Upscaling permeability for fractured concrete: meso–macro numerical approach coupled to strong discontinuities

A two‐scales numerical analysis is set up in order to upscale the permeability of fractured materials such as concrete. To that aim, we couple finite element (FE) kinematics enhancements (strong discontinuities) representing fine scale cracks to the fine scale permeability tensor. The latter may be split into two parts: the first one is isotropic and corresponds to flows within the porosity of the material; the second one, based upon a set of cracks with different orientations and openings, is anisotropic. For the latter, each crack is a path for mass flow according to the Poiseuille law considering two infinite planes. We show how the upscaling procedure leads both to the definition of macroscopic permeability tensors as well as the flow rate evaluation for components of concrete structures. Copyright © 2013 John Wiley & Sons, Ltd.     

Homogenization of a Darcy–Stokes system modeling vuggy porous media

We derive a macroscopic model for single-phase, incompressible, viscous fluid flow in a porous medium with small cavities called vugs. We model the vuggy medium on the microscopic scale using Stokes equations within the vugular inclusions, Darcy's law within the porous rock, and a Beavers–Joseph–Saffman boundary condition on the interface between the two regions. We assume periodicity of the medium and obtain uniform energy estimates independent of the period. Through a two-scale homogenization limit as the period tends to zero, we obtain a macroscopic Darcy's law governing the medium on larger scales. We also develop some needed generalizations of the two-scale convergence theory needed for our bimodal medium, including a two-scale convergence result on the Darcy–Stokes interface. The macroscopic Darcy permeability is computable from the solution of a cell problem. An analytic solution to this problem in a simple geometry suggests that: (1) flow along vug channels is primarily Poiseuille with a small perturbation related to the Beavers–Joseph slip, and (2) flow that alternates from vug to matrix behaves as if the vugs have infinite permeability.     

Multi-scale modelling of the permeability evolution of fine sands during cement suspension grouting with filtration

Fluid flow during permeation grouting of fine sands with a microcement-based grout is studied by assuming that the heterogeneous medium composed of the initial granular skeleton, filtered cement and the interstitial fluid phase can be replaced by a continuous equivalent medium at the macroscopic level. Consequently, the method of Homogenization of Periodic Structures (HPS) is used to identify the effective permeability tensor evolution under the effect of cement filtration. The expression of the macroscopic permeability tensor derived through the HPS procedure is shown to depend on the permeating fluid viscosity and the geometrical arrangement of the sand grains and cement deposit within the microstructure. Numerical computations are made using various two-dimensional and three-dimensional microstructures, and the model results are confronted with grouting experiments performed on small scale columns in the laboratory.     

Deformation band populations in fault damage zone—impact on fluid flow

Fault damage zones in highly porous reservoirs are dominated by deformation bands that generally have permeability-reducing properties. Due to an absence of sufficiently detailed measurements and the irregular distribution of deformation bands, a statistical approach is applied to study their influence on flow. A stochastic model of their distribution is constructed, and band density, distribution, orientation, and flow properties are chosen based on available field observations. The sensitivity of these different parameters on the upscaled flow is analyzed. The influence of a heterogeneous permeability distribution was also studied by assuming the presence of high permeability holes within bands. The fragmentation and position of these holes affect significantly the block-effective permeability. Results of local upscaling with a diagonal and full upscaled permeability tensor are compared, and qualitatively similar results for the flow characteristics are obtained. Further, the procedure of iterative local–global upscaling is applied to the problem.     

Nodal and Mixed Finite Elements for the Numerical Homogenization of 3D Permeability

The aim of upscaling is to determine equivalent homogeneous parameters at a coarse-scale from a spatially oscillating fine-scale parameter distribution. To be able to use a limited number of relatively large grid-blocks in numerical oil reservoir simulators or groundwater models, upscaling of the permeability is frequently applied. The spatial fine-scale permeability distribution is generally obtained from geological and geostatistical models. After upscaling, the coarse-scale permeabilities are incorporated in the relatively large grid-blocks of the numerical model. If the porous rock may be approximated as a periodic medium, upscaling can be performed by the method of homogenization. In this paper the homogenization is performed numerically, which gives rise to an approximation error. The complementarity between two different numerical methods – the conformal-nodal finite element method and the mixed-hybrid finite element method – has been used to quantify this error. These two methods yield respectively upper and lower bounds for the eigenvalues of the coarse-scale permeability tensor. Results of 3D numerical experiments are shown, both for the far field and around wells.     

An expression for the permeability of anisotropic granular media

There are many expressions proposed for the permeability of isotropic media based on flow channel and pore size distribution concepts, but there are no such expressions for anisotropic media. In this paper an expression for the permeability of an anisotropic medium is proposed, which has been verified in the laboratory. The mechanism behind fluid flow through soil was investigated using microscopic computer simulations to propose an expression for macroscopic permeability. The soil was assumed to be a spatially periodic porous medium, and the Navier-Stokes equation was solved using the FEM with appropriate boundary conditions for several different arrangements of the porous medium. The basic variables influencing flow through soil at the microscopic level were identified as specific surface area, void ratio, particle shape, material heterogeneity and the arrangement of particles in a porous medium. A sensitivity analysis was carried out to obtain an expression for the permeability in terms of the above variables. The corresponding macroscopic variables for the above microscopic variables are average specific surface area, average void ratio, anisotropy, tortuosity due to material heterogeneity, and the arrangement of particles respectively. An expression for the directional permeability is proposed in terms of these variables for the most common occurrence of particles in a porous medium. For the verification of the proposed equation, the permeability values of a fine-grained sand were measured at different void ratios and were compared with those predicted by the proposed equation. The results show that the predicted permeability values from the proposed equation are very close to the measured values.     

Permeability Upscaling Measured on a Block of Berea Sandstone: Results and Interpretation

To physically investigate permeability upscaling, over 13,000 permeability values were measured with four different sample supports (i.e., sample volumes) on a block of Berea Sandstone. At each sample support, spatially exhaustive permeability datasets were measured, subject to consistent flow geometry and boundary conditions, with a specially adapted minipermeameter test system. Here, we present and analyze a subset of the data consisting of 2304 permeability values collected from a single block face oriented normal to stratification. Results reveal a number of distinct and consistent trends (i.e., upscaling) relating changes in key summary statistics to an increasing sample support. Examples include the sample mean and semivariogram range that increase with increasing sample support and the sample variance that decreases. To help interpret the measured mean upscaling, we compared it to theoretical models that are only available for somewhat different flow geometries. The comparison suggests that the nonuniform flow imposed by the minipermeameter coupled with permeability anisotropy at the scale of the local support (i.e., smallest sample support for which data is available) are the primary controls on the measured upscaling. This work demonstrates, experimentally, that it is not always appropriate to treat the local-support permeability as an intrinsic feature of the porous medium, that is, independent of its conditions of measurement.     

Coupling between anisotropic damage and permeability variation in brittle rocks

In this paper, a coupled constitutive model is proposed for anisotropic damage and permeability variation in brittle rocks under deviatoric compressive stresses. The formulation of the model is based on experimental evidences and main physical mechanisms involved in the scale of microcracks are taken into account. The proposed model is expressed in the macroscopic framework and can be easily implemented for engineering application. The macroscopic free enthalpy of cracked solid is first determined by approximating crack distribution by a second‐order damage tensor. The effective elastic properties of damaged material are then derived from the free enthalpy function. The damage evolution is related to the crack growth in multiple orientations. A pragmatic approach inspired from fracture mechanics is used for the formulation of the crack propagation criterion. Compressive stress induced crack opening is taken into account and leads to macroscopic volumetric dilatancy and permeability variation. The overall permeability tensor of cracked material is determined using a micro–macro averaging procedure. Darcy's law is used for fluid flow at the macroscopic scale whereas laminar flow is assumed at the microcrack scale. Hydraulic connectivity of cracks increases with crack growth. The proposed model is applied to the Lac du Bonnet granite. Generally, good agreement is observed between numerical simulations and experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

Computation of the filtration laws through porous media for a non-Newtonian fluid obeying the power law

We consider a stationary flow of an incompressible non-Newtonian flow through a porous medium, induced by an injection velocity when inertial effects are negligible. At the pore scale, the governing equations are based on a nonlinear relation between the stress and the rate of deformation. In such a situation, the limit problem obtained when the pore size tends to zero, is called the homogenized problem that leads to the filtration law. This filtration law is given by a non-linear system coupling a local problem on a typical cell of the porous medium to a global problem at the scale of the whole porous medium. We propose, in this work, a numerical method to solve this homogenized problem and apply this method when the velocity dependent viscosity is given by the power law. Finally, we propose some numerical experiments to illustrate our approach.     

Modelling of steady‐state fluid flow in 3D fractured isotropic porous media: application to effective permeability calculation

In this paper, 3D steady‐state fluid flow in a porous medium with a large number of intersecting fractures is derived numerically by using collocation method. Fluid flow in the matrix and fractures is described by Darcy's law and Poiseuille's law, respectively. The recent theoretical development presented a general potential solution to model the steady‐state flow in fractured porous media under a far‐field condition. This solution is a hypersingular integral equation with pressure field in the fracture surfaces as the main unknown. The numerical procedure can resolve the problem for any form of fractures and also takes into account the interactions and the intersection between fractures. Once the pressure field and then the flux field in fractures have been determined, the pressure field in the porous matrix is computed completely. The basic problem of a single fracture is investigated, and a semi‐analytical solution is presented. Using the solution obtained for a single fracture, Mori‐Tanaka and self‐consistent schemes are employed for upscaling the effective permeability of 3D fractured porous media. Copyright © 2012 John Wiley & Sons, Ltd.     

Flow Simulations To Evaluate Upscaling of Permeability

We study upscaling of the permeability for porous media flow on a grid with one million blocks. The purpose is to illustrate how flow simulations can be used to evaluate upscaling methods.     

Adaptive hierarchical upscaling of flow in heterogeneous reservoirs based on an a posteriori error estimate

This paper treats the upscaling of the absolute permeability in a heterogeneous reservoir. By replacing the fine scale permeability tensor with an upscaled, or effective permeability tensor, a modelling error is introduced. An a posteriori error estimate on this modelling error is formulated and tested. An implementation of the theory, based on domain decomposition coupled with a hierarchical representation of the absolute permeability field, is given. As hierarchical basis functions we have chosen the Haar system, which leads to a wavelet representation of the permeability. The wavelet representation offers a natural upscaling technique which resembles the highcut filters commonly used in signal analysis. This procedure represents an adaptive upscaling method. The numerical results show that this method conserves both the dissipation and the mean velocity in the problem fairly well. The a posteriori error estimate on the modelling error coupled with domain decomposition methods constitutes a powerful modelling tool.     

Modeling permeability in imperfectly layered porous media. I. Derivation of block-scale permeability tensor for thin grid-blocks

Practical expressions are given for the nine components of the block-scale permeability tensor of a thin block. These expressions are derived from the local-scale continuity equation and Darcy's law in an anisotropic layered porous medium. The flow problem is separated in a bottom-flux problem and a top-flux problem, both of which can be solved in essentially the same way. The bottom-flux problem has been worked out in detail, and has been separated in two parts: a vertical potential difference and a horizontal potential difference part. Each is solved with a different approach specially designed for it. Depth-averaged expressions are obtained first, after which block-scale expressions are obtained by assuming a constant depth-averaged flux. In the zeroth order, this results in the well-known Dupuit approximation in geohydrology, and the vertical equilibrium (VE) approximation in petroleum reservoir engineering. The novelty of the theory presented here stems from the application of a perturbation technique to obtain first-order corrections to these well-known results. The local-scale laws are applied in the coordinate system coinciding with the principal axes of the local-scale permeability tensor. Only in this coordinate system the local-scale permeability tensor has zero off-diagonal components. However, since the porous medium is imperfectly layered, the first-order corrections show that the off-diagonal components of the block-scale permeability tensor are not zero. Furthermore, the block-scale permeability tensor is generally nonsymmetric, which implies that a coordinate system in which the off-diagonal terms disappear does not exist.     

Modified adaptive local–global upscaling method for discontinuous permeability distribution

The paper is devoted to the upscaling method appropriate for single-phase flow in media with discontinuous permeability distribution. The suggested algorithm is a modification of the iterative adaptive local–global upscaling developed by Chen and coauthors. The key feature of this method is a consistency between local and coarse global calculated characteristics. In this work, we apply a modified procedure to determine the boundary conditions used in the local fine-scale computation. To increase the accuracy of these boundary conditions on each iteration, we involve an additional preliminary step based on the results of coarse scale calculations from the previous iteration. Numerical tests show an essential improvement of the accuracy of upscaled flow rates for most of the realizations of statistical permeability distribution. Although the developed method is universal, its efficiency increases with increasing of permeability contrast.     

Numerical Calculation of Equivalent Cell Permeability Tensors for General Quadrilateral Control Volumes

A new method for upscaling fine scale permeability fields to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, applies a triangle-based finite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local fine scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of flow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a flow-based gridding technique. In this case, the approach is shown to provide results that are significantly more accurate than those obtained using standard techniques.     

Upscaling Hydraulic Conductivities in Cross-Bedded Formations

Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities.     

Groutability of cement‐based grout with consideration of viscosity and filtration phenomenon

The groutability depends on the properties of the grout, its injection processes, and on the mechanical properties of the soil formation. During the process of pouring cement‐based grouting into a porous medium, a variation with time occurs in the viscosity of grout suspension. In addition, the particle filtration phenomenon will limit the expansion of the grouted zone because cement particles are progressively stagnant within the soil matrix. In this paper, a closed‐form solution was derived by implementing the mass balance equations and the generalized phenomenological filtration law, which can be used to evaluate the deposition of cement‐based grout in the soil matrix. The closed‐form solution relevant to a particular spherical flow was modified by a step‐wise numerical calculation, considering the variable viscosity caused by a chemical reaction, and the decrease in porosity resulting from grout particle deposition in the soil pores. A series of pilot‐scale chamber injection tests was performed to verify that the developed step‐wise numerical calculation is able to evaluate the injectable volume of grout and the deposition of grout particles. The results of the chamber injection tests concurred well with that of the step‐wise numerical calculation. Based on the filtration phenomenon, a viable approach for estimating the groutability of cement‐based grout in a porous medium was also suggested, which might facilitate a new insight in the design of the grouting process. Copyright © 2009 John Wiley & Sons, Ltd.     

Upscaling permeability for three-dimensional fractured porous rocks with the multiple boundary method

Upscaling permeability of grid blocks is crucial for groundwater models. A novel upscaling method for three-dimensional fractured porous rocks is presented. The objective of the study was to compare this method with the commonly used Oda upscaling method and the volume averaging method. First, the multiple boundary method and its computational framework were defined for three-dimensional stochastic fracture networks. Then, the different upscaling methods were compared for a set of rotated fractures, for tortuous fractures, and for two discrete fracture networks. The results computed by the multiple boundary method are comparable with those of the other two methods and fit best the analytical solution for a set of rotated fractures. The errors in flow rate of the equivalent fracture model decrease when using the multiple boundary method. Furthermore, the errors of the equivalent fracture models increase from well-connected fracture networks to poorly connected ones. Finally, the diagonal components of the equivalent permeability tensors tend to follow a normal or log-normal distribution for the well-connected fracture network model with infinite fracture size. By contrast, they exhibit a power-law distribution for the poorly connected fracture network with multiple scale fractures. The study demonstrates the accuracy and the flexibility of the multiple boundary upscaling concept. This makes it attractive for being incorporated into any existing flow-based upscaling procedures, which helps in reducing the uncertainty of groundwater models.