Patent Publication Number: US-10323498-B2

Title: Methods, computer-readable media, and systems for applying 1-dimensional (1D) processing in a non-1D formation

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority from U.S. Provisional Application 61/885,215, filed Oct. 1, 2013, which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     This disclosure relates to evaluating geological formations and, more particularly, to the determination of formation parameters using electromagnetic measurements. 
     Multi-component directional electromagnetic tools and algorithms have been developed to obtain formation resistivity (e.g., horizontal resistivity—Rh; and vertical resistivity—Rv), anisotropy, and formation dips. In many processing methods, the earth is assumed to be a 1D (1-dimensional) layered mud cake model. 1D processing algorithms can be used for computing electromagnetic induction and propagation responses in 1D layered formation models. Generally, 1D processing provides a fast analytical solution within a reasonable amount of time, and thus inversions based on 1D processing are practical for solving for resistivity, anisotropy, formation dip, and/or layer thicknesses using a 1D layered mud cake model. However, in most real world instances, subsurface formations in the Earth are not a 1D structure, but rather 2D or 3D (non-1D). 
     SUMMARY 
     A summary of certain embodiments disclosed herein is set forth below. It should be understood that these embodiments are presented merely to provide the reader with a brief summary and that these are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of embodiments and associated aspects that may not be set forth below. 
     Embodiments of this disclosure relate to various methods, computer-readable media, and systems for applying 1-dimensional (1D) processing in a non-1D formation. In some embodiments, a method is provided that includes obtaining, by one or more processors, a 3D model or curtain section of a subsurface earth formation and determining, by one or more processors, a processing window within the 3D model or curtain section for 1D inversion processing. The method also includes building, by one or more processors, a local 1D model for the processing window and performing, by one or more processors, a 1D inversion on the local 1D model to generate an inverted 1D model having at least one formation parameter. The method further includes updating, by one or more processors, the 3D model or curtain section using the at least one formation parameter. 
     In some embodiments, a non-transitory computer-readable medium is provided. The computer-readable medium includes computer-executable instructions when executed by one or more processors, causes the one or more processors to perform operations that include obtaining a 3D model or curtain section of a subsurface earth formation and determining a processing window within the 3D model or curtain section for 1D inversion processing. The computer-readable medium includes computer-executable instructions when executed by one or more processors, causes the one or more processors to perform operations that also include building a local 1D model for the processing window and performing a 1D inversion on the local 1D model to generate an inverted 1D model having at least one formation parameter. The computer-readable medium includes computer-executable instructions when executed by one or more processors, causes the one or more processors to perform operations that further include updating the 3D model or curtain section using the at least one formation parameter. 
     In some embodiments, a system is provided that includes one or more processors and a non-transitory tangible computer-readable memory accessible by the one or more processors. The computer-readable memory includes computer-executable instructions that when executed by one or more processors, causes the one or more processors to perform operations that include obtaining a 3D model or curtain section of a subsurface earth formation and determining a processing window within the 3D model or curtain section for 1D inversion processing. The computer-readable memory includes computer-executable instructions that when executed by one or more processors, causes the one or more processors to perform operations that also include building a local 1D model for the processing window and performing a 1D inversion on the local 1D model to generate an inverted 1D model having at least one formation parameter. The computer-readable memory includes computer-executable instructions that when executed by one or more processors, causes the one or more processors to perform operations that further include updating the 3D model or curtain section using the at least one formation parameter. 
     Various refinements of the embodiments, aspects, and features noted above may be undertaken in relation to various embodiments, aspects, and features of the present disclosure. Further embodiments, aspects, and/or features may also be incorporated in these various embodiments, aspects, and/or features as well. These refinements and additional embodiments, aspects, and/or features may be determined individually or in any combination. For instance, various embodiments, aspects, and/or features discussed below in relation to the illustrated embodiments may be incorporated into any of the above-described embodiments, aspects, and/or features of the present disclosure alone or in any combination. The brief summary presented above is intended to familiarize the reader with certain embodiments, aspects, features, and contexts of embodiments of the present disclosure without limitation to the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various embodiments, aspects, and features of this disclosure may be better understood upon reading the following detailed description and upon reference to the drawings in which: 
         FIG. 1  is a schematic diagram of an example well site system in accordance with an embodiment of the disclosure; 
         FIG. 2  is a schematic diagram of an example electromagnetic measurement tool in accordance with an embodiment of the disclosure; 
         FIG. 3  is a block diagram of an example process for processing non-1D formation measurements using a 1D inversion model in accordance with an embodiment of the disclosure; 
         FIG. 4  is a diagram of an example curtain section obtained from modeling/interpretation software in accordance with embodiment of the disclosure; 
         FIGS. 5A and 5B  are block diagrams for defining example 1D processing windows in accordance with an embodiment of the disclosure; 
         FIG. 6  is a diagram of the example curtain section of  FIG. 4  showing 1D processing windows in accordance with embodiment of the disclosure; 
         FIG. 7  is a block diagram of an example process for constructing a 1D layered model from 3DP curtain section in accordance with an embodiment of the disclosure; 
         FIG. 8  is a diagram of the example curtain section of  FIG. 4  showing a resulting 1D layered model in accordance with an embodiment of the disclosure; 
         FIG. 9  is a diagram of the example curtain section of  FIG. 4  showing a resulting 1D layered mode after taking into account both crossed and non-crossed bed boundaries in accordance with an embodiment of the disclosure; and 
         FIG. 10  is a diagram of the example curtain section of  FIG. 4  showing inverted 1D models for the selected 1D processing in accordance with an embodiment of the disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Described herein are various embodiments related to applying 1-dimensional (1D) processing in a non-1D formation. A 3D earth model or curtain section of a non-1D formation may be obtained. Processing windows within the 3D earth model or curtain section that are suitable for 1D processing may be defined manually, via user input, or automatically. For example, in some embodiments, a processing window may be defined by selecting a base point and expanding a processing window until at least one stopping criterion is met. A sub-dataset for each processing window is created, and an initial local 1D model is generated for each processing window. An inversion is run on the local 1D model to generate an inverted 1D model having formation parameters such as a global dip, horizontal resistivity (Rh), vertical resistivity (Rv) and bed boundary locations. The inversion results may be used to update the 3D earth model or curtain section. 
     These and other embodiments of the disclosure will be described in more detail through reference to the accompanying drawings in the detailed description of the disclosure that follows. This brief introduction, including section titles and corresponding summaries, is provided for the reader&#39;s convenience and is not intended to limit the scope of the claims or the proceeding sections. Furthermore, the techniques described above and below may be implemented in a number of ways and in a number of contexts. Several example implementations and contexts are provided with reference to the following figures, as described below in more detail. However, the following implementations and contexts are but a few of many. 
       FIG. 1  depicts a simplified view of an example well site system in which various embodiments can be employed. The well site system depicted in  FIG. 1  can be deployed in either onshore or offshore applications. In this type of system, a borehole  11  is formed in subsurface formations by rotary drilling in a manner that is well known to those skilled in the art. Some embodiments can also use directional drilling. 
     A drill string  12  is suspended within the borehole  11  and has a bottom hole assembly (BHA)  100  which includes a drill bit  105  at its lower end. The surface system includes a platform and derrick assembly  10  positioned over the borehole  11 , with the assembly  10  including a rotary table  16 , kelly  17 , hook  18  and rotary swivel  19 . In a drilling operation, the drill string  12  is rotated by the rotary table  16  (energized by means not shown), which engages the kelly  17  at the upper end of the drill string. The drill string  12  is suspended from a hook  18 , attached to a traveling block (also not shown), through the kelly  17  and a rotary swivel  19  which permits rotation of the drill string  12  relative to the hook  18 . As is well known, a top drive system could be used in other embodiments. 
     Drilling fluid or mud  26  may be stored in a pit  27  formed at the well site. A pump  29  delivers the drilling fluid  26  to the interior of the drill string  12  via a port in the swivel  19 , which causes the drilling fluid  26  to flow downwardly through the drill string  12 , as indicated by the directional arrow  8  in  FIG. 1 . The drilling fluid exits the drill string  12  via ports in the drill bit  105 , and then circulates upwardly through the annulus region between the outside of the drill string  12  and the wall of the borehole, as indicated by the directional arrows  9 . In this known manner, the drilling fluid lubricates the drill bit  105  and carries formation cuttings up to the surface as it is returned to the pit  27  for recirculation. 
     The drill string  12  includes a BHA  100 . In the illustrated embodiment, the BHA  100  is shown as having one MWD module  130  and multiple LWD modules  120  (with reference number  120 A depicting a second LWD module  120 ). As used herein, the term “module” as applied to MWD and LWD devices is understood to mean either a single tool or a suite of multiple tools contained in a single modular device. Additionally, the BHA  100  includes a rotary steerable system (RSS) and motor  150  and a drill bit  105 . 
     The LWD modules  120  may be housed in a drill collar and can include one or more types of logging tools. The LWD modules  120  may include capabilities for measuring, processing, and storing information, as well as for communicating with the surface equipment. By way of example, the LWD module  120  may include an electromagnetic logging tool. 
     The MWD module  130  is also housed in a drill collar, and can contain one or more devices for measuring characteristics of the drill string and drill bit. In the present embodiment, the MWD module  130  can include one or more of the following types of measuring devices: a weight-on-bit measuring device, a torque measuring device, a vibration measuring device, a shock measuring device, a stick/slip measuring device, a direction measuring device, and an inclination measuring device (the latter two sometimes being referred to collectively as a D&amp;I package). The MWD tool  130  further includes an apparatus (not shown) for generating electrical power for the downhole system. For instance, power generated by the MWD tool  130  may be used to power the MWD tool  130  and the LWD tool(s)  120 . In some embodiments, this apparatus may include a mud turbine generator powered by the flow of the drilling fluid  26 . It is understood, however, that other power and/or battery systems may be employed. 
     The operation of the assembly  10  of  FIG. 1  may be controlled using control system  152  located at the surface. The control system  152  may include one or more processor-based computing systems. In the present context, a processor may include a microprocessor, programmable logic devices (PLDs), field-gate programmable arrays (FPGAs), application-specific integrated circuits (ASICs), system-on-a-chip processors (SoCs), or any other suitable integrated circuit capable of executing encoded instructions stored, for example, on tangible computer-readable media (e.g., read-only memory, random access memory, a hard drive, optical disk, flash memory, etc.). Such instructions may correspond to, for instance, workflows and the like for carrying out a drilling operation, algorithms and routines for processing data received at the surface from the BHA  100  (e.g., as part of an inversion to obtain one or more desired formation parameters), and so forth. 
       FIG. 2  depicts one example of an electromagnetic measurement tool  50 , which may be part of the LWD module  120  of  FIG. 1 . The tool  50  may be a multi-spacing directional electromagnetic propagation tool. In one embodiment, the tool  50  may be capable of making measurements at multiple frequencies, such as at 100 kHz, 400 kHz, and 2 MHz. In the depicted embodiment, the measurement tool  50  includes multiple transmitters T 1 , T 2 , T 3 , T 4 , T 5 , and T 6  depicted at  52 ,  54 ,  56 ,  58 ,  60 , and  62  and multiple receivers R 1 , R 2 , R 3 , and R 4  depicted at  64 ,  66 ,  68 , and  69  spaced axially along tool body  51 . In the depicted example, measurement tool  50  includes axial, transverse, and tilted antennas. As used herein, an axial antenna is one whose dipole moment is substantially parallel with the longitudinal axis of the tool, for example, as shown at  54 . Axial antennas are commonly wound about the circumference of the logging tool such that the plane of the antenna is orthogonal to the tool axis. Axial antennas produce a radiation pattern that is equivalent to a dipole along the axis of the tool (by convention the z-direction). Electromagnetic measurements made by axially oriented antennas may be referred to as conventional or non-directional measurements. 
     A transverse antenna is one whose dipole moment is substantially perpendicular to the longitudinal axis of the tool, for example, as shown at  62 . A transverse antenna may include a saddle coil (e.g., as disclosed in commonly owned U.S. Patent Publications 2011/0074427 and 2011/0238312) and generate a radiation pattern that is equivalent to a dipole that is perpendicular to the axis of the tool (by convention the x or y direction). A tilted antenna is one whose dipole moment is neither parallel nor perpendicular to the longitudinal axis of the tool, for example, as shown at  68  and  69 . Tilted antennas generate a mixed mode radiation pattern (i.e., a radiation pattern in which the dipole moment is neither parallel nor perpendicular with the tool axis). Electromagnetic measurements made by transverse or tilted antennas may be referred to as directional measurements. 
     In the particular embodiment depicted in  FIG. 2 , five of the transmitter antennas (T 1 , T 2 , T 3 , T 4 , and T 5 ) are axial antennas spaced along the axis of the tool. A sixth transmitter antenna (T 6 ) is a transverse antenna. First and second receivers (R 1  and R 2 ) located axially between the transmitters are axial antennas and may be used to obtain conventional non-directional type propagation resistivity measurements. Third and fourth receivers (R 3  and R 4 ) are tilted antennas located axially about the transmitters. Such a directional arrangement (including tilted and/or transverse antennas) produces a preferential sensitivity on one azimuthal side of the tool  50  that better enables bed boundaries and other features of the subterranean formations to be identified and located. 
     Accordingly, as the tool  50  provides both axial transmitters and axial receiver pairs as well as axial transmitter and tilted receiver pairs, the tool  50  is capable of making both directional and non-directional electromagnetic measurements. The example logging tool  50  depicted in  FIG. 2  may be a model of a tool available under the name PERISCOPE from Schlumberger Technology Corporation of Sugar Land, Tex. It will be understood, however, that the embodiments disclosed herein are not limited to any particular electromagnetic logging tool configuration, and that the tool depicted in  FIG. 2  is merely one example of a suitable electromagnetic logging tool. 
     As discussed above, the present disclosure relates to techniques and/or methods for processing non-1D formation measurements with a 1D inversion model. As described in more detail below, an embodiment of the method may include manually or automatically defining regions (“1D processing windows”) where 1D approximation can be applied and running 1D inversion processing in these regions. The results from the 1D inversion processing are then used to update the 2D/3D earth model. 
       FIG. 3  depicts a process for processing non-1D formation measurements using a 1D inversion model in accordance with an embodiment of the disclosure. As described in detail below, an initial 3D earth model or curtain section may be built based on a priori knowledge about the formation. Regions (also referred to as “processing windows”) where 1D processing is applicable may be defined manually, via user input, or automatically. For each identified window, measurement data and well trajectory information may be reformulated for 1D processing, and an initial local 1D model may be built based on the initial 3D earth model or curtain section. 1D inversion processing may be then be applied on each identified window to determine an inverted local 1D model that best fits the measurement data. The resulting inverted 1D model for that region of the formation is then used to update the 3D earth model or curtain section. 
     In some embodiments, the formation properties may include electromagnetic formation properties such as Rh, Rv, dip, azimuth, and bed boundary locations for each layer. In other embodiments, the formation properties may additionally include other suitable properties such as density, velocity, porosity, etc. 
     The process  300  illustrated in  FIG. 3  will now be described in further detail. As shown in  FIG. 3 , an initial 3D earth model or curtain section may be obtained (block  302 ), e.g., a 3D earth model or curtain section may be built using suitable techniques. As will be appreciated, a real earth model can be described with a 3D geometry model. Based on a priori knowledge, an initial 3D earth model or curtain section can be built as a starting point for the 1D processing described below. If the formation is a layered structure, the layer boundaries can be 3D surfaces in general, such that they are not flat planes and are not necessarily parallel to each other. 
     In other embodiments, such as where the layer boundaries are approximately plane shape, the formation can be expressed with curtain sections, such as used in Techlog/3DPetrophysics (3DP) modeling/interpretation software available from Schlumberger. A typical curtain section  400  is shown below in  FIG. 4  below with true horizontal length (THL) as the horizontal axis and true vertical depth (TVD) as the vertical axis. As shown in  FIG. 4 , a well trajectory  402  is shown crossing several layers. The layer boundaries are shown as lines in the curtain section. The boundary lines may be straight lines or any arbitrary 2D curves and not necessarily parallel to each other. The boundary lines may define the boundary position and dip angles within the curtain section plane (on the plane dip). The boundary plane can rotate around the boundary lines so that they become non-perpendicular to the curtain section. The rotation angle of the rotation may be defined as out of plane dip. The rotation angle when considered together with on the plane dip, defines the actual dip and azimuth of the bedding planes. For resistivity properties, horizontal resistivity (Rh) and vertical resistivity (Rv) can be assigned for each layer. 
     Next, processing windows where 1D processing is applicable may be defined (block  304 ). As will be appreciated, 1D processing may approximate the earth with a 1D layered structure with beddings parallel to each other. However, formations with non-1D structure generally may not be processed with 1D inversion algorithms to obtain accurate results. In some embodiments, defining the processing windows may include searching through the whole well and identifying regions where 1D processing is applicable. As described below, 1D inversion processing may be applied in the identified windows. The definition of processing windows for curtain sections is illustrated in  FIG. 5  and described in more detail below. Next, sub-datasets may be created for each defined window (block  306 ). For example, the sub-datasets may include measurement and well trajectory information. 
     As shown in  FIG. 3 , an initial local 1D model may be generated for each window (block  308 ). Next, 1D inversion control parameters for each window may be determined (block  310 ), and a 1D inversion is run for each window (block  312 ). The results of the 1D inversions (e.g., an inverted 1D model and formation parameters) may be used in subsequent processing (block  314 ), such as to update a 2D or 3D earth model. 
     As noted above, in order to run 1D processing within a local region, the formation within the region should be approximately a 1D layered structure. In embodiments having a 3D model, the 1D layered structure may be determined by checking the angle of each layer within the region and depth of investigation (DOI) of the measurement tool (e.g., tool  50  of  FIG. 2 ). The normal direction of the bed boundary surfaces are compared with each other, and the local 1D region (e.g., window) is defined so that the angle between the normal directions are below a cutoff value. 
     If the formation is described with a curtain section, then a 1D processing window may, in some embodiments, be defined (block  304  of process  300 ) in accordance with the process  500  shown below in  FIGS. 5A and 5B . As shown in  FIG. 5A , the process  500  may receive, as input, a curtain section  502 , a well trajectory  504  and a square log  506 . In some embodiments, the square log  506  may include measurement depth for each boundary crossing points, boundary surface dip and azimuth angle at each crossing, as well as Rh and Rv between the crossing points. 
     Next, as shown in  FIG. 5 , a base point of a 1D window may be selected (block  508 ). In some embodiments, the base point of a window may be selected manually, via input from a user, or automatically according to different rules, criteria, or both, depending on the application. In some embodiments, the full 1D model may be determined (e.g., formation properties Rh, Rv, dip and bed boundaries). In such embodiments, dip and bed boundaries inversion generally rely on resistivity contrast in different layers. Thus, in such embodiments, it may be desirable to include sufficient contrast within the window. Consequently, in such embodiments a base point may be selected by searching through all the bed crossing positions and selecting the crossing with the highest contrast as the base point of the window. 
     The window may be expanded to the left and right along the well trajectory (block  510 ) until a stopping criterion is met (decision block  510 ). In accordance with various embodiments, the stopping criterion my include but are not limited to the following:
         1. Bedding angles at each crossing point. The difference between the bedding angles and that of the base point may be compared to a cutoff value. The window may be expanded while the difference is below the cutoff value. In some embodiments, the difference between all the bedding angles may be compared to a second cutoff value, and the window may be expanded while the difference is below the second cutoff value;   2. The plane dip of all the bed boundaries within the window. The difference between these dips and that of the base point may be compared to a cutoff value, and the window may be expanded while the difference is below the cutoff value. The difference between all the plane dips may be compared to a second cutoff value, and the window may be expanded while the difference is below the second cutoff value;   3. The trajectory azimuth variation. The trajectory azimuth variation may be compared to a cutoff value, and the window may be expanded while the difference is below the cutoff value;   4. Total window length of the window. The total window length within the window may be compared to a cutoff value, and the window may be expanded while the difference is below the cutoff value. Satisfying this criterion will help to ensure accuracy of the inversion performance;   5. No (zero) property variation boundaries within the window;   6. No (zero) faults within the window;   7. The well trajectory does not cross the same layer bed boundary more than once;   8. Bed thickness variations. The bed thickness variations may be compared to a cutoff value, and the window may be expanded while the difference is below the cutoff value;   9. The number of layers within a window. The number of layers may be compared to a cutoff value, and the window may be expanded while the difference is below the cutoff value.       

     Once the resulting starting and ending measurement depth (MD) is determined, the actual formation region can be defined according to measurement sensor DOI. The part of the formation that the sensor has sensitivity when traveling from a starting MD and an ending MD may be defined as the 1D processing window. 
     As shown by connection block A, the process  500  is further illustrated in  FIG. 5B . As shown in  FIG. 5B , the window may be checked for multi-crossings (block  512 ) to determine if multi-crossings are present in the window (decision block  514 ). If multi-crossings are present, the window may be shrunk (block  516 ) and the multi-crossing rechecked (block  512 ). If no multi-crossings are present, the process  500  may record the current window as a 1D processing window (block  518 ). As shown in  FIGS. 5A and 5B  and by connection block B, the next possible window may be determined (block  520 ) by selecting the base point of a second window (block  508 ). In some embodiments, after a first window is defined, the base point for the next window may be determined by searching through all the crossing points outside of the first window and locating a crossing point with the highest contrast for use as the base point for searching for a second window. In some embodiments, this window defining process can continue until all the valid windows are defined. In cases of a highly non-1D formation, the expansion of a window to the left and right may be limited, such that the resulting window is relatively small. Because inversion results can be unreliable for very small windows, in some embodiments windows smaller than a selected cutoff size may be rejected. 
     For the curtain section shown above in  FIG. 4 , the 1D processing windows may be defined according to process  500  and as illustrated in  FIG. 6 . As shown in  FIG. 6 , Window I (indicated by  600 ) has a relatively large size as the formation in that region is close to 1D. However, a the Window II (indicated by  602 ) region contains non-parallel bed boundaries, and thus the expansion of the window to the left and right is stopped sooner than the expansion of Window I, thereby resulting in a smaller window as compared to Window I. As also shown in  FIG. 6 , the 1D processing windows do not necessarily cover all of the curtain section, as not all the formation may be suitable for 1D processing using the techniques described herein. 
     As mentioned above, because a curtain section or a 3D earth model is built based on a priori knowledge, the curtain section or 3D earth model may be the best candidate as the initial models for further processing, such as 1D inversion. For 1D inversion, a 1D layered model may be used as an initial starting point, which can be built according to curtain section or 3D earth model.  FIG. 7  shows an example process  700  for constructing a 1D layered model from 3DP curtain section in accordance with embodiments of the disclosure. As shown in  FIG. 7 , the process  700  may receive as input, a curtain section  702 , a well trajectory  704  and a square log  706 , similar to the process  500  for defining 1D processing windows illustrated in  FIG. 5 . The window size may be defined in terms of the well trajectory  704  within the window, which can be described by starting and ending MD of the trajectory, as discussed above. In some embodiments, the window size may be defined by the starting and ending index of the well trajectory log. In such embodiments, the starting and ending index  708  may be received as input by the process  700 . 
     As described above in process  300 , a sub-dataset may be created for each window (block  306 ) and an initial 1D layered model generated for each window (block  308 ). For example, taking the first window (Window I) depicted in  FIG. 6  as an example, for each crossed bed boundary, the bedding true dip and azimuth may be computed or input from the curtain section, the square log, or a combination thereof (block  710 ). In some embodiments, the true dip and azimuth of the 1D layered model may be computed by weighted averaging the dips of all crossed boundaries within the window. A 1D layered model may then be constructed for the window based on the crossed bed boundaries within the window and enforcing the newly computed dip and azimuth for all the layers (block  708 ). Next, bed thickness in true stratigraphic thickness (TST) may then be computed to remove potential multi-crossing layers (block  710 ), e.g., layers with multi-crossing (negative TST) or very thin layers (near zero TST thickness). 
       FIG. 8  depicts an example of the resulting 1D layered model  800  below. As can be seen, the resulting 1D layered model  800  for Window I may be a fairly accurate approximation of the curtain section, except, as shown in  FIG. 6 , the curtain section has an extra bed boundary  800  at the bottom. The bed boundary  800  is missing in the corresponding 1D layered model because it is not crossed by the well trajectory in this example. 
     In some embodiments, the non-crossed layer may be included as it is close enough to the well trajectory and can affect the response of the tool, i.e., it is within the tool DOI. As shown in  FIG. 7 , in order to include the non-crossed layers, the concept of extended (or imaginary) trajectory may be used. Using such techniques, the trajectory may be extended from both ends of the actual well trajectory (starting and end MD) to include non-crossed layers (block  712 ). The extension direction may be chosen such that it may only cross the originally non-crossed layers. The crossing MD on the extended trajectory may be computed (block  714 ). In the example depicted in  FIG. 8 , the formation is nearly horizontal and the first extended trajectory starts from the starting MD and goes up. The second extended trajectory starts from ending MD and goes down, which crosses the bottom bed boundary  800  in the curtain section. Next, the non-crossed layers may be added to the 1D layered model (block  716 ). For example, this crossing point at the bottom bed boundary  800  may be used to define the 1D layered model. The 1D layered model may be output with the extended trajectory and other indicators (e.g., multi-crossings, error codes, and the like).  FIG. 9  depicts an example of the 1D layered model  800  after taking into account both crossed and non-crossed bed boundaries as described above. As shown in  FIG. 9 , for example, the 1D layered model is extended to include the bottom bed boundary  800  in the curtain section. The initial Rh and Rv values may be taken from the curtain section and assigned to the layers in the local 1D layered model for Window I. 
     After a 1D layered model (also referred to as a “local 1D model”) has been obtained for Window I, the formation parameters may include, for example, a global dip, Rh, Rv, and bed boundary locations for each layer. An inversion algorithm may be used to invert for all or any subset of these parameters. In some embodiments, an inversion algorithm may also enable setting minimum and maximum values for each parameter to be inverted, assigning prior values, and applying regularization on the inversion. 
     As described above in process  300 , a 1D inversion may be performed (block  312 ) and the inversion may be used in subsequent processing (block  314 ). Thus, after the initial model, measurement and well trajectory information, and inversion settings are ready, a 1D inversion may be performed to obtain optimal model parameters that best fit the measurement data.  FIG. 10  depicts examples of inverted 1D models  1000  for the selected 1D processing windows of the curtain section of  FIG. 4  in accordance with an embodiment of the disclosure. In the example depicted in  FIG. 10 , all the model parameters are inverted. The inversion results may be used to update the curtain section and produce a more accurate 3D earth model. 
     After performing 1D inversion processing on the 1D processing windows, an original model may be updated to reflect the inversion results. For example, in some embodiments, the original Rh and Rv values may be replaced by the inverted values. In some embodiments, to avoid overwriting the Rh and Rv values from an inversion window with those from other windows, property variation boundaries can be inserted. In some embodiments, the bed boundary locations and dip angle may also be updated in the original model based on the parameters obtained from 1D inversion on the 1D layered models corresponding to the selected processing windows. After the model is updated, a synthetic resistivity log response may be computed using resistivity forward modeling to help ensure that the measured logs match with simulated logs throughout the entire model along the trajectory. 
     As will be understood, the various techniques described above and relating to applying 1D inversion processing in a non-1D formation are provided as example embodiments. Accordingly, it should be understood that the present disclosure should not be construed as being limited to only the examples provided above. Further, it should be appreciated that the log squaring techniques disclosed herein may be implemented in any suitable manner, including hardware (suitably configured circuitry), software (e.g., via a computer program including executable code stored on one or more tangible computer readable medium), or via using a combination of both hardware and software elements. Further, it is understood that the techniques described herein may be implemented on a downhole processor (e.g., a processor that is part of an electromagnetic logging tool, such as tool  50  of  FIG. 2 ), such that the processing is performed downhole, with the results sent to the surface by any suitable telemetry technique. Additionally, in other embodiments, directional and non-directional electromagnetic measurements may be transmitted uphole via telemetry, and the techniques for applying 1D inversion processing in a non-1D formation may be performed uphole on a surface computer (e.g., one that is part of control system  152  in  FIG. 1 ). 
     Conditional language, such as, among others, “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations could include, while other implementations do not include, certain features, elements, and/or operations. Thus, such conditional language is not generally intended to imply that features, elements, and/or operations are in any way used for one or more implementations or that one or more implementations necessarily include logic for deciding, with or without user input or prompting, whether these features, elements, and/or operations are included or are to be performed in any particular implementation. 
     Many modifications and other implementations of the disclosure set forth herein will be apparent having the benefit of the teachings presented in the foregoing descriptions and the associated drawings. Therefore, it is to be understood that the disclosure is not to be limited to the specific implementations disclosed and that modifications and other implementations are intended to be included within the scope of the appended claims. Although specific terms are employed herein, they are used in a generic and descriptive sense and not for purposes of limitation.