Gillian Elizabeth Pickup

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.

An assessment of steady-state scale-up for small-scale geological models

The calculation of pseudo-relative permeabilities can be speeded up considerably by using steady-state methods. The capillary equilibrium limit may be assumed at small scales (30 cm or less), when the flood rate is low. At high flow rates and larger distance scales, we may use a viscous-dominated steady-state method which assumes constant fractional flow. Steady-state pseudos may also be calculated at intermediate flow rates using fine-scale simulations, and allowing the flood to come into equilibrium at different fractional flow levels. The aim of this paper is to assess the accuracy of steady-state scale-up for small-scale sedimentary structures. We have tested steady-state scale-up methods using a variety of small-scale geological models. The success of steady-state scale-up depends not only on the flow rate, but also on the nature of the heterogeneity. If high permeability zones are surrounded by low permeability ones (e.g. low permeability laminae or bed boundaries), oil trapping may occur in a water-wet system. In this case pseudo-oil-relative permeabilities are very sensitive to flow rate, and care must be taken to upscale using the correct viscous/capillary ratio. However, in permeability models, where phase trapping may not occur (unconnected low permeability regions), the pseudos are similar, whatever the viscous/capillary ratio. The disadvantage of steady-state scale-up is that it cannot take account of numerical dispersion, in the manner in which dynamic methods can. However, we show examples of coarse-scale simulations with viscous-dominated steady-state pseudos which agree favourably with fine-scale simulations. Provided there are sufficient grid blocks in the coarse-scale model, the smearing of the flood front due to numerical effects is not serious.

A Comparison of Two-Phase Dynamic Upscaling Methods Based on Fluid Potentials

Although a number of methods for calculating dynamic pseudo-functions have been developed over the years, there is still a lack of understanding as to why a certain method will succeed in some cases but fail in others. In this paper, we describe the results of an assessment of several upscaling methods, namely the Kyte and Berry (KB) method, the Stone method, the Hewett and Archer (HA) method and the Transmissibility-Weighted (TW) method. We have analyzed the equations for deriving the methods and investigated the results of numerical simulations of gas displacing oil, in a variety of models to enable us to gain new insights into these, and related, upscaling methods. In particular, some novel observations on methods based on fluid potential are presented and the issue of using predicted fluid mobilities as a criterion of accuracy of an upscaling method is clarified.

The development of appropriate upscaling procedures

Permeability upscaling should be carried out with careful attention to the nature of rock heterogeneities. While there are many large-scale features which must be taken into account, there are also important heterogeneities at the small-scale. Many sedimentary structures contain laminae at the mm–cm scale, and beds at the m-scale, which give rise to strong contrasts in permeability. We use a 2D model derived from a photo-panel of an aeolian outcrop, along with permeability measurements from a North Sea oil field, to demonstrate the effects of small-scale heterogeneity. This model is similar in size to a typical cell of a reservoir geological model. We take imaginary probe and core plug measurements from the model, average them, and compare these with the effective permeability for the model computed from a finite difference flow calculation. Although this procedure is standard practice, we show that it can lead to biased estimates of the permeabilities used in flow simulation. As an alternative we suggest using models of representative beds, and performing flow simulation to calculate effective permeabilities for both single-phase and two-phase flow.

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.

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.

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.

Simulation of CO2 storage in a heterogeneous aquifer

Abstract The fate of carbon dioxide (CO2) injected into a deep saline aquifer depends largely on the geological structure within the aquifer. For example, low permeability layers, such as shales or mudstones, will act as barriers to vertical flow of CO2 gas, whereas high permeability channels may assist the lateral migration of CO2. It is therefore important to include permeability heterogeneity in models for numerical flow simulation As an example of a heterogeneous system, a model of fluvial-incised valley deposits was used. Flow simulations were performed using the generalized equation-of-state model—greenhouse gas software package from Computer Modelling Group, which is a compositional simulator, specially adapted for CO2 storage. The impacts of residual gas and water saturations, gas diffusion in the aqueous phase, hysteresis, and permeability anisotropy on the distribution of CO2 between the gaseous and aqueous phases were examined. Gas diffusion in the aqueous phase was found to significantly enhance solubility trapping of CO2, even when hysteretic trapping of CO2 as a residual phase is taken into account.

CO2 sequestration in a UK North Sea analogue for geological carbon storage

The Fizzy discovery, a southern North Sea (UK) gas accumulation with ∼50% natural CO 2 content, provides an opportunity to study the long-term quantity of CO 2 -related mineral reaction as an analogue for engineered CO 2 storage. The reservoir contains diagenetic dolomite typical of the formation; to identify and quantify any sequestration-related dolomite is challenging. To this end, CO 2 was extracted by stepwise extraction from dolomite from both the Fizzy discovery and equivalent sandstones from a low-CO 2 location. Between 0% and 22% of the dolomite in the Fizzy discovery precipitated due to the high CO 2 concentration. This corresponds to 11% ± 8% of the recent high-CO 2 charge sequestered as dolomite, a relatively low proportion after ∼50 m.y. of potential CO 2 -water-rock interaction.

Simulation of Near-Well Pressure Build-up in Models of CO2 Injection

Reservoir simulation plays an important role in predicting the outcome of a CO2 storage project, although it is challenging to simulate all the processes that arise. In particular, we need to predict the build-up of pressure in the near well region to be able to estimate the optimum injection rate whilst ensuring that the formation and overlying caprock are not fractured. In this work, we compare simulations of horizontal homogeneous models, with both 1D radial and 2D Cartesian grids, with analytical calculations of pressure build-up. Our results show that several inaccuracies arise when using too coarse a grid, due to the inability to resolve the shock fronts adequately. In a coarse cell, the amount of dissolution is over-estimated and the gas saturation builds up slowly. The presence of a large cell with intermediate gas saturation gives rise to a peak in the pressure build-up curve (due to low mobility). The pressure eventually reduces to the “correct” value when the dry-out region forms. However, if injection ceases before this time, the final pressure will be over-estimated. As the grid size is reduced, these effects become less severe.