**PRIOR RELATED APPLICATIONS**

This application is a non-provisional application which claims benefit under 35 USC §119(e) to U.S. Provisional Application Ser. No. 62/190,116 filed Jul. 8, 2015, entitled “IMPROVED GEOBODY CONTINUITY IN GEOLOGICAL MODELS BASED ON MULTIPLE POINT STATISTICS,” which is incorporated herein in its entirety.

**FIELD OF THE DISCLOSURE**

The disclosure generally relates to improved geobody continuity and hard data conditioning in Multiple Point Statistics (MPS) simulation.

**BACKGROUND OF THE DISCLOSURE**

Modeling subsurface heterogeneity is of crucial importance for the exploitation of subsurface resources and for the geological storage of nuclear wastes, CO_{2}, etc. For more than half a century, geostatistics has been developed for this purpose and widely used in practice.

Geostatistics is the branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture—especially in precision farming.

Geostatistics has gained popularity as a quantitative tool to generate multiple geological models, or “realizations,” that honor a given statistical structure and various types of measured and interpreted data. Two key approaches have traditionally been followed: a pixel based approach such as sequential indicator simulation based on variograms, and an object based (Boolean) approach in which large objects representing geobodies are inserted into the geological model. Both of these approaches are widely used, but have limitations in terms of reproducing realistic geobody shapes and conditioning to multiple types of data.

Multiple-point statistics (MPS) simulation is a more advanced spatial modeling technique because of its ability to reproduce complex geological patterns (e.g. sinuous channels) that cannot be modeled by two-point statistics (i.e., variograms). Unlike traditional geostatistics, MPS avoids the explicit definition of a random function, but directly infers the necessary multivariate distributions from “training images” or “TI”. This confers on MPS a potential applicability to any geological environment, provided that there is a training image representative of the geological heterogeneity and that the essential features of this training image can be characterized by statistics defined through a search template with a limited point configuration. The MPS approach appears flexible to data conditioning and to representing complex architectures of geological facies or petrophysical properties.

The goal of MPS approaches is to enable the ability to reproduce the geological “shapes” as object-based techniques do and maintain the data-conditioning flexibility as variogram-based techniques do. To achieve that, MPS needs to infer and reproduce multiple-point statistics way beyond the traditional two-point statistics (variogram). Because the available data are usually too sparse to infer such high-order statistics, Guardiano and Srivastava proposed the use of a training image, i.e. a three-dimensional numerical conceptual representation of the facies thought to be present in the reservoir to be modeled. The MPS simulation implementation is similar to Sequential Indicator Simulation, where the variogram is replaced with a training image, and Kriging is replaced with the following process to estimate local conditional facies probabilities:

1. Look for the n conditioning data (original well data or previously simulated values) closest to the grid node u to be simulated. These conditioning data form a data event d_{n }that is fully characterized by its geometrical configuration (the data locations relative to u), and its data values (the facies at the data locations).

2. Scan the training image to find all replicates of d_{n }(same geometric configuration and same data values as d_{n}). For each replicate, record the facies value at the central location of the training replicate. By “central location,” what is meant is the grid node corresponding to the same relative location as u in the data event d_{n}.

3. The estimated conditional probability of each facies at u is computed as the proportion of d_{n }replicates that have this facies at their central locations.

One major advantage of this implementation is that, as in any pixel-based sequential simulation method, in contrast to object-based methods, well data are honored exactly. In addition, by capturing multiple-point statistics from the training image through the estimation of facies probabilities conditional to multiple-point data events, the MPS model reproduces training image patterns. However, the repetitive scanning of the training image to estimate facies probabilities is very time-consuming.

To help solve this issue, Strebelle introduced a dynamic data structure named “search tree” to store, prior to the simulation, all the conditional probability distributions that could be inferred from the training image. He also used a multiple-grid simulation approach that consists in simulating increasingly finer nested grids to capture training image patterns at various scales. In this multiple-grid approach, the conditioning data search neighborhood is defined by a template that only consists of nodes from the nested grid currently simulated. One search tree is built per data template, or per level of nested multiple grids.

The introduction of the search tree was the technical breakthrough that made SNESIM—a program developed by Strebelle—the first practical implementation of MPS simulation. Further progress was made a few years ago to improve MPS simulation time by reducing the size of the data template used to store the multi-point statistics in the search tree. The search template was designed such that it mostly consists of data locations corresponding to previously simulated nodes, i.e. nodes belonging to grids coarser than the grid currently simulated. Also, intermediary sub-grids were added to the traditional multiple-grid simulation approach to increase the relative proportion of previously simulated nodes in each nested grid.

These SNESIM implementation enhancements not only allowed users to reduce MPS simulation time, but also helped alleviate memory demand to build search trees. Alternative implementations of MPS simulation were proposed to tackle memory demand issues by classifying training patterns into a limited number of representative clusters, e.g., SIMPAT or FILTERSIM, but those solutions were found at the expense of increased simulation time and data conditioning issues.

The definition of the search template thus plays an important part for MPS simulations. It contains the maximum allowable conditioning data when simulating a cell. In theory, the larger the search template, the better the reproduction of the geometric features of the training image. But in practice, the size of the search template is limited by the CPU and RAM costs, and also by the size of the training image. For a given TI, a large search template may result in few replicates of patterns of that template size, and this leads to non-representative statistics. However, a small search template size will not be able to capture long-range patterns of the TI.

The use of multiple grids allows one to use a smaller size search template to save CPU time and at the same time to reproduce long-range patterns. The idea is to start the simulation on the cells at the coarsest level of the multiple grids and to finish it on the cells at the finest level of multiple grids. The search template geometry is defined once for all grids, but rescaled to each multiple grid level and a search tree is built for each multiple grid level.

The use of multiple grids is recommended, but a very large number of multiple grids will not necessarily improve the reproduction of long-range patterns because of the limited TI size that resulting in a limited number of data event replicates.

Thus, the ability to model long range patterns continues to present issues in fully realizing the benefits of MPS simulation.

Therefore, there is a need for improving MPS methods to capture long-range patterns or continuities without the concomitant enormous CPU and RAM costs.

**SUMMARY OF THE DISCLOSURE**

A novel regular-random hybrid simulation path is proposed to improve geobody continuity and hard data conditioning in Multiple Point Statistics simulation. The method uses an anisotropy-specific regular path on the coarsest level of multiple grids to better capture the large-scale continuity and uses a random path at the lower levels of multiple grids to explore more uncertain variability in stochastic simulation.

Usually, MPS simulation is performed sequentially cell by cell following a random path. Consider that all data points are located on the simulation grid. Otherwise, they are relocated to their respective nearest grid cells. We can define a search template with certain data configuration constituted by n nodes (u_{0}, u_{1}, u_{2}, u_{n}), where u_{0 }is the central node, u_{1}=u_{0}+h_{1}, u_{2}=u_{0}+h_{2}, u_{n}=u_{0}+h_{n}, and h_{i }is the separate distance between u_{i }and u_{0 }

Then, scan the TI with the search template for identifying all the replicates of the data event {Z(u_{0}+h_{1})=z_{1}, Z(u_{0}+h_{2})=z_{2}, . . . , Z(u_{0}+h_{n})=z_{n}} in the TI. For each training replicate of the data event, we read the value at location u_{0}.

The ensemble of these values allows us to build a local distribution of Z(u) conditioned to the data event {Z(u_{0}+h_{1})=z_{1}, Z(u_{0}+h_{2})=z_{2}, . . . , Z(u_{0}+h_{n})=z_{n}}. From this local distribution, a value of z(u_{0}) is drawn and assigned to the corresponding grid cell, and then the simulation is moved to the next randomly selected cell on the simulation grid. This process is repeated until the all grid cells are filled.

In contrast, we have used a regular path—unilateral regular or bilateral regular path—for the cells at the coarsest level of the multiple grids, and used the typical random path for the cells at the rest of the multiple grid levels.

We have proven herein that in the unconditional simulation case, the simulated geobodies have equally improved continuity no matter whether the lateral regular path on the coarsest grid follows a horizontal I or J lateral direction and no matter what the major continuity direction of the geobodies is.

We also solved the hard data conditioning issue of the regular path by making the regular path follow the minor geobody continuity direction.

In the vertical direction of a 3D model, we proposed to reduce the number of multi-grid levels and use the regular path at the coarsest level of the multiple grids to achieve better geobody continuity.

The invention includes and one or more of the following embodiments, in any combination(s) thereof:

By “Multiple Point Statistics” what is meant is an alternate approach to two-point geostatistical methods that uses a training image instead of a variogram to estimate the conditional probability at the simulation location given the observed data and the previously simulated values.

A “search template” is defined by vectors radiating from a central node. The template is constituted by certain number of nodes. That template is used to scan the training image and record all training patterns.

By “grid,” what is meant herein is a discretization of a 3D space into a grid like pattern, such that the space can be simulated from cell to cell. Grid level refers to the spacing between the grid levels, thus a multi-grid level of 3 refers to a spacing of three cells in a single level, whereas a multi-grid level of 1 means that every cell starts a new grid level in that axis.

The simulation grid shape is typically “square” (e.g., with equidistance nodes), however, we can also use rectangular grid cells, where the grid size is reduced in one direction. Like other geostatistical algorithms, the MPS simulation is performed on the Cartesian grid.

By “coarsest” grid what is meant is the highest level of grid cells in the multiple grid system (e.g., the grid cells are the largest).

By “intermediate” grid, what is meant is the intermediate level(s) of grid cells in the multiple grid system.

By “finest” grid, what is meant is the lowest level of grid cells in the multiple grid system (e.g., the grid cells are the smallest).

Multiple levels of multiple grids can be used, but three or four is typical.

By “regular-random hybrid” or “hybrid” path, what is meant is that the simulation path at the coarsest level of the multiple grids is in a regular order, and that at the intermediate and the finest grid level is in a random order.

By “stationarity” what is meant is the condition in which the data do not exhibit any trend. This implies that a moving-window average shows hom*ogeneity in the mean and variance over the area of interest. As an example, the stationarity of a MPS facies model implies the facies proportions, geobody size and orientations remains statistically similar in the simulation region or the entire model.

By “regular” path what is meant is that the grid cells are simulated are in a predictable pattern. See e.g.,

By “unilateral regular” path what is meant is that the grids are simulated row-by-row or column-by-column or on the diagonal in a single direction. See e.g.,

By “bilateral regular” path what is meant is that the grids are simulated row-by-row or column-by-column or on the diagonal in a back-and-forth or two directional manner.

The use of the word “a” or “an” when used in conjunction with the term “comprising” in the claims or the specification means one or more than one, unless the context dictates otherwise.

The term “about” means the stated value plus or minus the margin of error of measurement or plus or minus 10% if no method of measurement is indicated.

The use of the term “or” in the claims is used to mean “and/or” unless explicitly indicated to refer to alternatives only or if the alternatives are mutually exclusive.

The terms “comprise”, “have”, “include” and “contain” (and their variants) are open-ended linking verbs and allow the addition of other elements when used in a claim.

The phrase “consisting of” is closed, and excludes all additional elements.

The phrase “consisting essentially of” excludes additional material elements, but allows the inclusions of non-material elements that do not substantially change the nature of the invention.

The following abbreviations are used herein:

**BRIEF DESCRIPTION OF THE DRAWINGS**

**DETAILED DESCRIPTION**

The MPS facies modeling method proposed by Strebelle is increasingly gaining attention in the petroleum geological modeling community. MPS models reproduce the geological patterns and also are flexible in data conditioning since they are based on a pixel-based algorithm. One issue of the current MPS algorithm, however, is that the long range or larger scale sinuous geobody (e.g. fluvial channel) continuity cannot be accurately reproduced. Discontinued channels are generated in MPS models even though they do not exist in the training image. The resulting blind end channels will negatively impact predictions of reservoir connectivity and reservoir flow performance.

To reproduce the large scale pattern continuity, Daly and Hu proposed to use a unilateral path on all grid cells. This method improved channel continuity, but reduced uncertainty ranges resulting from deterministic path and also it could not accurately condition model realizations to hard data.

As another possible solution, Strebelle and Remy proposed a re-simulating method. They identified the nodes simulated with a limited conditioning data and re-simulated them. This method required several iterations of re-simulation to complete a realization. Thus, it consumed more CPU time.

We propose a novel approach herein to address the long geobody continuity reproduction problem in MPS. This disclosure addresses the geobody continuity and hard data conditioning issues from one or more of the following aspects:

1. The regular path is used on the cells at the coarsest level of the multiple grid system and the random path is used on the rest of cells. This method significantly improves geobody continuity in MPS facies simulation. A unilateral or bilateral regular simulation path can be used, or the path can be regular in one axis and random in others, but this may be less preferred.

2. In the lateral direction, the regular unilateral path can follow I or J direction, and it has been proven that they produce equally much improved geobody continuity for unconditional simulations.

3. In the vertical direction, an improved vertical continuity can be achieved by reducing multiple grid level in the vertical axis; hence it increases the number of cells (along that axis) using the regular path since the regular path is only applied to the cells at the coarsest level of multiple grids and in the vertical direction there are now more cells.

4. Using the regular path following the minor geobody continuity direction resolves the hard data conditioning issue related to the regular simulation path.

5. We demonstrated the claimed method works for modeling 2D and 3D cases, modeling non-stationarity and modeling the geobodies with various major continuity directions.

The disclosure thus helps to build more geologically realistic facies model, which can provide better guidance in drilling new wells and production optimization.

In geostatistical simulation algorithms, the random simulation path is typically used. See

In geostatistical simulation algorithms, multiple grid schemes are usually used, as shown in **3**), yellow cells are at the intermediate multiple grid level (multiple grid level **2**) and grey ones are at the finest multiple grid level (multiple grid level **1**). Normally, the stochastic simulation is performed first at the coarsest level and then move to the cells at the next level of the multiple grids. At each grid level, the cells are simulated one by one in a random sequence.

Instead of the random simulation sequence, in this disclosure we used a regular unilateral path on the coarsest multi-grid and used a random path for the rest of multi-grid levels. This is referred to as a “regular-random-hybrid path” or just “hybrid path” herein. The intention is to capture the large scale continuity at the highest level of the multiple grids, in the meantime to maintain the uncertainty assessment capability on the lower level multi-grid cells.

Although the work presented herein used a regular unilateral path, such as in

Although not shown in

In

One issue of regular path is that it creates problems in conditioning the hard data (see the green ellipses in

We proposed that the regular path should therefore be along the minor continuity direction, which is along axis J in this case. We achieved perfect well data conditioning by following the minor continuity direction (perpendicular to channel direction) as well as much better channel continuity. Examples demonstrating this are shown in

If the channel orientation is not along I or J, but is instead at a 45 degree angle, both I and J regular paths give equally improved channel continuity. See e.g.,

For 3D cases, our studies indicated that we needed to include more cells in the regular path in order to improve the vertical geobody continuity. We proposed to reduce the number of multiple grid levels in the vertical direction, say from 3 to 2 or 1. Reducing the multi-grid level to 2 or 1 increases the numbers of cells in the vertical axis, thus improving continuity.

In

The following references are incorporated by reference in their entirety for all purposes:

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