WO2016089835A1

WO2016089835A1 - Spatial declustering of oilfield data using kernel density estimation

Info

Publication number: WO2016089835A1
Application number: PCT/US2015/063133
Authority: WO
Inventors: Vijaya Halabe; Robert Sauve
Original assignee: Schlumberger Canada Limited; Services Petroliers Schlumberger; Geoquest Systems B.V.; Schlumberger Technology Corporation
Priority date: 2014-12-01
Filing date: 2015-12-01
Publication date: 2016-06-09

Abstract

Systems and methods are described for declustering and estimating spatially distributed data, such as oilfield data. Spatial data from a density estimation grid, such as a geological model grid, can be binned, and a data density grid can be generated by applying kernel density estimation to the binned spatial data. One or more spatial data locations can be received, and a data density can be determined for each location in the data density grid corresponding to a received spatial data location. A declustering weight can be computed for each of the spatial data locations, and declustered data statistics can be computed using the spatial data and the declustering weights.

Description

SPATIAL DECLUSTERING OF OILFIELD DATA USING KERNEL DENSITY

ESTIMATION

Cross-Reference to Related Applications

[0001] This application claims priority to U.S. Provisional Patent Application serial no. 62/085,945, which was filed on December 1, 2014 and is incorporated herein by reference in its entirety.

Background

[0002] Spatially distributed data can represent the arrangement of phenomenon across the Earth's surface, such as, for example, oilfields, oil wells, rock layers, stratigraphic features, etc., as well as petrophysical properties of the earth, such as, for example, rock type, rock porosity, lithology, water saturation, permeability, density, etc. Spatially distributed data can be presented graphically (e.g., as reservoir models) and can be used to, for example, plan new oil wells or alter existing oil wells and/or can be used to estimate remaining oil reserves.

[0003] In order to obtain spatially distributed data, measurements are generally performed in scattered locations, and the collected data is extrapolated to larger areas with untested locations. Generally, locations with numerous existing oil wells are more thoroughly tested and/or have more measurement data associated with them, while other locations may have sporadic measurements performed in more isolated locations.

[0004] Accordingly, measured data can have different statistical weights for estimation of phenomenon arrangement and global petrophysical properties for an entire area or volume of interest, and calculating improved declustering weights can improve estimations of spatially distributed data.

Summary

[0005] Systems, apparatus, computer-readable media, and methods are disclosed for declustering and estimating spatially distributed data, such as geological and/or oilfield data. In an embodiment, a density estimation grid that includes spatially distributed geological data can be received, and the spatially distrusted geological data can be binned into multiple bins. A data density grid can be generated by applying kernel density estimation to the spatially distributed geological data in the bins. Additionally, a spatial data location can be received, and a data density at a location in the data density grid corresponding to the spatial data location can be determined. A declustering weight can be computed based on the data density, and declustered data statistics can be computed based on the spatially distributed geological data and the declustering weight.

[0006] It will be appreciated that this summary is intended merely to introduce some aspects of the present methods, systems, and media, which are more fully described and/or claimed below. Accordingly, this summary is not intended to be limiting.

Brief Description of the Drawings

[0007] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the present teachings and together with the description, serve to explain the principles of the present teachings. In the figures:

[0008] Figure 1 illustrates an example of a system that includes various management components to manage various aspects of a geologic environment, according to an embodiment.

[0009] Figure 2 illustrates an example of a method for computing declustering weights using binned kernel density estimation and computing declustered data statistics, according to an embodiment.

[0010] Figure 3 A illustrates an example of a grid segment of a 2d density estimation grid, according to an embodiment.

[0011] Figure 3B illustrates an example of a grid segment of a 2d density estimation grid, according to an embodiment.

[0012] Figure 4 illustrates an example of a grid segment of a 2d geological model, according to an embodiment.

[0013] Figure 5A illustrates an example schematic of a kernel density estimation and the effect of the bandwidth parameter, according to an embodiment.

[0014] Figure 5B illustrates an example schematic of a kernel density estimation and the effect of the bandwidth parameter, according to an embodiment.

[0015] Figure 6 depicts an illustrative computing system, according to an embodiment. Detailed Description

[0016] Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings and figures. In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of embodiments of the present disclosure. However, it will be apparent to one of ordinary skill in the art that embodiments of the present disclosure may be practiced without these specific details. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.

[0017] It will also be understood that, although the terms first, second, etc. may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first object or step could be termed a second object or step, and, similarly, a second object or step could be termed a first object or step, without departing from the scope of the disclosure. The first object or step, and the second object or step, are both, objects or steps, respectively, but they are not to be considered the same object or step.

[0018] The terminology used in the description herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description and the appended claims, the singular forms "a," "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term "and/or" as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms "includes," "including," "comprises" and/or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. Further, as used herein, the term "if may be construed to mean "when" or "upon" or "in response to determining" or "in response to detecting," depending on the context.

[0019] Attention is now directed to processing procedures, methods, techniques, and workflows that are in accordance with some embodiments. Some operations in the processing procedures, methods, techniques, and workflows disclosed herein may be combined and/or the order of some operations may be changed. [0020] Figure 1 illustrates an example of a system 100 that includes various management components 1 10 to manage various aspects of a geologic environment 150 (e.g., an environment that includes a sedimentary basin, a reservoir 151, one or more faults 153-1, one or more geobodies 153-2, etc.). For example, the management components 1 10 may allow for direct or indirect management of sensing, drilling, injecting, extracting, etc., with respect to the geologic environment 150. In turn, further information about the geologic environment 150 may become available as feedback 160 (e.g., optionally as input to one or more of the management components 110).

[0021] In the example of Figure 1, the management components 1 10 include a seismic data component 112, an additional information component 114 (e.g., well/logging data), a processing component 1 16, a simulation component 120, an attribute component 130, an analysis/visualization component 142 and a workflow component 144. In operation, seismic data and other information provided per the components 112 and 114 may be input to the simulation component 120.

[0022] In an example embodiment, the simulation component 120 may rely on entities 122. Entities 122 may include earth entities or geological objects such as wells, surfaces, bodies, reservoirs, etc. In the system 100, the entities 122 can include virtual representations of actual physical entities that are reconstructed for purposes of simulation. The entities 122 may include entities based on data acquired via sensing, observation, etc. (e.g., the seismic data 112 and other information 114). An entity may be characterized by one or more properties (e.g., a geometrical pillar grid entity of an earth model may be characterized by a porosity property). Such properties may represent one or more measurements (e.g., acquired data), calculations, etc.

[0023] In an example embodiment, the simulation component 120 may operate in conjunction with a software framework such as an object-based framework. In such a framework, entities may include entities based on pre-defined classes to facilitate modeling and simulation. A commercially available example of an object-based framework is the MICROSOFT^® .NET^® framework (Redmond, Washington), which provides a set of extensible object classes. In the .NET^® framework, an object class encapsulates a module of reusable code and associated data structures. Object classes can be used to instantiate object instances for use by a program, script, etc. For example, borehole classes may define objects for representing boreholes based on well data. [0024] In the example of Figure 1, the simulation component 120 may process information to conform to one or more attributes specified by the attribute component 130, which may include a library of attributes. Such processing may occur prior to input to the simulation component 120 (e.g., consider the processing component 116). As an example, the simulation component 120 may perform operations on input information based on one or more attributes specified by the attribute component 130. In an example embodiment, the simulation component 120 may construct one or more models of the geologic environment 150, which may be relied on to simulate behavior of the geologic environment 150 (e.g., responsive to one or more acts, whether natural or artificial). In the example of Figure 1, the analysis/visualization component 142 may allow for interaction with a model or model-based results (e.g., simulation results, etc.). As an example, output from the simulation component 120 may be input to one or more other workflows, as indicated by a workflow component 144.

[0025] As an example, the simulation component 120 may include one or more features of a simulator such as the ECLIPSE™ reservoir simulator (Schlumberger Limited, Houston Texas), the INTERSECT™ reservoir simulator (Schlumberger Limited, Houston Texas), etc. As an example, a simulation component, a simulator, etc. may include features to implement one or more meshless techniques (e.g., to solve one or more equations, etc.). As an example, a reservoir or reservoirs may be simulated with respect to one or more enhanced recovery techniques (e.g., consider a thermal process such as SAGD, etc.).

[0026] In an example embodiment, the management components 110 may include features of a commercially available framework such as the PETREL^® seismic to simulation software framework (Schlumberger Limited, Houston, Texas). The PETREL^® framework provides components that allow for optimization of exploration and development operations. The PETREL^® framework includes seismic to simulation software components that can output information for use in increasing reservoir performance, for example, by improving asset team productivity. Through use of such a framework, various professionals (e.g., geophysicists, geologists, and reservoir engineers) can develop collaborative workflows and integrate operations to streamline processes. Such a framework may be considered an application and may be considered a data-driven application (e.g., where data is input for purposes of modeling, simulating, etc.). [0027] In an example embodiment, various aspects of the management components 110 may include add-ons or plug-ins that operate according to specifications of a framework environment. For example, a commercially available framework environment marketed as the OCEAN^® framework environment (Schlumberger Limited, Houston, Texas) allows for integration of addons (or plug-ins) into a PETREL^® framework workflow. The OCEAN^® framework environment leverages .NET^® tools (Microsoft Corporation, Redmond, Washington) and offers stable, user- friendly interfaces for efficient development. In an example embodiment, various components may be implemented as add-ons (or plug-ins) that conform to and operate according to specifications of a framework environment (e.g., according to application programming interface (API) specifications, etc.).

[0028] Figure 1 also shows an example of a framework 170 that includes a model simulation layer 180 along with a framework services layer 190, a framework core layer 195 and a modules layer 175. The framework 170 may include the commercially available OCEAN^® framework where the model simulation layer 180 is the commercially available PETREL^® model-centric software package that hosts OCEAN^® framework applications. In an example embodiment, the PETREL^® software may be considered a data-driven application. The PETREL^® software can include a framework for model building and visualization.

[0029] As an example, a framework may include features for implementing one or more mesh generation techniques. For example, a framework may include an input component for receipt of information from interpretation of seismic data, one or more attributes based at least in part on seismic data, log data, image data, etc. Such a framework may include a mesh generation component that processes input information, optionally in conjunction with other information, to generate a mesh.

[0030] In the example of Figure 1 , the model simulation layer 180 may provide domain objects 182, act as a data source 184, provide for rendering 186 and provide for various user interfaces 188. Rendering 186 may provide a graphical environment in which applications can display their data while the user interfaces 188 may provide a common look and feel for application user interface components.

[0031] As an example, the domain objects 182 can include entity objects, property objects and optionally other objects. Entity objects may be used to geometrically represent wells, surfaces, bodies, reservoirs, etc., while property objects may be used to provide property values as well as data versions and display parameters. For example, an entity object may represent a well where a property object provides log information as well as version information and display information (e.g., to display the well as part of a model).

[0032] In the example of Figure 1 , data may be stored in one or more data sources (or data stores, generally physical data storage devices), which may be at the same or different physical sites and accessible via one or more networks. The model simulation layer 180 may be configured to model projects. As such, a particular project may be stored where stored project information may include inputs, models, results and cases. Thus, upon completion of a modeling session, a user may store a project. At a later time, the project can be accessed and restored using the model simulation layer 180, which can recreate instances of the relevant domain objects.

[0033] In the example of Figure 1, the geologic environment 150 may include layers (e.g., stratification) that include a reservoir 151 and one or more other features such as the fault 153-1, the geobody 153-2, etc. As an example, the geologic environment 150 may be outfitted with any of a variety of sensors, detectors, actuators, etc. For example, equipment 152 may include communication circuitry to receive and to transmit information with respect to one or more networks 155. Such information may include information associated with downhole equipment 154, which may be equipment to acquire information, to assist with resource recovery, etc. Other equipment 156 may be located remote from a well site and include sensing, detecting, emitting or other circuitry. Such equipment may include storage and communication circuitry to store and to communicate data, instructions, etc. As an example, one or more satellites may be provided for purposes of communications, data acquisition, etc. For example, Figure 1 shows a satellite in communication with the network 155 that may be configured for communications, noting that the satellite may additionally or alternatively include circuitry for imagery (e.g., spatial, spectral, temporal, radiometric, etc.).

[0034] Figure 1 also shows the geologic environment 150 as optionally including equipment 157 and 158 associated with a well that includes a substantially horizontal portion that may intersect with one or more fractures 159. For example, consider a well in a shale formation that may include natural fractures, artificial fractures (e.g., hydraulic fractures) or a combination of natural and artificial fractures. As an example, a well may be drilled for a reservoir that is laterally extensive. In such an example, lateral variations in properties, stresses, etc. may exist where an assessment of such variations may assist with planning, operations, etc. to develop a laterally extensive reservoir (e.g., via fracturing, injecting, extracting, etc.). As an example, the equipment 157 and/or 158 may include components, a system, systems, etc. for fracturing, seismic sensing, analysis of seismic data, assessment of one or more fractures, etc.

[0035] As mentioned, the system 100 may be used to perform one or more workflows. A workflow may be a process that includes a number of worksteps. A workstep may operate on data, for example, to create new data, to update existing data, etc. As an example, a workstep may operate on one or more inputs and create one or more results, for example, based on one or more algorithms. As an example, a system may include a workflow editor for creation, editing, executing, etc. of a workflow. In such an example, the workflow editor may provide for selection of one or more pre-defined worksteps, one or more customized worksteps, etc. As an example, a workflow may be a workflow implementable in the PETREL^® software, for example, that operates on seismic data, seismic attribute(s), etc. As an example, a workflow may be a process implementable in the OCEAN^® framework. As an example, a workflow may include one or more worksteps that access a module such as a plug-in (e.g., external executable code, etc.).

[0036] In some embodiments, system 100 may be used for spatial declustering of spatially distributed geological data. For example, measurement data in oilfield environments is usually clustered in particular areas as a consequence of the requirement to sample particular rock types such as good quality reservoir sands. Areas containing other rock types such as non-reservoir shales are therefore under-represented in the data samples. The dense horizontal sampling of multilateral horizontal wells, often targeting particular stratigraphic layers, also introduces further unrepresentative sampling of the reservoir as a whole. This effect is undesirable because the measurement data is used to derive statistical models that are the key input to various estimation and simulation methods for building numerical models of the entire reservoir.

[0037] For example, an area of interest can cover an oilfield which has been sampled along a number of horizontal and vertical wells. In this example, there may two dominant rock types: reservoir sand, which has been extensively targeted by horizontal wells, and, elsewhere, non- reservoir shale which has been sampled by a small number of vertical wells. The true distribution of sand and shale is unknown, but, if it were possible to exhaustively sample the area of interest, it could be determined that approximately 90% of the area is shale and 10% is sand. In reality, the knowledge of these rock type percentages is derived from the measured data which suggests an inversion of the percentages, i.e., 10% shale and 90% sand. These incorrect rock type percentages are due to the unrepresentative sampling of the measurement data and, if uncorrected, could lead to erroneous predictions of the economics and geographic features of such an oilfield.

[0038] Spatial declustering is the process by which weights are calculated and assigned to each measured data location in order to compensate for the preferential sampling of the measurement data in certain areas. These declustering weights are used together with the measured data values to calculate statistical parameters that are more representative of the area of interest than those derived by the standard calculations which implicitly assume that equal weights are assigned to all data locations.

[0039] For example, one important petrophysical property of a reservoir is the arithmetic mean porosity φ which can be calculated from n data samples as follows: φ = -∑₌₁ <Pi. Whereas the declustered arithmetic mean can be calculated from the same data samples, where

1

w_£ is the positive weight attached to i-th sample as follows: φ = =^ ∑?=i ^ΊΨι - ii= i ^wi

[0040] By comparison, the standard arithmetic mean is equivalent to assigning an equal weight (wi = to each data value.

[0041] Declustering weights can also be incorporated into the calculation of other important statistical parameters, such as the data variance, histograms, or regression models.

[0042] Declustering weights can be calculated using a process called "cell declustering." Cell declustering can be applied in d dimensions. The first step is to partition the area of interest into a regular mesh of rij cells of width hj in the jth dimension. For example, d can equal 2 and the mesh can contain 4x4 cells. The number of data samples are counted in each cell and the declustering weight assigned to any data sample is calculated to be inversely proportional to the total number of samples falling in same the cell in which it is located: w,- oc .

r ^{& L} # cell cou nt

[0043] Declustering weights can be assigned to each sample, and samples that fall in cell containing a smaller total number of samples can have declustering weights higher than the declustering weights assigned to samples that fall in cells containing a greater number of samples. In some embodiments, the declustering weights can be normalized to add up to one. [0044] The cell declustering weights can be sensitive to the choice of cell origin. The origin is often chosen subjectively by the user and yet relatively small changes in the origin can lead to large changes in the declustering weight assigned to particular samples. For example, shifting the origin by a value equal to half a cell width can cause a change in the cell counts and significant changes in the declustering weights.

[0045] In cell declustering, the impact of artefacts related to the cell origin choice can be reduced by averaging over a number of sets of declustering weights formed by progressively shifting the cell origin while maintaining a common cell width. In the jth dimension the origin can be shifted m,- times m steps equal to: 0,— ,— — .

¹ r j r j r j

[0046] As an example, cell mesh origins can be constructed such that m₁ = m₂ = 4 and the cell nearest the origin can subdivided into 14 alternative cell origins. The output declustering weights can represent average values determined from 16 offset cell meshes.

[0047] Figure 2 illustrates an example of a method for computing declustering weights using binned kernel density estimation and computing declustered data statistics. In some embodiments, the example method illustrated in Figure 2 can be performed using a computing device that includes the framework (e.g., framework 170) and the management components (e.g., management components 110) described above with reference to Figure 1.

[0048] Cell declustering is a form of declustering, which may be equivalent to a simple-form, multivariate probability density estimation known in computational statistics as the Average Shifted Histogram or ASH. The ASH approach is an attempt to circumvent the issues of where to choose the grid origin when computing a binned probability density estimate. Unless precautions are taken, a small change in the choice of origin may lead to a large change in the probability density estimate and, therefore, in the resulting cell declustering weights.

[0049] Kernel density estimation is a statistical technique which has advantages over ASH. For example, kernel density estimation may be less sensitive to the choice of grid origin than ASH, and may use a simpler parameterization scheme compared to ASH. Kernel density estimation uses smoothing parameters known as bandwidths that are intuitive for users to manipulate and may be optimized.

[0050] The example method can begin in 200, when the computing device receives or obtains a density estimation grid. In some embodiments, the density estimation grid can be a one- dimensional (Id) or a two-dimensional (2d) or a three-dimensional (3d) grid. In further embodiments, the density estimation grid can be or can correspond to a geological model grid that includes spatially distributed geological data that is used for reservoir simulation. In still further embodiments, the density estimation grid can be a coarsened version of a geological model grid.

[0051] The density estimation grid can be made up of cells, and each cell can be associated with one or more attributes. For example, in the case of a geological model grid, a cell can be associated with 3d coordinates defining a volume in a reservoir and an attribute can be geological data for the volume in the reservoir, such as rock porosity. The attribute can be assumed to be constant over the cell.

[0052] In some embodiments, the density estimation grid can include measured cells and unmeasured cells. For example, measured cells can correspond to volumes where measurements have been made, such as within oil wells. Additionally, for example, unmeasured cells can correspond to volumes where no measurements have been made and/or volumes where a value of a selected attribute is not populated for the cell.

[0053] In 210, the computing device can bin spatial data from the density estimation grid. For example, the spatial data can be spatially distributed geological data from a geological model grid, such as oil field data, oil well data, rock layer data, stratigraphic feature data, rock type data, rock porosity data, lithology data, water saturation data, permeability data, density data, etc. In some embodiments, the computing device can use a simple binning scheme. In other embodiments, the computing device can use a linear binning scheme.

[0054] In a simple binning scheme, a grid can be assigned x bins, with x being less than the number of cells in the grid. The bins can be associated with locations within the grid, and the spaces between the bins can be regular intervals. The space between bins and, accordingly, the number of assigned bins can be based on an input value from a user or a default value. Each cell is then associated with the bin with a location in closest proximity to the cell.

[0055] In some implementations, a value of a bin can be incremented for each cell associated with the bin that is a measured cell. In other implementations, one or more attributes can be selected, and the value of the bin can be incremented for each associated cell that has a measured value for the one or more attributes. [0056] Figure 3A illustrates an example of a grid segment of a 2d density estimation grid. Figure 3A is merely a simplified example of a simple binning scheme, for the sake of illustration, and is not intended to be limiting.

[0057] Indicators 300, 310, 320, and 330 can represent the locations associated with four bins that make up a square. Indicator 340 can represent a measured cell within the square. The data associated with the measured cell can be assigned to the nearest bin, which, in this example, is bin 320. Additionally, the valve of bin 320 can be incremented by one.

[0058] In a linear binning scheme, a grid can be assigned x bins, the bins can be associated with locations within the grid, and the spaces between the bins can be regular intervals, similar to a simple binning scheme. Unlike in a simple binning scheme, cells may not be associated with a single bin. Instead, a value of a bin can be incremented by a value computed as a function of a measured cell's location relative to the bin's location. Additionally, the data associated with the measured cell can be distributed amongst the nearest bins by a factor of their distance from the measured cell.

[0059] For example, in a 2d grid the location of four bins can make up a square with at least one measured cell located within the square. For each measured cell, the square can be split into four quadrants based on the position of the cell (a quadrant to the upper left, a quadrant to the upper right, a quadrant to the lower left, and a quadrant to the lower right). Each quadrant can be associated with a bin (e.g., upper left quadrant is associated with the bin in the upper left corner of the square). Each of the four bins can then be incremented by a value equal to the area of the opposite quadrant divided by the total area of the square. For example, the opposite quadrant of the upper left bin is the lower right quadrant, etc.

[0060] Figure 3B illustrates an example of a grid segment of a 2d density estimation grid. Figure 3B is merely a simplified example of a linear binning scheme, for the sake of illustration, and is not intended to be limiting.

[0061] Indicators 350, 360, 370, and 380 can represent the locations associated with four bins that make up a square. Indicator 390 can represent a measured cell within the square. The data associated with the measured cell can be distributed amongst the nearest bins by a factor of their distance from the measured cell, which, in this example, are bins 350, 360, 370, and 380. Additionally, the valve of the bins can be incremented by a value corresponding to a factor of their distance from the measured cell. [0062] In some embodiments, the factor of a bin's distance from a measured cell can be computed as described above. For example, for bin 370, the factor can be equal to the area of the opposite quadrant (quadrant B) divided by the area of the entire square. The value of bin 370 can be incremented by the factor. As an additional example, for bin 360, the factor can equal to the area of the opposite quadrant (quadrant C) divided by the area of the entire square. The value of bin 360 can be incremented by the factor.

[0063] The same principles applied above can be used for a 3d grid, where the location of eight bins make up a cube with at least one measured cell located within the cube. For each measured cell, the cube can be split into octants based on the position of the cell. Each octant can be associated with a bin, and each of the eight bins can be incremented by a value equal to the volume of the opposite octant divided by the total area of cube. Additionally, the data associated with the measured cell can be distributed amongst the nearest bins by a factor of their distance from the measured cell.

[0064] Application of a linear binning scheme can enable the use of a decimated grid without substantially affecting the accuracy of the density estimates. A compromise between performance and accuracy may be reached, for example, when the increment equals two. In a 3d declustering problem, using an increment of two may result in roughly a 2³ = 8 speedup.

[0065] When the grid is decimated, trilinear interpolation can be used to extract density at all (or a specific subset) of cell locations.

[0066] In 220, the computing device can generate a data density grid by applying kernel density estimation to the values of the bins. A kernel density estimation is related to a histogram, but, unlike a histogram, can include properties such as smoothness or continuity.

[0067] The application of the smoothing kernel to the bin values is effectively a convolution operation where the smoothing kernel may be convolved with the bin values (e.g., as an array of bin values). There are several numerical methods that could be employed to perform this convolution e.g., a direct method, a fast Fourier transform (FFT), etc. Embodiments of the method of the present disclosure may include (but are not limited to) a direct method that takes advantage of the structure of the array of bin values. In some embodiments, the grid of bin values may be sparse, with the majority of the bins being empty. The method thus uses a direct convolution algorithm that is structured so that it skips the contribution from all bins with a zero value, which can result in an increase compared to a naive algorithm that does not account for this bin value sparseness.

[0068] In 230, the computing device can receive a spatial data location for sampling the data density grid. The spatial data locations can be, for example, (x, y) coordinates (e.g., in a 2d graph), (x, y, z) coordinates or (r, Φ, z) radial coordinates (e.g., in a 3d graph), etc.

[0069] In 240, the computing device can determine the data density at a location in the data density grid corresponding the spatial data location received in 230.

[0070] In 250, the computing device can compute a declustering weight at the spatial data location based on the determined data density. In some embodiments, the declustering weight can be the inverse of the determined data density.

[0071] In some embodiments, in the case of a geological model grid, the computing device can normalize the declustering weights by the extent of each k-layer. The method may be applied in a grid framework defined by the structure of the geological model (or pillar) grid. Stratigraphic features such as erosion and truncation may be incorporated through the use of geological rules during the grid construction process. Such features may result in particular stratigraphic layers, or "k-layers," which have a limited lateral extent and/or may not be present over most portions of the grid. These variations in the spatial extent of individual k-layers may have implications for the application of the declustering algorithm. For example, a cell may contribute to an anomalously-low spatial data density in their local k-layer and, therefore, may be assigned an anomalously-high declustering weight. This assignment may result in unrepresentative statistics because data from a limited extent k-layer may receive more weight than data from a k-layer that is more laterally extensive.

[0072] In order to compensate for this effect, the method may iterate over the entire geological model grid and accumulate a count of the number of live grid cells in each k-layer. The method may also normalize the k-layer cell counts derived in by, for example, the maximum possible number of grid cells in a k-layer. The method may also, in some embodiments, apply the normalization factors, on a k-layer by k-layer basis, to the raw declustering weights derived by kernel density estimation.

[0073] In 260, the computing device can compute declustered data statistics using the declustering weight. For example, the computing device can compute declustered data statistics of the spatial data from the density estimation grid, such as the mean when the data represents a continuous property like porosity or a proportion when the data represents a discrete property like facies

[0074] In 270, the computing device can optimize the declustered data statistics by adjusting parameters that control the kernel density estimation. For example, the computing device can adjust a bandwidth parameter of the kernel density estimation based on user input such that a desired statistic is either minimized or maximized. In some embodiments, default parameters of the kernel density estimation may not be adjusted., and, accordingly, 270 is not performed

[0075] Although the selection of the bandwidth parameters may be optimized or otherwise enhanced according to a user-selected criterion, as described above, the method may be operated in an interactive mode, in which the user manually adjusts the bandwidth parameters and may immediately visualize the effect of the parameter changes in linked data displays. One example is a joint visualization of the declustering weights of the cells and the associated data density property in 3d displays. This interactive approach to declustering may enable the user to perform a quality control of the results and to introduce more geological knowledge into the process of selecting the optimal bandwidth parameters than might be achieved through the use of an automatic procedure alone.

[0076] In 280, the computing device can output the declustering weight computed in 250 (e.g., the declustering weight assigned to spatial data location received in 230), the data density grid generated in 220, and the computed declustered data statistics from 260. In various embodiments, 230-280 can be performed each time a new spatial data location for sampling the data density grid is received and/or perform an iteration (e.g., sequentially, in parallel, etc.) for each new spatial data location received.

[0077] In various embodiments, the output can be employed in geoscience workflows, such as, for example, in petrophysical property and facies modelling where the declustering process may impact the mean of a continuous property such as porosity or the global proportion of a facies in a reservoir model. Declustering technology may be employed in such petrophysical and facies modeling, as economic indicators derived from static and dynamic reservoir models, such as hydrocarbon volumetrics and flow-based connectivity, may be sensitive to these input statistics.

[0078] Accordingly, the present method may be tailored for use in reservoir property modelling, so as to drive such processes with accurate and representative statistical models. For example, in unconventional gas field appraisal and development, long sections of horizontal wells seeking sweet spots may be clustered both in a vertical sense (as they target stratigraphic levels) as well as spatially (as the wells target reservoir facies or fracture corridors). The present method may be tailored for use in this challenging and increasingly common scenario.

[0079] The present method, however, may be applied beyond standard petrophysical property modelling and to any geoscience workflow that involves aggregating spatially distributed data. A simple example is in rock physics workflows involving input data from multiple wells. In these situations it is often useful to compute regression models to calibrate petrophysical properties against seismic elastic properties. The present method may remove the effects of spatially clustered data in order to gain more a representative regression calibration and thereby improve quantitative interpretation of seismic data.

[0080] While the operations depicted in Figure 2 have been described as performed in a particular order, the order described is merely an example, and various different sequences of operations can be performed, consistent with certain disclosed embodiments. Additionally, the operations are described as discrete steps merely for the purpose of explanation, and, in some embodiments, multiple operations may be performed simultaneously and/or as part of a single computation. Further, the operations described are not intended to be exhaustive or absolute, and various operations can be inserted or removed.

[0081] Figure 4 illustrates an example of a grid segment of a 2d geological model. Figure 4 is merely a simplified example of a geological model, for the sake of illustration, and is not intended to be limiting.

[0082] Figure 4 depicts multilateral horizontal wells 400 and a cluster of vertical wells 410. Both multilateral horizontal wells 400 and the cluster of vertical wells 410 include indictors of measured data (e.g., indictors 420 and 430). In various embodiments, each indicator can represent a measured cell.

[0083] As depicted in Figure 4, multilateral horizontal wells 400 include many more measured data locations than the cluster of vertical wells 410. Without using declustering weights, the larger quantity of measured data locations for multilateral horizontal wells 400 compared to the cluster of vertical wells can result in computed data statistics that are unrepresentative of the entire area because the measured data locations for multilateral horizontal wells 400 are clustered together and the data they supply may not represent the area as whole. Contrarily, there are relatively few data locations for the cluster of vertical wells 410, and, accordingly, any differences between the data from the cluster of vertical wells 410 compared to multilateral horizontal wells 400 can be increasingly representative of the actual characteristics of unmeasured data locations in the grid. Accordingly, each individual data location in multilateral horizontal wells 400 can be downweighted and each individual data location in the cluster of vertical wells 410 can be upweighted as a result of computing declustering weights, as described with regard to Figure 2 above.

[0084] Figure 5A illustrates an example schematic of a kernel density estimation and the effect of the bandwidth parameter. The example schematic is merely a simplified example of kernel density estimation, and is not intended to be limiting.

[0085] As depicted in Figure 5 A, measured data locations are plotted in one-dimensional (Id) space along the x-axis of the chart. Each measured data location can represent a measured cell, as described above. Additionally, each measured data location can be associated with a kernel centered at the data location (e.g., kernel 500).

[0086] The density estimate for Figure 5A can be represented by the y-axis of the chart, and density estimate 510 can be plotted along the chart based on the density estimate for each location. As shown in Figure 5A, the density estimate at each point, computed using kernel density estimation, is the average contribution from each of the kernels combined with the bandwidth parameter. The bandwidth parameter controls the smoothness of the graph, and, in Figure 5A, can be set to a first value. The higher the value of the bandwidth, the smoother the graph.

[0087] In some embodiments, a graph, such as the graph in Figure 5A or a graph that represents 2d or 3d data locations, can be displayed for a user in 270 as described in Figure 2. The user can then elect to adjust the bandwidth parameter such that a desired statistic is either minimized or maximized. For example, if the user decreases the bandwidth parameter, the displayed graph can be adjusted, as described below with regard to Figure 5B.

[0088] Figure 5B illustrates an example schematic of a kernel density estimation and the effect of the bandwidth parameter. The example schematic is merely a simplified example of kernel density estimation, and is not intended to be limiting.

[0089] As depicted in Figure 5B, measured data locations are plotted in one-dimensional (Id) space along the x-axis of the chart. Each measured data location can represent a measured cell, as described above. Additionally, each measured data location can be associated with a kernel centered at the data location (e.g., kernel 520).

[0090] The density estimate for Figure 5B can be represented by the y-axis of the chart, and density estimate 530 can be plotted along the chart based on the density estimate for each location. As shown in Figure 5B, the density estimate at each point, computed using kernel density estimation, is the average contribution from each of the kernels combined with the bandwidth parameter. However, the bandwidth may have been decreased by a user after the graph in Figure 5A was displayed. Accordingly, density estimate 530 has an increased unevenness, compared to the density estimate shown in Figure 5A.

[0091] In some embodiments, a graph, such as the graph in Figure 5B or a graph that represents 2d or 3d data locations, can be displayed for a user in 270 as described in Figure 2. The user can then elect to use the current bandwidth parameter or further adjust the bandwidth parameter such that a desired statistic is either minimized or maximized.

[0092] In some embodiments, the methods of the present disclosure may be executed by a computing system. Figure 6 illustrates an example of such a computing system 600, in accordance with some embodiments. The computing system 600 may include a computer or computer system 601 A, which may be an individual computer system 601 A or an arrangement of distributed computer systems. The computer system 601 A includes one or more analysis modules 602 that are configured to perform various tasks according to some embodiments, such as one or more methods disclosed herein. To perform these various tasks, the analysis module 602 executes independently, or in coordination with, one or more processors 604, which is (or are) connected to one or more storage media 606. The processor(s) 604 is (or are) also connected to a network interface 607 to allow the computer system 601 A to communicate over a data network 609 with one or more additional computer systems and/or computing systems, such as 60 IB, 601C, and/or 60 ID (note that computer systems 60 IB, 601C and/or 60 ID may or may not share the same architecture as computer system 601 A, and may be located in different physical locations, e.g., computer systems 601A and 601B may be located in a processing facility, while in communication with one or more computer systems such as 601C and/or 60 ID that are located in one or more data centers, and/or located in varying countries on different continents). [0093] A processor may include a microprocessor, microcontroller, processor module or subsystem, programmable integrated circuit, programmable gate array, or another control or computing device.

[0094] The storage media 606 may be implemented as one or more computer-readable or machine-readable storage media. Note that while in the example embodiment of Figure 6 storage media 606 is depicted as within computer system 601 A, in some embodiments, storage media 606 may be distributed within and/or across multiple internal and/or external enclosures of computing system 601 A and/or additional computing systems. Storage media 606 may include one or more different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories, magnetic disks such as fixed, floppy and removable disks, other magnetic media including tape, optical media such as compact disks (CDs) or digital video disks (DVDs), BLURAY^® disks, or other types of optical storage, or other types of storage devices. Note that the instructions discussed above may be provided on one computer-readable or machine-readable storage medium, or alternatively, may be provided on multiple computer- readable or machine-readable storage media distributed in a large system having possibly plural nodes. Such computer-readable or machine-readable storage medium or media is (are) considered to be part of an article (or article of manufacture). An article or article of manufacture may refer to any manufactured single component or multiple components. The storage medium or media may be located either in the machine running the machine-readable instructions, or located at a remote site from which machine-readable instructions may be downloaded over a network for execution.

[0095] In some embodiments, computing system 600 contains one or more declustering module(s) 608. In the example of computing system 600, computer system 601 A includes the declustering module 608. In some embodiments, a single declustering module may be used to perform some or all aspects of one or more embodiments of the methods disclosed herein. In alternate embodiments, a plurality of declustering modules may be used to perform some or all aspects of methods herein.

[0096] It should be appreciated that computing system 600 is only one example of a computing system, and that computing system 600 may have more or fewer components than shown, may combine additional components not depicted in the example embodiment of Figure 6, and/or computing system 600 may have a different configuration or arrangement of the components depicted in Figure 6. The various components shown in Figure 6 may be implemented in hardware, software, or a combination of both hardware and software, including one or more signal processing and/or application specific integrated circuits.

[0097] Further, the aspects of the processing methods described herein may be implemented by running one or more functional modules in information processing apparatus such as general purpose processors or application specific chips, such as ASICs, FPGAs, PLDs, or other appropriate devices. These modules, combinations of these modules, and/or their combination with general hardware are all included within the scope of protection.

[0098] Geologic interpretations, models, and/or other interpretation aids may be refined in an iterative fashion; this concept is applicable to the methods discussed herein. This may include use of feedback loops executed on an algorithmic basis, such as at a computing device (e.g., computing system 600, Figure 6), and/or through manual control by a user who may make determinations regarding whether a given step, action, template, model, or set of curves has become sufficiently accurate for the evaluation of the subsurface three-dimensional geologic formation under consideration.

[0099] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit embodiments of the present disclosure to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. Moreover, the order in which the elements of the methods described herein are illustrated and described may be re-arranged, and/or two or more elements may occur simultaneously. The embodiments were chosen and described in order to best explain the principals of the present disclosure and its practical applications, to thereby enable others skilled in the art to best utilize embodiments of the present disclosure with various modifications as are suited to the particular use contemplated.

Claims

CLAIMS What is claimed is:

1. A method comprising:

receiving a density estimation grid comprising spatially distributed geological data;

binning the spatially distributed geological data into a plurality of bins;

generating a data density grid by applying kernel density estimation to the spatially distributed geological data in the plurality of bins;

receiving a spatial data location;

determining a data density at a location in the data density grid corresponding to the spatial data location;

computing a declustering weight based on the data density; and

computing declustered data statistics based on the spatially distributed geological data from the density estimation grid and the declustering weight.

2. The method of claim 1, wherein the density estimation grid is a one-dimensional grid.

3. The method of claim 1, wherein the density estimation grid is a two-dimensional grid.

4. The method of claim 1, wherein the density estimation grid is a three-dimensional grid.

5. The method of claim 4, wherein the density estimation grid is a geological model grid.

6. The method of claim 1 , wherein binning the spatially distributed geological data comprises using a simple binning scheme.

7. The method of claim 1 , wherein binning the spatially distributed geological data comprises using a linear binning scheme.

8. The method of claim 1, wherein the declustering weight is an inverse of the data density.

9. The method of claim 1, further comprising normalizing the declustering weight based on the extent of a k-layer associated with the spatial data location.

10. The method of claim 1 , wherein computing the declustered data statistics further comprises computing at least one of a mean or a proportion using the spatially distributed geological data and the declustering weight.

11. The method of claim 1 , further comprising:

displaying a visualization comprising indications of a plurality of declustering weights including the declustering weight;

receiving an adjustment to a bandwidth parameter from a user;

re-computing the declustering weight based on the data density and the bandwidth parameter from the user; and

displaying an adjusted visualization comprising indications of the re-computed declustering weight.

12. The method of claim 1, further comprising:

outputting at least one of the declustering weight, the data density grid, and the declustered data statistics, wherein the output is used in at least one geoscience workflow.

13. A system comprising :

a processing system of a device comprising one or more processors; and

a memory system comprising one or more computer-readable media, wherein the one or more computer-readable media contain instructions that, when executed by the processing system, cause the processing system to perform operations comprising:

binning the spatially distributed geological data into a plurality of bins;

receiving a spatial data location; determining a data density at a location in the data density grid corresponding to the spatial data location;

computing a declustering weight based on the data density; and

computing declustered data statistics based on the spatially distributed geological data from the density estimation grid and the declustering weight..

14. The system of claim 13, wherein the density estimation grid is a three-dimensional geological model grid.

15. The system of claim 13, wherein binning the spatially distributed geological data comprises using a linear binning scheme.

16. The system of claim 13, the operations further comprising normalizing the declustering weight based on the extent of a k-layer associated with the spatial data location.

17. The system of claim 13, wherein computing the declustered data statistics further comprises computing at least one of a mean or a proportion using the spatially distributed geological data and the declustering weight.

18. The system of claim 13, the operations further comprising:

receiving an adjustment to a bandwidth parameter from a user;

19. The system of claim 13, the operations further comprising:

20. A non-transitory computer readable storage medium comprising instructions for causing one or more processors to:

receive a density estimation grid comprising spatially distributed geological data;

bin the spatially distributed geological data into a plurality of bins;

generate a data density grid by applying kernel density estimation to the spatially distributed geological data in the plurality of bins;

receive a spatial data location;

determine a data density at a location in the data density grid corresponding to the spatial data location;

compute a declustering weight based on the data density; and

compute declustered data statistics based on the spatially distributed geological data from the density estimation grid and the declustering weight.