Abstract:
Systems and methods are disclosed for well logging using radiation detection and/or emission of gamma rays. A method according to the disclosure includes collecting data from the subterranean formation using a nuclear density tool, wherein the nuclear density tool is configured to collect data to form an azimuthal image. The method further includes characterizing a section of the subterranean formation comprising data and images acquired in a high angle wellbore section, a horizontal wellbore section, or a combination thereof. The method additionally includes performing a parallel inversion using apparent densities and volumetric photoelectric factor images to build a formation model, wherein the parallel inversion comprises a high angle workflow that models high angle wellbore sections and a horizontal workflow that models horizontal wellbore sections.

Description:
BACKGROUND 
       [0001]    This disclosure relates to methods for the interpretation of density images in high angle and horizontal wells. 
         [0002]    This section is intended to introduce the reader to various aspects of art that may be related to various aspects of the present techniques, which are described and/or claimed below. This discussion is believed to be helpful in providing the reader with background information to facilitate a better understanding of the various aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not as admissions of any kind 
         [0003]    When a well is drilled into a geological formation, a variety of measurements may be used to identify information about the geological formation. These may include density, porosity, lithology, and many others. Identifying this information for the geological formation can be useful to determine the likely presence or absence of hydrocarbons, as well as the amount of recoverable oil and/or gas reserves, for which the formation structure and the dip angle of the well through various layers of the geological formation are used. The nuclear density measurement, in particular, plays a large role in the interpretation of porosity. Logging-While-Drilling (LWD) borehole density images provide formation structural information including formation layer thickness and dip angle. However, automatic fast formation evaluation using density images in high-angle (HA) and horizontal (HZ) wells is still a challenge for petrophysicists. 
         [0004]    The first generation of imaging tools may use nuclear density inversion-based processing including two distinct workflows. Density images are first preprocessed, then, intervals where it is possible to extract sinusoids (e.g., sine-based patterns) and estimate relative dip are processed using a first HA workflow. In intervals where tool is oriented nearly parallel to boundaries, a second HZ workflow may be used. In addition to formation layering, the inversion process evaluates the borehole shape and properties. The HA workflow is based on interval processing assuming locally constant formation dip (e.g., a formation where the size of interval is dependent on relative dip). On the other hand, the HZ workflow assumes that the tool is parallel to formation boundaries, and thus the inversion process is able to take into account the non-crossed boundaries that may affect the tool readings. The approach allows for lateral variation of formation properties, layer thickness and dip. However, since in HZ workflows short segments (e.g., 2 ft. long segments) are processed at the time, the interpretation is not as robust and consistent as compared to the HA interpretation. It would be beneficial to improve HA and HZ inversion-based workflows. 
       SUMMARY 
       [0005]    A summary of certain embodiments disclosed herein is set forth below. It should be understood that these aspects are presented merely to provide the reader with a brief summary of these certain embodiments and that these aspects are not intended to limit the scope of this disclosure. Indeed, this disclosure may encompass a variety of aspects that may not be set forth below. 
         [0006]    Embodiments of the disclosure relate to well logging using gamma ray sources and detectors. A method according to the disclosure includes collecting data from the subterranean formation using a nuclear density tool, wherein the nuclear density tool is configured to collect data to form an azimuthal image. The method further includes characterizing a section of the subterranean formation comprising data and images acquired in a high angle wellbore section, a horizontal wellbore section, or a combination thereof. The method additionally includes performing a parallel inversion using apparent densities and volumetric photoelectric factor images to build a formation model, wherein the parallel inversion comprises a high angle workflow that models high angle wellbore sections and a horizontal workflow that models horizontal wellbore sections. 
         [0007]    In another example, a downhole measurement system includes a gamma ray source configured to generate gamma rays into a subterranean formation, a detector configured to detect a radiation from the subterranean formation, and a processor. The processor is configured to collect data from the subterranean formation using the detector to form an azimuthal image. The processor is additionally configured to characterize a section of the subterranean formation comprising data and images acquired in a high angle wellbore section, a horizontal wellbore section, or a combination thereof. The processor is further configured to perform an inversion using apparent densities and volumetric photoelectric factor images to build a formation model wherein the inversion comprises extracting model boundaries. 
         [0008]    The system is more particularly configured to carry out all the embodiments of the method as disclosed hereafter. 
         [0009]    Moreover, non-transitory, tangible computer readable storage medium, comprising instructions is described. The instructions are configured to collect data from a subterranean formation using a detector to form azimuthal images, and to characterize a section of the subterranean formation comprising data and images acquired in a high angle wellbore section, a horizontal wellbore section, or a combination thereof. The instructions are additionally configured to perform a parallel inversion using apparent densities and volumetric photoelectric factor images to build a formation model, wherein the parallel inversion comprises a high angle workflow that models high angle wellbore sections and a horizontal workflow that models horizontal wellbore sections. 
         [0010]    The instructions are configured to perform all the embodiments of the method as disclosed in this application. 
         [0011]    Another method according to the disclosure relates to perform a parallel inversion using apparent densities and volumetric photoelectric factor images to build a formation model, wherein the parallel inversion comprises a horizontal workflow that models horizontal wellbore sections. The horizontal workflow comprises performing a horizontal well pre-processing workflow, wherein the pre-processing workflow comprises: 
         [0012]    obtaining a curtain-section model from azimuthal squaring compensated density image data; 
         [0013]    identifying points defining a formation boundary of the subterranean formation by comparing boundary true vertical depths (TVDs) and density contrast in adjacent sections of data sequentially; 
         [0014]    filtering out hanging points; 
         [0015]    sampling formation boundaries within a predefined interval; and 
         [0016]    building up a topological model for each of the horizontal wellbore sections by identifying geometry and zones of the formation model. 
         [0017]    This method may also comprise other features described in this disclosure and particularly features in relation with the previous method. 
         [0018]    Various refinements of the features noted above may be undertaken in relation to various aspects of the present disclosure. Further features may also be incorporated in these various aspects as well. These refinements and additional features may exist individually or in any combination. For instance, various features discussed below in relation to one or more of the illustrated embodiments may be incorporated into any of the above-described aspects of the present disclosure alone or in any combination. The brief summary presented above is intended to familiarize the reader with certain aspects and contexts of embodiments of the present disclosure without limitation to the claimed subject matter. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0019]    Various aspects of this disclosure may be better understood upon reading the following detailed description and upon reference to the drawings in which: 
           [0020]      FIG. 1  illustrates and embodiment of a drilling system that may apply the techniques described herein; 
           [0021]      FIG. 2  is a diagram of an embodiment of a final formation model that may be derived using inversion workflow techniques; 
           [0022]      FIG. 3  is a diagram of an embodiment of a second final formation model that may be derived using inversion workflow techniques; 
           [0023]      FIG. 4  is a diagram of embodiments of azimuthal compensated density images that may be processed using the techniques described herein; 
           [0024]      FIG. 5  depicts an embodiment of a r-θ grid image that may be derived using the techniques described herein; 
           [0025]      FIG. 6  is perspective view of an embodiment of a 3D grid extending from approximately −45° to 45° that may be based on the images of  FIG. 5 ; 
           [0026]      FIG. 7  is a schematic drawing of an embodiment of a high-angle (HA) well model parameterization; 
           [0027]      FIG. 8  is a schematic drawing of an embodiment of a borehole model parameterization; 
           [0028]      FIG. 9  is a flowchart depicting an embodiment of a process  130  suitable for executing an HA well inversion workflow; 
           [0029]      FIG. 10  depicts an embodiment of a log-squaring chart (left figure), sinusoids that may be obtained from sinusoid extraction process (middle figure), and interval segmentation (right figure); 
           [0030]      FIG. 11  depicts embodiments of images that may be processed via HA inversion, and results obtained before and after processing; 
           [0031]      FIG. 12  illustrates an embodiment of a borehole model parameterization for horizontal (HZ) sections; 
           [0032]      FIG. 13  depicts embodiments of images that may be processed via horizontal (HZ) inversion, and results obtained before and after processing; 
           [0033]      FIG. 14  depicts an embodiment of a curtain-section model obtainable via HZ inversion; 
           [0034]      FIG. 15  depicts embodiments of images that may be applied to obtain a final borehole geometry image; 
           [0035]      FIG. 16  depicts a curtain-section model that may be obtained by combining HZ and HA images; 
           [0036]      FIG. 17  depicts embodiments of a serial and a parallel process, including a parallel inversion process suitable for deriving visualizable formation models; 
           [0037]      FIG. 18  illustrates an embodiment of a model parameterization derived via interval processing in HZ wells; 
           [0038]      FIG. 19  illustrates a variety of images that may be derived at various activities of an enhanced pre-process workflow; 
           [0039]      FIG. 20  illustrates an embodiment of an enhanced inversion workflow; 
           [0040]      FIG. 21  illustrates a full 2D HZ model parameterization; 
           [0041]      FIG. 22  illustrates an improved curtain-section model that may be derived via the techniques described herein; 
           [0042]      FIGS. 23-27  illustrate embodiments of various curtain-section models that may be used to validate the techniques described herein via synthetic data; and 
           [0043]      FIGS. 28-31  illustrate embodiments of various curtain-section models that may be used to validate the techniques described herein via field data. 
       
    
    
     DETAILED DESCRIPTION 
       [0044]    One or more specific embodiments of the present disclosure will be described below. These described embodiments are examples of the presently disclosed techniques. Additionally, in an effort to provide a concise description of these embodiments, features of an actual implementation may not be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions may be made to achieve the developers&#39; specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure. 
         [0045]    When introducing elements of various embodiments of the present disclosure, the articles “a,” “an,” and “the” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than 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 additional embodiments that also incorporate the recited features. 
         [0046]    The techniques described herein include a parallelized workflow for inversion-based interpretation of nuclear density images in high-angle (HA) as well as horizontal (HZ) wells. The workflow applies interval processing and extended parameterization that may improve consistency of interpreted structure and density profiles compared to, for example, point-by-point inversion techniques. A model may be defined, as described in more detail below, and the model definition may extend beyond 1D. The model described herein is able to define formations with non-parallel boundaries, variable dip, and lateral property changes, which are functionalities useful for HZ well interpretation. That is, the boundaries may be non-parallel curves and the boundaries can start and end at any location in the interval, thus, the boundaries do not have to start from the beginning of the interval and terminate at the end of the interval. A process described herein may automatically extract model boundaries starting from azimuthal squaring followed by an “advancing-front” to construct boundary contours in a “curtain-section” plot. The parameterizations described herein also enable definitions of 2D zones that take into account internal layering and cross-bedding structures. 
         [0047]    In a preprocessing workflow and formation boundary extraction, an image segment is chosen to be large enough to cover a complete HZ image feature, such as ‘bull&#39;s eye’ feature. An approximate curtain-section image is obtained after squaring the azimuthal compensated density (COMP) response within small intervals. Formation boundaries are extracted by comparing the true vertical depth (TVD) of the bed boundaries between neighbor intervals and the densities above and below each boundary. A manual adjustment may be used so that the automatic extraction is more accurate. Afterwards, the layout of the boundaries is found by comparing the center location of the boundaries in the curtain-section plane. Each formation layer between two or more boundaries is assumed to have a unique density and photoelectronic factor. If desired, the layer can be further divided into sub-layers to add more details into the formation model. The implementation of sub-layers is similar to those of main layers. 
         [0048]    A graphical user interface may be used so that it would be easier to adjust the formation boundaries graphically. Once the boundaries are defined and the initial formation model is constructed, layer density and photoelectric factor may be found in the inversion workflow. 
         [0049]    Inversion workflow and post-processing may include inverting layer density and other parameters by fixing the boundaries position from pre-processing. Gauss-Newton optimization with line search adaptive regularization and parameter constraints may be used, but other similar techniques (e.g., non-linear lest square solving techniques), may be used. A cost function may include a weighted L 2 -norm (e.g., Euclidean vector norm) error between the measurements and the modeled responses over log points in each segment. The inversion workflow may include the following three activities: 1) Invert borehole average radius and mud properties using short spacing density (SS), differential density (DRHO) and apparent volumetric photoelectric factor (UAPP) channels for sectors and log points. In each segment the mud properties are assumed to be uniform and the borehole is assumed to have cylindrical geometry. 2) Invert borehole geometry with staircase or harmonic model using SS and DRHO channels for each sector and log points. Borehole geometry is defined in terms of borehole radius from tool axis to the borehole wall. For staircase geometry, the borehole radius is assumed to be piecewise constant within each sector, while harmonic expansion over tool azimuth is used in the harmonic model. 3) Invert formation layer density and photoelectric factors based on the inverted mud properties and borehole geometry using long spacing density (LS), COMP and UAPP channels for sectors and log points. 
         [0050]    Parallelization may be used. Indeed, a serial process may be parallelized based on the segments, since the inversion for each segment is independent to the others. Grid Services Lite (GSL) parallelization scheme shows good performance for uniform segmentation. For non-uniform segmentation scheme, the parallelization provides reasonable additional speed. The GSL is a task scheduler for parallel execution of computing jobs, so any other similar scheduler may be used. 
         [0051]    It may be useful to illustrate certain systems that apply the techniques described herein. Accordingly,  FIG. 1  illustrates a drilling system  610  that includes a Logging-While-Drilling (LWD) system  611 . The drilling system  610  may be used to drill a well into a geological formation  612  and obtain measurements useful to identify characteristics of the well. In the drilling system  610 , a drilling rig  614  at the surface  616  may rotate a drill string  618  having a drill bit  620  at its lower end. The drill bit  620  may traverse horizontal sections  615  as well as high angle sections  617  of the formation  612 . As the drill bit  620  is rotated, a drilling fluid pump  622  is used to pump drilling fluid  623 , which may be referred to as “mud” or “drilling mud,” downward through the center of the drill string  618  in the direction of the arrow to the drill bit  620 . The drilling fluid  623 , which is used to cool and lubricate the drill bit  620 , exits the drill string  618  through the drill bit  620 . The drilling fluid  623  then carries drill cuttings away from the bottom of a wellbore  626  as it flows back to the surface  616 , as shown by the arrows through an annulus  630  between the drill string  618  and the formation  612 . However, as described above, as the drilling fluid  623  flows through the annulus  630  between the drill string  618  and the formation  612 , the drilling mud  623  may begin to invade and mix with the fluids stored in the formation, which may be referred to as formation fluid (e.g., natural gas or oil). At the surface  616 , return drilling fluid  624  is filtered and conveyed back to a mud pit  632  for reuse. 
         [0052]    As illustrated in  FIG. 1 , the lower end of the drill string  618  includes a bottom-hole assembly (BHA)  634  that may include the drill bit  620  along with various downhole tools. The downhole tools may collect a variety of information relating to the geological formation  612  and/or the state of drilling of the well. For instance, a measurement-while-drilling (MWD) tool  636  may measure certain drilling parameters, such as the temperature, pressure, orientation of the drilling tool, and so forth. Likewise, a logging-while-drilling (LWD) tool  638  may measure the physical properties of the geological formation  612 , such as density, porosity, resistivity, lithology, and so forth. The MWD tool  636  and/or the LWD tool  638  may include a gamma logging tool that emits and/or detects radioactivity. For example, the MWD tool  636  and/or the LWD tool  638  may include a radiation detector  637  and a radiation generator (e.g., neutron generator)  639 . In certain embodiments, the generator  639  may produce neutrons continuously (e.g., continuous mode of operations) or in pulses (e.g., pulsed mode of operations). It is to be noted that the tool  114  as described above may be included in the MWD tool  636  and/or the LWD tool  638 . 
         [0053]    The MWD tool  636  and/or the LWD tool  638  may collect a variety of data  640  that may be stored and processed in the BHA or, as illustrated in  FIG. 1 , may be sent to the surface for processing. The data  640  may include azimuthal images (e.g., long spacing density [LS], short spacing density [SS], compensated density [COMP], apparent volumetric photoelectric factor [UAPP], and Delta Rho differential density (DRHO) images), density data (e.g., apparent density data), volumetric photoelectric factor data, and so on. The data  640  that is collected may include counts and/or detected energies gamma rays that may contain information relating to characteristics of the geological formation  612 . The data  640  may be sent via a control and data acquisition system  642  to a data processing system  644 . The control and data acquisition system  642  may receive the data  640  in any suitable way. In one example, the control and data acquisition system  642  may transfer the data  640  via electrical signals pulsed through the geological formation  612  or via mud pulse telemetry using the drilling fluid  624 . In another example, the data  640  may be retrieved directly from the MWD tool  636  and/or the LWD tool  638  upon return to the surface. 
         [0054]    The data processing system  644  may include a processor  646 , memory  648 , storage  650 , and/or a display  652 . The data processing system  644  may use the data  640  to determine various properties of the well using any suitable techniques. To process the data  640 , the processor  646  may execute instructions stored in the memory  648  and/or storage  650 . As such, the memory  648  and/or the storage  650  of the data processing system  644  may be any suitable article of manufacture that can store the instructions. The memory  646  and/or the storage  650  may be ROM memory, random-access memory (RAM), flash memory, an optical storage medium, or a hard disk drive, to name a few examples. The display  652  may be any suitable electronic display that can display the logs and/or other information relating to properties of the well as measured by the MWD tool  636  and/or the LWD tool  638 . It should be appreciated that, although the data processing system  644  is shown by way of example as being located at the surface, the data processing system  644  may be located in the MWD tool  636  and/or the LWD tool  638 . In such embodiments, some of the data  640  may be processed and stored downhole, while some of the data  640  may be sent to the surface in real time. This may be the case particularly in LWD, where a limited amount of the data  640  may be transmitted to the surface during drilling or reaming operations. 
         [0055]    Turning now to  FIG. 2 , the figure is a chart view  10  of an embodiment of a proposed formation model  12  for HZ wells in a curtain cross-section plane  14 . The model  12  is described by formation boundaries  16  (solid lines) and the topology of formation layers. There are six formation layers  18 ,  20 ,  22 ,  24 ,  26 ,  28  in which the third layer is further divided into three sub-layers. A dashed line  30  is the tool trajectory and solid lines  32  show the borehole geometry.  FIG. 2  additionally depicts a HZ inversion of noisy synthetic data. From top to bottom: a measured COMP image  34 , final borehole-corrected image  36 , and final formation model  12  in curtain-section plane  14 . 
         [0056]      FIG. 3  depicts an embodiment of a chart or curtain-section model  50  showing results of a HZ inversion on field data. The figure also shows, from top to bottom, a measured COMP image  52 , a final borehole-corrected image  54 , and the final visual formation model  50  in curtain-section plane  56 , showing various layers  58  (e.g., density layers). The black curves represent trajectory  60  (dashed line) and borehole geometry  62  (solid lines). The techniques described herein include an inversion-based workflow for interpretation of nuclear density images (e.g., images  34 ,  36 ,  52 ,  54 ) in HZ wells. 
         [0057]    The method, as described in more detail below applies interval processing and extended parameterization to advantageously include non-parallel and curved boundaries as shown in the curtain-section plot  56 . Parallelized workflows may include sliding window processing to allow lateral property discontinuity. Workflows may also include an automatic extraction of model  50  boundaries starting from azimuthal squaring followed by an “advancing-front” to construct the model  50  boundary contours in the curtain-section plot  56 . The parameterization also allows definition of 2D zones to take into account internal layering and cross-bedding structures. The model  50  definition may be beyond 1D and is able to define formation with non-parallel boundaries, variable dip and lateral property changes. The boundaries may be non-parallel curves and the boundaries can start and end at any location in the interval, thus, the boundaries do not have to start from the beginning of the interval and terminate at the end of the interval. Indeed, inversion parameterization may include 2D and 3D formation information (cross-bedding with changes in layer that are not conforming to original layering). 
         [0058]    As shown in  FIG. 4 , a variety of images  70 ,  72  derived from formation analysis techniques, such as nuclear gamma-gamma density measurement techniques, may be processed to build the 2D and 3D formation information. More specifically, images  70 ,  72  may include azimuthal compensated density images  70 ,  72  that may include an ordinal axis  74  with measurements in feet and an abscissa axis  76  with measurements representative of formation density. It is to be noted that the axes  74 ,  76  may have different scales for each image  70 ,  72 . The images  70 ,  72  may be processed via inversion workflows, as described in more detail below, to derive the formation&#39;s geophysics. Inversion workflows may include HA wells as well as HZ wells, and may be parallelized for increased modeling speed. HA and HZ sections may be integrated, further increasing accuracy. That is, HA and HZ image sections may be analyzed together to derive an improved model of the formation undergoing analysis. In certain embodiments, nuclear density fast modeling may be applied to produce synthetic images in corresponding formation  70 ,  72 , as shown in  FIG. 5 , to enable a faster and more accurate modeling of the formation being analyzed. 
         [0059]    More specifically,  FIG. 5  depicts an embodiment of an r-θ grid image  80  for θ approximately between approximately −45° and +45° that may be derived for fast-forward modeling. The depicted image shows a circle  82  representative of a borehole and a circle  84  representative of nuclear gamma-gamma density measurement tool, such as an Ecoscope™ tool available from Schlumberger Limited, of Houston, Tex. The tool  84  may provide for a series of imaging windows  86  representative of a formation density. In one embodiment the windows  86  may be stacked, as shown in  FIG. 6 , to derive a three-dimensional (3D) view  90  representative of the formation being analyzed. Also shown in 3D are the borehole  82  and the measurement path  84  taken by the measurement tool. The view  90  includes a 3D grid  92  of the segments  86  derived, for example, by stacking windows  86  on top of each other. A series of sensitivity functions may be applied to the 3D grid  92  to pre-compute measurement responses for density and photoelectric factor properties defined on the same grid  92 . In one embodiment, 5 cells may be disposed in the azimuthal direction, with grid nodes located at ±5°, ±15° and ±45° with respect to the detector face normal. In other embodiments, less than or more than 5 cells may be used, with grid node angles correlative to the number of cells chosen. 
         [0060]    A fast-forward model may use the pre-computed first and second order sensitivity functions to compute responses for log points and sectors, such as linear approximation and 2D flux sensitivity functions. By using the pre-computed functions, the forward modeling may achieve speeds over one million times faster than Monte Carlo N-Particle (MCNP) simulations, thus providing for real-time or near real-time view of the formation. Further speed improvements may be achieved via parallelization, as described in more detail below. 
         [0061]    Turning now to  FIG. 7 , is a schematic drawing of an embodiment of a high-angle (HA) formation model parameterization  100  at true vertical depth (TVD) and true horizontal length (THL), including set of dipping layers  102 ,  104 ,  106 ,  108  with trajectory of varying inclination, strictly up-section or down-section. Borehole geometry  110  is represented with dashed lines. The HA well model  100  may assume that a trajectory  112  can be discretized into segments such that the formation in each segment is 1D layering with constant dip and azimuth. The trajectory azimuth is assumed to be constant in each segment, and the trajectory inclination is assumed to be “down-section”, with layers crossed at relative dip less than 90°, or “up-section”, when layers are crossed at relative dip greater than 90°.  FIG. 7  additionally shows the formation for a typical down-section segment when viewed in the “curtain-section” plane, e.g., a vertical surface defined by the trajectory. 
         [0062]    Free parameters for the segment are an apparent formation dip α, TVD location z of each boundary layer, layer densities ρ, and layer photoelectric factors PEF. The segmented nature of the HA model  100  allows the model  100  to capture variations in dip and layer properties along the trajectory of a density measurement tool  114 . The high-angle parameterization may degrade for relative dip close to 90°, where a separate approach, such as a horizontal approach, may be used. The HA modeling may be applied to images with sinusoids, such that segment size is approximately equal to sinusoid amplitude. 
         [0063]      FIG. 8  depicts an embodiment of a borehole model parameterization  120  having a borehole geometry  122  and homogeneous mud  124  properties. The density measurement tool  114  is also depicted, with an arrow  126  (e.g., r tool  arrow) representative of measured depth (MD). The mud  124  properties, ρ mud  and PEF mud , may be defined in the same trajectory segments used for the formation model  100 , and may be assumed to be uniform in each segment. The geometry of the borehole is defined in terms of the radial distance from the tool axis to the borehole wall, denoted by r, where θ is the tool  114  azimuth. Cylindrical geometry algorithms, 3D staircase geometry algorithms, and/or 3D harmonic geometry algorithms may be applied to images  86  to derive the geometry  122 . Accordingly, an inversion-based approach and a caliper-based approach may be used, which may be more robust and accurate by modeling the true geometry  122  as well as a layered background, more effectively decoupling a borehole effect from a boundary effect. 
         [0064]      FIG. 9  is a flowchart of an embodiment of a workflow or process  130  suitable for executing a HA well inversion workflow to derive, for example, the formation model  100  shown in  FIG. 7 . The process  130  may be implemented as computer instructions executable via a processor of the tool  114  and may be stored in a memory of the tool  114 , and/or may be stored in memory  648  to be executed via processor  646 , or in other external systems to be stored and processed by the external systems. In the depicted embodiment, the process  130  may, for example, analyze nuclear gamma-gamma density data acquired via the tool  114  beginning at a bottom quadrant density log (ROBB) and applying a log squaring approach (block  132 ). ROBB may be particularly useful because standoff (e.g., distance between tool  114  and borehole) may be minimized. The log squaring approach may yield an initial guess for layer properties, e.g., ρ i , PEF i . The process  130  may then derive an initial guess for dip and azimuth of the formation (block  134 ), for example, via sinusoid extraction described in more detail below. Adaptive trajectory segmentation (block  136 ) may subdivide images into sub-domains or sub-images for further processing based on, for example, the dip derivations. 
         [0065]    An initial borehole model (e.g., model  82 ) may be defined (block  138 ) by deriving a cylinder based on a diameter of a bit size used to drill through the formation. The borehole model may be further approximated via a density caliper tool data, and user specified mud properties (e.g., mud density and type of mud used during drilling) may be additionally used during the definition of the borehole model (block  138 ). Each segment  86  may then be inverted (block  140 ) for mud properties. That is, available mud property methods may be reproduced numerically via forward models that incorporate specific measurement response functions for numerical simulation of LWD measurements (block  140 ). Shallow depth data, e.g., shallow sensing channels included in the tool  114 , may provide more accuracy as compared deep sensing channels. 
         [0066]    Inversion may also be used to derive borehole geometry (block  140 ). Shallow sensing channels may also be used, and sector standoff or harmonic coefficient processing may be used during the inversion (block  142 ) derivation of the borehole geometry. Likewise, inversion may also be used to derive layer properties, boundary positions, and dip (block  144 ). However, deep sensing channels may provide more accuracy to derive the layer properties, boundary positions, and dip (block  144 ). The process  130  may then provide for a visualization of the inverted data, and an error and uncertainty calculation (block  146 ). It may be beneficial to illustrate imaging derived by applying the process  130 . 
         [0067]    Accordingly,  FIG. 10  shows a squared log chart  150  including a ROBB bottom quadrant data portion  152  obtained at block  132 . The chart  150  may be used to obtain images suitable for inversion. More specifically, a log-squaring algorithm is applied to the ROBB log  152 . Image  154  may be an original image that may include sinusoids  156 , and the sinusoids  156  may then be extracted (as described at block  134 ), using, for example, image processing techniques that identify curved structures in images. An amplitude and a phase for each of the sinusoids  156  may correspond to relative dip and relative azimuth of a layer boundary. 
         [0068]    Adaptive segmentation (as described at block  136 ) may then be used to divide the image  154  into an image  158  subdivided into multiple segments  160 . Adaptive segmentation may change segment  160  sizes so that at least one sinusoid  156  is inside of each segment  160 . The ROBB log  152  may then be combined with segments  160  during analysis. Using the process  130  as shown, high angles of up to 88° may be more accurately imaged. While angles between 88° and 92° may also be imaged via the process  130 , it may be preferable to apply a horizontal well approach (HZ) described in more detail with respect to  FIG. 12  below. The segments  160  may then be inverted to derive a variety of parameters as depicted in  FIG. 11 . 
         [0069]    More specifically,  FIG. 11  depicts embodiments of a first set of COMP images  170  and a second set of short spacing density (SS) images  172  to illustrate HA derivation of density parameters ρ based on COMP channels (e.g., ρ comp -based parameters) and SS channels (e.g., ρ ss -based parameters). As illustrated, image  174  may be derived by using measurements taken via the tool  114  (e.g., nuclear gamma-gamma measurement tool) using COMP channels. An initial guess image  176  may then be derived (as described at block  132 ). As mentioned above, squared log of ROBB data  152  and sinusoids  156  may be used to derive the initial guess image  176 . For example, squared log  152  and sinusoids  156  are used to derive the initial model ( 100  and  120 ), then the initial tool responses,  176  and  186 , are calculated using density fast-forward model, e.g.,  90 . HA inversion may then be applied to certain models, e.g.,  100  and/or  120 , resulting in a final COMP image  178  (as obtained at block  146 ). A fast-forward model may use pre-computed first and second order sensitivity functions to compute responses for log points and sectors, such as linear approximation and 2D flux sensitivity functions during inversion. That is the fast-forward model may be used to compute initial and final tool response. Image  180  shows a relative error for the initial guess image  176  compared to the measurement image  174 , while image  182  shows a relative error for the final COMP image  178  compared to the measurement image  174 . As shown, image  182  is less noisy when compared to image  180 , depicting how inversion may provide for a more efficient and accurate measure of density properties ρ comp . 
         [0070]    Also shown is a measured image  184  measured using SS channels. An initial guess image  186  (as described at block  132 ) may then be derived. Inversion may then be applied to certain models e.g.,  100 ,  120 , resulting in a final SS image  186  (as obtained at block  146 ). A fast-forward model may use pre-computed first and second order sensitivity functions to compute responses for log points and sectors, such as linear approximation and 2D flux sensitivity functions during inversion. Image  190  shows a relative error for the initial guess image  186  compared to the measurement image  184 , while image  192  shows a relative error for the final SS image  188  compared to the measurement image  184 . As shown, image  192  is less noisy when compared to image  190 , depicting how HA inversion may provide for a more efficient and accurate measure of density properties ρ ss . 
         [0071]    Turning now to  FIG. 12 , the figure depicts an embodiment of a borehole model parameterization  200  useful for modeling horizontal (HZ) wells. In the depicted embodiment, the HZ borehole model parameterization  200  includes layers  202 ,  204 ,  206 , and  208  with a trajectory of substantially horizontal inclination. Indeed, the HZ borehole model parameterization  200  may be particularly useful at angles between 88° and 92°. Mud  210  properties, ρ mud  and PEF mud , may be defined in the same trajectory segments used for the formation model, and are assumed to be uniform in each segment. A geometry  212  of the borehole is defined in terms of the radial distance from the tool axis to the borehole wall, denoted by r, where θ is the tool azimuth and r tool  is the tool  114  radius. The HZ well model  200  also assumes discrete segments such that the formation in each segment is 1D layering with dip and azimuth, and such that the trajectory azimuth is constant in each segment. However, the HZ well model  200  constrains the layer boundaries in each segment to be parallel to the trajectory, e.g., constant relative dip 90°. Shorter segments would be used to account for the more horizontal tool orientation and to better capture lateral changes in the formation. The HZ borehole model parameterization  200  may also be particularly applicable to images with “bull&#39;s eye”, “reverse bull&#39;s eye”, “railroad tracks”, and similar features. 
         [0072]      FIG. 13  depicts embodiments of a first set of COMP images  220  and a second set of short spacing density (SS) images  222  to illustrate HZ derivation of density parameters ρ based on COMP channels (e.g., ρ comp -based parameters) and SS channels (e.g., ρ ss -based parameters). As illustrated, image  224  may be derived by using measurements taken via the tool  114  (e.g., nuclear gamma-gamma measurement tool) using COMP channels. An initial guess image  226  may then be derived. As mentioned above, log squaring of ROBB data may be used to derive the initial guess image  226 . HZ inversion may then be applied to certain models, e.g.,  110 ,  120 , resulting in a final COMP image  228 . A fast-forward model may use pre-computed first and second order sensitivity functions to compute responses for log points and sectors, such as linear approximation and 2D flux sensitivity functions during inversion. Image  230  shows a percent error for the initial guess image  226 , while image  232  shows a percent error for the final COMP image  228 . As shown, image  232  is less noisy when compared to image  230 , depicting how inversion may provide for a more efficient and accurate measure of density properties ρ comp . 
         [0073]    Also shown is an initial measured image  234  measured using SS channels. Log squaring of ROBB data may additionally be used to derive an initial guess image  236 . HZ inversion may then be applied to certain model types, e.g.,  100 ,  120 , resulting in a final SS image  238 . A fast-forward model may use pre-computed first and second order sensitivity functions to compute responses for log points and sectors, such as linear approximation and 2D flux sensitivity functions during inversion. Image  240  shows a percent error for the initial guess image  236 , while image  242  shows a percent error for the final SS image  238 . As shown, image  242  is less noisy when compared to image  240 , depicting how HZ inversion may provide for a more efficient and accurate measure of density properties ρ ss . 
         [0074]    HZ inversion workflows (e.g., for relative dip between 88°-92°) may take into account non-crossed layers and may enable building a 2D curtain-section model  248  as superposition of 1D formation models defined in short segments (e.g., approximately between 1 and 3 ft) along the tool  114  trajectory  250 , with layer boundaries in each segment oriented parallel to the trajectory  250 , as shown in  FIG. 14 . More specifically,  FIG. 14  depicts an embodiment of the 2D curtain-section model  248  that may have been derived, for example, via images  252 ,  254 . Image  252  is an initially acquired image using COMP channels, while image  254  is a final image that includes borehole and shoulder-bed corrections. It is to be noted that in the depicted embodiment, the images  252 ,  254  include THL in excess of 300 feet while TVD varies from horizontal by less than 3 feet. Indeed, longer sections of formations, including longer sections with small horizontal variations, may be more accurately modeled with the techniques described herein. The 2D curtain-section model  248  may then be derived via superposition of small segments.  FIG. 14  additionally shows a borehole geometry  256  in the 2D curtain plane. 
         [0075]    Certain imaging, as depicted in  FIG. 15 , may be used to more accurately derive borehole geometry and stand-off. Because the techniques described herein do not assume a homogeneous formation but rather a variant formation, borehole effects may be decoupled from a final image. For example, image  260  may be an image captured via a density caliper tool. A section  262  may include sinusoidal artifacts  264 , whose locations correlate with layers in an initial image  266  (e.g., COMP image). As shown, the sinusoids are not present in the initial image  266 . Accordingly, a final image  268  shows a more accurate borehole geometry, for example, in section  270 , and other sections. 
         [0076]    HA and HZ workflows may be integrated, for example, as shown in  FIG. 16 . A sliding window approach may be used, where a sliding window  280  may move or otherwise “slide” through sections of an image to analyze the image sections. In the depicted embodiment, the HA workflow may be used to analyze a COMP image  282  at sections that clearly show sinusoids, such as sections  284  and  286 . A certain color, on the greyscale bar above, such as dark color  288  here, may be indicative of sinusoids, thus providing for identification of sinusoids based on color. These portions may be analyzed with the HA workflow. A middle portion  290  of the image  282  may not include many, if any sinusoids, and may include a preponderance of another color, such fair color  294  here. Accordingly, the middle portion  290  may be analyzed via the HZ workflow with smaller segments. A resultant borehole and shoulder-bed corrected image  296  may then include more accurate formation density properties and borehole geometries. A curtain-section model  298  may be derived from the borehole and shoulder-bed corrected image  296 . As shown, sections  300  and  302  may have been derived via the HA workflow, while middle portion  304  may have been derived via the HZ worflow. These sections  300 ,  302  on one side and  304  on the other side are processed separately with a different workflow. 
         [0077]      FIG. 16  also depicts in zoom detail a section  306 . The section  306  may show certain discontinuities because of the HZ workflow. To improve the HZ workflow used to derive portion  304 , certain techniques may be applied as follows. For speed improvements, Grid Services Lite (GSL) or other parallelization may be used to parallelize the previous serial inversion code. Parallelization may thus split inversion into multiple smaller workflows. Log squaring may be applied to true stratigraphic thickness (TST) space to improve accuracy. Inversion of bed locations may be done in measured depth (MD), instead of thickness in TVD. Sliding windows  280  may be adjusted, for example by increasing or decreasing window size, by overlapping windows  280  with lesser or greater overlap distances, and so on. Adjustments may be made based on inclination angle of the tool  114 , sinusoids, observed boundaries, and so on. For example, smaller windows  280  and/or greater overlap may be used to observe certain formation features in more detail. In certain embodiments, layers may be kept consistent along tool  114  trajectory, thus removing inconsistencies. 
         [0078]    The handling of transitions between neighboring density zones may also be improved. For example, a slope may be derived between density zones and used to insert a transition density layer if the slope exceeds a certain amount. Weighting may also be applied along MD. For example, data near or on a middle region of a layer may be given more weight as compared to data near boundary edges. Edge detection may be applied to derive boundary edges, and data closer to a middle of the layer may be giving more weight based on the edge detection. Accordingly, data that may be more accurate may be given greater weight during analysis. A modified model with HZ sections modified based on boundary positions, formation properties, and/or layering sequence may also be used, as described in more detail below. 
         [0079]      FIG. 17  depicts a serial workflow process  310  suitable for creating vizualizable models (e.g., curtain-section models) showing various layers, such as model  50  and layers  58 . The figure also depicts a parallel workflow or process  312 , equivalent to the serial process  310 , for comparison. These workflows include HA workflows that can be used for sections  300 ,  302  as depicted on  FIG. 16 . The parallel workflow  312  may in addition to processing different windows in parallel following, for example, image segmentation, include having multiple passes connecting different windows to build a more consistent formation model. It should be noted that the process  312  above may be applicable to both HA and HZ workflows. The processes  310  and  312  may be implemented as computer instructions executable via a processor of the tool  114  and may be stored in a memory of the tool  114 , and/or may be stored in memory  648  to be executed via processor  646 , or in other external systems to be stored and processed by the external systems. As depicted, the process  310  may preprocess tool  114  data, apply a log-squaring (e.g., ROBB  152  log squaring), execute a sinusoid extraction, and then provide for a trajectory segmentation to derive one or more segmented images, such as image  158  (block  314 ) as described in more detail in  FIG. 9 . Each window may be inverted to derive 1) borehole radius r bh , standoff S bot , and mud properties; 2) borehole geometry; and 3) formation (e.g., layers) and dip (block  316 ). The block  316  may execute twice (e.g., two refinements) for enhanced accuracy. The process  310  may begin at a bottom and then extend a top of the formation for the next window via a sliding window approach (block  318 ). If the windows have been processed (decision  320 ), the process  310  may then provide for a visualization (block  322 ) including a graphical view of the formation, such as the graphical model  50 . If not every window has been processed (decision  320 ), the process  310  may continue at block  316 . 
         [0080]    The parallel process  312  may process the windows in parallel. Accordingly, the windows may undergo pre-processing, log squaring, sinusoid extraction, and trajectory segmentation in parallel (block  324 ). The process  324  may then process each window in parallel to invert the window to derive 1) borehole radius r bh , standoff S bot , and mud properties; 2) borehole geometry; and 3) formation (e.g., layers) and dip (block  326 ). Process  312  may work from a middle section, rather than start at a bottom section like process  310 . Accordingly, a top of the formation may be extended from a middle (e.g., preceding) window having more accurate data (block  328 ), and a bottom of the formation may be extended from the middle (e.g., succeeding) window (block  330 ). The process  312  may then go through multiple passes or refinements, as desired. For example, 2, 3, 4, 5, 6, 7, or more passes may be desired. Accordingly, if the number of passes or refinements has not been reached (decision  332 ), the process  312  may iterate back to block  326 . Otherwise the process  312  may provide for a visualization (block  334 ) including a graphical view of the formation, such as the graphical model  50 . By incorporating parallelization system (with automated task scheduling and load balancing) such as GSL, the process  312  may more quickly derive the visualizations  334 , and may additionally improve accuracy. 
         [0081]    Horizontal wells may stay within a few layers over a long trajectory segment and may cross the same boundary multiple times, creating artifacts such as railroad tracks, bull&#39;s eyes and/or reverse bull&#39;s eye on borehole images. These artifacts cause difficulty in extraction of dip and geometry from borehole images, and therefore create difficulties for practical use of automated inversion-based workflows. The reasons for the difficulty include variation of geometry, and the curvature of boundaries and their non-planar shape, in addition to lateral changes in formation density. The serial workflow or process  310  handles these by reducing the size of the processing window (e.g., to less than 3 feet). However, this may cause lateral inconsistency in the resulting interpretation, e.g., as seen in section  306 . One way of handling structures with non-parallel boundaries is to identify a set of points in the image belonging to the same boundary, and then construct the boundary surface in 3D. The difficulty with this approach is in automating the process, as it currently uses manual (labor-intensive) interactions with the images and model. 
         [0082]      FIGS. 18-22  illustrate an HZ workflow that may be used for processing zone  304  of the formation. Of course, other workflows as described in relation to  FIG. 13-15  may also be used as HZ workflow. For example,  FIG. 18  is a parameterization,  FIG. 19  is a representation of applying pre-processing and the advanced front concept to the model building,  FIG. 20  is a flowchart of certain of the techniques used,  FIG. 21  is an extended 2D model parameterization, and  FIG. 22  is an example of results with geometry of formation model after points belonging to the same boundaries are identified and grouped. 
         [0083]      FIG. 18  illustrates embodiments of a new model parameterization  340 , based on interval processing in HZ wells. Within each segment (e.g., segment  160 ), the formation model  340  is described by formation boundaries  342 ,  344 ,  346 ,  348 ,  350  and layer properties shown in circled numbers. The formation boundaries can be any arbitrary curves, thus improving accuracy. The layer properties are assumed to be uniform within each layer. Formation layers (e.g., six layers are shown with circled numbers inside the layers) do not have to be strictly parallel to each other, and can be truncated on any boundary  342 ,  344 ,  346 ,  348 ,  350 . For non-uniform layers, sub-layers can be added to allow property variation within a layer. A tool trajectory  362  and a borehole geometry  364  are also shown. 
         [0084]    An enhanced pre-processing and formation boundary extractions for the HZ workflow will be described hereinafter. In the HZ sections, the approximate curtain-section model  340  is obtained after squaring the azimuthal compensated density image (COMP) over small intervals. Formation boundaries  342 ,  344 ,  346 ,  348 ,  350  are extracted by comparing the true vertical depth (TVD) of bed boundaries between neighboring intervals and the densities above and below each boundary  342 ,  344 ,  346 ,  348 ,  350 . A manual adjustment may be used to ensure boundary consistency. After local models are built, a topological model for the entire HZ interval is constructed. Each formation layer, defined by two or more boundaries, is assumed to have a unique density and photoelectric factor value. If desired, the layer can be further divided into sub-layers to add finer detail. The sub-layers are similar to main layers, allowing 2D structures. The initial structure can also be built from image contouring, used to construct boundaries from 3D contours. 
         [0085]    The automated process can be followed by an interactive one, driven by a graphical user interface, with options to adjust the boundaries, or apply some geometry-healing techniques. Once the boundaries are defined and the initial model is constructed, layer density and photoelectric factor are found in the inversion workflow. An enhanced pre-processing workflow may be defined as follows: 
         [0086]    1. Obtain curtain-section model from squaring azimuthal COMP data. 
         [0087]    2. Identify points defining a formation boundary by comparing boundary TVDs and density contrast. 
         [0088]    3. Filter out hanging points. 
         [0089]    4. Manually adjust where appropriate to ensure consistency. 
         [0090]    5. Sample formation boundaries within the predefined interval. 
         [0091]    6. Build up the topological model for the entire HZ interval. 
         [0092]    7. Each layer defined by two or more boundaries may be assumed to have unique properties. 
         [0093]    The enhanced pre-processing workflow may be implemented as computer instructions executable via a processor of the tool  114  and may be stored in a memory of the tool  114 , and/or may be stored in memory  648  to be executed via processor  646 , or in other external systems to be stored and processed by the external systems. The images correlative to the activities in the enhanced pre-process workflow are illustrated in  FIG. 19 . For example, an image  370  may be squared (e.g., squaring of azimuthal COMP data) to obtain a curtain-section model  372 . Points identifying a formation boundary may be identified, for example, by comparing boundary TVDs and density contrasts, as shown in more detail in section  374  of image  376 . Hanging points may additionally be filtered, and image  376  may be manually adjusted, if desired, to derive image  378 . Formation boundaries may be sampled within a predefined interval, as shown in image  380 . A final topological model for the HZ interval may then be build up, so that each layer defined by two or more boundaries is assumed to have unique properties, as depicted in image  382 . 
         [0094]    The parameterization used to derive model  340  (shown in  FIG. 18 ) may be further generalized to handle full 2D curtain-section geometry that may be used in HZ wells, as shown in  FIG. 21 . However, it is to be understood that the parameterization described can also be used for HA wells. Accordingly, an extended full 2D model parameterization is defined via polygonal shapes, and can be used to model any 2D shape, including lateral change of properties or cross-bedding in addition to the original model with curved boundaries. To further define and adjust this class of model a user friendly and intuitive interface may also be provided. More descriptions on the extended full 2D model parameterization is described below with reference to  FIG. 21 . 
         [0095]    The inverted parameters are layer and zone densities and photoelectric factors, for the defined model geometry. Standard Gauss-Newton optimization with line search adaptive regularization and parameter constraints is used. The cost function is the weighted L 2 -norm error between the measurements and the modeled responses over log points in each segment. An enhanced inversion workflow may include the following activities: 
         [0096]    1. Invert borehole average radius and mud properties using the short-spacing density (SS), density correction (DRHO) and apparent volumetric photoelectric factor (UAPP) channels for every sector over the entire interval (e.g., every log points). Cylindrical borehole and uniform mud properties are assumed. 
         [0097]    2. Invert for borehole geometry based on a staircase or harmonic geometry model using SS and DRHO channels for each sector and log points. 
         [0098]    3. Invert densities and photoelectric factors of formation layers and zones using long-spacing density (LS), COMP and UAPP channels, using sector data for the entire interval. 
         [0099]    4. Repeat activities numbered 1-3 above until parameters converge to a specified tolerance or until the specified maximum number of repetitions is reached. The inversion may be implemented as computer instructions executable via a processor of the tool  114  and may be stored in a memory of the tool  114 , and/or may be stored in memory  648  to be executed via processor  646 , or in other external systems to be stored and processed by the external systems. The inversion can also be run using a sliding window scheme to allow lateral property changes. It may be useful to depict a flowchart including embodiment of the activities above, as shown in  FIG. 20 . More specifically,  FIG. 20  shows an embodiment of an enhanced inversion workflow  400  that includes the enhanced pre-processing workflow, a boundary extraction, and an initial model derivation (block  402 ). The workflow  400  may be implemented as computer instructions executable via a processor of the tool  114  and may be stored in a memory of the tool  114 , and/or may be stored in memory  648  to be executed via processor  646 , or in other external systems to be stored and processed by the external systems. The model may then include the activities  1 ,  2  and  3  described above (block  404 ). If parameters do not converge to the specified tolerance or if the specified maximum number of repetitions is not reached (decision  406 ), the process  400  may iterate back to block  404 . If the parameters do converge to the specified tolerance or if the specified maximum number of repetitions is reached (decision  406 ), the process  400  may then provide a visualization and an error analysis (block  408 ). 
         [0100]      FIG. 21  depicts an embodiment of an extended full 2D model parameterization  390 . The extended full 2D model parameterization  410  may be derived by applying the activities above, including the enhanced pre-processing and/or process  400  to the same data used to generate the model  340  shown in  FIG. 18 . As illustrated, the extended full 2D model parameterization  410  is described by formation boundaries  342 ,  344 ,  346 ,  348 ,  350  and the topology of formation layers. There are six formation layers numbered with circular numbers shown, in which the second layer is further divided into five sub-layers (layers 2.1, 2.2, 2.3, 2.4, 2.5 show in dash dot lines) and the third layer is further divided into three sub-layers (layers 3.1, 3.2, 3.3 also shown in dash dot lines). The tool trajectory  362  and the borehole geometry  364  are also shown. As depicted model  410  may thus include certain sub-layers, for enhanced accuracy. 
         [0101]      FIG. 22  shows an embodiment of a curtain-section model  410  derived from using field data and the application of the enhanced techniques described herein, such as the enhanced pre-process modeling used to derive  FIG. 19  and/or the enhanced full 2D HZ model parameterization (e.g., process  400 ) used to derive  FIG. 21 . In the depicted embodiment, various boundaries  412 ,  414 ,  416 ,  418 ,  420 ,  422 ,  424 ,  426 ,  428  are displayed more accurately and efficiently, which may provide more accurate formation density measures in a more efficient manner. Accordingly, a geographical formation may be explored more accurately and in a faster manner. 
         [0102]      FIGS. 23-26  show embodiments of various curtain-section models that may be used to validate the techniques described herein via synthetic data. More specifically,  FIG. 23  depicts a curtain-section model  440  that may have been derived without using the techniques described herein. According, a variety of blocks or artifacts  442  may be included. An improved 1D curtain-section model  450  is depicted in  FIG. 24 . The model  450  includes 10 ft. window interval processing, and shows less artifacts  442 . Increasing processing to a 20 ft. window interval is shown in  FIG. 25 . A model  460 , as depicted in  FIG. 25 , includes less artifacts  442  than models  440 ,  450 .  FIG. 26  shows a model  470  at 40 ft. window intervals, with minimal or artifacts  442 . A model  480 , as depicted in  FIG. 27 , may apply the 2D techniques described herein. As shown, the model  480  substantially minimizes or eliminates artifacts  442  in the synthetic data when 2D techniques are applied, even by using larger window intervals. By providing for more accurate and faster workflows, including workflows  130 ,  310 ,  312 , and  400 , the techniques described herein may enable an improved geophysical analysis of a formation. 
         [0103]      FIGS. 28-31  show embodiments of various curtain-section models that may be used to validate the techniques described herein via field data as opposed to synthetic data. More specifically,  FIG. 28  depicts a model  480  derived without using the improved techniques described herein. The model  480  shows blocky boundaries and unwanted artifacts  482 . In comparison,  FIG. 29  shows a model  490  that may be derived using the same data as that used for model  480  but applying the improved techniques described herein, including workflows  130 ,  310 ,  312 , and  400 . As illustrated, the model  490  shows improved boundary visualizations and the elimination of the artifacts  482 .  FIG. 30  depicts a model  500  derived without using the improved techniques described herein. The model  500  also shows blocky boundaries and unwanted artifacts  482 , similar to the model  480 . In comparison,  FIG. 31  depicts a model  510  that may be derived using the same data as that used for model  500  but applying the improved techniques described herein, including workflows  130 ,  310 ,  312 , and  400 . As illustrated, the model  510  shows improved boundary visualizations and the elimination of the artifacts  482 . Indeed, field data validation, as shown in  FIGS. 27-30 , shows improvements in both accuracy as well as speed for the techniques described herein. 
         [0104]    The specific embodiments described above 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 be further understood that the claims are not intended to be limited to the particular forms disclosed, but rather to cover modifications, equivalents, and alternatives falling within the spirit and scope of this disclosure.