John G. Manchuk

A flexible sequential Gaussian simulation program: USGSIM

Sequential Gaussian simulation is a widely used algorithm for the stochastic characterization of properties from various earth science disciplines. Many variants have been developed to deal with the increasing complexity of modeling applications. The program described in this paper is a flexible, tested, and documented implementation. Multiple variables can be cosimulated within different rock types simultaneously. The stepwise transform is integrated into the program as are collocated cokriging, collocated cokriging with the intrinsic model, and cokriging with a linear model of coregionalization for the cosimulation of multiple variables. Multiple secondary data can be incorporated using locally varying means, collocated cokriging, and Bayesian updating. The search options and other parameters are flexible within rock types. Fortran source code and a compiled executable are provided.

Implementation aspects of sequential Gaussian simulation on irregular points

The increasing use of unstructured grids for reservoir modeling motivates the development of geostatistical techniques to populate them with properties such as facies proportions, porosity and permeability. Unstructured grids are often populated by upscaling high-resolution regular grid models, but the size of the regular grid becomes unreasonably large to ensure that there is sufficient resolution for small unstructured grid elements. The properties could be modeled directly on the unstructured grid, which leads to an irregular configuration of points in the three-dimensional reservoir volume. Current implementations of Gaussian simulation for geostatistics are for regular grids. This paper addresses important implementation details involved in adapting sequential Gaussian simulation to populate irregular point configurations including general storage and computation issues, generating random paths for improved long range variogram reproduction, and search strategies including the superblock search and the k-dimensional tree. An efficient algorithm for computing the variogram of very large irregular point sets is developed for model checking.

Direct Geostatistical Simulation on Unstructured Grids

Unstructured grids are commonly used in reservoir modeling and are being increasingly considered in complex mining engineering applications. Block kriging of the attributes can be easily implemented; however, this implicitly assumes linear averaging, which is not the case after Gaussian transformation or with variables such as permeability. Direct simulation has been proposed as a solution; however, there are a number of important implementation considerations. This paper addresses the following considerations: (1) search for nearby relevant block and point data, (2) stabilization of the kriging equations and weights in presence of complex screening, (3) correction of the homoscedastic kriging variance to account for realistic proportional effect, (4) determination of valid conditional distribution shapes, (5) accounting for geological controls including stratigraphic surfaces and mixture of multiple facies within an unstructured grid block, and (6) accounting for directional permeability that does not average linearly. Direct simulation on unstructured grids is made practical by addressing these six considerations.

Accounting for non-exclusivity in sequential indicator simulation of categorical variables

Facies models are used to better capture heterogeneity in mineral deposits and petroleum reservoirs. Facies are often considered as mutually exclusive and exhaustive at the scale of the geological model. These two assumptions are needed for sequential indicator simulation and most other facies modeling techniques; however, the assumption that an entire grid block consists of one facies type becomes unreasonable as the scale increases. Most geological models are built at a scale that is larger than the scale of variation of facies. Mixing of multiple facies types within a grid cell is common, especially in zones of transition between different facies. This paper develops a new technique to address the issue of non-exclusivity of facies within grid cells. The approach quantifies the uncertainty resulting from majority-vote upscaling of facies from core or well log scale to the grid cell scale and utilizes this uncertainty to build better models of continuous reservoir properties such as bitumen grade. Uncertainty is quantified using a measure of entropy that is capable of handling situations where there may be similarity between different facies types. A methodology to implement entropy in geo-modeling is introduced and demonstrated with several small examples. An example involving real data from the McMurray formation of Totals Joslyn lease is used to demonstrate the improvement in accuracy compared with traditional modeling workflows.

Micro-modeling for Enhanced Small Scale Porosity-Permeability Relationships

Accurate modeling of vertical and horizontal permeability in oil sands is difficult due to the lack of representative permeability data. Core plug data could be used to model permeability through the inference of a porosity-permeability relationship. The drawbacks of this approach include: variability and uncertainty in the porosity-permeability scatter plot as a result of sparse sampling, and biased core plug data taken preferentially from sandy or homogeneous intervals. A two-step process can be used where core photographs and core plug data are used to assess small scale permeability followed by upscaling to a representative geomodeling cell size. This paper expands on a methodology that utilizes core photographs to infer porosity-permeability relationships. This methodology is robust because there is abundant core photograph data available compared to core plug permeability samples and the bias due to preferential sampling can be avoided. The proposed methodology entails building micro-scale models with 0.5 mm cells conditional to 5 cm×5 cm sample images extracted from core photographs. The micro-models are sand/shale indicator models with realistic permeability values (k sand≈7 000 mD, k shale≈0.5 mD). The spatial structure of the micro-model controls the resulting porosity-permeability relationships that are obtained from upscaling. Previously, these models were generated with sequential indicator simulation (SIS). However, SIS may not capture the spatial structure of the complex facies architecture observed in core photographs. Models based on multiple point statistics and object based techniques are proposed to enhance realism. Micro-models are upscaled to the scale of the log data (5 cm in this case) with a steady-state flow simulation to determine the porosity-permeability relationship. The porosity-permeability relationships for geomodeling, or flow simulation, can be determined with subsequent mini-modeling and further upscaling. The resulting porosity-permeability relationship can be used to populate reservoir models and enhance traditional core data. Wells from the Nexen Inc. Long Lake Phase 1 site in the Alberta Athabasca oil sands region are used to demonstrate the methodology.

Reproduction of secondary data in projection pursuit transformation

A challenge when working with multivariate data in a geostatistical context is that the data are rarely Gaussian. Multivariate distributions may include nonlinear features, clustering, long tails, functional boundaries, spikes, and heteroskedasticity. Multivariate transformations account for such features so that they are reproduced in geostatistical models. Projection pursuit as developed for high dimensional data exploration can also be used to transform a multivariate distribution into a multivariate Gaussian distribution with an identity covariance matrix. Its application within a geostatistical modeling context is called the projection pursuit multivariate transform (PPMT). An approach to incorporate exhaustive secondary variables in the PPMT is introduced. With this approach the PPMT can incorporate any number of secondary variables with any number of primary variables. A necessary alteration to the approach to make this numerically practical was the implementation of a continuous probability estimator that relies on Bernstein polynomials for the transformation that takes place in the projections. Stopping criteria were updated to incorporate a bootstrap t test that compares data sampled from a multivariate Gaussian distribution with the data undergoing transformation.

Applications of Data Coherency for Data Analysis and Geological Zonation

A measure of coherency in facies between nearby wells is developed to aid in the processes of quality control of well data and geological zonation. Coherency measures the agreement between a well and its immediate neighbors based on structural markers and facies interpretations. The calculation can be done incrementally on sets of wells to identify incoherencies caused by errors in interpretation, measurement differences, data acquisition problems or actual changes caused by geological differences. Attributes may include the year a well was interpreted or included in a database, the interpreter, or what logging tool was used. Coherency is also used as a similarity metric in a hierarchical clustering algorithm for geological zonation. Results are applied in several examples that demonstrate the use of coherency and clustering for quality control and grouping data into geologically similar objects.

Upscaling permeability to unstructured grids using the multipoint flux approximation

Grids used for flow simulation are often at a much coarser scale than that of grids for geological modelling due to computational demand. Unstructured grids offer increased flexibility for the flow grid design; however, solving the flow equations and upscaling from high resolution geological grids to the coarse flow grid is more complex than using coarse regular grids. The multipoint flux approximation (MPFA) is one technique applied to discretize the flow equations on unstructured grids. This paper develops an upscaling technique that uses the MPFA method to solve the flow equations on the fine- and coarse-scale grids. Unlike most cases where the fine-scale grid is regular or structured, this work utilizes a high resolution triangular grid that conforms to the coarse-scale grid. The triangular grid is generated using the coarse-scale interaction regions as constraints. Upscaling leads to transmissibility matrices of the coarse-scale interaction regions. Two different types of local boundary conditions for the MPFA upscaling approach are developed, including linear varying pressures and pressures computed by solving the flow equations around the element boundary. The method is tested using flow simulation on several cases. Results are comparable with flow using a high resolution regular grid.

Multivariate Geostatistical Simulation of Full Permeability Tensors on Unstructured Grids

Unstructured grids are being used more frequently in reservoir modeling; however, the tools for populating them are not fully developed. This work will introduce methods of geostatistical simulation on unstructured grids specifically for full permeability tensors. Modeling and drawing samples from a multivariate distribution will be required. There will be no assumptions of normality for this distribution.