MI-ECku: A novel methodology for estimating unsaturated hydraulic conductivity of porous media

Acta Geotechnica - Numerous applications of hydraulic conductivity of porous media (e.g., soils, clay liners, rocks, concrete, ceramic filters, etc.) in their unsaturated state are well established...     

A new mechanism for pressure solution in porous quartzose sandstone

The mechanism of pressure solution, a source of controversy for years, must be understood before we can evaluate the effectiveness of pressure solution during geological processes. The water film diffusion (WFD) mechanism proposed by Weyl (1959) and Rutter (1976, 1983) is believed by many to be the primary mechanism responsible for intergranular pressure solution (IPS) in non-porous metamorphic rocks as well as porous sedimentary rocks. Tada and Siever (1986), experimenting with halite single crystals, suggested the new plastic deformation plus free-face pressure solution (PD + FFPS) mechanism.The effectiveness of PD + FFPS as an IPS mechanism is theoretically evaluated for porous quartzose sandstone and compared with WFD. The result suggests that, though the driving force of the reaction (relative activity increase) is 4 to 5 orders of magnitude larger in WFD, the ease of diffusion (diffusion path width times the diffusion coefficient) is 7 to 9 orders of magnitude larger in PD + FFPS. Consequently. PD + FFPS yields diffusion rates 2 to 5 orders of magnitude faster than WFD.In WFD, diffusion is always the rate-controlling process, whereas either dissolution at IPS contacts or precipitation on free grain surfaces may be the rate-controlling process in PD + FFPS, when temperatures are low and/or grain sizes are small. The dissolution or precipitation rate of PD + FFPS is faster than the diffusion rate of WFD except when the total free grain surface area is very small. In final stages of compaction, when the total free grain surface area has become very small, WFD replaces PD + FFPS.     

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper, we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential/capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes, the resulting systems of nonlinear algebraic equations are solved with Newton’s method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust, and efficient. In particular, no postprocessing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1,000 processors.     

Variational inequalities for modeling flow in heterogeneous porous media with entry pressure

One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the saturation. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple models. Hence, we consider in this paper two extensions to standard capillary pressure relationships. Firstly, to correct the nonphysical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous porous media, we apply a model with a capillary threshold pressure that controls the penetration process. Mathematically, we rewrite this model as inequality constraint at the interfaces, which allows discontinuities in the saturation and pressure. For the standard model, often finite-volume schemes resulting in a nonlinear system for the saturation are applied. To handle the enhanced model at the interfaces correctly, we apply a mortar discretization method on nonmatching meshes. Introducing the flux as a new variable allows us to solve the inequality constraint efficiently. This method can be applied to both the standard and the enhanced capillary model. As nonlinear solver, we use an active set strategy combined with a Newton method. Several numerical examples demonstrate the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence of the retardation term. This work was supported in part by IRTG NUPUS.     

A method for predicting swelling pressure of compacted bentonites

An approach for predicting swelling pressure of bentonites based on thermodynamic relationships between swelling pressure and suction is presented in this paper. The proposed method requires sorption isotherm data of the bentonites. A series of swelling pressure tests were performed on compacted specimens of bentonite-sand mixtures with different bentonite contents, water contents, and dry densities. The sorption isotherm of the pure bentonite was measured using a chilled-mirror hygrometer. It is found that the method works well for the bentonite-sand mixtures tested. Several published data on bentonites that have been proposed to be used as buffer and sealing material for nuclear waste repository were collected and used to verify the method. The proposed method is found to be also applicable for other bentonites of different types and therefore, can be used to predict swelling pressure of bentonites.     

A general reduction scheme for reactive transport in porous media

We present a method to transform the governing equations of multispecies reactive transport in porous media. The reformulation leads to a smaller problem size by decoupling of equations and by elimination of unknowns, which increases the efficiency of numerical simulations. The reformulation presented here is a generalization of earlier works. In fact, a whole class of transformations is now presented. This class is parametrized by the choice of certain transformation matrices. For specific choices, some known formulations of reactive transport can be retrieved. Hence, the software based on the presented transformation can be used to obtain efficiency comparisons of different solution approaches. For our efficiency tests, we use the MoMaS benchmark problem on reactive transport.     

A semi-implicit method for incompressible three-phase flow in porous media

In this paper, we present a semi-implicit method for the incompressible three-phase flow equations in two dimensions. In particular, a high-order discontinuous Galerkin spatial discretization is coupled with a backward Euler discretization in time. We consider a pressure-saturation formulation, decouple the pressure and saturation equations, and solve them sequentially while still keeping each equation implicit in its respective unknown. We present several numerical examples on both homogeneous and heterogeneous media, with varying permeability and porosity. Our results demonstrate the robustness of the scheme. In particular, no slope limiters are required and a relatively large time step may be taken.     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

A prefetching technique for prediction of porous media flows

In many applications in flows through porous media, one needs to determine the properties of subsurface to detect, monitor, or predict the actions of natural or induced forces. Here, we focus on two important subsurface properties: rock permeability and porosity. A Bayesian approach using a Markov Chain Monte Carlo (MCMC) algorithm is well suited for reconstructing the spatial distribution of permeability and porosity, and quantifying associated uncertainty in these properties. A crucial step in this approach is the computation of a likelihood function, which involves solving a possibly nonlinear system of partial differential equations. The computation time for the likelihood function limits the number of MCMC iterations that can be performed in a practical period of time. This affects the consistency of the posterior distribution of permeability and porosity obtained by MCMC exploration. To speed-up the posterior exploration, we can use a prefetching technique, which relies on the fact that multiple likelihoods of possible states into the future in an MCMC chain can be computed ahead of time. In this paper, we show that the prefetching technique implemented on multiple processors can make the Bayesian approach computationally tractable for subsurface characterization and prediction of porous media flows.     

A model for moisture and heat flow in fractured unsaturated porous media

The present study investigates propagation of a cohesive crack in non‐isothermal unsaturated porous medium under mode I conditions. Basic points of skeleton deformation, moisture, and heat transfer for unsaturated porous medium are presented. Boundary conditions on the crack surface that consist of mechanical interaction of the crack and the porous medium, water, and heat flows through the crack are taken into consideration. For spatial discretization, the extended finite element method is used. This method uses enriched shape functions in addition to ordinary shape functions for approximation of displacement, pressure, and temperature fields. The Heaviside step function and the distance function are exploited as enrichment functions for representing the crack surfaces displacement and the discontinuous vertical gradients of the pressure and temperature fields along the crack, respectively. For temporal discretization, backward finite difference scheme is applied. Problems solved from the literature show the validity of the model as well as the dependency of structural response on the material properties and loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multivalued phase pressures and a notion of weak solution for the flow which have been introduced in Cancès and Pierre (SIAM J Math Anal 44(2):966–992, 2012). We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil-trapping phenomenon.     

A robust method to tackle pressure boundary conditions in porous media flow: application to biogrout

Biogrout is a soil improvement method in which microorganisms are used to produce the solid calcium carbonate, which strengthens the soil by connecting soil particles. Microorganisms in the soil are supplied with some nutrients, which they convert into calcium carbonate. These nutrients and the side product of the reaction are dissolved in water. Because of these chemicals, the fluid is denser than water. Moreover, the density changes as a result of the varying composition. This changing density has a significant impact on the flow. Since the composition and hence, the density is not known beforehand, a careful choice of the (pressure) boundary conditions, especially on the outflow boundary, is needed. In this article, several methods to approximate the pressure on the outflow boundary are compared. The method that we propose also works for an unstructured mesh, which gives a large freedom in the mesh generation.     

A mixed solution for two-dimensional unsteady flow in fractured porous media

A mixed finite element–boundary element solution for the analysis of two-dimensional flow in porous media composed of rock blocks and discrete fractures is described. The rock blocks are modelled implicitly by using boundary elements whereas finite elements are adopted to model the discrete fractures. The computational procedure has been implemented in a hybrid code which has been validated first by comparing the numerical results with the closed-form solution for flow in a porous aquifer intercepted by a vertical fracture only. Then, a more complex problem has been solved where a pervious, homogeneous and isotropic matrix containing a net of fractures is considered. The results obtained are shown to describe satisfactorily the main features of the flow problem under study. © 1997 by John Wiley & Sons, Ltd.     

A boundary integral method for steady flow in unsaturated porous media

The governing equation for steady flow in a homogeneous, partially saturated, porous medium can be written in a linear form if one adopts a hydraulic conductivity function which varies exponentially with capillary-pressure head. The resulting linear field equation is a steady Fokker–Planck equation and is well-suited to numerical solution by the boundary integral equation method (BIEM). The exponential conductivity function is often used in soil physics and is known to be a reasonable approximation over limited ranges of pressure head. A computer code based on the BIEM for obtaining numerical solutions is described and tested. The BIEM is found to exhibit quadratic convergence with element size reduction on smooth solutions and on singular problems, if mesh grading is used. Agreement between results from the BIEM code and a finite element code that solves the fully non-linear problem is excellent, and is achieved at a substantial advantage in computer processing time. As an illustrative example, the code is applied to determine the distribution of moisture in the vicinity of a tunnel.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

A developed technique for measuring water content in oil-contaminated porous media

Water content is an important physical parameter for soil, vadose zone, and porous aquifer. Accurate measurement of water content in oil-contaminated porous media is critical for the research on oil pollution process and remediation in soil and groundwater systems. In this study, an improved water content calculation formula for oil-contaminated porous media was proposed based on the theory of oven-drying method, and laboratory experiments were conducted to test the applicability and accuracy of the formula for several types of manually prepared oil-contaminated porous media with different water contents. Furthermore, the measuring method and calculation formula, which can be used to determine the water content of porous media sampled from the oil-contaminated sites, were proposed for the first time in this study based on the improved formula. The experimental results showed that the improved formula was very accurate when used to calculate the water contents of diesel-contaminated sand, gasoline-contaminated mild clay, and engine oil-contaminated sand, indicating that it was widely applicable to oils with different volatile ability as well as porous media with different texture. This study meets the urgent need for accurate determination of water content in oil-contaminated porous media, and it solves the technical problem that the existing water content measuring methods cannot be applied directly in the field study.     

Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media

In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two‐phase and three‐phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three‐phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well‐known element free Galerkin mesh‐free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh‐free methods. Different problems of multimaterial poro‐elasticity including fully saturated, partially saturated one, and two‐phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

多孔介质两相系统毛细压力与饱和度关系试验研究

两相系统毛细压力-饱和度(h~S)关系曲线的确定是多孔介质多相流动研究的基础。采用简易试验装置对理想和实际介质中水-气和油-水两相系统中的h~S关系曲线进行了测定。试验结果表明,对于相同两相系统,多孔介质孔隙度愈小,同一毛细压力对应的饱和度相应愈大;对于不同两相系统,理想介质的关系曲线在一定毛细压力以下平缓,较大毛细压力时陡直,实际介质关系曲线走势相对较陡。分析结果表明,水-气和油-水两相系统的实测数据符合Parker等提出的基于van Genuchten(1980)关系式的折算理论;应用折算理论,可以在同一多孔介质某一两相系统h~S关系已知的情况下较好地估计同一孔隙度条件下其它两相系统的h~S关系曲线。     

A robust mesh optimisation method for multiphase porous media flows

Flows of multiple fluid phases are common in many subsurface reservoirs. Numerical simulation of these flows can be challenging and computationally expensive. Dynamic adaptive mesh optimisation and related approaches, such as adaptive grid refinement can increase solution accuracy at reduced computational cost. However, in models or parts of the model domain, where the local Courant number is large, the solution may propagate beyond the region in which the mesh is refined, resulting in reduced solution accuracy, which can never be recovered. A methodology is presented here to modify the mesh within the non-linear solver. The method allows efficient application of dynamic mesh adaptivity techniques even with high Courant numbers. These high Courant numbers may not be desired but a consequence of the heterogeneity of the domain. Therefore, the method presented can be considered as a more robust and accurate version of the standard dynamic mesh adaptivity techniques.