Discrete Fracture Matrix Research Articles

Two-phase Discrete Fracture Matrix models with linear and nonlinear transmission conditions

This work deals with two-phase Discrete Fracture Matrix models coupling the two-phase Darcy flow in the matrix domain to the two-phase Darcy flow in the network of fractures represented as co-dimension one surfaces. Two classes of such hybrid-dimensional models are investigated either based on nonlinear or linear transmission conditions at the matrix–fracture interfaces. The linear transmission conditions include the cell-centred upwind approximation of the phase mobilities classically used in the porous media flow community as well as a basic extension of the continuous phase pressure model accounting for fractures acting as drains. The nonlinear transmission conditions at the matrix–fracture interfaces are based on the normal flux continuity equation for each phase using additional interface phase pressure unknowns. They are compared both in terms of accuracy and numerical efficiency to a reference equi-dimensional model for which the fractures are represented as full-dimensional subdomains. The discretization focuses on Finite Volume cell-centred Two-Point Flux Approximation which is combined with a local nonlinear solver allowing to eliminate efficiently the additional matrix–fracture interfacial unknowns together with the nonlinear transmission conditions. 2D numerical experiments illustrate the better accuracy provided by the nonlinear transmission conditions compared to their linear approximations with a moderate computational overhead obtained thanks to the local nonlinear elimination at the matrix–fracture interfaces. The numerical section is complemented by a comparison of the reduced models on a 3D test case using the Vertex Approximate Gradient scheme.

A simple embedded discrete fracture–matrix model for a coupled flow and transport problem in porous media

Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

Numerical analysis of a two-phase flow discrete fracture matrix model

We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since pressure discontinuities at the matrix-fracture interfaces are permitted. Additionally, a layer of damaged rock at the matrix-fracture interfaces is accounted for. The numerical analysis is carried out in the general framework of the Gradient Discretisation Method. Compactness techniques are used to establish convergence results for a wide range of possible numerical schemes; the existence of a solution for the two phase flow model is obtained as a byproduct of the convergence analysis. A series of numerical experiments conclude the paper, with a study of the influence of the damaged layer on the numerical solution.

Advanced computation of steady-state fluid flow in Discrete Fracture-Matrix models: FEM–BEM and VEM–VEM fracture-block coupling

In this note the issue of fluid flow computation in a Discrete Fracture-Matrix (DFM) model is addressed. In such a model, a network of percolative fractures delimits porous matrix blocks. Two frameworks are proposed for the coupling between the two media. First, a FEM–BEM technique is considered, in which finite elements on non-conforming grids are used on the fractures, whereas a boundary element method is used on the blocks; the coupling is pursued by a PDE-constrained optimization formulation of the problem. Second, a VEM–VEM technique is considered, in which a 2D and a 3D virtual element method are used on the fractures and on the blocks, respectively, taking advantage of the flexibility of VEM in using arbitrary meshes in order to ease the meshing process and the consequent enforcement of the matching conditions on fractures and blocks.

Analysing the limitations of the dual-porosity response during well tests in naturally fractured reservoirs

Geological reservoirs can be extensively fractured but the well-test signatures observed in the wells may not show a pressure transient response that is representative of naturally fractured reservoirs (NFRs): for example, one that indicates two distinct pore systems (i.e. the mobile fractures and immobile matrix). Yet, the production behaviour may still be influenced by these fractures. To improve the exploitation of hydrocarbons from NFRs, we therefore need to improve our understanding of fluid-flow behaviour in fractures. Multiple techniques are used to detect the presence and extent of fractures in a reservoir. Of particular interest to this work is the analysis of well-test data in order to interpret the flow behaviour in an NFR. An important concept for interpreting well-test data from an NFR is the theory of dual-porosity model. However, several studies pointed out that the dual-porosity model may not be appropriate for interpreting well tests from all fractured reservoirs. This paper therefore uses geological well-testing insights to explore the limitations of the characteristic flow behaviour inherent to the dual-porosity model in interpreting well-test data from Type II and III NFRs of Nelson's classification. To achieve this, we apply a geoengineering workflow with discrete fracture matrix (DFM) modelling techniques and unstructured-grid reservoir simulations to generate synthetic pressure transient data in both idealized fracture geometries and real fracture networks mapped in an outcrop of the Jandaira Formation. We also present key reservoir features that account for the classic V-shape pressure derivative response in NFRs. These include effects of fracture skin, a very tight matrix permeability and wells intersecting a minor, unconnected fracture close to a large fracture or fracture network. Our findings apply to both connected and disconnected fracture networks.

Coupled fluid flow and geomechanics modeling of stress-sensitive production behavior in fractured shale gas reservoirs

Compared with conventional reservoirs, gas flow in shale formation is affected by additional nonlinear coupled processes such as matrix/fracture deformations. We present a fully-coupled fluid flow and geomechanics model to accurately characterize the complex production behaviors of fractured shale gas reservoirs. The flow equations are discretized using a mimetic finite difference method, and poro-elasticity equations by a Galerkin finite-element approximation. A unified apparent-permeability model is implemented to quantify the combined impacts of the non-Darcy flow, adsorbed layer and pore-structure alterations on matrix permeability.The discrete fracture-matrix (DFM) model based on a conformal unstructured grid is employed to explicitly represent fractures. The nonlinear contact problem between the two fracture surfaces is introduced to describe the fracture mechanics behavior. A splitting-node technique is used to deal with the discontinuities in the displacement field across the fracture interface. Under the effects of pressure decline and high confining stresses on the fracture faces, proppant compaction and embedment may occur, causing fracture closure and thus substantial production loss. Hence we also develop a comprehensive proppant-fracture model which is based on the theories of elasto-plastic contact mechanics, to capture the complex interactions between proppant and fracture.The multiphysics numerical model enables us to investigate which factors have the most influential effect on the gas recovery of shale formations. High fidelity numerical solutions are provided to characterize the rate-transient signatures in the presence of the different flow and geomechanical mechanisms.

An improved projection-based embedded discrete fracture model (pEDFM) for multiphase flow in fractured reservoirs

Abstract The discrete fracture-matrix (DFM) approaches based on conforming grids become popular for modeling fractured reservoirs in the last decade. However, the application of conforming DFMs at field scale is limited due to its prohibitive computational cost. In recent years, embedded discrete fracture model (EDFM) has received considerable attention as a promising alternative. EDFM incorporates the effect of each fracture explicitly without requiring the simulation grid to conform to the fracture geometry. A compromise between accuracy and efficiency could be achieved in EDFM by enabling the use of standard corner-point grids for the background matrix domain. Although many works confirm the high accuracy of EDFM for the solutions of pressure and velocity field, very few results have been presented to examine its accuracy for the saturation solutions from multiphase flow problems. This paper shows that EDFM can induce large errors for multiphase displacement processes, due to its incapability to capture the proper flux split through a fracture. For the first time in the literature we present a systematic evaluation on the performances of EDFM for multiphase flow and provide a detailed analysis to illuminate when and why the method fails. The analysis motivates us to exploit the projection-based extension of EDFM (pEDFM) as an effective method to resolve the limitations associated with EDFM. pEDFM is recently developed by Tene et al. (2017) to address the issue of flow barriers, and is based on the introduction of extended fracture-matrix fluxes. Moreover, we make several improvements upon the original pEDFM method. A physical constraint on the preprocessing stage is proposed to overcome the limitation in a ‘naive implementation’ of pEDFM. A number of test cases with different fracture geometries are presented to benchmark the performances of the improved pEDFM method for multiphase flow. Grid convergence studies are conducted for different numerical schemes. The results show that improved pEDFM significantly outperforms the original EDFM method.

Assisted history matching for the inversion of fractures based on discrete fracture-matrix model with different combinations of inversion parameters

Accurate prediction of fracture distribution in fractured reservoirs is important in the development process. Considering that assisted history matching technology is an effective method for the inversion of reservoir parameters, the technology can also be applied for the inversion of fractures. Because applying assisted history matching technology for the inversion of fractures has an inherent defect of multiplicity of solution, it is therefore necessary to alleviate the multiplicity for the success of inversion. Although there are many factors affecting the multiplicity, the paper focuses on the study of the inversion results of different combinations of inversion parameters which are all representative parameters of fractures and determine the distribution of fractures. Firstly, we simulate the flow behavior in fractured media based on the discrete fracture matrix (DFM) module of Matlab Reservoir Simulation Toolbox (MRST) to explicitly describe the effect of fractures on flow behavior. Secondly, history matching objective function is established based on Bayesian theory and different kinds of representative parameters of fractures are chosen as inversion parameters. Thirdly, simultaneous perturbation stochastic approximation (SPSA) algorithm is adopted to minimize the objective function to achieve the inversion of fractures corresponding to different inversion parameters. Finally, theoretical cases verify that the inversion method is effective for the accurate prediction of fracture distribution and proper inversion parameters are crucial to the success of fracture inversion.

CVD-MPFA full pressure support, coupled unstructured discrete fracture–matrix Darcy-flux approximations

Two novel control-volume methods are presented for flow in fractured media, and involve coupling the control-volume distributed multi-point flux approximation (CVD-MPFA) constructed with full pressure support (FPS), to two types of discrete fracture–matrix approximation for simulation on unstructured grids; (i) involving hybrid grids and (ii) a lower dimensional fracture model. Flow is governed by Darcy's law together with mass conservation both in the matrix and the fractures, where large discontinuities in permeability tensors can occur. Finite-volume FPS schemes are more robust than the earlier CVD-MPFA triangular pressure support (TPS) schemes for problems involving highly anisotropic homogeneous and heterogeneous full-tensor permeability fields. We use a cell-centred hybrid-grid method, where fractures are modelled by lower-dimensional interfaces between matrix cells in the physical mesh but expanded to equi-dimensional cells in the computational domain. We present a simple procedure to form a consistent hybrid-grid locally for a dual-cell. We also propose a novel hybrid-grid for intersecting fractures, for the FPS method, which reduces the condition number of the global linear system and leads to larger time steps for tracer transport. The transport equation for tracer flow is coupled with the pressure equation and provides flow parameter assessment of the fracture models. Transport results obtained via TPS and FPS hybrid-grid formulations are compared with the corresponding results of fine-scale explicit equi-dimensional formulations. The results show that the hybrid-grid FPS method applies to general full-tensor fields and provides improved robust approximations compared to the hybrid-grid TPS method for fractured domains, for both weakly anisotropic permeability fields and very strong anisotropic full-tensor permeability fields where the TPS scheme exhibits spurious oscillations. The hybrid-grid FPS formulation is extended to compressible flow and the results demonstrate the method is also robust for transient flow. Furthermore, we present FPS coupled with a lower-dimensional fracture model, where fractures are strictly lower-dimensional in the physical mesh as well as in the computational domain. We present a comparison of the hybrid-grid FPS method and the lower-dimensional fracture model for several cases of isotropic and anisotropic fractured media which illustrate the benefits of the respective methods.

An efficient fully-implicit multislope MUSCL method for multiphase flow with gravity in discrete fractured media

The first-order methods commonly employed in reservoir simulation for computing the convective fluxes introduce excessive numerical diffusion leading to severe smoothing of displacement fronts. We present a fully-implicit cell-centered finite-volume (CCFV) framework that can achieve second-order spatial accuracy on smooth solutions, while at the same time maintain robustness and nonlinear convergence performance.A novel multislope MUSCL method is proposed to construct the required values at edge centroids in a straightforward and effective way by taking advantage of the triangular mesh geometry. In contrast to the monoslope methods in which a unique limited gradient is used, the multislope concept constructs specific scalar slopes for the interpolations on each edge of a given element. Through the edge centroids, the numerical diffusion caused by mesh skewness is reduced, and optimal second order accuracy can be achieved. Moreover, an improved smooth flux-limiter is introduced to ensure monotonicity on non-uniform meshes. The flux-limiter provides high accuracy without degrading nonlinear convergence performance.The CCFV framework is adapted to accommodate a lower-dimensional discrete fracture-matrix (DFM) model. Several numerical tests with discrete fractured system are carried out to demonstrate the efficiency and robustness of the numerical model.

Impact of eliminating fracture intersection nodes in multiphase compositional flow simulation

AbstractAlgebraic elimination of nodes at discrete fracture intersections via the star‐delta technique has proven to be a valuable tool for making multiphase numerical simulations more tractable and efficient. This study examines the assumptions of the star‐delta technique and exposes its effects in a 3‐D, multiphase context for advective and dispersive/diffusive fluxes. Key issues of relative permeability‐saturation‐capillary pressure (kr‐S‐Pc) and capillary barriers at fracture‐fracture intersections are discussed. This study uses a multiphase compositional, finite difference numerical model in discrete fracture network (DFN) and discrete fracture‐matrix (DFM) modes. It verifies that the numerical model replicates analytical solutions and performs adequately in convergence exercises (conservative and decaying tracer, one and two‐phase flow, DFM and DFN domains). The study culminates in simulations of a two‐phase laboratory experiment in which a fluid invades a simple fracture intersection. The experiment and simulations evoke different invading fluid flow paths by varying fracture apertures as oil invades water‐filled fractures and as water invades air‐filled fractures. Results indicate that the node elimination technique as implemented in numerical model correctly reproduces the long‐term flow path of the invading fluid, but that short‐term temporal effects of the capillary traps and barriers arising from the intersection node are lost.

Production forecasting and uncertainty quantification for naturally fractured reservoirs using a new data-space inversion procedure

A new method for production forecasting and uncertainty quantification, applicable for realistic naturally fractured reservoirs (NFRs) represented as general discrete-fracture-matrix (DFM) models, is developed and applied. The forecasting procedure extends a recently developed data-space inversion (DSI) technique that generates production predictions using only prior-model simulation results and observed data. The method does not provide posterior (history-matched) geological models. Rather, the DSI method treats production data as random variables. The prior distribution is estimated from the flow simulations performed on prior geological models, and the posterior data-variable distribution is sampled using a data-space randomized maximum likelihood method. The DSI treatment requires the parameterization of data variables to render them approximately multivariate Gaussian. The complex production data considered here (resulting from frequent well shut-ins) is treated using a new reparameterization that involves principal component analysis combined with histogram transformation. The DSI method is first applied for two-dimensional DFM systems involving multiple fracture scenarios. In this case, comparison with a rejection sampling procedure is possible, and we show that the DSI results for P10, P50, and P90 statistics are consistent with rejection sampling results. The DSI method is then applied to a realistic NFR that has undergone 15 years of primary production and is under consideration for waterflooding. To construct the DSI representation, around 400 prior DFM models, which correspond to different geologic concepts and properties, are simulated. Two different reference ‘true’ models, along with different data-assimilation durations, are considered to evaluate the performance of the DSI procedure. In all cases, the DSI predictions are shown to be consistent with the forecasts from the ‘true’ model and to provide reasonable quantification of forecast uncertainty.

Inversion of fractures with combination of production performance and in-situ stress analysis data

Successful identification of the fractures in fractured reservoirs is important to guarantee an effective development. Considering that the production performance of fractured reservoirs contains important information of the distribution of fractures, the production performance can be applied for the inversion of fractures. However, the inversion of fractures is difficult to achieve because it is an inverse problem with an inherent defect of multiplicity of solutions. In order to alleviate the defect, we estimate the possible distribution ranges of fractures which can be obtained by analyzing the correlation of fractures and in-situ stress, and the ranges are used as constraint conditions when the production performance is applied for obtaining the accurate distribution of fractures. Firstly, we choose the geometric parameters of fractures such as midpoint coordinate, azimuth and extension length of fractures as inversion parameters and use production data as inversion indexes. Secondly, we simulate the flow behavior in fractured reservoirs based on the Discrete Fracture Matrix (DFM) module of Matlab Reservoir Simulation Toolbox (MRST) to explicitly describe the effect of fractures on the flow behavior of fluid. Thirdly, the possible distribution ranges of fractures which can be obtained by analyzing the correlation between fractures and in-situ stress based on Griffith failure criterion are used as the constraint conditions of inversion parameters. Finally, Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm is adopted to minimize the inversion objective function to obtain the accurate distribution of fractures. Theoretical cases verify that the method is effective for the accurate inversion of fractures while the inversion results of more fractures become worse because more fractures make the sensitivities of production performance to individual fractures decrease greatly.

Hybrid Coupled Discrete-Fracture/Matrix and Multicontinuum Models for Unconventional-Reservoir Simulation

SummaryUnconventional reservoirs are the focus of considerable attention as a primary energy source. Numerical simulation is a core kernel of reservoir-engineering work flows for reservoir evaluation, optimization, and management. Accurate and efficient numerical simulation of unconventional reservoirs is challenging. There is substantial physical complexity involving a number of tightly coupled mechanisms in the modeling of these reservoirs. The complexity is further amplified by the multicontinuum nature of the stimulated formation, and the complex fracture networks with a wide range of fracture-length scales and topologies.To adequately capture the effects of the multiscaled fracture system, we develop two alternative hybrid approaches that are aimed at combining the advantages of multicontinuum and discrete-fracture/matrix (DFM) representations. During the development of unconventional resources, geological and geophysical information may be available in some cases to suggest a prior characterization, whereas in many other cases, this prior model may be incomplete and limited to hydraulic fractures. The two hybrid approaches could be used for different applications depending on the available characterization data and different requirements for efficiency and accuracy considerations.The first hybrid model couples an embedded-discrete-fracture model (EDFM) with multiple interacting continua (MINC) into EDFM/MINC, which simulates the fracture network characterized by stimulated-reservoir-volume (SRV) concept. This optimized model can reduce the computational cost that is associated with the widely applied logarithmically spaced/locally refined (LS/LR) DFM technique, while improving the flexibility to model the complex geometry of hydraulic fractures. The MINC concept allows the hybrid model to handle the extreme contrast in conductivity between the small-scale fracture network and the ultratight matrix that results in steep potential gradients.For the second class of hybrid model (unstructured DFM/continuum), the primary fractures are described by use of DFM with unstructured gridding, and the small-scale fractures are simulated by continuum-type approaches in a fully coupled manner. Optimized local-grid refinement is used to accurately handle the transient-flow regime around primary fractures. An upscaling technique that applies EDFM on the detailed realization of the discrete-fracture network by use of the target unstructured grid to generate an appropriate dual-permeability model is also developed. The upscaling technique is suitable for cases where a detailed prior model for the complete fracture network is available.Simulation studies demonstrate the applicability of the developed hybrid-fracture models. Model verification is conducted against several reference solutions.

Integration of Pressure-Transient Data in Modeling Tengiz Field, Kazakhstan—A New Way To Characterize Fractured Reservoirs

Summary A systematic work-flow to integrate pressure-transient data collected from single-well buildup tests in numerical reservoir-simulation models for a fracture/matrix system is presented. The results of its application in a sector model in the southeast region of Tengiz field in Kazakhstan are also discussed. The procedure starts with a selected numerical-simulation model, either a discrete fracture/matrix (DFM) model or a dual-porosity dual-permeability (DPDK) model, and follows with the analysis of a numerically generated buildup test to calculate the fracture spacing and shape factor of the model. Then, following the correlations between pressure-transient-analysis results and the average or representative values of the model-input parameters near the well, which contain the previously obtained fracture spacing and shape factor, the numerical-model parameters are adjusted in each iteration to match the pressure-transient behavior observed in the buildup test including the interporosity flow between matrix and fracture and the radial flow in the total system. Before field application, the numerical-simulation results from both DFM and DPDK models were validated against analytical pressure-transient solutions for a dual-porosity system. The gridding and time-steps were calibrated to reproduce the analytical transient behavior. Finally, the new work-flow was applied to a sector model of Tengiz field in the southeast region focusing on two wells. Following the developed work-flow, a DFM model was constructed, and its fracture and matrix properties were adjusted to honor buildup-test data at both wells and the transient data collected during a pulse test conducted between them. The study results show that the key factors of a DFM model on buildup transient response are the fracture permeability, fracture aperture, and matrix permeability in the well-drainage area, and the dominant parameters on pulse-test response are the fracture permeability and matrix porosity in the influence area between the two wells. Using the correlations quantitatively for each simulation step could reduce the total number of iterations needed to converge to the numerical solution. The modified model also generated flow distribution along the wellbore, consistent with production-logging data at one well. The resulting sector map of pressure change during buildup test indicates the area with well-connected fracture network. Dynamic transient data contain rich information about reservoirs, and the effective integration of dynamic and static data would have a big impact on reservoir management by potentially minimizing the number of wells to be drilled, maximizing the production, and optimizing recovery. The novelty of this study is the quantitative use of the correlations between pressure-transient-analysis results and the representative values of the input parameters in a numerical model to reduce the number of simulation iterations. Its application in Tengiz is also one of the rare examples in which single-well and multiple-well transient data, production logging, and image-log data are all available.

Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture–matrix simulations on unstructured grids in three-dimensions (3D). The grid is aligned with fractures and barriers which are then modelled as lower-dimensional surface interfaces located between the matrix cells in the physical domain. The three-dimensional pressure equation is solved in the matrix domain coupled with a two-dimensional (2D) surface pressure equation solved over fracture networks via a novel surface CVD-MPFA formulation. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix–fracture fluxes are expressed in terms of matrix and fracture pressures and define the transfer function, which is added to the lower-dimensional flow equation and couples the three-dimensional and surface systems. An additional transmission condition is used between matrix cells adjacent to low permeable fractures to couple the velocity and pressure jump across the fractures. Convergence and accuracy of the lower-dimensional fracture model is assessed for highly anisotropic fractures having a range of apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approximation for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Pressure continuity is built into the model across highly conductive fractures, leading to reduced local degrees of freedom in the CVD-MPFA approximation. The formulation is applied to geologically complex fracture networks in three-dimensions. The effects of the fracture permeability, aperture and grid resolution are also assessed with respect to convergence and computational cost.

Comparing Discrete Fracture and Continuum Models to Predict Contaminant Transport in Fractured Porous Media

We used the FRAC3Dvs numerical model (Therrien and Sudicky 1996) to compare the dual-porosity (DP), equivalent porous medium (EPM), and discrete fracture matrix diffusion (DFMD) conceptual models to predict field-scale contaminant transport in a fractured clayey till aquitard. The simulations show that the DP, EPM, and DFMD models could be equally well calibrated to reproduce contaminant breakthrough in the till aquitard for a base case. In contrast, when groundwater velocity and degradation rates are modified with respect to the base case, the DP method simulated contaminant concentrations up to three orders of magnitude different from those calculated by the DFMD model. In previous simulations of well-characterized column experiments, the DFMD method reproduced observed changes in solute transport for a range of flow and transport conditions comparable to those of the field-scale simulations, while the DP and EPM models required extensive recalibration to avoid high magnitude errors in predicted mass transport. The lack of robustness with respect to variable flow and transport conditions suggests that DP models and effective porosity EPM models have limitations for predicting cause-effect relationships in environmental planning. The study underlines the importance of obtaining well-characterized experimental data for further studies and evaluation of model key process descriptions and model suitability.

An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations

We consider a control volume discretization with a multi-point flux approximation to model Discrete Fracture–Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions. Inspired by a recently introduced approach based on a two-point flux approximation, we explicitly account for the fractures by representing them as hybrid cells between the matrix cells. As well as simplifying the grid generation, our hybrid approach excludes small cells in the intersection of the fractures and hence avoids severe time-step restrictions associated with small cells. Excluding the small cells also reduces the condition number of the discretization matrix. For examples involving realistic anisotropy ratios in the permeability, numerical results show significant improvement compared to existing methods based on two-point flux approximations. We also investigate the hybrid method by studying the convergence rates for different apertures and fracture/matrix permeability ratios. Finally, the effect of removing the cells in the intersections of the fractures are studied. Together, these examples demonstrate the efficiency, flexibility and robustness of our new approach.

Transport and reduction of nitrate in clayey till underneath forest and arable land

Transport and reduction of nitrate in a typically macroporous clayey till were examined at variable flow rate and nitrate flux. The experiments were carried out using saturated, large diameter (0.5 m), undisturbed soil columns (LUC), from a forest and nearby agricultural sites. Transport of nitrate was controlled by flow along the macropores (fractures and biopores) in the columns. Nitrate reduction (denitrification) determined under active flow mainly followed first order reactions with half-lives ( t 1/2) increasing with depth (1.5–3.5 m) from 7 to 35 days at the forest site and 1–7 h at the agricultural site. Nitrate reduction was likely due to microbial degradation of accumulated organic matter coupled with successive consumption of O 2 and NO 3 − in the macropore water followed by reductive dissolution of Fe and Mn from minerals along the macropores. Concentrations of total organic carbon measured in soil samples were near identical at the two study sites and consequently not useful as indicator for the observed differences in nitrate reduction. Instead the high reduction rates at the agricultural site were positively correlated with elevated concentration of water-soluble organic carbon and nitrate-removing bacteria relative to the forest site. After high concentrations of water-soluble organic carbon in the columns from the agricultural site were leached they lost their elevated reduction rates, which, however, was successfully re-established by infiltration of new reactive organics represented by pesticides. Simulations using a calibrated discrete fracture matrix diffusion (DFMD) model could reasonably reproduce the denitrification and resulting flux of nitrate observed during variable flow rate from the columns.

Monitoring well interception with fractures in clayey till.

When using monitoring wells for investigation of contaminant sources in clayey till, there is a high risk that fractures may cause mobile contaminants to bypass the monitoring wells. This paper indicates that the probability of interception between monitoring wells and hydraulic conductive fractures is often significantly less than 50%. Based on a field experiment and application of a calibrated discrete fracture matrix diffusion numerical model (FRAC3Dvs), the paper also evaluates pesticide-monitoring results for different positions of monitoring well screen relative to fractures. For well screens situated 0.25 and 2 m from a conductive fracture, the first concentrations of the pesticide metabolite (2,6 dichlorobenzamide, "BAM") would be measured two years and 18 years, respectively, after the contaminant had been transported into an underlying aquifer. In this way, underlying aquifers may be subjected to contamination by downward moving contamination without being observed in monitoring wells in the till.