Ivan Lunati

Multiscale finite-volume method for compressible multiphase flow in porous media

The Multiscale Finite-Volume (MSFV) method has been recently developed and tested for multiphase-flow problems with simplified physics (i.e. incompressible flow without gravity and capillary effects) and proved robust, accurate and efficient. However, applications to practical problems necessitate extensions that enable the method to deal with more complex processes. In this paper we present a modified version of the MSFV algorithm that provides a suitable and natural framework to include additional physics. The algorithm consists of four main steps: computation of the local basis functions, which are used to extract the coarse-scale effective transmissibilities; solution of the coarse-scale pressure equation; reconstruction of conservative fine-scale fluxes; and solution of the transport equations. Within this framework, we develop a MSFV method for compressible multiphase flow. The basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure. The effects of compressibility are taken into account in the solution of the coarse-scale pressure equation and, if necessary, in the reconstruction of the fine-scale fluxes. We consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows (tracer transport in gas flow, imbibition and drainage of partially saturated reservoirs, depletion of gas-water reservoirs, and flooding of oil-gas reservoirs). We demonstrate that the MSFV method provides accurate solutions for compressible multiphase flow problems. Whereas slightly compressible flows can be treated with a very simple model, a more sophisticate flux reconstruction is needed to obtain accurate fine-scale saturation fields in highly compressible flows.

Evolution of superficial lake water temperature profile under diurnal radiative forcing

In lentic water bodies, such as lakes, the water temperature near the surface typicallyincreases during the day, and decreases during the night as a consequence of the diurnalradiative forcing (solar and infrared radiation). These temperature variations penetratevertically into the water, transported mainly by heat conduction enhanced by eddy diffusion,which may vary due to atmospheric conditions, surface wave breaking, and internaldynamics of the water body. These two processes can be described in terms of an effectivethermal diffusivity, which can be experimentally estimated. However, the transparency of thewater (depending on turbidity) also allows solar radiation to penetrate below the surface intothe water body, where it is locally absorbed (either by the water or by the deployed sensors).This process makes the estimation of effective thermal diffusivity from experimental watertemperature profiles more difficult. In this study, we analyze water temperature profiles in alake with the aim of showing that assessment of the role played by radiative forcing isnecessary to estimate the effective thermal diffusivity. To this end we investigate diurnalwater temperature fluctuations with depth. We try to quantify the effect of locally absorbedradiation and assess the impact of atmospheric conditions (wind speed, net radiation) on theestimation of the thermal diffusivity. The whole analysis is based on the results of fiber opticdistributed temperature sensing, which allows unprecedented high spatial resolutionmeasurements ( 4 mm) of the temperature profile in the water and near the water surface.

Challenges in modeling unstable two‐phase flow experiments in porous micromodels

The simulation of unstable invasion patterns in porous media flow is very challenging because small perturbations are amplified, so that slight differences in geometry or initial conditions result in significantly different invasion structures at later times. We present a detailed comparison of pore-scale simulations and experiments for unstable primary drainage in porous micromodels. The porous media consist of Hele-Shaw cells containing cylindrical obstacles. By means of soft lithography, we have constructed two experimental flow cells, with different degrees of heterogeneity in the grain size distribution. As the defending (wetting) fluid is the most viscous, the interface is destabilized by viscous forces, which promote the formation of preferential flow paths in the form of a branched finger structure. We model the experiments by solving the Navier-Stokes equations for mass and momentum conservation in the discretized pore space and employ the Volume of Fluid (VOF) method to track the evolution of the interface. We test different numerical models (a 2-D vertical integrated model and a full-3-D model) and different initial conditions, studying their impact on the simulated spatial distributions of the fluid phases. To assess the ability of the numerical model to reproduce unstable displacement, we compare several statistical and deterministic indicators. We demonstrate the impact of three main sources of error: (i) the uncertainty on the pore space geometry, (ii) the fact that the initial phase configuration cannot be known with an arbitrarily small accuracy, and (iii) three-dimensional effects. Although the unstable nature of the flow regime leads to different invasion structures due to small discrepancies between the experimental setup and the numerical model, a pore-by-pore comparison shows an overall satisfactory match between simulations and experiments. Moreover, all statistical indicators used to characterize the invasion structures are in excellent agreement. This validates the modeling approach, which can be used to complement experimental observations with information about quantities that are difficult or impossible to measure, such as the pressure and velocity fields in the two fluid phases.

Modeling complex wells with the multi-scale finite-volume method

In this paper, an extension of the multi-scale finite-volume (MSFV) method is devised, which allows to simulate flow and transport in reservoirs with complex well configurations. The new framework fits nicely into the data structure of the original MSFV method and has the important property that large patches covering the whole well are not required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial (the well-bore reference pressure is explicitly specified), additional equations have to be solved to obtain the unknown well-bore pressure of rate-constraint wells. Numerical simulations of test cases with multiple complex wells demonstrate the ability of the new algorithm to capture the interference between the various wells and the reservoir accurately.

An iterative multiscale finite volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).

An Operator Formulation of the Multiscale Finite-Volume Method with Correction Function

The multiscale finite-volume (MSFV) method has been derived to efficiently solve large problems with spatially varying coefficients. The fine-scale problem is subdivided into local problems that can be solved separately and are coupled by a global problem. This algorithm, in consequence, shares some characteristics with two-level domain decomposition (DD) methods. However, the MSFV algorithm is different in that it incorporates a flux reconstruction step, which delivers a fine-scale mass conservative flux field without the need for iterating. This is achieved by the use of two overlapping coarse grids. The recently introduced correction function allows for a consistent handling of source terms, which makes the MSFV method a flexible algorithm that is applicable to a wide spectrum of problems. It is demonstrated that the MSFV operator, used to compute an approximate pressure solution, can be equivalently constructed by writing the Schur complement with a tangential approximation of a single-cell overlapping grid and incorporation of appropriate coarse-scale mass-balance equations.

Treating Highly Anisotropic Subsurface Flow with the Multiscale Finite‐Volume Method

The multiscale finite‐volume (MSFV) method has been designed to solve flow problems on large domains efficiently. First, a set of basis functions, which are local numerical solutions, is employed to construct a fine‐scale pressure approximation; then a conservative fine‐scale velocity approximation is constructed by solving local problems with boundary conditions obtained from the pressure approximation; finally, transport is solved at the fine scale. The method proved very robust and accurate for multiphase flow simulations in highly heterogeneous isotropic reservoirs with complex correlation structures. However, it has recently been pointed out that the fine‐scale details of the MSFV solutions may be lost in the case of high anisotropy or large grid aspect ratios. This shortcoming is analyzed in this paper, and it is demonstrated that it is caused by the appearance of unphysical “circulation cells.” We show that damped‐shear boundary conditions for the conservative‐velocity problems or linear boundaryco...

An adaptive multiscale method for density-driven instabilities

Accurate modeling of flow instabilities requires computational tools able to deal with several interacting scales, from the scale at which fingers are triggered up to the scale at which their effects need to be described. The Multiscale Finite Volume (MsFV) method offers a framework to couple fine- and coarse-scale features by solving a set of localized problems which are used both to define a coarse-scale problem and to reconstruct the fine-scale details of the flow. The MsFV method can be seen as an upscaling-downscaling technique, which is computationally more efficient than standard discretization schemes and more accurate than traditional upscaling techniques. We show that, although the method has proven accurate in modeling density-driven flow under stable conditions, the accuracy of the MsFV method deteriorates in case of unstable flow and an iterative scheme is required to control the localization error. To avoid large computational overhead due to the iterative scheme, we suggest several adaptive strategies both for flow and transport. In particular, the concentration gradient is used to identify a front region where instabilities are triggered and an accurate (iteratively improved) solution is required. Outside the front region the problem is upscaled and both flow and transport are solved only at the coarse scale. This adaptive strategy leads to very accurate solutions at roughly the same computational cost as the non-iterative MsFV method. In many circumstances, however, an accurate description of flow instabilities requires a refinement of the computational grid rather than a coarsening. For these problems, we propose a modified iterative MsFV, which can be used as downscaling method (DMsFV). Compared to other grid refinement techniques the DMsFV clearly separates the computational domain into refined and non-refined regions, which can be treated separately and matched later. This gives great flexibility to employ different physical descriptions in different regions, where different equations could be solved, offering an excellent framework to construct hybrid methods.

Hybrid Multiscale Finite Volume method for two-phase flow in porous media

We present a novel hybrid (or multiphysics) algorithm, which couples pore-scale and Darcy descriptions of two-phase flow in porous media. The flow at the pore-scale is described by the Navier-Stokes equations, and the Volume of Fluid (VOF) method is used to model the evolution of the fluid-fluid interface. An extension of the Multiscale Finite Volume (MsFV) method is employed to construct the Darcy-scale problem. First, a set of local interpolators for pressure and velocity is constructed by solving the Navier-Stokes equations; then, a coarse mass-conservation problem is constructed by averaging the pore-scale velocity over the cells of a coarse grid, which act as control volumes; finally, a conservative pore-scale velocity field is reconstructed and used to advect the fluid-fluid interface. The method relies on the localization assumptions used to compute the interpolators (which are quite straightforward extensions of the standard MsFV) and on the postulate that the coarse-scale fluxes are proportional to the coarse-pressure differences. By numerical simulations of two-phase problems, we demonstrate that these assumptions provide hybrid solutions that are in good agreement with reference pore-scale solutions and are able to model the transition from stable to unstable flow regimes. Our hybrid method can naturally take advantage of several adaptive strategies and allows considering pore-scale fluxes only in some regions, while Darcy fluxes are used in the rest of the domain. Moreover, since the method relies on the assumption that the relationship between coarse-scale fluxes and pressure differences is local, it can be used as a numerical tool to investigate the limits of validity of Darcys law and to understand the link between pore-scale quantities and their corresponding Darcy-scale variables.

Effects of pore volume–transmissivity correlation on transport phenomena

The relevant velocity that describes transport phenomena in a porous medium is the pore velocity. For this reason, one needs not only to describe the variability of transmissivity, which fully determines the Darcy velocity field for given source terms and boundary conditions, but also any variability of the pore volume. We demonstrate that hydraulically equivalent media with exactly the same transmissivity field can produce dramatic differences in the displacement of a solute if they have different pore volume distributions. In particular, we demonstrate that correlation between pore volume and transmissivity leads to a much smoother and more homogeneous solute distribution. This was observed in a laboratory experiment performed in artificial fractures made of two plexiglass plates into which a space-dependent aperture distribution was milled. Using visualization by a light transmission technique, we observe that the solute behaviour is much smoother and more regular after the fractures are filled with glass powder, which plays the role of a homogeneous fault gouge material. This is due to a perfect correlation between pore volume and transmissivity that causes pore velocity to be not directly dependent on the transmissivity, but only indirectly through the hydraulic gradient, which is a much smoother function due to the diffusive behaviour of the flow equation acting as a filter. This smoothing property of the pore volume-transmissivity correlation is also supported by numerical simulations of tracer tests in a dipole flow field. Three different conceptual models are used: an empty fracture, a rough-walled fracture filled with a homogeneous material and a parallel-plate fracture with a heterogeneous fault gouge. All three models are hydraulically equivalent, yet they have a different pore volume distribution. Even if piezometric heads and specific flow rates are exactly the same at any point of the domain, the transport process differs dramatically. These differences make it important to discriminate in situ among different conceptual models in order to simulate correctly the transport phenomena. For this reason, we study the solute breakthrough and recovery curves at the extraction wells. Our numerical case studies show that discrimination on the basis of such data might be impossible except under very favourable conditions, i.e. the integral scale of the transmissivity field has to be known and small compared to the dipole size. If the latter conditions are satisfied, discrimination between the rough-walled fracture filled with a homogeneous material and the other two models becomes possible, whereas the parallel-plate fracture with a heterogeneous fault gouge and the empty fracture still show identifiability problems. The latter may be solved by inspection of aperture and pressure testing.