a smoothness term, as discussed in more detail below. In some embodiments, one or more terms (e.g., a data mismatch term, an entropy term, and a smoothness term) may be formulated based on horizontal resistivity, vertical resistivity, dip, or azimuth, or any combination thereof, as defined by the model. In addition, process 72 may also include generating (process box 78) at least one of a horizontal resistivity value, a vertical resistivity value, a horizontal conductivity value, a vertical resistivity value, a dip value, an azimuth value, a data mismatch, or any combination thereof based on the anisotropic resistivity formation model.

[0042] Although described in a particular order that represents a particular embodiment, it should be noted that process 72 may be performed in any suitable order. Additionally, embodiments of process 72 may omit process blocks and / or include additional process blocks. Furthermore, in some embodiments, process 72 may be implemented at least in part by executing instructions stored in a tangible, non-transitory computer-readable medium (memory 32 implemented in data processing system 28) using processing circuitry (e.g., processor 30 implemented in data processing system 28).

[0043] As discussed herein, the techniques of the present disclosure may include inverting EM logging measurements using a cost function. In some embodiments, the cost function minimized by the inversion may be given by:

[0044]

[0045] in:

[0046]

[0047]

[0048]

[0049] where σ h (=1 / R h ) and σ v (=1 / R v ) are the horizontal and vertical conductivity of the formation and are to be minimized by the cost function The cost function of equation (1) generally includes three terms, which are respectively represented by equations (2), (3) and (4) and are discussed in more detail below.

[0050] The first term χ 2 , on the right side of equation (1), here called the “data mismatch term”, is a measure of the difference between the simulated data and the measured data, where and are the measured in-phase and quadrature components of the apparent conductivity, respectively; and are their respective counterparts in the simulation. It should be noted that although in the current formula the data is assumed to be the apparent conductivity, the data may also be a measured voltage, or any other measurement that can be converted from a measured voltage, such as phase shift and attenuation. Here, the indices p, j, and k in the superscript are used for spacing and / or frequency, transmitter orientation, and receiver orientation, respectively, where p = 1, ..., N p ,j=1,…N j , and k=1,…Nk The fast forward solver for one-dimensional formations can quickly obtain simulation data. and and the Jacobian. and are scaling factors. In the current implementation, they are given by:

[0051]

[0052]

[0053] The second item, The "entropy term" referred to in this paper as equation (1) is given in equation (3), which describes the entropy of the horizontal and vertical conductivity models. Here, Tσ h and Tσ v is h and σ v The average value of σ h , p , and σ v,p is h and σ v It should be noted that, at least in some embodiments of the present disclosure, in the inversion, let Tσ h =σ h,p and Tσ v =σ v,p Using the maximum entropy term can make the solution as close as possible to the existing model, making the iterative process more stable.

[0054] The third item, The "smoothness term" in equation (1) is given in equation (4) and is configured to make the inversion preferentially seek a smooth model to avoid getting stuck in a local minimum. The term μ in equation (4) is h and μ v (e.g., as discussed in more detail in Equation 35) can be accomplished by applying the h and σ v The data sensitivity is determined in order to retain σ v It should be understood that although the first-order derivative is used for the smoothness term, the inversion can use other properties of the model to achieve the same effect. In one embodiment, the variance of the model can be used instead of the first-order derivative. In another embodiment, the second-order derivative can also be used to impose smoothness on the model.

[0055] For numerical implementation, the cost function of equation (1) can be discretized to obtain:

[0056]

[0057] Where mv and m are two N-dimensional vectors consisting of the horizontal and vertical conductivities of all pixels in the solution domain. Here, it is assumed that the solution domain is first truncated into a finite range of depth regions, which are then subdivided into N pixels of equal thickness, as Figure 4 The two vectors are given by:

[0058] m h =(σ h,1 ,σ h,2 ,…,σ h,N ) T (8)

[0059] m v =(σ v,1 ,σ v,2 ,…,σ v,N ) T (9)

[0060] In this embodiment, the subscript T specifies the operation of matrix transposition. The discrete forms of the three terms in equation (7) are:

[0061]

[0062]

[0063]

[0064] On top, and are the real and imaginary parts of the measured apparent conductivity at M depths,

[0065]

[0066]

[0067] Where M = N z N p N j N k , where N z is the number of depth points. R and d X are the real and imaginary parts of the simulated apparent conductivity at the same depth,

[0068] d R (m σ ,m ε )=[d R,1 (m σ ,m ε )d R,2 (m σ,m ε ) … d R,M (m σ ,m ε )] T , (15)

[0069] d X (m σ ,m ε )=[d X,1 (m σ ,m ε )d X,2 (m σ ,m ε ) … d X,M (m σ ,m ε )] T , (16)

[0070] matrix and is the inverse of the standard deviation of the diagonal line, which may contain real and imaginary apparent conductivity noise,

[0071]

[0072]

[0073] In equation (10):

[0074]

[0075]

[0076] The vector l∈R in equation (11) Nx1 is a constant vector, l=(1 1…1) T The matrix in equation (12) is the difference operator, for example:

[0077]

[0078] In some embodiments, the Gauss-Newton method may be used to reduce (eg, minimize) the cost function in equation (6) to find the horizontal conductivity m h and vertical conductivity m v For simplicity of the formula, the following symbols can be used:

[0079]

[0080]

[0081]

[0082]

[0083] In equation (25), is a zero matrix. When the current iteration step is l, the solution of this step is:

[0084] m l =m l-1 +v l-1 q l-1 , (26)

[0085] where q l-1 is the Newton search direction; v l-1 is the step size to reduce the impact of the approximation error caused by the quadratic approximation under the current step size. The search vector can be given by

[0086]

[0087] where g l-1 is the gradient of the cost function, is its Hessian. The gradient of the cost function and Hessian can be expressed as:

[0088]

[0089]

[0090] In the above two equations, d l-1 is the model m obtained in the previous step l-1 Corresponding simulation data; J l-1 is the data term χ of the cost function 2 The Jacobian determinant of and are evaluated at and are the gradients of the maximum entropy and smoothness terms in Eq. (6), respectively. and are their Hessians respectively. The forms of these four gradients and Hessians can be obtained from equations (10) and (11): and Export. You can use χ in the iterative process 2 Dynamically adjust the two regularization terms γ_P and γ_P so that:

[0091]

[0092]

[0093] where χ 2 (m l-1) is the model obtained in the previous step, when m=m l-1 The data evaluated at the location do not match. Numerical experiments show that P and δ S Setting it to 1 is an appropriate choice for synthetic and field data processing. Once the search direction is determined from equation (27), a linear search is then performed to determine the step size v l-1 .

[0094] Jacobian Included in m=m l-1 The evaluation R and d X The first derivatives of the horizontal and vertical conductivity and the inclination and azimuth angles. The Jacobian can be expressed as:

[0095]

[0096] in:

[0097]

[0098]

[0099] In some embodiments, the conductivity d about a pixel may be calculated using analytical methods or using a finite difference approximation. R and d X The former can speed up the inversion.

[0100] The two additional terms μ in equation (12) are h and μ v To illustrate σ h and σ v At least in some cases, the difference in data sensitivity between σ h The data sensitivity of v As a result, the same regularization term γ is used for the two sub-terms of the smoothness term L_s. s This may result in a resolution-limited σ v Oversmoothing. By using two relaxation factors μ h and μ alleviate this undesirable effect. These two relaxation factors are defined so that the smoothing term The sensitivity of a given term is related to the data term c in equation (11) 2 To this end, the horizontal conductivity σ h The horizontal relaxation factor μ h Set to 1, Vertical conductivity σ v The vertical relaxation factor μ v Given by:

[0101]

[0102] In at least some embodiments, the vertical relaxation factor may be set to 1, and the horizontal relaxation factor may be an equation generally similar to equation (35).

[0103] On top, and They are respectively in m=m l-1 Evaluated The first and second items. and They are About σ h and σ v The Hessians. and They are respectively about σ h and σ v The data item χ 2 Four Hessians in m = m l-1 The operator "tr()" in equation (35) gives the trace of the matrix. A rigorous computation of the two Hessian matrices for the data items can be very expensive. Therefore, the following are approximations of the two Hessians:

[0104]

[0105]

[0106] in:

[0107]

[0108] The stopping criterion for the inversion is χ 2 < tol and l>l max , where l is the index of the iteration step. tol is the number of degrees of freedom. If all data are independent random variables, then χ tol =2M, scaling factor Δ R and Δ X is the standard deviation of the in-phase and quadrature components of the apparent conductivity. In the current implementation, χ tol Set to a small positive number. In inversion, the maximum number of iterations l max Set to 30.

[0109] To help illustrate the above discussion, Figure 6An example process 80 for determining horizontal resistivity, vertical resistivity, dip, and azimuth according to the present disclosure is described in . In general, process 80 includes providing an initial guess for horizontal resistivity, vertical resistivity, dip, and azimuth (process block 82), simulating EM data (process block 84), calculating χ based on received field EM data 88, and 2 ) (process box 86), and determining whether the inversion meets the stopping criteria (process box 90). Process 80 includes calculating the Jacobian of horizontal conductivity or resistivity, vertical conductivity or resistivity, dip, azimuth, or any combination thereof, and dip and azimuth when the inversion does not meet the stopping criteria (process box 92), determining the terms (e.g., γ S and γ P ) and / or relaxation factors (e.g., μ S and μ P )(process box 94), calculate the search direction and determine the step size (process box 96) and update the horizontal conductivity or resistivity, vertical conductivity or resistivity, dip, azimuth or any combination thereof and dip and azimuth (process box 98), and continue with process box 84.

[0110] When the inversion does meet the stopping criteria, process 80 includes applying a low pass filter to the horizontal and vertical conductivities 100, calculating the horizontal and vertical resistivities from the filtered horizontal and vertical conductivities (process box 102) to output horizontal and vertical resistivities 104, outputting horizontal and vertical conductivities 106, and outputting dip and azimuth, as well as data mismatch 108.

[0111] Although described in a particular order representing a particular embodiment, it should be noted that process 80 may be performed in any suitable order. Additionally, embodiments of process 80 may omit process blocks and / or include additional process blocks. Furthermore, in some embodiments, process 80 may be implemented using processing circuitry (processor 30 implemented in data processing system 28) at least in part by executing instructions stored in a tangible, non-transitory computer-readable medium (e.g., memory 32 implemented in data processing system 28).

[0112] In some embodiments, some variations can be derived from the above formula to further enhance the performance of the inversion. For example, instead of h and σ v By inverting, we can get σ h and σ v Invert the logarithm of . Thus, the vector m h and m v becomes:

[0113] m h =(lnσ h,1 ,lnσ h,2,…,lnσ h,N ) T (39)

[0114] m v =(lnσ v,1 ,lnσ v,2 ,…,lnσ v,N ) T (40)

[0115] To accommodate the transformation, the maximum entropy term (e.g., as shown in Equation 3) can be modified as:

[0116] in and are two positive numbers to prevent the denominator from gradually decreasing. Therefore, the smoothness term (e.g., as shown in Equation 4) can be modified as:

[0117]

[0118] In some embodiments, μ and v can be inverted, and σ h and σ v The transformation can be expressed as:

[0119] μ≡σ v (43)

[0120] v≡σ h -σ v (44)

[0121] Given data dη,η=R,X,d, the derivative with respect to the transformed variable can be expressed as:

[0122]

[0123]

[0124] If the condition is consistent throughout the model, then the transformation using equations (41) and (42) leads to a much better solution than directly solving for σ h and σ v A more balanced inversion problem. When using μ and v, the unknown vector m h and m v Given by the following formula

[0125] m h =(μ 1 ,μ 2 ,…,μ N ) T (47)

[0126] mv =(v 1 ,v 2 ,…,v N ) T (48)

[0127] By using σ in equations (1)-(38) h Replace μ and σ v Substituting v for μ and v yields the inversion method. With σ h and σ v Similarly, instead of inverting μ and v directly, we can choose to invert the logarithms of μ and v, which leads to the third variant of the invention. When the logarithms of μ and v are used as unknowns, the two vectors m h and m v becomes: m h =(lnμ 1 ,lnμ 2 ,…,lnμ N ) T (49)

[0128] m v =(lnv 1 ,lnv 2 ,…,lnv N ) T (50)

[0129] The maximum entropy and smoothness terms of the logarithms of μ and v can be obtained by replacing σ in equations (39)-(42) h Replace μ and σ v Replace v to find.

[0130] For ease of numerical implementation, if the depth region to be processed is long, the region is first subdivided into multiple short intervals. The inversion is then run separately at each interval. The results of all intervals are combined to create a single output. In one embodiment, each interval is set to 30 feet with a 25-foot transition region on each side. In the event of any undesirable artifacts, the σ obtained in the last iteration can be used as the inversion result. h and σ v is low-pass filtered before being delivered as the final solution. A Gaussian filter with a standard deviation of 0.25 ft is usually used as a low-pass filter. h and σ v , and also provide the horizontal and vertical resistivity R h and R v , as σ h and σ v The reciprocal of .

[0131] Numerical results

[0132] 7-9 show example EM logs obtained using the inversion process discussed herein. As discussed further below, the inversion process of the present disclosure is successful in many cases.

[0133] I. Chirp Stratigraphic Model

[0134] Fig. 7A and 7B Panels 110, 112, 114, and 116 are shown showing well log data associated with EM well log measurements based on a chirp formation model. The well log data for each panel (e.g., panel 110, panel 112, panel 114, and panel 116) is a set of corresponding well logs shown at depth (e.g., axis 118) relative to axes 120, 124, 126, and 128. Panel 110 depicts well logs of depth versus horizontal and vertical resistivity obtained using the inversion process discussed herein and a set of Rt scanner data, panel 112 depicts well logs of depth versus horizontal and vertical resistivity obtained using another set of Rt scanner data, panel 114 depicts depth versus dip and azimuth obtained using a first set of Rt scanner data, and panel 116 depicts depth versus dip and azimuth obtained using a second set of Rt scanner data.

[0135] The chirped formation model includes an alternating sequence of resistive and conductive layers with increasing thickness from top to bottom (e.g., depth along axis 118). In this example, the thickness of the first layer (e.g., at approximately 100 feet) is 0.5 feet. The thickness of the last layer (e.g., between approximately 130 and 140 feet) is 6 feet. The other layers in between are 1, 2, 3, 4, and 5 feet from top to bottom. All resistive layers of panels 110 and 112 have a horizontal resistivity of 100 ohm.m and a vertical resistivity of 200ohm.mm, and all conductive layers have a horizontal resistivity of 2ohm.m and a vertical resistivity of 4ohm.m. The anisotropy ratio is set to 2 throughout the model (i.e., all layers). The Rt scanner data is simulated for the chirped model and then sent as input data to the inversion. The operating frequency is 26kHz. Two sets of results are shown in true vertical depth (TVD) in Figure 7. TVD is the projection of the measured depth (i.e., depth along the well path) onto the normal to the bedding plane. The first set of results (e.g., panels 110 and 114) including RH60, RV60, DPAP60, and DPAA60 were obtained using 39-inch and 54-inch data. The second set uses 54-inch and 72-inch array data, including RH80, RV80, DPAP80, and DPAA80. The square log curves labeled RH and RV in panels 110 and 112 are the true horizontal and vertical resistivities. The two lines labeled DPAP and DPAA in panels 114 and 116 are the true dip and azimuth. In this case, the true relative dip and azimuth are 30 degrees and 200 degrees, respectively.

[0136] for Fig. 7A and 7B The results in the figure were set to 3 inches in the inversion to resolve the thin layer on the top. The entire area is divided into 2 intervals of 30 feet each, with a transition area of ​​25 feet on each side. The dip and azimuth obtained by inversion are shown in the middle of each interval. No filtering was performed on the inversion output. It is clear that all layers, including the two thin layers on the top, can be clearly seen on the inverted horizontal resistivity logs RH60 and RH80. The horizontal resistivity is slightly underestimated in the 0.5-foot thin layer. On the inverted vertical resistivity logs RV60 and RV80, the two thin layers are still well resolved, although the readings in the shoulder layer are not as accurate as the horizontal resistivity.

[0137] II. Anisotropic Oklahoma Model.

[0138] An anisotropic Oklahoma model is typically made by adding anisotropy to the Oklahoma model that is commonly used to test the performance of inversion methods. An example of an anisotropic Oklahoma model is shown in Table 1. The location of the layer boundaries is defined on the TVD, or at depth along the normal to the bedding plane in tool coordinates.

[0139]

[0140] Table 1. Anisotropic Oklahoma model

[0141] Fig. 8A and 8B Panels 130, 132, 134, and 136 are shown, which display well log data related to EM log measurements based on an anisotropic Oklahoma formation model. The well log data of panels 130, 132, 134, and 136 are depth (e.g., axis 138) and a corresponding set of well logs shown in axes 140, 142, 144, and 146. Panel 130 depicts depth versus horizontal and vertical resistivity well logs obtained with a set of Rt scanner data using an inversion process as discussed herein, panel 132 depicts depth versus horizontal and vertical resistivity well logs obtained with another set of Rt scanner data, panel 134 depicts depth versus dip and azimuth obtained with a first set of Rt scanner data, and panel 136 depicts depth versus dip and azimuth obtained with a second set of Rt scanner data.

[0142] exist Fig. 8A and 8B In the example shown, the true relative dip and azimuth of the model are 30 degrees and 200 degrees, respectively. The Rt scanner data is simulated for the true Oklahoma model and then sent as input data to the inversion. The operating frequency is 26kHz. Two sets of results are shown on the TVD of Figure 8. As in the case of the chirp model, the first set contains RH60, RV60, DPAP60 and DPAA60, which are obtained using 39-inch and 54-inch array data. The second set of data, including RH80, RV80, DPAP80 and DPAA80, was obtained using 54-inch and 72-inch array data. The square log curves marked RH and RV in panels 130 and 132 are the true horizontal resistivity and vertical resistivity. The two lines marked DPAP and DPAA in panels 134 and 136 are the true dip and azimuth.

[0143] In the inversion, the pixel height was set to 3 inches. The entire area was divided into 9 intervals of 30 feet each, with a transition area of ​​25 feet on each side on both sides. The dip and azimuth obtained by inversion are shown in the middle of each interval. Before output, the inverted resistivity was filtered with a Gaussian filter with a standard deviation of 0.25 feet. It is obvious that all resistive layers, including the two thin layers at the bottom of 139-145 feet, are well resolved in the inverted horizontal and resistivity logs. Relatively speaking, the accuracy of the inverted horizontal resistivity RH60 and RH80 is better than that of the inverted vertical resistivity RV60 and RV80. This is especially true for thin conductive layers, such as those at 50 feet, 118 feet, and 142 feet.

[0144] Fig. 9A and 9B Panels 148, 150, 152, and 154 are shown showing well log data associated with EM log measurements based on anisotropic Oklahoma formation at 60 degrees relative dip. The well log data of panels 148, 150, 152, and 154 are depth (e.g., axis 156) and a corresponding set of well logs shown in axes 158, 160, 162, and 164. Panel 148 depicts depth versus horizontal and vertical resistivity well logs obtained with one set of Rt scanner data using an inversion process as discussed herein, panel 150 depicts depth versus horizontal and vertical resistivity well logs obtained with another set of Rt scanner data, panel 152 depicts depth versus dip and azimuth with a first set of Rt scanner data, and panel 154 depicts depth versus dip and azimuth with a second set of Rt scanner data.

[0145] exist Fig. 9A and 9BIn the example shown, the true relative dip and azimuth of the model are 60 degrees and 200 degrees, respectively. Because the data is acquired along the measured depth, a longer data interval may be required to cover the same stratigraphic interval for the lower dip model. With this in mind, the entire area is subdivided into 9 intervals of 60 feet each, with a 50-foot transition zone on each of the two sides. In addition, the pixel height is set to 6 inches in the inversion. The dip and azimuth obtained by inversion are displayed in the middle of each interval. Before output, the inverted resistivity is filtered with a Gaussian filter with a standard deviation of 0.25 feet. Overall, the observations are the same as in the 30-degree case. That is, all the resistive layers, including the two thin layers at the bottom below 139-145 feet, are well resolved on the inverted horizontal and resistivity logs. Relatively speaking, the accuracy of the inverted horizontal resistivity RH60 and RH80 is better than the inverted vertical resistivity RV60 and RV80. Another observation is that the artificial spikes or overshoots that appear around the layer boundaries are slightly more important than in the 30 degree case, indicating that the high-frequency components of the formation are missing at 60 degree dip and cannot be recovered by inversion.

[0146] Thus, the present disclosure relates to techniques for generating and analyzing anisotropic properties of geological formations using electromagnetic logging measurements. In some embodiments, resistivity logging data is inverted based on a cost function that includes multiple terms related to horizontal resistivity and vertical resistivity. As discussed herein, the cost function may include data mismatch terms (e.g., x 2 ), entropy terms (e.g., ) and smoothness terms (e.g., In some embodiments, the inversion may include determining two regularization terms (e.g., γ P and γ S ) to avoid potential bias caused by the regularization term during the iterations of the inversion. In some embodiments, the smoothness term may include a relaxation factor (e.g., μ S and μ P ), which is configured to take into account σ v and σ h In this way, the techniques of the present disclosure improve methods of determining physical properties of geological formations in which anisotropy of conductivity and / or resistivity may exist by including vertical and horizontal terms in the cost function so that the resolution of anisotropy (e.g., vertical resistivity variations) is not suppressed during inversion.

[0147] The above specific embodiments have been shown by way of example, and it should be understood that these embodiments may be susceptible to various modifications and alternative forms. It should also be understood that the claims are not intended to be limited to the specific forms disclosed, but to cover modifications, equivalents and alternative forms that fall within the spirit and scope of the present disclosure.

Claims

1. A method for identifying the properties of geological formations using downhole electromagnetic measurements, characterized in that include: obtaining, via a processor, multi-axis electromagnetic measurements in a wellbore penetrating a geological formation using one or more multi-axis electromagnetic downhole logging tools; inverting, via a processor, the multi-axis electromagnetic measurements based on a formation model to determine horizontal conductivity, vertical conductivity, dip, and azimuth of the formation, wherein inverting the multi-axis electromagnetic measurements based at least in part on the formation model includes minimizing a cost function having a data mismatch term, an entropy term, and a smoothness term, wherein the smoothness term includes a horizontal smoothness term and a vertical smoothness term, wherein the formation model includes a plurality of geological layers associated with a plurality of regions of the geological formation, and wherein each of the plurality of geological layers includes a corresponding vertical resistivity value and a corresponding horizontal resistivity value, wherein the horizontal smoothness term is based at least in part on a horizontal relaxation factor associated with the corresponding horizontal resistivity value of each of the plurality of geological layers; wherein the vertical smoothness term is based at least in part on a vertical relaxation factor associated with the corresponding vertical resistivity value of each of the plurality of geological layers; and wherein the horizontal relaxation factor and the vertical relaxation factor are each configured to account for data sensitivity differences between the corresponding horizontal resistivity value and the corresponding vertical resistivity value of each of the plurality of geological layers, A horizontal resistivity and conductivity log or a vertical resistivity and conductivity log, or both, and dip and azimuth logs of the geological formation are generated via a processor based at least in part on the output of the inversion of the multi-axis electromagnetic measurements.

2. The method according to claim 1, characterized in that At least one geological layer of the plurality of geological layers is associated with a region of the geological layer having a specified height.

3. The method according to claim 1, characterized in that: The vertical relaxation factor or the horizontal relaxation factor is based at least in part on a ratio of the Hessians of the smoothness term and the data mismatch term.

4. The method according to claim 1, characterized in that: The vertical relaxation factor, the horizontal relaxation factor, or both are dynamically adjusted during iterations of the inversion.

5. The method according to claim 1, characterized in that include: A dip log, an azimuth log, or a data mismatch log, or any combination thereof, is generated based at least in part on the output of the inversion of the multi-axis electromagnetic measurements.

6. The method according to claim 1, characterized in that Inverting the multi-axis electromagnetic measurement includes dynamically adjusting one or more regularization terms during the inversion based at least in part on the data mismatch term, wherein dynamically adjusting the one or more regularization terms modifies weights of a smoothing term, an entropy term, or both.

7. The method according to claim 1, characterized in that include: filtering and outputting a horizontal conductivity log and a vertical conductivity log based at least in part on a low pass filter; and Based at least in part on the horizontal conductivity and the vertical conductivity, a horizontal resistivity log, a vertical resistivity log, or both are generated and output.

8. The method according to claim 1, wherein: The stopping criteria for the inversion is based at least in part on the mismatches.

9. An article of manufacture for identifying properties of geological formations using downhole electromagnetic measurements, characterized in that A tangible, non-transitory machine-readable medium comprising instructions that, when executed by a processor, cause the processor to: receiving multi-axis electromagnetic measurements associated with a geological formation obtained by one or more multi-axis electromagnetic logging tools; Based at least in part on the formation model, inverting the multi-axis electromagnetic measurements to determine horizontal conductivity, vertical conductivity, dip, and azimuth of the formation, wherein inverting the multi-axis electromagnetic measurements based at least in part on the formation model comprises minimizing a cost function having a data mismatch term, an entropy term, and a smoothness term, wherein the smoothness term comprises: a horizontal smoothness term based at least in part on a horizontal relaxation factor; and a vertical smoothing term based at least in part on a vertical relaxation factor, wherein the vertical relaxation factor, the horizontal relaxation factor, or both are based at least in part on a ratio of the Hessian of the smoothing term and the data mismatch term; and A horizontal conductivity log, a vertical conductivity log, or both of the geological formation is generated based at least in part on the output of the inversion of the multi-axis electromagnetic measurements.

10. The product according to claim 9, characterized in that The formation model includes a plurality of geological layers associated with a plurality of regions of the geological formation, and wherein each geological layer of the plurality of geological layers includes a corresponding vertical resistivity value and a corresponding horizontal resistivity value.

11. The product according to claim 10, characterized in that The cost function is discretized based at least in part on a respective vertical resistivity value and a respective horizontal resistivity value for each of the plurality of geological layers.

12. The article according to claim 9, characterized in that The cost function includes horizontal entropy term and vertical entropy term.

13. The article according to claim 9, characterized in that The cost function is minimized based at least in part on a Gauss-Newton method.

14. The article according to claim 9, characterized in that The cost function is iteratively minimized until a threshold is reached, wherein the threshold is based at least in part on the data mismatches.

15. A system for identifying properties of geological formations using downhole electromagnetic measurements, characterized in that include: one or more multi-axis electromagnetic logging tools configured to obtain one or more multi-axis electromagnetic measurements from a geological formation; processor; and A memory storing instructions configured to be executed by the processor, the instructions causing the processor to: receiving multi-axis electromagnetic measurements from one or more multi-axis electromagnetic logging tools; Inverting the multi-axis electromagnetic measurements based at least in part on a formation model, wherein the inversion includes minimizing a cost function having a data mismatch term, a smoothness term, and an entropy term to determine horizontal conductivity, vertical conductivity, dip, and azimuth of the formation, wherein the formation model includes a plurality of geological layers associated with a plurality of regions of the geological formation, and wherein each of the plurality of geological layers includes a corresponding vertical resistivity value and a corresponding horizontal conductivity value, wherein the smoothness term includes a horizontal smoothness term and a vertical smoothness term, wherein the horizontal smoothness term is based at least in part on a horizontal relaxation factor associated with the corresponding horizontal conductivity value of each of the plurality of geological layers; the vertical smoothness term is based at least in part on a vertical relaxation factor associated with the corresponding vertical conductivity value of each of the plurality of geological layers; and the horizontal relaxation factor and the vertical relaxation factor are configured to account for data sensitivity differences between a sensitivity of the corresponding horizontal resistivity value and a sensitivity of the corresponding vertical resistivity value of each of the plurality of geological layers, wherein inverting the multi-axis electromagnetic measurements includes: dynamically adjusting one or more regularization terms during inversion based at least in part on the data mismatch term, wherein dynamically adjusting the one or more regularization terms modifies a weight of a smoothness term, an entropy term, or both; and A plurality of horizontal conductivity values, a plurality of vertical conductivity values ​​associated with the geological formation are generated based at least in part on the output of the inversion of the multi-axis electromagnetic measurements.

16. The system according to claim 15, characterized in that The instructions cause the processor to: A horizontal resistivity log is output based at least in part on the plurality of horizontal conductivity values; a vertical resistivity log is output based at least in part on the plurality of vertical conductivity values, or both.

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