Upscaling of Flow in Porous Media from a Tracer Perspective
Cas Berentsen · Research Repository (Delft University of Technology) · 2003
Most of our knowledge of flow in porous media is obtained at the pore and the macro scale. For reservoir scale modelling it is not practical to model the flow at these fine scales. Considering the usual objectives (e.g. large scale flow pattern and production forecast) it is undesirable to have to gather the tremendous amount of fine scale data that is required to model an entire reservoir. Moreover, present computational resources are simply not able to handle flow simulations of this size. Hence we resort to models that describe the essential physical behaviour in an averaged sense at the mega scale without modelling all finer scale details. Unfortunately, fine scale details are often correlated over large distances and appear to be part of the essential behaviour at a larger scale. As a consequence the reservoir scale flow is not just a function of some average properties of the finer scale structures. For these cases the scales in the problem are non-separable. The major challenge in upscaling is to account for this non-separability of scales. Going from single phase single component, via single phase multi component to multi phase fluid flow, upscaling becomes progressively more complex. In this Thesis we treat several aspects of upscaling of single and multi phase flow in porous media approached from a single phase context. Single phase upscaling The simplest flow type in porous media is single phase single component flow. For this type of flow the general assumption is that the upscaled megascopic pressure flux relation is the same as at the macro scale, namely Darcyâs law, albeit with effective coefficients. When the spatial length scales are non-separable the upscaled quantity depends on choices we make at the fine scale. As a consequence, single phase upscaling is mainly concerned with the proper estimation of the fine scale flux boundaries. In this thesis a method is developed that allows us to investigate the pressure and flux response of each individual flux boundary independent of the other boundaries. Fourier decomposition enables us further to investigate the effect of each individual frequency a flux boundary consists of. The major observation is that the significance of the flux response in the far field generally decreases with increasing frequency of the flux boundary that is applied. This does not generally hold for the pressure response; In case of long correlated streaks of especially low permeability the far field pressure response may be significantly larger than one would a priori expect. Upscaling of tracer dispersion At the next level of complexity we arrive at the upscaling of tracer dispersion. Traditionally dispersion has been modelled using Fickian models. However the Fickian approach does not satisfactorily explain all of the experimental observations. For instance the Fickian model cannot describe effective dispersion coefficients that vary with the Péclet number or with the length scale of observation. Moreover the Fickian description cannot account for the partial reversibility of dispersion when the flow direction is reversed. We focus on (Taylor) dispersion of a tracer released in a unidirectional velocity field belonging to a two dimensional (2D) porous medium. The tracer concentration is described by the two dimensional unidirectional classical convection dispersion equation (2D uCCDE), in which we initially restrict the small scale dispersive mechanisms to isotropic molecular diffusion. Our goal is to describe the (non-Fickian) evolution of the height averaged tracer concentration in time. The analysis starts with the relaxation concept; transverse diffusion causes the particle cloud to describe a transition from a correlated convective behaviour for short times towards uncorrelated Fickian behaviour for asymptotic long times. From a particle viewpoint this process describes the convergence of the velocity auto correlation function (VACF) of tracer particles to zero in time, as a result of the transverse displacement by diffusion. The correlation of the velocity of a particle in time results from the spatial correlation of the velocity field in the transverse direction and the limited (averaged) transverse displacement of particles by diffusion over short times. This relaxation process is characterised by a height averaged concentration profile that initially reflects the velocity profile and that converges in time towards a Gaussian profile typical for Fickian displacements. The corresponding spatial variance is initially proportional to t2 and becomes asymptotically with time proportional to t. The time that characterises this relaxation process is called the relaxation time. To analyse the model behaviour we transform the 2D uCCDE into an equivalent spectral representation using Fourier transformation. Each non-zero mode contributes to the mass balance of the height averaged concentration (zero-th mode) via a dispersive modal flux term. The evolution of each non-zero mode is described by a relaxation equation. It is characterised by a modal relaxation time that originates from the transverse molecular diffusion term in the 2D uCCDE. This modal relaxation time is proportional to the total height of the velocity field squared and decreases inversely proportional to the molecular diffusion and to the modal number squared. From the spectral representation we are able to derive the exact behaviour of the spatial moments belonging to the height averaged concentration. Moreover it forms the base of the upscaling approach we developed. To obtain the essential behaviour of the height averaged concentration, it is impractical to solve all modal equations. Hence we need to reduce the set of equations to a manageable size. A common approach is to describe the evolution of the dispersive Taylor flux as a single valued quantity. An important consequence of this approach is that the multi-scale character of the full model is lost. Classically this Taylor flux has been modelled in a Fickian way. However, such an approach ignores the relaxation process of the tracer and only covers the tracer behaviour for asymptotic long times. Here we follow the upscaling approach of Camacho. By summing all evolution equations of the modal fluxes, an approximate evolution equation of the Taylor flux is obtained. The result is a linear parabolic relaxation equation that is characterised by an approximate effective relaxation time and a closure term that accounts for all higher order modal interactions. Combining this equation with the mass balance for the averaged concentration results in the so-called c-J model. In case the spreading by diffusion in the longitudinal direction is negligible compared to the spreading induced by the velocity field, the c-J model simplifies to a generalised hyperbolic Telegraph equation. The c-J model describes the variance correctly in both the short and the long time limit. As it incorporates the relaxation process in a single scale sense, it is also able to show qualitatively the proper development of the variance for intermediate times. Quantitatively, we show that the c-J model improves significantly when scale separation is applied; The smaller Fourier scales that are relaxed with respect to the time scale of observation are described by a Fickian term and only the relaxation of the large(r) scales is modelled explicitly. In this way part of the multi scale character of the full model is again retrieved. Special attention is paid to the closure term in the c-J model that accounts for the modal interactions of the higher order concentration modes. As mentioned the multi scale character of the 2D uCCDE is lost by describing the dispersive flux as a single valued quantity. A consequence of this is that the c-J model is unable to show a sign change in the 3-rd moment that may occur in the full 2D model. To remedy this we consider alternative descriptions of the closure term in the upscaled model. We restrict our focus to the proper description of the third spatial moment and require that the present description of the mean and variance remains unaltered. Moment analysis demonstrates that the present form of the term that accounts for the modal interactions in the c-J modal is the only allowable linear form that does not affect the present description of the mean and variance. Alternatively, the coefficient in front of the closure term may be replaced by an empirical time varying function. The result describes qualitatively the proper behaviour of the 3-rd moment and is exact in the convective and dispersive limit. Overall it produces smaller errors in the 3-rd moment than the original c-J model while the lower moments remain unchanged. Until now we discussed the behaviour for particle distributions that are initially uniform over the height. Distributions that are initially non-uniform converge asymptotically in time towards a uniform distribution. This is shown by the convergence of the mean particle velocity towards the mean fluid velocity in time. Another consequence of non-uniform distributions is that the modal interactions of the higher order concentrations modes also affect the variance. Moreover, each moment is characterised by its "own" effective relaxation time. In general it is not longer adequate to describe the behaviour of the modal fluxes as a single valued quantity. We develop a truncation model in which, up to a separation scale, the evolution of each mode is modelled individually while the smaller scales are combined in an effective upper mode. The separation scale is determined such that the mean and variance are correctly described. The detail required in the truncation model increases as the initial distribution deviates more from uniformity and increases for decreasing observation times. However, even for quite small observation times, the truncation model is able to describe the proper behaviour with only little detail. We compare the 1D CCDE