P.S. Ringrose

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.

Capillary trapping mechanisms in water-wet laminated rocks

Most flooding experiments in sandstone cores are carried out either in almost homogeneous samples or in core samples of uncertain heterogeneity. As a result, the interaction of small-scale sedimentary heterogeneity with the fluid mechanics of water-oil displacement cannot be adequately understood or quantified. Because most clastic sediments show some degree of lamination, this might be expected to have a significant influence on both oil displacement efficiency and residual/remaining oil saturation. This paper reports results from low-rate, drainage/imbibition floods in a 20x10x1-cm water-wet slab of cross-laminated heterogeneous eolian sandstone. The distribution of porosity, initial water saturation and residual oil saturation were monitored with computerized-tomography (CT) scanning techniques. The low-rate inhibition floods show that between 30% and 55% of original oil may be trapped in isolated high-permeability laminae. This work shows the importance of recognizing the role of core-scale heterogeneity in the laboratory measurement of waterflood behavior (i.e., the interaction of capillary forces with rock structure, particularly lamination). The practice of performing high-rate floods on rock samples assumed to be homogeneous is unwise and can lead to erroneous conclusion. The results of this work have major implications for (1) two-phase petrophysical measurements; (2) assessment of residual/remaining oil, and (3) multiphase-flow scaleup.

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.

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.

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.

The Use of Correlation Statistics for Modelling Immiscible Displacements in Petroleum Reservoirs

Most statistical models which are used when simulating oil reservoir performance employ a correlation function. We have found that fields in which correlation, as a function of direction, is represented only as a positive variable do not give an adequate representation of immiscible displacements in realistic geological formations. A better representation can be achieved by introducing negative correlation in one or more directions. We describe the water/oil displacement efficiency in a selection of deterministic fields, based on typical sediment bedform structures, and then show how this performance may be reproduced in random correlated fields with varying amounts of positive and negative correlation. Negative correlation needs to be considered when the heterogeneity displays significant periodicity; for example, in layered systems. Positive correlation, which represents the tendency for (local) similarity, results in favourable viscous-capillary interactions and better displacement efficiency. Negative correlation results in poorer displacement efficiency. Clastic sedimentary formations, which are characterised by contrasting layers, are better represented, statistically, by anisotropic positively/negatively correlated permeability fields in which displacement efficiency is strongly directiondependent. The implications of negative correlation for numerical flow models are also assessed.

The Dispersive-Fingering Transition and Flow Upscaling for Miscible Displacements in Heterogeneous Systems

The nature of fluid flow through porous media is determined by the interaction between the fluid mechanics of the displacement process and the rock heterogeneity. The interplay between the various forces may be described by dimensionless scaling groups e.g. mobility ratio, capillary number, viscous/gravity ratio etc. A quantitative measui~ of rock heterogeneity is more difficult to define in real geological systems but it is central to the prediction of the fluid displacement process. In this work, the permeability heterogeneity is described by a correlated random field which is the simplest model into which we can introduce both variability and structure in a systematic way. We consider mainly viscous displacements at adverse mobility ratios in first contact miscible floods. The various flow regimes that are possible in such fluid displacement-heterogeneity combinations are described and, in particular, we focus on the fingering-dispersive transition. The results presented here extend previously reported work on this topic. In particular, the importance of certain “shape” scaling groups in detennining the global flow regime within the system are demonstrated. These scaling groups are related to the vertical equilibrium concept. The significance of the flow regime is discussed in the light of how the flow parameters in miscible displacement may be scaled up. The pseudo-isation technique, which is commonly used in two phase flow, is applied and some observations are made on the resulting averaged or “upscaled” parameters which relate to the overall flow regime in the displacement process.

Scaled Physical Model of Secondary Oil Migration: Discussion

Thomas and Clouse have presented results of a detailed laboratory model of secondary oil migration using a sand pack, and then used scaling group theory to extrapolate their observations to field-scale source rock/carrier-bed/reservoir systems. They conclude that for water-wet homogeneous systems, the charging of traps is not controlled by secondary migration rates or losses but by the rate of primary migration from the source rock. Although this conclusion is compatible with the observation that estimates of field charging times are broadly similar to estimates of the time interval for a source rock to mature and expel oil, we feel that several of their arguments about real secondary migration processes are misleading and require further discussion.

The effects of hydrodynamic flows on secondary oil migration efficiency

The efficiency of secondary oil migration is a key parameter in determining the prospectivity of potential traps. Under hydrostatic conditions, oil migrates as a result of the buoyancy forces created by the density differences between the oil phase and the surrounding pore water (Berg, 1975 ; Schowalter, 1979). The direction of the movement will be vertical until a cap rock is reached, whereby the oil will migrate updip for a distance proportional to the dip angle of the seal and the volume of free oil present in the system (Thomas and Clouse, 1995). However, it is unlikely that oil migrates in exclusively hydrostatic conditions. The fluids in a sedimentary basin will be subjected to hydrodynamic flows which occur as a result of meteoric influx, sediment compaction, zones of overpressure, or thermal convection (e.g. Corbet & Bethke, 1992; Hitchon, 1984). The presence of hydrodynamic flows in the North Sea petroleum system are evident from a number of cases of tilted oil-water contacts in oil reservoirs (e.g. Thomasen and Jacobsen, 1994).