Two Artifacts of Probability Field Simulation

Probability field simulation is being used increasingly to simulate geostatistical realizations. The method can be faster than conventional simulation algorithms and it is well suited to integrate prior soft information in the form of local probability distributions. The theoretical basis of probability field simulation has been established when there are no conditioning data; however, no such basis has been established in presence of conditioning data. Realizations generated by probability field simulation show two severe artifacts near conditioning data. We document these artifacts and show theoretically why they exist. The two artifacts that have been investigated are (1) local conditioning data appear as local minima or maxima of the simulated values, and (2) the variogram model in range of conditioning data is not honored; the simulated values have significantly greater continuity than they are supposed to. These two artifacts are predicted by theory. An example flow simulation study is presented to illustrate that they affect more than the visual appearance of the simulated realizations. Notwithstanding the flexibility of the probability field simulation method, these two artifacts suggest that it be used with caution in presence of conditioning data. Future research may overcome these limitations.     

Kriging with strings of data

The concept of a random function and, consequently, the application of kriging cells for the implicit assumption that the data locations are embedded within an infinite domain. An implication of this assumption is that, all else being equal, outlying data locations will receive greater weight because they are seen as less redundant, hence, more informative of the infinite domain. A two- step kriging procedure is proposed for correcting this siring effect. The first step is to establish the total kriging weight attributable to each string. The distribution of that total weight to the samples in the string is accomplished by a second stage of kriging. In the second stage, a spatial redundancy measure r_(n) is used in place of the covariance measure in the data-data kriging matrix. This measure is constructed such that each datum has the same redundancy with the (n)data of the string to which it belongs. This paper documents the problem of kriging with strings of data, develops the redundancy measure r_(n),and presents a number of examples.     

A new precambrian paleomagnetic pole for northern Europe

A paleomagnetic study was made of the granitic rock farsundite, exposed in southern Norway. An objective was to test the contemporaneity of this body with the neighbouring Egersund anorthosite of presumed age about 900 m.y. Two of the nine sites sampled were rejected, as the magnetization was dominantly unstable. At the seven other sites, this unstable component was either absent or it could be equally well removed by AF or thermal demagnetization: after AF treatment, all samples from these sites were left with a very stable remanence, directed steeply upwards. This magnetization was probably acquired at the time of either emplacement or recrystallization of the farsundite. A magnetic test for anisotropy indicated that the stable remanence is misaligned with the ancient Earth's field direction by about 3°, apparently due to layering of the rock fabric. After correction for this anisotropy, the mean direction from the seven sites is D = 341°, I = 82.2°, k = 142, α = 5.0°, corresponding to a paleomagnetic north pole at 43.3°S, 166.0°W, dp = 9.3°, dm = 9.7°, which lies on Spall's European polar wandering curve. The farsundite pole is not significantly different from a pole position based on the Egersund anorthosite, which supports the supposition that the two rock formations are cogenetic.     

On the complex magnetic behaviour of titanomagnetites

The bulk magnetic properties, including Curie temperature, susceptibility and hysteresis at various temperatures, and the Mössbauer spectra of coarse synthetic members of the titanomagnetite series are compared as a function of titanium content, using some previously reported and some new results. It has been noted previously that the titanium-rich members fail to show the expected “true multidomain” behaviour observed in pure magnetite. One possible explanation is that the behaviour observed may be due to a tendency to inhibit domain wall formation in some titanomagnetites. In this paper we discuss some other possible mechanisms to account for the magnetic properties of such titanomagnetites observable even when domain-wall-related structures do form in them. These mechanisms suggest that magnetocrystalline anisotropy and its temperature dependence in titanomagnetites may be related to local magnetic inhomogeneities, and do not depend only on crystal structure as in the case of classical ferromagnetic materials.     

Reservoir Characterization of McMurray Formation by 2D Geostatistical Modeling

There is a need to estimate reserve uncertainty for large lease areas. Detailed 3D models of heterogeneity are not necessarily required, but there is a need to integrate all available data into an in-situ reserve estimate with uncertainty. A 2D mapping approach is presented to appraise reserves accounting for multiple variables, multiple data sources, and uncertainty. The approach can be considered in three primary steps: (1) Bayesian updating is used to determine local distributions of uncertainty for each primary variable while integrating multiple secondary information, (2) matrix simulation is employed to jointly and simultaneously simulate multiple collocated variables to determine a derived variable such as OOIP, and (3) probability field simulation then is applied to permit joint simulation of multiple locations. This methodology permits local and global uncertainty assessment while integrating multiple sources of information. An application to the McMurray Formation in Alberta, Canada is demonstrated.     

Author index 1984 (volumes 67–71)

Magnetic properties of samples from Bell Island sedimentary rocks have been studied. X-ray analysis indicates that the main magnetic mineral is hematite in all samples. The other iron-bearing minerals identified are siderite and chamosite. Microscope observations of thin sections suggest that the rocks consist of oolitic hematite in a matrix of siderite or calcite. The intensity of natural remanent magnetization (NRM) varies in the range of (0.03–0.4 A m^?1), depending on the percentage of hematite. The thermal demagnetization curves of NRM show in some cases a sharp increase in magnetization at temperatures in the range 500–600°C. The peaks that occur in these demagnetization curves are due to a chemical change of siderite during repeated laboratory heating. X-ray analysis confirmed that the newly formed material is magnetite. Since the original NRM has been masked by the new intergrown material, this would result in a serious error in the determination of paleomagnetic pole positions. The samples showing this behaviour were not considered for paleomagnetic study. The samples containing oolitic hematite in a calcite matrix exhibit very high stability of NRM, including directional stability until almost 670°C. For these samples, a virtual pole position based on N = 6 samples (32 specimens) demagnetized to 665°C is 34°N, 114°E, not far from published Ordovician poles for the North American craton.     

Young Alpine dykes south of the Tauern Window (Austria): a K-Ar and Sr isotope study

Fortyfive new K-Ar ages and Sr isotope data on amphiboles, biotites, clinopyroxenes and whole rock samples from subvolcanic dykes south of the Tauern Window establish, that alkalibasaltic dykes were intruded 30 m.y. ago and shoshonitic volcanism occured between 30 and 24 m.y. ago. Two calc-alkaline rocks of high-potassium composition yielded ages of 40 and 26 m.y. resp., a spread which may or may not be real. Calc-alkaline dykes with medium and low potassium contain excess argon and are hence undatable. Alkalibasaltic dykes have ⁸⁷Sr/⁸⁶Sr ratios of 0.7056–0.7070, shoshonitic rocks 0.7075–0.7133, potassium rich calc-alkaline dykes 0.7077–0.7100. ⁸⁷Sr/⁸⁶Sr of all other calc-alkaline rocks scatter between 0.7074 and 0.7150. Sr data indicate that dykes studied do not represent closed Sr systems, but that Sr characteristics result from selective strontium assimilation en route to surface. Primary Sr isotopic ratios of alkalibasaltic dykes point to an origin of these rocks in enriched sub-continental upper mantle. The source region of shoshonitic and high-potassium calcalkaline rocks could have ⁸⁷Sr/⁸⁶Sr around 0.707, which is assigned to the input of a component rich in alkalies, LREE and LIL elements. Genetic relationships with other Tertiary magmatites of similar geotectonic position are explained in terms of plate tectonic models of the Eastern Alps.     

Kriging in a finite domain

Adopting a random function model {Z(u),u ε study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed.     

Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil-water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.