Jerry Lee Jensen

Permeability tensors for sedimentary structures

Accurate modeling of fluid flow through sedimentary units is of great importance in assessing the performance of both hydrocarbon reservoirs and aquifers. Most sedimentary rocks display structure from the mm or cm scale upwards. Flow simulation should therefore begin with grid blocks of this size in order to calculate effective permeabilities for larger structures. In this paper, we investigate several flow models for sandstones, and examine their impact on the calculation of effective permeability for single phase flow. Crossflow arises in some structures, in which case it may be necessary to use a tensor representation of the effective permeability. We establish conditions under which tensors are required, e.g., in crossbedded structures with a high bedding angle, high permeability contrast, and laminae of comparable thickness. Cases where the off-diagonal terms can be neglected, such as in symmetrical systems, are also illustrated. We indicate how the method of calculating tensor permeabilities may be extended to model multiphase flow in sedimentary structures.

Immiscible flow behaviour in laminated and cross-bedded sandstones

Abstract In this paper, we describe models of water/oil displacement in typical, geologically-structured media. We focus specifically on laminated and cross-bedded structures, since these are almost ubiquitous in clastic sedimentary reservoirs. The importance, for field-scale models, of properly representing the interaction of viscous, capillary and gravitational forces with small-scale heterogeneity is clearly demonstrated. Because the length-scale, δχ, of sedimentary lamination is of order 10−3 to 10−2 m, capillary forces, which are inversely proportional to δχ, may play a very significant role in determining the effective flow behaviour at larger scales. We show that there are important differences in oil recovery between cross-layer flow and along-layer flow. Differences of up to a factor of two in ultimate recovery may be produced by different representations of realistic clastic sedimentary structure (such as parallel lamination, cross-lamination and small-scale faulting). The significance of these findings is in determining the correct scale-up procedure for the multiphase effective flow parameters. We use the term geopseudo to describe correctly-scaled, multi-phase, pseudofunctions which capture the effects of small-scale sedimentary structure. Field-scale reservoir models must take account of these small-scale effects in order to lay claim to reasonable accuracy in production forecasts.

Permeability semivariograms, geological structure, and flow performance

Clastic sediments may have a strong deterministic component to their permeability variation. This structure may be seen in the experimental semivariogram, but published geostatislical studies have not always exploited this feature during data analysis and covariance modeling. In this paper, we describe sedimentary organization, its importance for flow modeling, and how the semivariogram can be used for identification of structure. Clastic sedimentary structure occurs at several scales and is linked to the conditions of deposition. Lamination, bed, and bedset scales show repetitive and trend features that should be sampled carefully to assess the degree of organization and levels of heterogeneity. Interpretation of semivariograms is undertaken best with an appreciation of these geological units und how their features relate to the sampling program. Sampling at inappropriate intervals or with instruments having a large measurement volume, for example, may give misleading semivariograms. Flow simulations for models which include and ignore structure show that the repetitive features in permeability can change anisotropy and recovery performance significantly. If systematic variation is present, careful design of the permeability fields therefore is important particularly to preserve the structure effects.

Use of the geometric average for effective permeability estimation

The geometric average is often used to estimate the effective (large-scale) permeability from smaller-scale samples. In doing so, one assumes that the geometric average is a good estimator of the geometric mean. Problems with this estimator arise, however, when one or more of the samples has a very low value. The estimate obtained becomes very sensitive to the small values in the sample set, while the true effective permeability may be only weakly dependent on these small values. Several alternative methods of estimating the geometric mean are suggested. In particular, a more robust estimator of the geometric mean, the jth Winsorized mean, is proposed and several of its properties are compared with those of the geometric average.

A Method for Calculating Permeability Tensors using Perturbed Boundary Conditions

For reservoir simulation, it is usually necessary to represent fine-scale permeability heterogeneities by larger scale effective permeabilities. The effective permeability of a heterogeneous medium is a tensor and depends on the boundary conditions which dictate the direction of flow through the medium. We have reviewed current methods for determining effective permeability tensors, and find that existing methods either apply one type of boundary conditions, or give approximate results for a range of boundary conditions. This paper presents a new method for calculating the effective permeability tensors for single phase flow. The method is based on a pressure perturbation scheme which uses two flow cases. The first case uses boundary conditions which reflect the actual flow conditions for the medium. The second case uses a perturbation of the pressures calculated from the first case. This perturbation is applied first in the horizontal and then in the vertical directions. By using perturbed pressures, the flow is not distorted by unrepresentative boundary conditions. Each term of the effective permeability tensor is proportional to the ratio of the increment in flow to the increment in pressure gradient. This method has been tested using a variety of 2D permeability fields, both stochastic and deterministic, and gives good agreement with analytical results. We have applied the method to study the effects of no-flow boundaries in deterministic fields, representing certain types of elementary bedform. We have also investigated the effect of coarse-block size on the effective permeability in correlated random fields.

The Influence of Sample Size and Permeability Distribution on Heterogeneity Measures

This study evaluates the influence of sample size on the Dykstra-Parsons and Lorenz measures of heterogeneity. Because either coefficient is determined for a reservoir from a finite number of data, only an estimate is made of the true coefficient. The authors show that, on average, the estimate is less than the true value. They give the relationship between estimate error and number of samples and show how significant errors may arise when too few data are used. The influence of the permeability distribution on heterogeneity measures is also studied. The authors show that a variety of distributions exhibiting different reservoir performance can have the same Dykstra-Parsons coefficient. They propose a heterogeneity measure that includes an indication of the permeability distribution. The new measure requires fewer data than the Dykstra-Parsons estimator. The relationships between the new measure and the Dykstra-Parsons and Lorenz coefficients are given.

Application of probe permeametry to the prediction of two-phase flow performance in laminated sandstones (lower Brent Group, North Sea)

Abstract The Rannoch Formation (lower Brent Group, North Sea) is an important laminated, oil-bearing reservoir in the northern North Sea. Recent work has shown that, at the estimated field frontal advance rates for the Rannoch Formation, capillary effects in laminated sandstones might significantly affect the two-phase (oil and water) flow characteristics. Previous published reservoir simulation models of the Rannoch have not accounted for this. Newly acquired probe permeameter measurements have been used to map the fine scale permeability structure and develop geologically reasonable effective properties. Permeability and capillary pressure data were combined for geologically meaningful groups of laminae to define dynamic pseudo-properties (absolute permeability, relative permeability and capillary pressure) by numerical simulation. A geological model for packages of lamination was then used to combine these lamina groups at a scale of the grid block. The flow performance of this geologically reasonable grid block (containing the laminated structure) is compared with that of ‘simple’ models based on conventional procedure — i.e. the arithmetic average (in a horizontal direction) or harmonic average (vertical) permeability and rock curves. At this scale, the formation is highly anisotropic using the more detailed model, with quite different flow characteristics in the horizontal and vertical directions. The simple homogeneous models show more isotropic recovery characteristics and give significantly higher recoveries for vertical displacements. From these findings, it is suggested that previous models of the Rannoch Formation should be modified to include the capillary effects associated with lamination. The procedure outlined here is referred to as the ‘geopseudo’ approach. The approach is widely applicable to laminated reservoirs.

Use of geology in the interpretation of core-scale relative permeability data

A number of factors, such as wettability, pore-size distribution, and core-scale heterogeneity, are known to affect the measured relative permeability in core plug samples. This paper focuses on the influence of geological structure at the laminaset scale on water-oil imbibition relative permeability curves. The endpoint positions and curve shapes vary as a function of the type of internal heterogeneity, the flow rate, and the assumptions on the pore-scale petrophysics (e.g. wettability). Interaction between the capillary forces and heterogeneity can occur at the cm-dm scale, which results in widely varying two-phase flow behavior for rocks with the same single-phase permeability. The geometry of heterogeneity as expressed in standard geological descriptions (e.g., cross-laminated, ripple-laminated, plane-laminated) can be translated into features of the expected relative permeability behavior for each rock type, thus aiding the interpretation of relative permeability data. The authors illustrate how their findings can help to interpret sets of relative permeability data from the field, using some examples from the Admire sand, El Dorado Field, Kansas.

A New Method for Estimating the Dykstra-Parsons Coefficient To Characterize Reservoir Heterogeneity

The authors propose a new method to estimate the Dykstra-Parsons coefficient that leads to a more statistically reliable indication of the true heterogeneity level. The new method extracts more information from the data to produce an estimate that gives half the error of the traditional approach. Several cases are considered to demonstrate the effects of more reliable Dykstra-Parsons coefficients on predicted reservoir performance.

A review of up-scaling and cross-scaling issues in core and log data interpretation and prediction

Abstract In a heterogeneous geological formation, each rock petrophysical property (e.g., permeability, porosity, and electrical conductivity) reflects the heterogeneity and varies in a manner related to the underlying changes in fabric (grainsize, mineralogy, lamination, wettability, etc.). However, measurements, both laboratory and downhole, are made at certain volume scales dictated by the size of the core plug used or the wireline log resolution. The comparison of core and log data needs to account for both the scale and physics of the particular measurements and how these relate to the underlying scale of the geological heterogeneity of the formation. In this review, these two fundamental issues are addressed as follows: (a) measurement scale and how it relates to the ‘true’ or ‘required’ petrophysical properties of the formation is defined as ‘up-scaling’; (b) measurement physics and how we relate the physics of one measurement (e.g. permeability) to that of another (e.g. density, electrical, or acoustic properties) is termed ‘cross-scaling’. We illustrate how these two issues arise in the comparison and prediction of permeability using several published studies. We also outline an approach to petrophysical measurement reconciliation termed ‘genetic petrophysics’. This combines all three elements—measurement scale, measurement physics, and geology—to provide an integrated and robust model. We illustrate this approach for permeability to provide fit-for-purpose models of anisotropy in the near-well region of a reservoir.