Effects of pore-scale geometry and wettability on two-phase relative permeabilities within elementary cells

We study the relative role of the complex pore space geometry and wettability of the solid matrix on the quantification of relative permeabilities of elementary cells of porous media. These constitute a key element upon which upscaling frameworks are typically grounded. In our study we focus on state immiscible two-phase flow taking place at the scale of elementary cells. Pressure-driven two-phase flow following simultaneous co-current injection of water and oil is numerically solved for a suite of regular and stochastically generated two-dimensional explicit elementary cells with fixed porosity and sharing main topological/morphological features. We show that the relative permeabilities of the randomly generated elementary cells are significantly influenced by the formation of preferential percolation paths, called principal pathways, giving rise to a strongly nonuniform distribution of fluid fluxes. These pathways are a result of the spatially variable resistance that the random pore structures exert on the fluid. The overall effect on relative permeabilities of the diverse organization of principal pathways, as driven by a given random realization at the scale of the elementary cell, is significantly larger than that of the wettability of the host rock. In contrast to what can be observed for the random cells analyzed, the relative permeabilities of regular cells display a clear trend with contact angle at the investigated scale

Three-phase Permeabilities: upscaling, analytical solutions and uncertainty analysis in elementary pore structures

Immiscible three-phase flow in a rigid porous medium is upscaled from the pore to the continuum (Darcy) scale through homogenization relying on multiple scale expansion. We solve the Stokes flow problem at the pore level upon imposing continuity of velocity and shear stress at the fluid-fluid interfaces. This enables one to explicitly account for the momentum transfer between the moving phases. A macroscopic model describing the system at the Darcy scale is then rigorously obtained. This allows defining a tensor of three-phase effective relative permeabilities, , as a function of the distribution of the fluids in the system, phase saturations and fluid viscosity ratios. We present an analytical solution for corresponding to a scenario where three-phase fluid flow takes place within (a) a plane channel and (b) a capillary tube with circular cross-section. These geometrical settings are typical of microfluidics applications and are archetypal to the analysis of key processes occurring in topologically complex porous or fractured systems. Our results show the relevance of the viscous coupling effects between the three phases on the continuum-scale system behavior and demonstrate that the traditional extension of Darcy's law to model multiphase relative permeabilities might be inadequate. We then exploit our analytical solutions to investigate the way the uncertainty associated with the characterization of the phase viscosities propagates to through a global sensitivity analysis approach. We quantify the relative contribution of the considered uncertain parameters to the total variability of by relying on the variance-based Sobol indices which are derived analytically for the investigated settings

Analytical expressions for three-phase generalized relative permeabilities in water- and oil-wet capillary tubes

We analyze three-phase flow of immiscible fluids taking place within an elementary capillary tube with circular cross-section under water- and oil-wet conditions. We account explicitly for momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, by imposing continuity of velocity and shear stress at fluid-fluid interfaces. The macroscopic flow model which describes the system at the Darcy scale includes three-phase effective relative permeabilities, K (i j,r) , accounting for the flux of the ith phase due to the presence of the jth phase. These effective parameters strongly depend on phase saturations, fluid viscosities, and wettability of the solid matrix. In the considered flow setting, K (i j,r) reduce to a set of nine scalar quantities, K (i j,r) . Our results show that K (i j,r) of the wetting phase is a function only of the fluid phase own saturation. Otherwise, K (i j,r) of the non-wetting phase depends on the saturation of all fluids in the system and on oil and water viscosities. Viscous coupling effects (encapsulated in K (i j,r) with i not equal j) can be significantly relevant in both water- and oil-wet systems. Wettability conditions influence oil flow at a rate that increases linearly with viscosity ratio between oil and water phases

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

none5We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity Y = ln K, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete ‘‘pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.M. Riva; A. Guadagnini; S.P. Neuman; E. Bianchi Janetti; B. MalamaRiva, Monica; Guadagnini, Alberto; S. P., Neuman; BIANCHI JANETTI, Emanuela; B., Malam

Regional evaluation of three day snow depth for avalanche hazard mapping in Switzerland

The distribution of the maximum annual three day snow fall depth H72, used for avalanche hazard mapping according to the Swiss procedure (Sp), is investigated for a network of 124 stations in the Alpine part of Switzerland, using a data set dating back to 1931. Stationarity in time is investigated, showing in practice no significant trend for the considered period. Building on previous studies about climatology of Switzerland and using an iterative approach based on statistical tests for regional homogeneity and scaling of H72 with altitude, seven homogenous regions are identified. A regional approach based on the index value is then developed to estimate the T-years return period quantiles of H72 at each single site i, H72i(T). The index value is the single site sample average μH72i. The dimensionless values of H*72i=H72i / μH72i are grouped in one sample for each region and their frequency of occurrence is accommodated by a General Extreme Value, GEV, probability distribution, including Gumbel. The proposed distributions, valid in each site of the homogeneous regions, can be used to assess the T-years return period quantiles of H*72i. It is shown that the value of H72i(T) estimated with the regional approach is more accurate than that calculated by single site distribution fitting, particularly for high return periods. A sampling strategy based on accuracy is also suggested to estimate the single site index value, i.e. the sample average μH72i, critical for the evaluation of the distribution of H72i. The proposed regional approach is valuable because it gives more accurate snow depth input to dynamics models than the present procedure based on single site analysis, so decreasing uncertainty in hazard mapping procedure.ISSN:1561-8633ISSN:1684-998