Exact Solution for Coupled Reactive Flow and Dissolution with Porosity Changes

We derive exact solution for mineral-dissolution reactive flows in porous media with porosity variations. These conditions are relevant to injection of incompatible liquids into aquifers for disposal or waste storage, rock alteration during well stimulation by acidising or invasion of corrosive, far-from-equilibrium fluids related to ore deposit formation and heap or in situ leaching in mineral processing. Despite the porosity change making the one-dimensional flow equations nonlinear, the problem allows for exact integration, and a novel analytical model is developed. It allows presenting typical curves for breakthrough concentrations and porosity evolution. The exact solution provides a tool for predictive testing of reactive models that account for porosity creation. The analytical model derived exhibits high agreement with laboratory data, which validate the model.     

Reactive Solute Transport in a Nonstationary,Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.     

A Structured Approach to the Derivation of Effective Properties for Combined Water and Gas Flow in the EDZ

The generation, accumulation, and release of corrosion gases is an important issue in the assessment of long-term repository performance. For repository concepts in clay-rich rock formations such as the Opalinus Clay of Northern Switzerland the transport path through the Excavation Damage Zone (EDZ) around the emplacement tunnels is of particular interest because the gas transport capacity of the host rock is limited and therefore a significant fraction of the produced gas could be released along the EDZ. This article describes the development of a structured approach to abstract complex geoscientific models of two-phase flow through the EDZ to simplified models suitable for use within a Probabilistic Safety Assessment (PSA). The approach utilizes three different models: a discrete fracture network (DFN) model of the EDZ, an equivalent heterogeneous continuum porous medium (CPM) model and a simplified CPM model suitable for use within PSA. Equivalent properties of the elements of the heterogeneous CPM models are upscaled from DFN realizations. Results from gas injection simulations with the heterogeneous CPM models are then used to derive appropriate parameters for the simplified CPM model. The modeling presented in this article represents the first step in the development of a structured methodology for treatment of gas, solute, and water flow through the EDZ. The emphasis is on methodology development, and both input data and structural models used in this study are of a generic nature and would have to be adapted to the actual conditions at a real repository site.     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.     

Stochastic Analysis of Reactive Solute Transport in Heterogeneous,Fractured Porous Media: A Dual-Permeability Approach

A nonlocal, first-order, Eulerian stochastic theory is developed for reactive chemical transport in a heterogeneous, fractured porous medium. A dual-permeability model is adopted to describe the flow and transport in the medium, where the solute convection and dispersion in the matrix are considered. The chemical is under linear nonequilibrium sorption and first-order degradation. The hydraulic conductivities, sorption coefficients, degradation rates in both fracture and matrix regions, and interregional mass transfer coefficient are all assumed to be random variables. The resultant theory for mean concentrations in both regions is nonlocal in space and time. Under spatial Fourier and temporal Laplace transforms, the mean concentrations are explicitly expressed. The transformed results are then numerically inverted to the real space via Fast Fourier Transform method. The theory developed in this study generalizes the stochastic studies for a reactive chemical transport in a one-domain flow field (Hu et al., 1997a) and in a mobile/immobile flow field (Huang and Hu, 2001). In comparison with one-domain transport, the dual-permeability model predicts a larger second moment in the longitudinal direction, but smaller one in the transverse direction. In addition, various simplification assumptions have been made based on the general solution. The validity of these assumptions has been tested via the spatial moments of the mean concentration in both fracture and matrix regions.     

Fracture-Flow-Enhanced Matrix Diffusion in Solute Transport Through Fractured Porous Media

Over the past few decades, significant progress of assessing chemical transport in fractured rocks has been made in laboratory and field investigations as well as in mathematic modeling. In most of these studies, however, matrix diffusion on fracture–matrix surfaces is considered as a process of molecular diffusion only. Mathematical modeling based on this traditional concept often had problems in explaining or predicting tracer transport in fractured rock. In this article, we propose a new conceptual model of fracture-flow-enhanced matrix diffusion, which correlates with fracture-flow velocity. The proposed model incorporates an additional matrix-diffusion process, induced by rapid fluid flow along fractures. According to the boundary-layer theory, fracture-flow-enhanced matrix diffusion may dominate mass-transfer processes at fracture–matrix interfaces, where rapid flow occurs through fractures. The new conceptual model can be easily integrated with analytical solutions, as demonstrated in this article, and numerical models, as we foresee. The new conceptual model is preliminarily validated using laboratory experimental results from a series of tracer breakthrough tests with different velocities in a simple fracture system. Validating of the new model with field experiments in complicated fracture systems and numerical modeling will be explored in future research.     

Numerical study on the shear-flow behavior and transport process in rough rock fractures

This paper presents a systematic research for understanding mechanical shearing effects on the fluid flow and the solute transport behavior of rough fractures through a numerical simulation approach. The aperture fields were modeled based on a real rock fracture geometry and the normal displacement obtained from the shear-flow test. The fluid flow through the rough fracture under shear was simulated using a finite element code that solves the Reynolds equation, and the transport behavior through the rough fracture under shear was simulated calculating the advection–dispersion equation. The results show that the fracture apertures increase as the shear displacement increases, with a few major flow channels detected through the fracture. The shear-induced flow channels increase both flow connectivity and transport connectivity, which accelerate the movement of solutes in a particular direction and lead to early breakthrough of the contaminants. Adsorption, acting as a retardation term, has a decisive influence on the transport process. These results can give a basic knowledge of the hydromechanical and solute transport progress through fracture, and will be helpful to safety assessment for high-level radioactive waste disposal facilities.     

Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.     

Coupled Diffusion-Dissolution Around a Fracture Channel: The Solute Congestion Phenomenon

The paper studies the coupled diffusion-dissolution process in reactive porous media, separated by a fracture channel. An aggressive solute, corresponding for e.g., to a complete demineralization that dissolves the solid skeleton of the surrounding porous material, is prescribed at the inlet of the fracture. By means of asymptotic dimensional analysis it is shown that for large times the diffusion length in the fracture develops with the quadratic root of time. In comparison with the 1D-Stefan Problem, in which the dissolution front evolves with the square root of time, this indicates that the overall solute evacuation through the structure slows down in time. This phenomenon is referred to as a diffusive solute congestion in the fracture. This asymptotic behavior is confirmed by means of model-based simulation, and the relevant material parameters, related to only the chemical equilibrium condition, are identified. The obtained results suggest that the presence of a small crack does not significantly increase the propagation of the dissolution front in the porous bulk, and hence the overall chemical degradation of the porous material. The same applies to other diffusion induced demineralization, mineralization, sorption and melting processes, provided that the convective transport of the solute in the crack is small in comparison with the solute diffusion. The result is relevant for several problems in durability mechanics: calcium leaching of concrete in nuclear waste containment, mineralization and demineralization in bone remodeling, chloride penetration, etc.     

裂隙岩体中非饱和渗流与运移的概念模型及数值模拟

探讨了裂隙岩体中非饱和地下水渗流与溶质运移的几种概念模型的构造及数值模拟问题 ,如裂隙网络模型、连续体模型、等效连续体模型、双孔隙度 (单渗透率 )模型、双渗透率模型、多组份连续体模型等。在裂隙岩体中 ,非饱和地下水的渗流可能只局限于岩体中的岩石组份、或裂隙网络 ,也可能在裂隙和岩石中同时发生 ;对前一种情形只需考虑单一连续体中的流动 ,而后一种情况则需要包括地下水在岩石和裂隙之间的交换。岩体中的裂隙网络往往是溶质运移的主要通道 ;但当溶质在裂隙与岩石之间的渗透和扩散是重要的运移机制时 ,就需要考虑岩石与裂隙界面处的溶质交换。为了模拟岩石与裂隙之间地下水和溶质的交换 ,就需要了解岩石与裂隙之间相互作用的模式和范围 ,使得这类问题的概念模型较单一连续体模型多了一层不确定性、其数值模拟也变得更为困难。因为在实际问题中不易、甚至根本不能判别非饱和渗流的实际形态 ,具体采用哪种模型主要取决于分析的目的和对现场数据的掌握程度。不论哪种模型都会受到模型及参数不确定性的影响 ,因此必须考虑与其他辅助模型的比较.     

A Decomposition Method for Solving Coupled Multi–Species Reactive Transport Problems

Concerns over the problems associated with mixed waste groundwater contamination have created a need for more complex models that can represent reactive contaminant fate and transport in the subsurface. In the literature, partial differential equations describing solute transport in porous media are solved either for a single reactive species in one, two or three dimensions, or for a limited number of reactive species in one dimension. Those solutions are constrained by many simplifying assumptions. Often, it is desirable to simulate transport in two or three dimensions for a more practical system that might have multiple reactive species. This paper presents a decomposition method to solve the partial differential equations of multi–dimensional, multi–species transport problems that are coupled by linear reactions. A matrix method is suggested as a tool for describing the reaction network. In this way, the level of complexity required to solve the multi–species reactive transport problem is significantly reduced.     

Matrix Heterogeneity Effects on Gas Transport and Adsorption in Coalbed and Shale Gas Reservoirs

In coalbeds and shales, gas transport and storage are important for accurate prediction of production rates and for the consideration of subsurface greenhouse gas sequestration. They involve coupled fluid phenomena in porous medium including viscous flow, diffusive transport, and adsorption. Standard approach to describe gas–matrix interactions is deterministic and neglects the effects of local spatial heterogeneities in porosity and material content of the matrix. In this study, adopting weak-noise and mean-field approximations and using a statistical approach in spectral domain, matrix heterogeneity effects are investigated in the presence of non-equilibrium adsorption with random partition coefficient. It is found that the local heterogeneities can generate non-trivial transport and kinetic effects which retard gas release from the matrix and influence the ultimate gas recovery adversely. Macro-transport shows 1/[1 + N _Pe /(1 + N _Pe )] dependence on the Péclet number, and persists at the diffusive ultra-low permeability limit. Macro-kinetics is directly related to Thiele modulus by the following expression: N _Th /(1 + 2N _Pe ). It leads to trapping of gas in the adsorbed phase during its release from the matrix, and to an adsorption threshold during the gas uptake by the matrix. Both effects are proportional to the initially available adsorbed gas amount and becomes more pronounced with the increasing variance of the porosity field. Consequently, a new upscaled deterministic gas mass balance is proposed for practical purposes. Numerical results are presented showing free and adsorbed gas distributions and fractional gas sorption curves for unipore coal matrix exhibiting Gaussian porosity distribution. This study is a unique approach for our further understanding of the coalbeds and gas shales, and it is important for the development of sound numerical gas production and sequestration models.     

Artificial neural network modeling of fixed bed biosorption using radial basis approach

In modern day scenario, biosorption is a cost effective separation technology for the removal of various pollutants from wastewater and waste streams from various process industries. The difficulties associated in rigorous mathematical modeling of a fixed bed bio-adsorbing systems due to the complexities of the process often makes the development of pure black-box artificial neural network (ANN) models particularly useful in this field. In this work, radial basis function network has been employed as ANN to model the breakthrough curves in fixed bed biosorption. The prediction has been compared to the experimental breakthrough curves of Cadmium, Lanthanum and a dye available in the literature. Results show that this network gives fairly accurate representation of the actual breakthrough curves. The results obtained from ANN modeling approach shows the better agreement between experimental and predicted breakthrough curves as the error for all these situations are within 6%.     

Modeling Water Leak-off Behavior in Hydraulically Fractured Gas Shale under Multi-mechanism Dominated Conditions

Fracturing-fluid leak-off in fractured gas shale is a complex process involving multiple pore/fluid transports and interactions. However, water leak-off behavior has not been modeled comprehensively by considering the multi-pores and multi-mechanisms in shale with existing simulators. In this paper, we present the development of a comprehensive multi-mechanistic, multi-porosity, and multi-permeability water/gas flow model that uses experimentally determined formation properties to simulate the fracturing-fluid leak-off of hydraulically fractured shale gas wells. The multi-mechanistic model takes into account water transport driven by hydraulic convection, capillary and osmosis, gas transport caused by hydraulic convection, and salt ion transport caused by advection and diffusion. The multi-porosity includes hydraulic fracture millipores, organic nanopores, clay nanopores, and other inorganic micropores. The multi-permeability model accounts for all the important processes in shale system, including gas adsorption on the organics’ surface, multi-mechanistic clay/other inorganic mineral mass transfer, inorganic mineral/hydraulic fracture mass transfer, and injection from a hydraulically fractured wellbore. The dynamic water saturation and pressure profiles within clay and other inorganic matrices are compared, revealing the leak-off behavior of water in rock media with different physicochemical properties. In sensitivity analyses, cases with different clay membrane efficiency, volume proportion of source rock, connate water salinity, and saturation are considered. The impacts of shale properties on water fluxes through wellbore, hydraulic fracture and matrix, and the total injection and leak-off volumes of the well during the treatment of hydraulic fracturing are investigated. Results show that physicochemical properties in both organic and inorganic matrices affect the water leak-off behavior.     

Simulation of Solute Transport Through Fractured Rock: A Higher-Order Accurate Finite-Element Finite-Volume Method Permitting Large Time Steps

Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure–(semi)-implicit transport scheme for the advection–diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2–4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a highly accurate overall solution. Numerical and fracture-related mechanical dispersions are compared in the realistic DFM models using second-order accurate runs as reference cases. With a CFL histogram, we establish target error criteria for CFL overstepping. This analysis indicates that for extreme fracture heterogeneity, only a few transport steps can be sufficient to analyse macro-dispersion. This makes our implicit method attractive for quick analysis of transport properties on multiple realisations of DFM models.     

Upscaling in Vertically Fractured Oil Reservoirs Using Homogenization

Flow modeling in fractured reservoirs is largely confined to the so-called sugar cube model. Here, however, we consider vertically fractured reservoirs, i.e., the situation that the reservoir geometry can be approximated by fractures enclosed columns running from the base rock to the cap rock (aggregated columns). This article deals with the application of the homogenization method to derive an upscaled equation for fractured reservoirs with aggregated columns. It turns out that vertical flow in the columns plays an important role, whereas it can be usually disregarded in the sugar cube model. The vertical flow is caused by coupling of the matrix and fracture pressure along the vertical faces of the columns. We formulate a fully implicit three-dimensional upscaled numerical model. Furthermore, we develop a computationally efficient numerical approach. As found previously for the sugar cube model, the Peclet number, i.e., the ratio between the capillary diffusion time in the matrix and the residence time of the fluids in the fracture, plays an important role. The gravity number plays a secondary role. For low Peclet numbers, the results are sensitive to gravity, but relatively insensitive to the water injection rate, lateral matrix column size, and reservoir geometry, i.e., sugar cube versus aggregated column. At a low Peclet number and sufficiently low gravity number, the effective permeability model gives good results, which agree with the solution of the aggregated column model. However, ECLIPSE simulations (Barenblatt or Warren and Root (BWR) approach) show deviations at low Peclet numbers, but show good agreement at intermediate Peclet numbers. At high Peclet numbers, the results are relatively insensitive to gravity, but sensitive to the other conditions mentioned above. The ECLIPSE simulations and the effective permeability model show large deviations from the aggregated column model at high Peclet numbers. We conclude that at low Peclet numbers, it is advantageous to increase the water injection rate to improve the net present value. However, at high Peclet numbers, increasing the flow rate may lead to uneconomical water cuts.     

Non-Fickian Transport in Transparent Replicas of Rough-Walled Rock Fractures

We present an experimental investigation and modeling analysis of tracer transport in two transparent fracture replicas. The original fractures used in this work are a Vosges sandstone sample with nominal dimensions approximately 26 cm long and 15 cm wide, and a granite sample with nominal dimensions approximately 33 cm long and 15.5 cm wide. The aperture map and physical characteristics of the fractures reveal that the aperture map of the granite fracture has a higher spatial variability than the Vosges sandstone one. A conservative methylene blue aqueous solution was injected uniformly along the fracture inlets, and exited through free outlet boundaries. A series of images was recorded at known time intervals during each experiment. Breakthrough curves were subsequently determined at the fracture outlets and at different distances, using an image processing based on the attenuation law of Beer–Lambert. These curves were then interpreted using a stratified medium model that incorporates a permeability distribution to account for the fracture heterogeneity, and a continuous time random walk (CTRW) model, as well as the classical advection–dispersion equation (ADE). The stratified model provides generally satisfactory matches to the data, while the CTRW model captures the full evolution of the long tailing displayed by the breakthrough curves. The transport behavior is found to be non-Fickian, so that the ADE is not applicable. In both stratified and CTRW models, parameter values related to the aperture field spatial variability indicate that the granite fracture is more heterogeneous than the Vosges sandstone fracture.     

Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.     

The Dual-Reciprocity Boundary Element Analysis for Hydraulically Fractured Shale Gas Reservoirs Considering Diffusion and Sorption Kinetics

This work presents a new application of boundary element method (BEM) to model fluid transport in unconventional shale gas reservoirs with discrete hydraulic fractures considering diffusion, sorption kinetics and sorbed-phase surface diffusion. The fluid transport model consists of two governing partial differential equations (PDEs) written in terms of effective diffusivities for free and sorbed gases, respectively. Boundary integral formulations are analytically derived using the fundamental solution of the Laplace equation for the governing PDEs and Green’s second identity. The domain integrals arising due to the time-dependent function and nonlinear terms are transformed into boundary integrals employing the dual-reciprocity method. This transformation retains the domain-integral-free, boundary-integral-only character of standard BEM approaches. In the proposed solution, the free- and sorbed-gas flow in the shale matrix is solved simultaneously after coupling the fracture flow equation of free gas. Well production performance under the effect of relaxation phenomenon due to delayed responses of sorbed gas under nonequilibrium sorption condition is rigorously captured by imposing the zero-flux condition at fracture–matrix interface for the sorbed-gas transport equation. The validity of proposed solution is verified using several case studies through comparison against a commercial finite-element numerical simulator.

     

Transport,Dilution, and Dispersion of Contaminant in a Leaky Karst Conduit

Often the water flowing in a solution conduit is a combination of contaminated water entering at sinkholes and cleaner water released from the limestone matrix. The concentration of contaminants flowing in a conduit is reduced by this dilution and also by longitudinal conduit dispersion. This article seeks to quantify relative importance of these two mechanisms. Water entering the conduit from the matrix causes the conduit flow speed to increase with distance. This in turn causes the strength of dispersion to increase in the downstream direction. The breakthrough curve at a spring, resulting from transport, dilution, and dispersion, has been obtained for the initial-value problem using a modification of the standard Green’s function approach, employing characteristic curves. The predicted breakthrough curves are skewed, with a rapid rise and slow decay. This feature does not result from the ordinary advection–dispersion model. Applying the new model to a dye-tracing experiment between Ames Sink and Indian Spring, Northwest Florida yields a value of dispersivity at 400 m. It is concluded that variable dispersion provides a possible explanation for the skewness and tailing that are often observed in spring breakthrough curves. It is demonstrated that the new model, in conjunction with the measured spring breakthrough curve, can extract the parameters of the conduit and flow.