Modeling of Gas Migration Through Low-Permeability Clay Rock Using Information on Pressure and Deformation from Fast Air Injection Tests

The characterization of gas migration through low-permeability clay formations has been a focus of R&D programs for radioactive waste disposal, which is also of great importance for shale gas exploration, cap-rock behavior of hydrocarbon reservoirs, and \(\hbox {CO}_{2}\) sequestration. Laboratory tests have been performed on Opalinus Clay, a Mesozoic claystone that is being investigated in Switzerland as a potential host rock for the storage of nuclear waste. The laboratory program included specific water and air injections tests, as well as oedometer and isotropic compression tests. Undisturbed core samples have been retrieved from a shallow borehole in the Mont Terri Underground Research Laboratory (URL) and from a deep borehole in northern Switzerland. For the shallow cores from Mont Terri URL, largely linear-elastic deformations associated with the gas injection test could be inferred and the change in void ratio was accounted for by the pore compressibility. The corresponding change in permeability was obtained from the results of the water tests, indicating a log-linear relation between permeability and porosity. The derived porosity change and the corresponding change in permeability were implemented in the standard TOUGH2 code, which reproduced the measured gas test results using fitted water retention data derived from laboratory measurements. Similar air injection tests performed on Opalinus Clay cores from the borehole at greater depth showed overall similar behavior, but at lower porosities, lower permeability values, and lower compressibility. These cases indicated nonlinear behavior which was implemented using an effective stress-dependent porosity change and the associated change in permeability. In addition, the anisotropy associated with the bedding planes of the clay formation was considered by assuming different properties for “soft” and “hard” layers to account for storage capacity for the injected gas prior to gas breakthrough. The computed change in the overall porosity could be compared to the measured axial deformation during the gas injection test and was used for calibration of the parameters describing the relationship between the effective stress and porosity, as well as the corresponding change in permeability and capillary pressure.     

Modeling and Application of Gas Pressure Measurement in Water-Saturated Coal Seam Based on Methane Solubility

The unconfined seepage problem is a classic moving boundary problem, in which the position of phreatic surface is unknown at the beginning of solution and should be determined through iteration. Mesh-free methods are especially suitable for solving this problem. In this work, the moving Kriging mesh-free method with Monte Carlo integration is proposed. Additionally, a corresponding procedure for handling material discontinuity is presented, which extends the approach to inhomogeneous medium. The present method is a true mesh-free method, which does not require a mesh for either shape function construction or numerical integration. Another advantage of the present method is the convenient numerical implementation. Numerical examples show that the present method can achieve better convergence and higher accuracy with rational computation cost when compared with the original mesh-free method. The present method is also verified to be applicable in analyzing transient seepage through homogeneous and inhomogeneous media.     

Dissipation Rate Estimation in a Highly Turbulent Isotropic Flow Using 2D-PIV

Flow, Turbulence and Combustion - In experimental turbulent flows, the estimation of the dissipation rate of turbulent kinetic energy, $$\varepsilon$$ , is a challenge. The dimensional analysis...     

Break-Up of Aerosol Agglomerates in Highly Turbulent Gas Flow

Agglomerate aerosols in a turbulent flow may be subjected to very high turbulent shear rates which through the generation of lift and drag can overcome the adhesive forces binding the constituents of an agglomerate together and cause it to break-up. This paper presents an analysis of the experimental measurements of the breakup of agglomerates between 0.1?C10???m in size, in a turbulent pipe flow followed by an expansion zone with a Reynolds numbers in the range 10⁵ to 10⁷. The analysis shows that even in wall bounded turbulence, the high turbulent shear stresses associated with the small scales of turbulence in the core can be the main source of breakup preceding any break-up that may occur by impaction at the wall. More importantly from these results, a computationally fast and efficient solution is obtained for the General Dynamic Equation (GDE) for agglomerate transport and breakup in highly turbulent flow. Furthermore the solution for the evolution of the aerosol size distribution is consistent with the experimental results. In the turbulent pipe flow section, the agglomerates are exposed continuously to turbulent shear stresses and experience more longer term breakup than in the expansion zone (following the pipe flow) where the exposure time is much less and break-up occurs instantaneously under the action of very high local turbulent shear stresses. The validity of certain approximations made in the model is considered. In particular, the inertia of the agglomerates characterised by a Stokes Number from 0.001 for the smallest particles up to 10 for 10???m particles and the fluctuations of the turbulent shear stresses are important physical phenomena which are not accounted for in the model.     

LBM Investigation of Immiscible Displacement in a Channel with Regular Surface Roughness

We investigated immiscible displacement in a channel considering regular surface roughness at walls. A color-fluid LBM code is developed and validated against the Hagen–Poiseuille flow before it is employed for simulating the displacement process. The dimensionless roughness height and roughness spacing ratio are defined to characterize the surface roughness. The simulation results show that the presence of surface roughness obviously impels a finger formation in a channel. The impelling effect is more significant at larger roughness heights and medium roughness spacing ratios. The critical capillary number and viscosity ratio of a finger formation is reduced with increasing roughness height. The obvious effect of wettability on the finger development in a smooth channel is attenuated in rough channels. The attenuation magnitude increases with increasing roughness height.     

Evidence for a Second Transport Porosity for the Diffusion of Tritiated Water (HTO) in a Sedimentary Rock (Opalinus Clay - OPA): Application of Through- and Out-Diffusion Techniques

The diffusion of tritiated water (HTO) in Opalinus clay (OPA) samples from bore cores from the Benken area (Northern Switzerland) was studied using the radial through- and out-diffusion technique. Results from inverse modelling of out-diffusion data for HTO indicated the presence of two preferential diffusion pathways: a fast and a slow one. Analysing through-diffusion data, however, provides hardly any information concerning a second transport-relevant porosity. Only by also analysing the out-diffusion phase can finer details of the diffusion process and information on sample heterogeneity be recognised. The extracted values for the effective diffusion coefficient are in the order of 3 × 10^–11 m² s^–1 for the faster transport porosity and roughly an order of magnitude smaller for the slower type of porosity. We had to account for tritium sorption on the clay minerals by a small but non-zero K_d-value in the order of 10^–5 m³ kg^–1 in order to reproduce the data with acceptable precision. In the model applied both porosities are considered as being independent from each other. Roughly 30% of the tracer diffused through the second, slower porosity; such a fact might be interesting for future performance assessments for radioactive waste repositories hosted by clay formations. Based on our present picture from water-saturated OPA, on a microscopic scale three different kind of waters can be discriminated: free water, double layer water and interlayer water. However, using HTO as tracer only, it could not be deduced which type of water-filled pores finally account for the transport-relevant porosity.Author for correspondence: Tel.: +41-56-3102257; Fax: +41-56-3104438; E-mail: luc.vanloon@psi.ch     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

Kinematic Parameters of a Two-Phase Flow in a Rectangular Channel

A kinematic two-phase flow pattern formed in a rectangular channel due to the interaction of a gas flow with an initially stationary or moving water layer is investigated. Using laser diagnostics and hot-wire methods, the velocity distributions in the water and the air are found for a stratified flow regime.     

Effective Diffusivity Prediction of Radionuclides in Clay Formations Using an Integrated Upscaling Workflow

Transport in Porous Media - The effective diffusivity is a key parameter in the diffusive transport calculations, thus decisive for predicting the radionuclide migration in low-permeable clay-rich...     

Two-Phase Models of Formation of Cavitating Spalls in a Liquid

The dynamics of the structure of a liquid layer structure (with microbubbles of a free gas) behind a rarefaction wave front is studied numerically using the two-phase Iordansky–Kogarko–van Wijngaarden model and the frozen mass-velocity field model. An analysis of the initial stage of cavitation by the Iordansky–Kogarko–van Wijngaarden model showed that tensile stresses behind the rarefaction wave front relax quickly and the mass-velocity field in the cavitation zone turns out to be frozen. This effect is used to describe the late stage of the development of the cavitation zone. These models were combined to study the formation of cavitating spalls in a free-surface liquid under shock-wave loading.     

Mesoscopic Modeling of Capillarity-Induced Two-Phase Transport in a Microfluidic Porous Structure

Mesoscopic modeling at the pore scale offers great promise in exploring the underlying structure transport performance of flow through porous media. The present work studies the fluid flow subjected to capillarity-induced resonance in porous media characterized by different porous structure and wettability. The effects of porosity and wettability on the displacement behavior of the fluid flow through porous media are discussed. The results are presented in the form of temporal evolution of percentage saturation and displacement of the fluid front through porous media. The present study reveals that the vibration in the form of acoustic excitation could be significant in the mobilization of fluid through the porous media. The dependence of displacement of the fluid on physicochemical parameters like wettability of the surface, frequency along with the porosity is analyzed. It was observed that the mean displacement of the fluid is more in the case of invading fluid with wetting phase where the driving force strength is not so dominant.     

Observation of Light Non-Aqueous Phase Liquid Migration in Aggregated Soil Using Image Analysis

Physical model experiments were conducted to observe the migration of light non-aqueous phase liquids (LNAPL) in a double-porosity soil medium. The double-porosity characteristics of the soil were simulated through aggregation of kaolin which resulted in well-defined intra-aggregate and inter-aggregate pores. Digital images were collected to monitor LNAPL (modeled by toluene) migration. A special experimental setup was developed to enable the instantaneous capture of the LNAPL migration around the whole soil column using a single digital camera. An image processing module was applied to the captured images and the results plotted using a surface mapping programme. Events observed during the duration of the experiments were discussed. It was found that the LNAPL flowed much faster in the aggregated soil as compared to a single-porosity soil. The wettability of the fluid and the capillary pressure characteristics were demonstrated to be influential factors in immiscible fluids migration when the soil fabric showed highly contrasting porosity values.     

Analysis of reflective and non-reflective boundary conditions in LBM simulations of flow past an obstacle in a channel

This paper analyses the effects of considering non-reflective (characteristic boundary conditions) versus reflective boundary conditions (BCs) on the flow past a square object. We observe a clear dependence of the force exerted over the obstacle on the choice of BCs. Recirculation lengths, lift and drag coefficients, phase diagrams and streamlines for several angles of incidence are compared for a range of low Reynolds numbers (50–150) and two different values of the ratio of the object cross section to the channel width, 1/8 and 1/16. We remark distinct effects depending on the combination of BCs used at the inlet and at the outlet.     

Modeling Approaches for Investigating Gas Migration from a Deep Low/Intermediate Level Waste Repository (Switzerland)

In low/intermediate-level waste (L/ILW) repositories, anaerobic corrosion of metals and degradation of organic materials produce mainly hydrogen, methane, and carbon dioxide. The Swiss reference concept for the L/ILW repository consists of parallel caverns sealed off from a single access tunnel in a deep low-permeability claystone formation. The potential buildup of excess gas pressures in the backfilled emplacement caverns was investigated in a series of two-phase flow models. In the first step, a large-scale model was constructed, implementing the 3D radial tunnel and cavern geometry with a simplified rectangular geometry. In the second step, the potential impact of the detailed geometry of the engineered barrier system (EBS) and the associated heterogeneity inside the cavern was examined using detailed models of the repository caverns, tunnel seals, access tunnel, and surrounding host rock. The simulation results from the large-scale 3D repository model show that during the early post-closure period simulated pressures can vary significantly between different parts of the repository. The simulated pressure increase in the emplacement caverns remained below the fracture pressure of the rock for realistic assumptions. Gas flow is largely limited to the EBS and the excavation disturbed zone (EDZ); thus, gas flows through and around the repository seal into the adjacent tunnel system, which is also demonstrated in the detailed repository-cavern model. The repository seal model described the detailed two-phase flow pattern of early time resaturation of the repository by water inflow from the ramp and subsequent counter flow associated with the gas flow from the repository cavern. Overall, the results of the detailed models complement and confirmed the results of the large-scale 3D model in terms of the timing of the pressure peaks and the migration of gas from the cavern into the surrounding host rock and through the repository seal.     

Two-Phase Filtration in a Transversally Isotropic Porous Medium: Experiment and Theory

The results of an experimental determination of the relative phase permeabilities during flow of two immiscible fluids in stratified sandstone with transversally isotropic characteristics are presented. The measurements were performed on samples oriented in three directions: along, perpendicular to and at an angle of 45° to the stratification plane. An approximate solution of the problem of steady two-phase flow toward a finite gallery in an anisotropic porous medium for arbitrary relative orientation of the gallery and the principal axes of the absolute permeability tensor is given. This solution was tested against the experimental results. The good agreement between the theoretical and experimental results makes it possible to recommend for engineering calculations both the relations between the absolute and phase permeabilities for transversally isotropic and orthotropic porous media and the approximate solution obtained.     

Influence of Macro-pores on DNAPL Migration in Double-Porosity Soil Using Light Transmission Visualization Method

Double porosity is a substantial microstructure characteristic in a wide range of geomaterials. It is a natural phenomenon that can be found in many types of soil, and it can result from biological, chemical or mechanical damage. In this paper, the influence of macro-pores on dense non-aqueous phase liquid (DNAPL) migration in double-porosity medium was investigated using light transmission visualization technique. Three experiments were carried out in two-dimensional flow chambers filled with a double-porosity medium composed of a mixture of local sand and sintered kaolin clay spheres arranged in a periodic manner. In each experiment, a different volumetric fraction of macro-pores and micropores was used. Tetrachloroethylene (PCE) was used to simulate DNAPL, and it was dyed using Oil-Red-O for better visualization. A predetermined amount of PCE was injected into the flow chambers and this amount was re-calculated using image analysis. A very strong correlation was found between the PCE amount injected and the amount calculated from image analysis in each experiment. The experiment was repeated by filling the flow chamber with silica sand to represent single-porosity medium. The results show that the macro-pores have a considerable effect on the PCE migration in double-porosity soil as the PCE movement was the fastest in the third experiment which contained the largest macro-pores volume. The accuracy of the method was validated using statistical analysis. The results show a slight difference between the means of the three experiments, indicating that the method is viable for monitoring NAPL migration in double-porosity medium under different volumetric fractions of macro-pores and micropores.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Simulation of Gas Dipole Tests in Fractures at the Intermediate Scale Using a New Upscaling Method

A high-level radioactive waste disposal site may lead to gas generation by different physical mechanisms. As these sites are to be located in areas with low water flow, any small amount of gas can lead to relative high gas pressures, so that multiphase flow analysis becomes relevant. The movement of gas and water through the system has two important implications. Firstly, water flow takes place in unsaturated conditions, and thus travel times of the radioactive particles transported are affected; and secondly, gas can also carry radioactive particles. Therefore, one of the key points in such studies is the time when gas would break through the biosphere under a number of different flow conditions. In fractured zones, gas would flow preferentially through the most conductive features. We consider a two-dimensional system representing an isolated fracture. In each point we assign a local porosity and permeability and a local pressure-saturation relationship. A dipole (injector-producer) gas flow system is generated and the variation in water saturation is studied. A simple method is proposed for obtaining upscaled values for several parameters involved in two-phase flow. It is based on numerical simulation on a block scale assuming steady-state conditions and absence of capillary pressure gradients. The proposed method of upscaling is applied to simulate a dipole test using a coarser grid than that of the reference field. The comparison between the results in both scales shows an encouraging agreement.