Jeff B. Boisvert

Programs for kriging and sequential Gaussian simulation with locally varying anisotropy using non-Euclidean distances

Geological deposits display nonlinear features such as veins, channels or folds that result in complex spatial anisotropies that are difficult to model with currently available geostatistical techniques. The methodology presented in this paper for incorporating locally varying anisotropy in kriging or sequential Gaussian simulation is based on modifying how locations in space are related. Normally, the straight line path is used; however, when nonlinear features exist the appropriate path between locations follows along the features. The Dijkstra algorithm is used to determine the shortest path/distance between locations and a conventional covariance or variogram function is used. This nonlinear path is a non-Euclidean distance metric and positive definiteness of the resulting kriging system of equations is not guaranteed. Multidimensional scaling (landmark isometric mapping) is used to ensure positive definiteness. In addition to the variogram, the only parameters required for the implementation of kriging or sequential Gaussian simulation with locally varying anisotropy are (1) the local orientation and magnitude of anisotropy and (2) the number of dimensions required for multidimensional scaling. This paper presents a suite of programs that can be used to krige or simulate practically sized geostatistical models with locally varying anisotropy. The programs kt3d_LVA, SGS_LVA and gamv_LVA are provided.

A library of training images for fluvial and deepwater reservoirs and associated code

Geostatistical algorithms that consider multiple-point statistics are becoming increasingly popular. These methods allow for the reproduction of complicated features beyond the commonly implemented variogram. In practice, it is not possible to infer many multiple-point statistics directly from the available data; therefore, it is common to borrow statistics from training images. A library of training images is developed for fluvial and deepwater depositional settings. These training images are based on object-based models, surface-based models and pseudo-genetic process mimicking (event-based) models. The training images represent a range of net-to-gross fractions and depositional styles. Associated code provides the ability to modify, format and tailor the training images and to extract multiple-point statistics. The training image library provides a source for multiple-point statistics, can be used in comparative flow studies and as an aid in scenario-based uncertainty studies.

Geometallurgical Modeling at Olympic Dam Mine, South Australia

Modeling of geometallurgical variables is becoming increasingly important for improved management of mineral resources. Mineral processing circuits are complex and depend on the interaction of a large number of properties of the ore feed. At the Olympic Dam mine in South Australia, plant performance variables of interest include the recovery of Cu and U3O8, acid consumption, net recovery, drop weight index, and bond mill work index. There are an insufficient number of pilot plant trials (841) to consider direct three-dimensional spatial modeling for the entire deposit. The more extensively sampled head grades, mineral associations, grain sizes, and mineralogy variables are modeled and used to predict plant performance. A two-stage linear regression model of the available data is developed and provides a predictive model with correlations to the plant performance variables ranging from 0.65–0.90. There are a total of 204 variables that have sufficient sampling to be considered in this regression model. After developing the relationships between the 204 input variables and the six performance variables, the input variables are simulated with sequential Gaussian simulation and used to generate models of recovery of Cu and U3O8, acid consumption, net recovery, drop weight index, and bond mill work index. These final models are suitable for mine and plant optimization.

Geostatistics with Locally Varying Anisotropy

Since the origin of geostatistics with Krige (1951) and Matheron’s (1962) pioneering work, numerous new techniques have been developed to generate distributions of properties of interest at unsampled locations. Many of these techniques are based on the theory of kriging; introductory discussions of kriging can be found in geostatistical text books such as Journel and Huijbregts (1978), Deutsch (2002) or Isaaks and Srivastava (1989). One advantage of using kriging to generate estimates is the incorporation of anisotropy into the modeling process. Anisotropy is the concept that the properties in a geological deposit are often more continuous in one direction than another. Consider the locally varying anisotropy shown in these cross sections:

A methodology to construct training images for vein-type deposits

Fracture models of vein formation can produce realistic training images (TIs) for use in multiple-point geostatistics. Vein formation is modeled by applying flow simulation to a fracture model to mimic the flow of an ore-bearing fluid through fractured rock. TIs are generated by assuming that veins form in areas of high flow where there would be preferential deposition of the mineral of interest. We propose a methodology to simulate mineralized veins by constructing a fracture model within the deposit, modeling the permeability and simulating the flow of ore-bearing fluids. The veins are defined by considering the areas of high flow. The methodology is implemented with a fracture model of the Whiteshell area in Manitoba. To assess the reasonableness of the TIs, comparisons are made to geological models of gold deposits in Quebec and Nova Scotia that display similar geometric characteristics such as braiding, thickening and thinning. A FORTRAN program TIGEN, based on GSLIB program formats is included and can be used to generate TIs from fracture models.

Inference of locally varying anisotropy fields from diverse data sources

A locally varying anisotropy (LVA) field characterizes the magnitude and direction of anisotropy in a modelling domain and can provide valuable information to geostatistical modeling methodologies that account for non-linear spatial features, in particular, second order non-stationarities that are observed in complex geological formations. In this framework, the primary variable to model (i.e. grade, porosity, concentration, etc) is measured at discrete spatial locations and there may or may not be exhaustively sampled gridded secondary information. Existing geostatistical techniques require knowledge of the magnitude of anisotropy (correlation lengths) and local orientations (strike, dip, and plunge) exhaustively in the domain. Inference of these parameters from samples of the primary variable is difficult when anisotropy varies locally; the correlation lengths and orientations of the discrete primary variable must be estimated at all locations in a domain and are not globally constant. Estimation of the required LVA field parameters depends on the available data and is unique to the type of information available. This work proposes three different methodologies for LVA field inference, including (1) use of a correlated exhaustively sampled secondary variable (2) use of direct point measurements of the orientation of the primary variable, and (3) use of a geological model that displays the desired LVA characteristics. Often a gridded exhaustively sampled secondary variable is available from geophysical surveys and displays similar spatial features as the primary variable. An adaptive moving window technique is proposed to infer the required LVA field from correlated exhaustive secondary information. The local anisotropy orientation is obtained from the gradient of the secondary information and the appropriate window size is calculated automatically, which is important when the scale of features vary in the domain. Secondly, direct angle measurements of orientation may be available from dip meter or borehole images, however, these data are axial in nature (when plunge is assumed to be 0?) or they represent a 3D coordinate system (when considering strike, dip and plunge); preprocessing is required before the LVA field can be estimated from point measurements of orientation because of the wrapping effect at 0?. The proposed technique treats point measurements of orientation as quaternions and assigns weights to nearby samples to estimate the exhaustive LVA field. In addition to the proposed methodologies for exhaustive and point data, a skeletonization technique is implemented to determine orientations from geological interpretations. The geological object is reduced to a spline curve and the direction of the curve is used as the local LVA field orientation. The suite of proposed LVA inference techniques are implemented in both 2D and 3D and illustrated on several data sets. Modeling non-stationary features is important, tools are provided to help with this.We model locally varying anisotropy fields for spatial data.Input data could include orientation samples, exhaustive data and geological models.Tools provided can be applied todifferent non-stationary methodologies.Programs are provided for all techniques discussed.

Using a Discrete Fracture Network and Spatial Point Processes to Populate Veins and Model Grade in a Coarse Gold Deposit

Vein-hosted gold deposits are characterized by mineralization, which is spatially restricted to narrow vein structures. Drillholes intersecting a mineralized vein can lead to unreliable and biased assay values compared to selective mining unit scale block grades. In this work, a discrete fracture network is simulated and adapted to model gold mineralization within the veins. Veins are assumed planar and the required inputs are distributions of vein orientation, vein length, and vein intensity (i.e., density). These inputs are collected from drillhole data, geological mapping, and expert knowledge of the deposit. A spatial point process is then applied to model gold grade as discrete events or “nuggets,” which are spatially restricted to the simulated quartz veins for the case of incomplete mineralization of the veins; when the vein is completely mineralized, a vein thickness distribution is required. The methodology is applied to an epithermal gold deposit in northwestern British Columbia, Canada and shows improvement in restricting the influence of the high-grade gold samples without resorting to ad-hoc manipulation of input assays through capping or cutting. The final output of this methodology is a block model of gold grade, which better honors the spatial structure of the veins in the deposit and is suitable for use in mine planning or resource estimation.

Conditioning 3D Object Based Models to a Large Number of Wells: A Channel Example

Object based modeling is commonly used for generating facies or rock type models that better reproduce complex realistic geology. A drawback of object based modeling is the difficulty of conditioning to dense data. Object based models have other uses, but their use as training images (TI) has become very prevalent in multiple point simulation (MPS) workflows; however, if the object could be conditioned to dense data, they would be used directly as facies models for complex deposits. The proposed methodology is to consider an object as defined by a set of parameters. Optimization of this object is based on the mismatch with data at the well locations. No gradients are used and any object that can be defined by a finite number of parameters could be conditioned to well data. This does not preclude the use of process based models rather than object based models; in this framework, process based models are more complex, yet fully parametric, models. Four different optimization schemes are reviewed for conditioning. An example of fluvial channels with crevasse splays is presented. Conditioning is considered on dense data up to 100 wells and performs well, requiring seconds to condition.

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.

Impact of Heterogeneous Geomechanical Properties on Coupled Geomechanical-Flow Simulation of SAGD

In the modern oil industry, geostatistical property models are built for different purposes such as resource estimation and flow simulation. Processing of multiple realizations, obtained from geostatistical simulation techniques, helps assess uncertainty analysis which is important for development planning and decision-making processes. Each geological model is a combination of structural, facies, and attributes models. In the case of conventional flow simulation (i.e. without considering geomechanical simulation), the petrophysical properties porosity, permeability and saturation, are the only attributes necessary to model. These parameters are included in the fluid flow governing equations. But in the case of dealing with coupled geomechanical-flow simulation, rock mechanical properties are also required. In the case of conventional simulation process, geostatistical property models have been used widely, but in the case of coupled geomechanical-flow simulation processes, geostatistical modeling for geomechanical attributes has yet to be incorporated. Therefore, uncertainty assessment could be underestimated according to the spatial distribution of these parameters. In this work, the effect of heterogeneous geomechanical properties on coupled geomechanical-flow simulation process was investigated for a steam assisted gravity drainage (SAGD) process for a heavy oil reservoir in Alberta-Canada. Cumulative oil Production (COP), Steam Oil Ratio (SOR) and Vertical Deformation Profile (VDP) of the top of reservoir is considered as three simulation output variables. Consideration of heterogeneous models for both flow and geomechanical properties in coupled geomechanical flow simulation of the SAGD process resulted in a range of uncertainties for these three variables. The importance of considering geomechanical properties as heterogeneous models is illustrated by comparing these ranges with the ranges obtained from coupled simulations in which geomechanical properties are considered as homogeneous models. Representative synthetic data of a sand/shale spatial distribution of McMurray formation in Alberta-Canada is considered for the case study.