Methods of selecting an earth model from a plurality of earth models

A logging system includes an electromagnetic logging tool that collects measurements of a subterranean formation as the tool is conveyed along a borehole through the formation. The system further includes a processing system that: generates a first plurality of models of the formation based on at least one first measurement of the formation, the at least one first measurement collected by the tool at a first location of a plurality of locations located along a drilling axis; generates a second plurality of models of the formation based on at least one second measurement of the formation, the at least one second measurement collected by the tool at a second location of the plurality of locations, wherein the second location is adjacent the first location; and selects a model of the first plurality of models based on a spatial continuity of the model with respect to the second plurality of models.

CROSS-REFERENCE TO RELATED APPLICATION

This application is the National Stage of, and therefore claims the benefit of, International Application No. PCT/US2016/018009 filed on Feb. 16, 2016, entitled “METHODS OF SELECTING AN EARTH MODEL FROM A PLURALITY OF EARTH MODELS,” which was published in English under International Publication Number WO 2017/142508 on Aug. 24, 2017. The above application is commonly assigned with this National Stage application and is incorporated herein by reference in its entirety.

BACKGROUND

In the field of well drilling and logging, resistivity logging tools are used to provide an indication of the electrical resistivity of rock formations surrounding an earth borehole. Such information regarding resistivity is useful in ascertaining the presence or absence of fluids, such as hydrocarbons. A typical electromagnetic propagation resistivity logging tool includes at least one transmitting antenna and multiple receiving antennas located at different distances from the transmitting antenna along the axis of the tool.

The transmitting antenna is used to generate electromagnetic fields in the surrounding formation. In turn, the electromagnetic fields in the formation induce a voltage in each receiving antenna. The response of the formation is converted into a set of inversion parameters, which are then used to estimate the anisotropic properties of the formation.

Inversion can be performed on a point-by-point basis during logging of a borehole. At each of two or more points (or locations) along a length (or stretch) of the borehole, the process described in the above paragraph is performed. A whole-space or “zero-dimensional” or “0D” inversion takes into account the tool measurements at only a single point and ignores formation heterogeneity. To deal with formation heterogeneity, such as shoulder effects from formation layer boundaries, a layered earth or “one-dimensional” or “1D” inversion takes into account the tool measurements from at least a single point to find a layered formation model that matches those measurements. Accordingly, a 1D inversion can be used to determine the locations of boundaries between formation layers.

DETAILED DESCRIPTION

Disclosed herein are methods and systems for quantitatively interpreting logging-while-drilling (LWD) data (e.g., resistivity LWD data). Particular embodiments relate to selecting a formation model from two or more generated formation models. In at least some embodiments, a method includes generating a first plurality of models of the subterranean formation based on at least one first measurement of the subterranean formation. The at least one first measurement corresponds to a first location of a plurality of locations located along a drilling axis. The method also includes generating a second plurality of models of the subterranean formation based on at least one second measurement of the subterranean formation. The at least one second measurement corresponds to a second location of the plurality of locations. The second location is adjacent to the first location. The method also includes selecting a model of the first plurality of models based on a spatial continuity of the model with respect to the second plurality of models.

A related system includes an electromagnetic logging tool that collects measurements of a subterranean formation as the tool is conveyed along a borehole through the subterranean formation. The logging system further includes a processing system that generates a first plurality of models of the subterranean formation based on at least one first measurement of the subterranean formation. The at least one first measurement is collected by the electromagnetic logging tool at a first location of a plurality of locations located along a drilling axis. The processing system also generates a second plurality of models of the subterranean formation based on at least one second measurement of the subterranean formation. The at least one second measurement is collected by the electromagnetic logging tool at a second location of the plurality of locations. The second location is adjacent to the first location. The processing system selects a model of the first plurality of models based on a spatial continuity of the model with respect to the second plurality of models.

An illustrative LWD environment is shown inFIG. 1. A drilling platform102is equipped with a derrick104that supports a hoist106for raising and lowering a drill string108. The hoist106suspends a top drive110that is used to rotate the drill string108and to lower the drill string through the well head112. Sections of the drill string108are connected by threaded connectors107. Connected to the lower end of the drill string108is a drill bit114. Rotation of bit114creates a borehole120that passes through various formations121. A pump116circulates drilling fluid through a supply pipe118to top drive110, downhole through the interior of drill string108, through orifices in drill bit114, back to the surface via the annulus around the drill string, and into a retention pit124. The drilling fluid transports cuttings from the borehole120into the pit124and aids in maintaining the integrity of the borehole.

A logging tool126is integrated into the bottom-hole assembly near the bit114. The logging tool126may take the form of a drill collar, e.g., a thick-walled tubular that provides weight and rigidity to aid the drilling process. In at least one embodiment, the logging tool126is an electromagnetic resistivity LWD tool. For example, the logging tool126may be an Azimuthal Deep Resistivity (ADR) service offered by Halliburton Energy Services, Inc., operating in a rotating (drilling) mode. As the bit114extends the borehole120through the formations121, the logging tool126collects measurements relating to various formation properties as well as the tool orientation and position and various other drilling conditions.

In wells employing mud pulse telemetry for LWD, downhole sensors (including resistivity logging tool126) are coupled to a telemetry module128including a mud pulse telemetry transmitter that transmits telemetry signals in the form of pressure variations in the tubing wall of drill string108. A mud pulse telemetry receiver array130(including, e.g., one or more pressure transducers) may be coupled to tubing below the top drive110to receive transmitted telemetry signals. Other telemetry techniques can be employed including acoustic telemetry (using, e.g., one or more repeater modules132, to receive and retransmit telemetry signals), electromagnetic telemetry, and wired drill pipe telemetry. Many telemetry techniques also offer the ability to transfer commands from the surface to the tool, thereby enabling adjustment of the tool's configuration and operating parameters. In at least some embodiments, the telemetry module128additionally, or alternatively, stores measurements for later retrieval when the tool returns to the surface.

A computer system (or processing system)140collects measurements from the logging tool126(e.g., via the receiver array130), and includes computing facilities for processing and storing the measurements gathered by the logging tool. In at least some embodiments, the computer system140includes a processor142that performs formation modeling analysis operations by executing software or instructions obtained from a local or remote non-transitory computer-readable medium148. The processor142may be, for example, a general purpose microprocessor, a microcontroller, a digital signal processor, an application specific integrated circuit, a field programmable gate array, a programmable logic device, a controller, a state machine, a gated logic, discrete hardware components, an artificial neural network, or any like suitable entity that can perform calculations or other manipulations of data. In at least some embodiments, computer hardware can further include elements such as, for example, a memory (e.g., random access memory (RAM), flash memory, read only memory (ROM), programmable read only memory (PROM), erasable read only memory (EPROM)), registers, hard disks, removable disks, CD-ROMS, DVDs, or any other like suitable storage device or medium. The computer system140also may include input device(s)146(e.g., a keyboard, mouse, touchpad, etc.) and output device(s)144(e.g., a monitor, printer, etc.). Such input device(s)146and/or output device(s)144provide a user interface that enables an operator to interact with the logging tool126and/or software executed by the processor142. For example, the computer system140may enable an operator to select resistivity analysis options, to view collected resistivity data, to view resistivity analysis results, and/or to perform other tasks.

FIG. 2is a block diagram illustrating a resistivity inversion according to an embodiment. The inversion may be a distance-to-bed-boundary (DTBB) inversion for analysis and interpretation. In this situation, a position of a well logging instrument (e.g., logging tool126) with respect to a bed boundary (e.g., a formation layer discontinuity) is determined by inversion processing.

An initial formation model (or earth model)202is used. The initial formation model202carries an initial estimate of the geometry and/or properties of the earth formations (e.g., formations121ofFIG. 1) surrounding a wellbore in which the well logging instrument is positioned. For example, the initial formation model202may be characterized by particular layer boundaries and/or particular isotropic or anisotropic values (e.g., resistivity values). Electromagnetic (EM) attributes of the initial formation model202may include resistivity, conductivity, permittivity, permeability, chargeability, and/or other induced polarization (IP) parameters. The EM attributes may be isotropic or anisotropic. A layer dip may be recovered from the orientation of the well logging instrument with respect to the 1D resistivity model.

The initial formation model202may represent the earth formations surrounding the wellbore as a series of layers or strata, demarcated by boundaries between contiguous layers. In the model202, physical properties of the individual layers in the model may include, e.g., resistivity (or conductivity) of each layer, a thickness of each layer, and a selected number of layers above and/or below a layer of interest. In at least some situations, the layer of interest is the layer in which the well logging instrument is positioned in the wellbore.

To refine the model202, measurement data204collected by the well logging instrument is input to the model. The measurement data204reflects a response of the earth formations to transmissions by the well logging instrument. For example, the measurement data204may include measured resistivity LWD data. According to at least some embodiments, other information is input to the model202. The additional information may include a priori geological information206, such as surfaces interpreted from seismic analysis (e.g., 3D seismics), well ties, and/or adjacent wells. According to at least some embodiments, the information206regards the model202as derived from interpretation and/or analysis of prior EM surveys (e.g., marine controlled-source EM surveys, borehole-to-surface EM surveys, cross-well EM surveys). Although the resolution of such information may be lower than the resolution of well logs, such information may still provide useful information regarding general structural trends. In general, the information206may be imposed on the model202(e.g., in a selective manner) as data weights, model weights, regularization, model constraints and/or a priori models.

Based on the measurement data204(and, in at least some embodiments, the a priori information206), a predicted formation model208is generated. The predicted formation model208provides a predicted response of the earth formations. The predicted response is converted into a set of inversion parameters, which are then used to estimate (or predict) data210of the formations. For example, the estimated data210may include resistivity characteristics of the formations.

The resistivity LWD inversion may be based on one or more stochastic optimization algorithms including, e.g., Monte Carlo (MC), Markov Chain Monte Carlo (MCMC), Nearest Neighbor (NN), Genetic Algorithm (GA), or Simulated Annealing (SA) algorithms. Stochastic optimization algorithms extensively search the solution space for global minima and provide statistical information about the earth model parameters. These algorithms are essentially “physics free,” in that models are guided on the basis of statistics only, and are not guided by any model sensitivity analysis.

As another example, the resistivity LWD inversion may be based on one or more deterministic optimization algorithms including, but not limited to, Conjugate Gradient (CG), Non-linear Conjugate Gradient (NLCG), and Gauss-Newton (GN) algorithms. Deterministic optimization algorithms are “physics based,” in that models are guided by model sensitivity analysis. Deterministic optimization algorithms may also yield statistical information about the earth model parameters. However, such algorithms are dependent upon their initial models, and may converge upon local and not global minima.

With continued reference toFIG. 2, at block212, the estimated data210are compared against the measurement data204. As described earlier, the measurement data204reflect the measured response of the earth formations. A difference(s) between the estimated data210and the measured response204is referred to as a misfit. At block212, the difference(s) is compared against a particular threshold(s) (e.g., a preselected threshold(s)). In at least some embodiments, the value of the threshold corresponds to a level of noise that is present in the measurement data204. If it is determined that the misfit is below the threshold, then the predicted model208is used (or adopted) as a final predicted model214.

However, if it is determined that the misfit is equal to or above the threshold, then one or more parameters of the predicted model208are adjusted. For example, a level of the misfit is used to update (or adjust) parameters of the predicted model208, such that adjustments216to the predicted model208are generated. The predicted model208is updated accordingly. The updated model208provides a predicted response of the earth formations. The predicted response is converted into a set of inversion parameters, which are then used to estimate data210of the formations. The estimated data210is then compared against the measurement data204. As illustrated inFIG. 2, the described adjustment of block216and comparison of block212are repeated, until the misfit is below the threshold.

The resistivity inversion illustrated inFIG. 2may be performed on a “point by point” basis. In more detail, in the wellbore, the well logging instrument may measure data at (or around) two or more locations located in the wellbore (e.g., along a drilling axis). For each of the locations in the wellbore, the resistivity inversion ofFIG. 2is performed using the data measured at the location. Further, for each location, a predicted model (e.g., model214) providing estimated data that is sufficiently close to the measured data is determined. Accordingly, for a particular number of locations, an equal number of final predicted models214are generated. The final models may be 1D resistivity models. These 1D resistivity models may then be stitched together to form a 2D resistivity image of the formation. This 2D image is commonly referred to as a “curtain plot.”

According to the block diagram ofFIG. 2, a single initial model202is considered. More specifically, only a single initial model is used (and perhaps adjusted) for each logging point, resulting in a single predicted model for the logging point. According to a further example, two or more initial models are considered for each logging point. In this situation, two or more resistivity inversions are performed independently of one another. Accordingly, two or more predicted models are generated for each logging point.

FIG. 3is a block diagram showing selection of a formation model from among multiple generated formation models. As illustrated inFIG. 3, two or more initial models302-1. . .302-N are considered. The initial models302-1. . .302-N are different from each other. For example, each of the initial models302-1. . .302-N may reflect a different combination of, e.g., resistivity models, tool placement with respect to layers of the model, and/or predicted model parameters defined from apparent logs or other a priori information. In other aspects, each of the models302-1. . .302-N is similar to model202ofFIG. 2, and therefore, will not be described in further detail below.

Based on the initial models302-1, . . . ,302-N, respectively, final predicted models314-1, . . . ,314-N are generated. For example, just as blocks208,210,212,216are performed to generate the final predicted model214ofFIG. 2, blocks308-1,310-1,312-1,316-1are performed to generate a final predicted model314-1. Similarly, blocks308-N,310-N,312-N,316-N are performed to generate a final predicted model314-N. Generation of each of the final predicted models314-1, . . . ,316-N may include performing a resistivity inversion. The resistivity inversion may be based on a stochastic optimization algorithm and/or a deterministic optimization algorithm.

Each of the final predicted models314-1, . . . ,316-N is characterized by a corresponding misfit. The final predicted models314-1, . . . ,316-N are said to be equivalent (or non-unique), in that the respective misfits that characterize the models all fall below a particular threshold (e.g., the threshold corresponding to blocks312-1, . . . ,312-N). The threshold may correspond to the level of noise that is present in the measurement data204. The equivalency of the models may be due to, e.g., a lack of sufficient measurement sensitivity, modeling errors, and/or noise in the data.

When analysis of a formation results produces multiple models that are equivalent, one of the models may be selected as being optimal (e.g., better than the remaining models in at least one aspect, such as geological accuracy). In the embodiment ofFIG. 3, the particular values of the misfits are utilized to select an optimal model. At block318, the values of the misfits that characterize the models314-1, . . . ,316-N are analyzed. The model that has the lowest misfit is selected as the optimal predicted model320.

Similar to the resistivity inversion illustrated inFIG. 2, the selection illustrated inFIG. 3may also be performed on a “point by point” basis. Accordingly, for a number of multiple locations, an equal number of optimal predicted models320are generated. These 1D resistivity models may then be stitched together to form a 2D resistivity image (or “curtain plot”) of the formation.

When the underlying models (e.g., model320) are chosen strictly based on a degree of misfit, the resulting 2D resistivity images may contain artefacts that are geologically unrealistic. This may occur, e.g., because the selection of block318does not consider a degree of spatial continuity of the models (e.g., models314-1, . . . ,314-N) with respect to at least one different location (or point) in the wellbore. When 2D resistivity images contain such artefacts, the images may have little (or insufficient) resemblance to actual earth models. The appearance of a large number of artefacts erodes confidence in the quality of the modeling results. For example, a large number of artefacts in a well log erodes an interpreter's confidence in the resistivity LWD inversion.

For each of the locations, the model characterized by the lowest misfit is selected as the optimal predicted model. This is similar to the situation described earlier with respect toFIG. 3, in which the model320is selected from models314-1, . . . ,314-N. As illustrated inFIG. 4, for locations 1, 2, 3, 4, . . . , Y, the models402-2,404-1,406-3,408-1, . . . ,410-X are selected, respectively.

For example, for location 2, the model404-1is selected from the models404-1,404-2,404-3, . . . ,404-X because the model404-1is characterized by a misfit that is lower than the misfits that characterize the remaining models (models404-2,404-3, . . . ,404-X). In this situation, the selection of the model404-1does not take into consideration a spatial continuity of the models404-1,404-2,404-3, . . . ,404-X with respect to models generated for at least one other location. For example, the model404-1is selected without considering any of the models generated for adjacent location 1 (models402-1,402-2,402-3, . . . ,402-X). Also for example, the model404-1is selected without considering any of the models generated for adjacent location 3 (models406-1,406-2,406-3, . . . ,406-X).

The above-described nature of the selection ofFIG. 3may increase the likelihood that a resulting 2D resistivity image will contain geologically unrealistic artefacts. For example, with respect to location 2, such a likelihood may be increased, e.g., if unselected model404-2bears a high degree of similarity to one or more models generated for location 3 (e.g., model406-3). In this situation, a degree of spatial smoothness between unselected model404-2and model406-3is likely stronger than degrees of spatial smoothness between selected model404-1and any of models406-1,406-2,406-3, . . . ,406-X. Accordingly, the selection of model404-2as an optimal model for location 2 may be preferable over the selection of model404-1.

In at least some situations, properties of a formation are generally continuous. For example, the lithological interfaces and physical properties of sedimentary formations generally exhibit lateral variations that vary slowly. This may be particularly true in the case of a horizontal well that extends generally parallel to the formation boundaries. Accordingly, for a particular location, selecting an optimal model without considering earth models generated for at least one other location (e.g., an adjacent or nearby location) may lead to an increased number of artefacts in a resulting 2D resistivity image.

As will be described in further detail below, according to various aspects of the disclosure, two or more equivalent earth models are generated (e.g., based on resistivity LWD inversions) for a particular location in a borehole. The generated models are analyzed based on at least one metric in order to select one of the models as an optimal model. According to particular aspects, the metric is based on spatial coherency (e.g., of resistivity LWD data), and the observation that that most earth formations are continuous yet smoothly varying (due to, e.g., the notion that the earth is generally continuous along the lateral direction). The metric may include a degree of spatial continuity (e.g., spatial smoothness) of the generated models with respect to models that are generated for at least one other location (e.g., an adjacent or nearby location) in the borehole. Accordingly, from among the equivalent earth models generated for the particular location, one model is selected based at least on a determination that the model has a sufficiently high degree of spatial continuity with respect to at least one model generated for a different location. This increases the likelihood that the selected model is more geologically reasonable (or plausible) than unselected models. Accordingly, the likelihood that the selected model is more geologically accurate than other model(s) (e.g., any or at least one of the unselected models) is improved, relative to selecting a model based on misfit alone (e.g., as described earlier with reference toFIG. 4).

FIG. 5is a block diagram showing selection of a formation model from among multiple generated formation models according to an embodiment. As described earlier with reference toFIG. 3, based on the initial models302-1, . . . ,302-N, final predicted models314-1, . . . ,314-N are generated, respectively, for a particular location in a borehole. The final predicted models314-1, . . . ,314-N are equivalent in that the respective misfits that characterize the models all fall below a particular threshold. At block522, one of the models314-1, . . . ,314-N is selected based on, at least, a constraint relating to spatial smoothness. The selection of block522uses information524regarding one or more models that are generated for an adjacent location. According to particular embodiments, one of the models314-1, . . . ,314-N is selected as an optimal model520based on a degree of spatial continuity of one or more parameters of the model with respect to the one or more models corresponding to information524. Unlike the comparison of block318ofFIG. 3, the selection522is based not merely on misfits that characterize the models314-1, . . . ,314-N.

For a particular model (e.g., any of models314-1, . . . ,314-N), a vector m may denote one or more parameters of the model. For another model (e.g., any model corresponding to information524), a second vector momay denote one or more corresponding parameters of this other model.

According to at least one embodiment, the selection at block522involves comparing m and mo(e.g., computing a difference between m and mo). This comparison may be performed for each of models314-1, . . . ,314-N. Accordingly, for a given one of the models314-1, . . . ,314-N, the comparison is performed with respect to one or more models generated for the adjacent location. Therefore, for the models314-1, . . . ,314-N, at least N comparison results are obtained.

According to a further embodiment, the selection at block522involves normalizing the N or more comparison results, based on one of the comparison results. For example, if differences between m and moare determined, then the differences may be normalized based on a model that results in a largest difference M. As captured in Expression (1), a minimum normalized difference is identified:
∥m−mo∥M2→min.  (1)

From among the models314-1, . . . ,314-N, the model that results in the minimum normalized difference is selected as the optimal model520.

The vector m may denote EM attributes of the earth models314-1, . . . ,314-N. As described earlier, these EM attributes may include resistivity, conductivity, permittivity, permeability, chargeability, and/or other induced polarization (IP) parameters. The EM attributes may be either isotropic or anisotropic. According to further embodiments, the vector m may, additionally or alternatively, denote one or more other attributes. For example, these other attributes may include the depth to a layer boundary, the depths to each of a pair of layer boundaries, and/or the dip of the boundaries.

According to at least one embodiment, the vector m may denote a function of one or more attributes (e.g., one or more of the attributes that are noted above). For example, the vector m may denote the thickness of a layer, as derived from the difference between the depths to each of a pair of two boundaries. As a further example, the vector m may denote the resistivity (conductivity) of a layer, as derived from the product of the resistivity and the anisotropy coefficient. Also, the vector m may denote both the thickness and the resistivity, and potentially one or more other attributes.

As described earlier with reference to Expression (1), differences between m and moare computed. According to other examples, other formulations may be determined. For example, a normalized difference between a first derivative of m and a first derivative of momay be determined. As captured in Expression (2), a minimum normalized difference is identified:
∥∇m−∇mo∥M2→min.  (2)

From among the models314-1, . . . ,314-N, the model that results in the minimum normalized difference (e.g., in a minimum rate of change of the underlying attribute) is selected as the optimal model520.

According to another example, a normalized difference between a second derivative (e.g., Laplacian) of m and a second derivative of momay be determined. As captured in Expression (3), a minimum normalized difference is identified:
∥∇2m−∇2mo∥M2→min.  (3)

From among the models314-1, . . . ,314-N, the model that results in the minimum normalized difference (e.g., in a minimum rate of change of the rate of change of the underlying attribute) is selected as the optimal model520.

According to another example, a combination (e.g., a linear combination) of formations that are captured in Expressions (1), (2), and/or (3) is considered. For example, a sum of (i) a normalized difference between m and mo, (ii) a normalized difference between a first derivative of m and a first derivative of mo, and (iii) a normalized difference between a second derivative of m and a second derivative of momay be determined. As captured in Expression (4), a minimum sum is identified:
∥m−mo∥M2+β∥∇m−∇o∥M2+γ∥∇2m−∇2mo∥M2→min.  (4)

In Expression (4) and other expression(s) presented in this disclosure, β and γ denote non-negative scalar parameters that provide balance (or bias) between the normalized values ∥m−mo∥M2, ∥∇m−∇o∥M2and ∥∇2m−∇2mo∥M2.

As described earlier with reference to Expressions (1), (2), (3) and (4), a minimum result based on a parameter of a model is identified. According to further embodiments, the minimum result may be further based on an indicator of inversion quality (e.g., misfit, signal-to-noise ratio, importance). For example, the result may be based on a value of a misfit ϕ. As described earlier with reference to blocks212and312ofFIGS. 2 and 3, respectively, a misfit refers to a difference(s) between estimated data (e.g., estimated data210ofFIG. 2) and a measured response (e.g., measured response204ofFIG. 2).

For example, a minimum result based on a combination (e.g., linear combination) of the misfit ϕ and formulations that appear in Equations (1), (2), (3) and/or (4) may be identified. For example, a sum of: (i) the misfit ϕ and (ii) a normalized difference between m and momay be determined. As captured in Expression (5), a minimum sum is identified:
ϕ+α∥m−mo∥M2→min.  (5)

From among the models314-1, . . . ,314-N, the model that results in the minimum result is selected as the optimal model520.

In Expression (5) and other expression(s) presented in this disclosure, α denotes a scalar quantity that provides a balance (or bias) between the misfit ϕ and the noted formulations. According to particular embodiments, α denotes a noise-to-signal ratio (NSR) for the measured data (e.g., measured data204ofFIG. 2).

According to another example, a sum of (i) the misfit ϕ and (ii) a normalized difference between a first derivative of m and a first derivative of momay be determined. As captured in Expression (6), a minimum sum is identified:
ϕ+α∥∇m−∇mo∥M2→min.  (6)

From among the models314-1, . . . ,314-N, the model that results in the minimum result is selected as the optimal model520.

According to another example, a sum of (i) the misfit ϕ and (ii) a normalized difference between a second derivative (e.g., Laplacian) of m and a second derivative of momay be determined. As captured in Expression (7), a minimum sum is identified:
ϕ+α∥∇2m−∇2mo∥M2→min.  (7)

From among the models314-1, . . . ,314-N, the model that results in the minimum result is selected as the optimal model520.

As another example, a sum of (i) the misfit ϕ, (ii) a normalized difference between m and mo, (iii) a normalized difference between a first derivative of m and a first derivative of mo, and (iv) a normalized difference between a second derivative of m and a second derivative of momay be determined. As captured in Expression (8), a minimum sum is identified:
ϕ+α∥m−mo∥M2+β∥∇m−∇mo∥M2+γ∥∇2m−∇2mo∥M2→min.  (8)

From among the models314-1, . . . ,314-N, the model that results in the minimum result is selected as the optimal model520.

In Expressions (1) to (8), the spatial formulations effectively serves as filters upon the earth models (e.g., models314-1, . . . ,314-N). These filters may be selected to act upon the earth models based on a scale length that is typical of the geological formation (e.g., 5 to 10 feet) rather than a scale length corresponding to a distance between measurement points (e.g., 0.5 feet).

The parameters of the models in Expressions (1) to (8) may be weighted by a model weighting matrix Wm, as captured in Expression (9):
ϕ+α∥Wm(m−mo)∥M2→min,  (9)

Elements of the model weighting matrix Wmprovide spatial weighting for directionality, e.g., a dip known a priori based on seismic interpretation or borehole imaging.

According to another embodiment, the equivalent earth models (e.g., models314-1, . . . ,314-N) are assembled in a 2D (pixel) resistivity model in coordinates of measured depth (MD) and true vertical depth (TVD), e.g., ρ(MD, TVD), where ρ denotes resistivity, and (MD, TVD) denote coordinates of a trajectory. In this situation, a degree of spatial continuity may be evaluated on the basis of Expression (10) below:

Here MDjfor j=1, . . . , T denotes the MD point in processed interval where MD1and MDTspecify the minimal and maximal MD range that the models are supposed to be continuous.

To solve Expression (10), the derivative of the resistivity with respect to measured depth may be approximated with a finite-difference approximation from the 2D resistivity model, as captured in Expression (11) below:

Alternatively—to solve Expression (10), the 2D resistivity model may be fit to a spatially continuous 1D function with respect to measured depth (e.g., with a cubic spline) such that the derivatives of the spatially continuous 1D function with respect to measured depth may be analytically evaluated, and summed per Expression (10).

FIG. 6illustrates a scenario in which the selections ofFIG. 5are performed for multiple locations. For example, as described earlier with reference toFIG. 4, one of the models generated for location 2 (e.g., models404-1,404-2,404-3, . . . ,404-X) is selected as an optimal model.

For each of the locations, the model that meets a particular constraint (e.g., the constraint of block522) is selected as the optimal predicted model. As illustrated inFIG. 6, for locations 1, 2, 3, 4, . . . , Y, the models402-1,404-2,406-3,408-3, . . . ,410-2are selected, respectively.

For example, for location 2, the model404-2is selected from the models404-1,404-2,404-3, . . . ,404-X. The selection of the model404-2considers at least a spatial continuity of the model with respect to models generated for at least one other location. For example, in selecting the model404-2, the models generated for adjacent location 1 (models402-1,402-2,402-3, . . . ,402-X) are considered. Alternatively, or addition, in selecting the model404-2, the models generated for adjacent location 3 (models406-1,406-2,406-3, . . . ,406-X) are considered. Alternatively, or in addition, the models generated for one or more other adjacent locations (e.g., location 4) may be considered. To account for factors relating to directionality, parameters of such models may be weighted (e.g., by the model weighting matrix Wmof Expression (9)).

The nature of the above-described selection may decrease the likelihood that a resulting 2D resistivity image will contain geologically unrealistic artefacts. For example, with respect to location 2, such a likelihood may be decreased, e.g., if the selected model404-2bears a high degree of similarity to one or more models generated for location 1 (e.g., model402-1). Accordingly, a degree of spatial smoothness between selected model404-2and model402-1is stronger than degrees of spatial smoothness between an unselected model (e.g., any of models404-1,404-3, . . . ,404-X) and any of models406-1,406-2,406-3, . . . ,406-X. As a further or additional example, with respect to location 2, such a likelihood may be decreased, e.g., if the selected model404-2bears a high degree of similarity to one or more models generated for location 3 (e.g., model406-3). Accordingly, a degree of spatial smoothness between selected model404-2and model406-3is stronger than degrees of spatial smoothness between an unselected model (e.g., any of models404-1,404-3, . . . ,404-X) and any of models406-1,406-2,406-4, . . . ,406-X. In the noted regards, the selection of model404-2as an optimal model for location 2 is preferable over the selection of any of models404-1,404-3, . . . ,404-X.

FIG. 7is a flowchart showing an illustrative selection method700employing LWD measurements. At block702, a first plurality of models of the subterranean formation (e.g., models314-1, . . . ,314-N) are generated based on at least one first measurement of the subterranean formation. The at least one first measurement corresponds to a first location (e.g., location 2 ofFIG. 6) of a plurality of locations located along a drilling axis. At block704, a second plurality of models of the subterranean formation (e.g., models314-1, . . . ,314-N) are generated based on at least one second measurement of the subterranean formation. The at least one second measurement corresponds to a second location (e.g., location 1 ofFIG. 6) of the plurality of locations. The second location is adjacent to the first location. At block706, a model (e.g., a particular model) of the first plurality of models is selected based on a spatial continuity of the model with respect to the second plurality of models. For example, the selection is based on a spatial constraint described earlier with reference to the selection of block522ofFIG. 5.

According to a further embodiment, at block708, a third plurality of models (or at least a third plurality of models) of the subterranean formation (e.g., models314-1, . . . ,314-N) are generated based on at least one third measurement of the subterranean formation. The at least one third measurement corresponding to a third location (e.g., location 3 ofFIG. 6) of the plurality of locations. The third location is adjacent to the first location and/or the second location. At block710, the selection of the model (from among the first plurality of models) is further based on a spatial continuity of the model with respect to the third plurality of models.

A: A logging system includes an electromagnetic logging tool that collects measurements of a subterranean formation as the tool is conveyed along a borehole through the subterranean formation. The logging system further includes a processing system that: generates a first plurality of models of the subterranean formation based on at least one first measurement of the subterranean formation, the at least one first measurement collected by the electromagnetic logging tool at a first location of a plurality of locations located along a drilling axis; generates a second plurality of models of the subterranean formation based on at least one second measurement of the subterranean formation, the at least one second measurement collected by the electromagnetic logging tool at a second location of the plurality of locations, wherein the second location is adjacent to the first location; and selects a model of the first plurality of models based on a spatial continuity of the model with respect to the second plurality of models.

B. A method of modeling a subterranean formation includes generating a first plurality of models of the subterranean formation based on at least one first measurement of the subterranean formation, the at least one first measurement corresponding to a first location of a plurality of locations located along a drilling axis. The method also includes generating a second plurality of models of the subterranean formation based on at least one second measurement of the subterranean formation, the at least one second measurement corresponding to a second location of the plurality of locations, wherein the second location is adjacent to the first location. The method also includes selecting a model of the first plurality of models based on a spatial continuity of the model with respect to the second plurality of models.

Each of the embodiments, A and B, may have one or more of the following additional elements in any combination. Element 1: wherein: a misfit of each model of the first plurality of models is less than a particular threshold value, the misfit based on a difference between the at least one first measurement of the subterranean formation and a predicted measurement based on the corresponding model; and the selection based on the spatial continuity improves a likelihood that the selected model is more geologically accurate than at least one other of the first plurality of models, relative to selecting based on the misfit alone. Element 2: wherein the selected threshold is based on a noise level corresponding to the at least one first measurement. Element 3: wherein a measure of the spatial continuity is determined based at least in part on a difference between a first parameter associated with the at least one first measurement, and a second parameter associated with the at least one second measurement. Element 4: wherein the second position is adjacent to the first position along the drilling axis. Element 5: wherein: the at least one first measurement comprises a first resistivity measurement; and the at least one second measurement comprises a second resistivity measurement. Element 6: wherein a measure of the spatial continuity is determined based at least in part on a difference between the first resistivity measurement and the second resistivity measurement. Element 7: wherein: the processing system generates the first plurality of models by performing a plurality of resistivity inversions based on the at least one first measurement; and the processing system generates the second plurality of models by performing a plurality of resistivity inversions based on the at least one second measurement. Element 8: wherein, among the first plurality of models, the selected model has a highest degree of spatial continuity with respect to the second plurality of models. Element 9: wherein: the processing system further generates at least a third plurality of models of the subterranean formation based on at least one third measurement of the subterranean formation, the at least one third measurement corresponding to a third location of the plurality of locations, wherein the third location is adjacent to the first location, wherein the selection of the model is further based on a spatial continuity of the model with respect to the at least a third plurality of models.

Element 10: wherein: a misfit of each model of the first plurality of models is less than a particular threshold value, the misfit based on a difference between the at least one first measurement of the subterranean formation and a predicted measurement based on the corresponding model; and the selection based on the spatial continuity improves a likelihood that the selected model is more geologically accurate than at least one other of the first plurality of models, relative to selecting based on the misfit alone. Element 11: wherein the selected threshold is based on a noise level corresponding to the at least one first measurement. Element 12: wherein a measure of the spatial continuity is determined based at least in part on a difference between a first parameter associated with the at least one first measurement, and a second parameter associated with the at least one second measurement. Element 13: wherein the second position is adjacent to the first position along the drilling axis. Element 14: wherein: the at least one first measurement comprises a first resistivity measurement; and the at least one second measurement comprises a second resistivity measurement. Element 15: wherein a measure of the spatial continuity is determined based at least in part on a difference between the first resistivity measurement and the second resistivity measurement. Element 16: wherein: generating the first plurality of models comprises performing a plurality of resistivity inversions based on the at least one first measurement; and generating the second plurality of models comprises performing a plurality of resistivity inversions based on the at least one second measurement. Element 17: wherein, among the first plurality of models, the selected model has a highest degree of spatial continuity with respect to the second plurality of models. Element 18: generating at least a third plurality of models of the subterranean formation based on at least one third measurement of the subterranean formation, the at least one third measurement corresponding to a third location of the plurality of locations, wherein the third location is adjacent to the first location, wherein the selection of the model is further based on a spatial continuity of the model with respect to the at least a third plurality of models.

Numerous variations and modifications will become apparent to those skilled in the art once the above disclosure is fully appreciated. The methods and systems can be used for drilling, logging and/or other operations where a particular formation model is to be selected from two or more formation models (e.g., equivalent formation models). The ensuing claims are intended to cover such variations where applicable.