A Method to Represent a Well in a Three-Dimensional Discrete Fracture Network Model

In discrete fracture network (DFN) modeling, fractures are randomly generated and placed in the model domain. The rock matrix is considered impermeable. Small fractures and isolated fractures are often ignored to reduce computational expense. As a result, the rock matrix between fractures could be large and intersections may not be found between a well introduced in the model and the hydraulically connected fracture networks (fracture backbones). To overcome this issue, this study developed a method to conceptualize a well in a three-dimensional (3D) DFN using two orthogonal rectangular fractures oriented along the well's axis. Six parameters were introduced to parameterize the well screen and skin zone, and to control the connectivity between the well and the fracture backbones. The two orthogonal fractures were discretized using a high-resolution mesh to improve the quality of flow and transport simulations around and along the well. The method was successfully implemented within dfnWorks 2.0 (Hyman et al. 2015) to incorporate a well in a 3D DFN and to track particles leaving an injection well and migrating to a pumping well. Verification of the method against MODFLOW/MODPATH found a perfect match in simulated hydraulic head and particle tracking. Using three examples, the study showed that the method ensured the connectivity between wells and fracture backbones, and honored the physical processes of flow and transport along and around wells in DFNs. Recommendations are given for estimating the values of the six introduced well parameters in a real-world case study.     

FracKfinder: A MATLAB Toolbox for Computing Three-Dimensional Hydraulic Conductivity Tensors for Fractured Porous Media

Fractures in porous media have been documented extensively. However, they are often omitted from groundwater flow and mass transport models due to a lack of data on fracture hydraulic properties and the computational burden of simulating fractures explicitly in large model domains. We present a MATLAB toolbox, FracKfinder, that automates HydroGeoSphere (HGS), a variably saturated, control volume finite-element model, to simulate an ensemble of discrete fracture network (DFN) flow experiments on a single cubic model mesh containing a stochastically generated fracture network. Because DFN simulations in HGS can simulate flow in both a porous media and a fracture domain, this toolbox computes tensors for both the matrix and fractures of a porous medium. Each model in the ensemble represents a different orientation of the hydraulic gradient, thus minimizing the likelihood that a single hydraulic gradient orientation will dominate the tensor computation. Linear regression on matrices containing the computed three-dimensional hydraulic conductivity (K) values from each rotation of the hydraulic gradient is used to compute the K tensors. This approach shows that the hydraulic behavior of fracture networks can be simulated where fracture hydraulic data are limited. Simulation of a bromide tracer experiment using K tensors computed with FracKfinder in HGS demonstrates good agreement with a previous large-column, laboratory study. The toolbox provides a potential pathway to upscale groundwater flow and mass transport processes in fractured media to larger scales.     

Understanding of hydraulic properties from configurations of stochastically distributed fracture networks

A systematic investigation of the effect of configurations of stochastically distributed fracture networks on hydraulic behaviour for fractured rock masses could provide either quantitative or qualitative correlation between the structural configuration of the fracture network and its corresponding hydraulic behaviour, and enhance our understanding of appropriate application of groundwater flow and contaminant transport modelling in fractured rock masses. In this study, the effect of block sizes, intersection angles of fracture sets, standard deviations of fracture orientation, and fracture densities on directional block hydraulic conductivity and representative elementary volume is systematically investigated in two dimensions by implementing a numerical discrete fracture fluid flow model and incorporating stochastically distributed fracture configurations. It is shown from this investigation that the configuration of a stochastically distributed fracture network has a significant quantitative or qualitative effect on the hydraulic behaviour of fractured rock masses. Compared with the deterministic fracture configurations that have been extensively dealt with in a previous study, this investigation is expected to be more practical and adequate, since fracture geometry parameters are inherently stochastically distributed in the field. Moreover, the methodology and approach presented in this study may be generally applied to any fracture system in investigating the hydraulic behaviours from configurations of the fracture system while establishing a ‘bridge’ from the discrete fracture network flow modelling to equivalent continuum modelling in fractured rock masses. Copyright © 2007 John Wiley & Sons, Ltd.     

Static and dynamic analysis on slope stability using a DFN-DEM approach on the right abutment of the Karun 4 dam

Uncertainty in input fracture geometric parameters during analysis of the stability of jointed rock slopes is inevitable and therefore the stochastic discrete fracture network (DFN) — distinct element method (DEM) is an efficient modeling tool. In this research, potentially unstable conditions are detected in the right abutment of the Karun 4 dam and downstream of the dam body as a case study. Two extreme states with small and relatively large block sizes are selected and a series of numerical DEM models are generated using a number of validated DFN models. Stability of the rock slope is assessed in both static and dynamic loading states. Based on the design basis earthquake (DBE) and maximum credible earthquake (MCE) expected in the dam site, histories of seismic waves are applied to analyze the stability of the slope in dynamic earthquake conditions. The results indicate that a MCE is likely to trigger sliding of rock blocks on the rock slope major joint. Furthermore, the dynamic analysis also shows a local block failure by the DBE, which can consequently lead to slope instability over the long term. According to the seismic behavior of the two models, larger blocks are prone to greater instability and are less safe against earthquakes.

     

Characterizing production‐induced anisotropy of fractured reservoirs having multiple fracture sets

Since natural fractures in petroleum reservoirs play an important role in determining fluid flow during production, knowledge of the orientation and density of fractures is required to optimize production. This paper outlines the underlying theory and implementation of a fast and efficient algorithm for upscaling a Discrete Fracture Network (DFN) to predict the fluid flow, elastic and seismic properties of fractured rocks. Potential applications for this approach are numerous and include the prediction of fluid flow, elastic and seismic properties for fractured reservoirs, model‐based inversion of seismic Amplitude Versus Offset and Azimuth (AVOA) data and the optimal placement and orientation of infill wells to maximize production. Given that a single fracture network may comprise hundreds of thousands of individual fractures, the sheer size of typical DFNs has tended to limit their practical applications. This paper demonstrates that with efficient algorithms, the utility of Discrete Fracture Networks can be extended far beyond mere visualization.     

3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations

 3D groundwater flow at the fractured site of Asp? (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.     

Extension of observed flood series by combining a distributed hydro-meteorological model and a copula-based model

Long flood series are required to accurately estimate flood quantiles associated with high return periods, in order to design and assess the risk in hydraulic structures such as dams. However, observed flood series are commonly short. Flood series can be extended through hydro-meteorological modelling, yet the computational effort can be very demanding in case of a distributed model with a short time step is considered to obtain an accurate flood hydrograph characterisation. Statistical models can also be used, where the copula approach is spreading for performing multivariate flood frequency analyses. Nevertheless, the selection of the copula to characterise the dependence structure of short data series involves a large uncertainty. In the present study, a methodology to extend flood series by combining both approaches is introduced. First, the minimum number of flood hydrographs required to be simulated by a spatially distributed hydro-meteorological model is identified in terms of the uncertainty of quantile estimates obtained by both copula and marginal distributions. Second, a large synthetic sample is generated by a bivariate copula-based model, reducing the computation time required by the hydro-meteorological model. The hydro-meteorological modelling chain consists of the RainSim stochastic rainfall generator and the Real-time Interactive Basin Simulator (RIBS) rainfall-runoff model. The proposed procedure is applied to a case study in Spain. As a result, a large synthetic sample of peak-volume pairs is stochastically generated, keeping the statistical properties of the simulated series generated by the hydro-meteorological model. This method reduces the computation time consumed. The extended sample, consisting of the joint simulated and synthetic sample, can be used for improving flood risk assessment studies.     

Where Does Water Go During Hydraulic Fracturing?

During hydraulic fracturing millions of gallons of water are typically injected at high pressure into deep shale formations. This water can be housed in fractures, within the shale matrix, and can potentially migrate beyond the shale formation via fractures and/or faults raising environmental concerns. We describe a generic framework for producing estimates of the volume available in fractures and undamaged shale matrix where water injected into a representative shale site could reside during hydraulic fracturing, and apply it to a representative site that incorporates available field data. The amount of water that can be stored in the fractures is estimated by calculating the volume of all the fractures associated with a discrete fracture network (DFN) based on real data and using probability theory to estimate the volume of smaller fractures that are below the lower cutoff for the fracture radius in the DFN. The amount of water stored in the matrix is estimated utilizing two distinct methods—one using a two‐phase model at the pore‐scale and the other using a single‐phase model at the continuum scale. Based on these calculations, it appears that most of the water resides in the matrix with a lesser amount in the fractures.     

Numerical evaluation of the shear stimulation effect in naturally fractured formations

In shale gas fracking, the stimulated natural fracture system is often critical to the gas production. In this paper, we present the results of state-of-the-art modeling of a detailed parametric evolution of the shear stimulation effect in discrete fracture network (DFN) formations. Two-dimensional computational modeling studies have been used in an attempt towards understanding how naturally fractured reservoirs response in hydraulic fracturing. Simulations were conducted as a function of: (1) the in-situ stress ratio; (2) internal friction angle of DFN; (3) DFN orientation with the stress field; and (4) operational variables such as injection rate. A sensitivity study reveals a number of interesting observations resulting from these parameters on the shear stimulation in natural fracture system. This work strongly links the production technology, geomechanical evaluation and aids in the understanding and optimization of hydraulic fracturing simulations in naturally fractured reservoirs.     

Numerical evaluation of the effect of fracture network connectivity in naturally fractured shale based on FSD model

In this paper, the effect of pre-existing discrete fracture network (DFN) connectivity on hydraulic fracturing is numerically investigated in a rock mass subjected to in-situ stress. The simulation results show that DFN connectivity has a significant influence on the hydraulic fracture (HF) & DFN interaction and hydraulic fracturing effectiveness, which can be characterized by the total interaction area, stimulated DFN length, stimulated HF length, leak-off ratio, and stimulated total length. In addition, even at the same fluid injection rate, simulation models exhibit different responses that are strongly affected by the DFN connectivity. At a low injection rate, total interaction area decreases with increasing DFN connectivity; at a high injection rate, total interaction area increases with the increase of DFN connectivity. However, for any injection rate, the stimulated DFN length increases and stimulated HF length decreases with the increase of connectivity. Generally, this work shows that the DFN connectivity plays a crucial role in the interaction between hydraulic fractures, the pre-existing natural fractures and hydraulic fracturing effectiveness; in return, these three factors affect treating pressure, created microseismicity and corresponding stimulated volume. This work strongly relates to the production technology and the evaluation of hydraulic fracturing effectiveness. It is helpful for the optimization of hydraulic fracturing simulations in naturally fractured formations.     

基于统计岩石物理模型的各向异性页岩储层参数反演

提出了各向异性页岩储层统计岩石物理反演方法.通过统计岩石物理模型建立储层物性参数与弹性参数的定量关系,使用测井数据及井中岩石物理反演结果作为先验信息,将地震阻抗数据定量解释为储层物性参数、各向异性参数的空间分布.反演过程在贝叶斯框架下求得储层参数的后验概率密度函数,并从中得到参数的最优估计值及其不确定性的定量描述.在此过程中综合考虑了岩石物理模型对复杂地下介质的描述偏差和地震数据中噪声对反演不确定性的影响.在求取最大后验概率过程中使用模拟退火优化粒子群算法以提高收敛速度和计算准确性.将统计岩石物理技术应用于龙马溪组页岩气储层,得到储层泥质含量、压实指数、孔隙度、裂缝密度等物性,以及各向异性参数的空间分布及相应的不确定性估计,为页岩气储层的定量描述提供依据.     

Constructing a discrete fracture network constrained by seismic inversion data

Rock fractures are of great practical importance to petroleum reservoir engineering because they provide pathways for fluid flow, especially in reservoirs with low matrix permeability, where they constitute the primary flow conduits. Understanding the spatial distribution of natural fracture networks is thus key to optimising production. The impact of fracture systems on fluid flow patterns can be predicted using discrete fracture network models, which allow not only the 6 independent components of the second‐rank permeability tensor to be estimated, but also the 21 independent components of the fully anisotropic fourth‐rank elastic stiffness tensor, from which the elastic and seismic properties of the fractured rock medium can be predicted. As they are stochastically generated, discrete fracture network realisations are inherently non‐unique. It is thus important to constrain their construction, so as to reduce their range of variability and, hence, the uncertainty of fractured rock properties derived from them. This paper presents the underlying theory and implementation of a method for constructing a geologically realistic discrete fracture network, constrained by seismic amplitude variation with offset and azimuth data. Several different formulations are described, depending on the type of seismic data and prior geologic information available, and the relative strengths and weaknesses of each approach are compared. Potential applications of the method are numerous, including the prediction of fluid flow, elastic and seismic properties of fractured reservoirs, model‐based inversion of seismic amplitude variation with offset and azimuth data, and the optimal placement and orientation of infill wells to maximise production.     

Optimization of groundwater-monitoring networks for identification of the distribution of a contaminant plume

A new methodology is proposed to optimize monitoring networks for identification of the extent of contaminant plumes. The optimal locations for monitoring wells are determined as the points where maximal decreases are expected in the quantified uncertainty about contaminant existence after well installation. In this study, hydraulic conductivity is considered to be the factor that causes uncertainty. The successive random addition (SRA) method is used to generate random fields of hydraulic conductivity. The expected value of information criterion for the existence of a contaminant plume is evaluated based on how much the uncertainty of plume distribution reduces with increases in the size of the monitoring network. The minimum array of monitoring wells that yields the maximum information is selected as the optimal monitoring network. In order to quantify the uncertainty of the plume distribution, the probability map of contaminant existence is made for all generated contaminant plume realizations on the domain field. The uncertainty is defined as the sum of the areas where the probability of contaminant existence or nonexistence is uncertain. Results of numerical experiments for determination of optimal monitoring networks in heterogeneous conductivity fields are presented.     

Stochastic ground water flow simulation with a fracture zone continuum model

A method is presented for incorporating the hydraulic effects of vertical fracture zones into two-dimensional cell-based continuum models of ground water flow and particle tracking. High hydraulic conductivity features are used in the model to represent fracture zones. For fracture zones that are not coincident with model rows or columns, an adjustment is required for the hydraulic conductivity value entered into the model cells to compensate for the longer flowpath through the model grid. A similar adjustment is also required for simulated travel times through model cells. A travel time error of less than 8% can occur for particles moving through fractures with certain orientations. The fracture zone continuum model uses stochastically generated fracture zone networks and Monte Carlo analysis to quantify uncertainties with simulated advective travel times. An approach is also presented for converting an equivalent continuum model into a fracture zone continuum model by establishing the contribution of matrix block transmissivity to the bulk transmissivity of the aquifer. The methods are used for a case study in west-central Florida to quantify advective travel times from a potential wetland rehydration site to a municipal supply wellfield. Uncertainties in advective travel times are assumed to result from the presence of vertical fracture zones, commonly observed on aerial photographs as photolineaments.     

Assessing hydrological model predictive uncertainty using stochastically generated geological models

In distributed and coupled surface water–groundwater modelling, the uncertainty from the geological structure is unaccounted for if only one deterministic geological model is used. In the present study, the geological structural uncertainty is represented by multiple, stochastically generated geological models, which are used to develop hydrological model ensembles for the Norsminde catchment in Denmark. The geological models have been constructed using two types of field data, airborne geophysical data and borehole well log data. The use of airborne geophysical data in constructing stochastic geological models and followed by the application of such models to assess hydrological simulation uncertainty for both surface water and groundwater have not been previously studied. The results show that the hydrological ensemble based on geophysical data has a lower level of simulation uncertainty, but the ensemble based on borehole data is able to encapsulate more observation points for stream discharge simulation. The groundwater simulations are in general more sensitive to the changes in the geological structure than the stream discharge simulations, and in the deeper groundwater layers, there are larger variations between simulations within an ensemble than in the upper layers. The relationship between hydrological prediction uncertainties measured as the spread within the hydrological ensembles and the spatial aggregation scale of simulation results has been analysed using a representative elementary scale concept. The results show a clear increase of prediction uncertainty as the spatial scale decreases. Copyright © 2015 John Wiley & Sons, Ltd.     

不同尺度裂缝对弹性波速度和各向异性影响的实验研究

裂缝广泛分布于地球介质中并且具有多尺度的特点,裂缝尺度对于油气勘探和开发有着重要的意义.本文制作了一组含不同长度裂缝的人工岩样,其中三块含裂缝岩样中的裂缝直径分别为2 mm、3 mm和4 mm,裂缝的厚度都约为0.06 mm,裂缝密度大致相同(分别为4.8%、4.86%和4.86%).在岩样含水的条件下测试不同方向上的纵横波速度,实验结果表明,虽然三块裂缝岩样中的裂缝密度大致相同,但是含不同直径裂缝岩样的纵横波速度存在差异.在各个方向上,含数量众多的小尺度裂缝的岩样中纵横波速度都明显低于含少量的大尺度裂缝的岩样中纵横波速度.尤其是对纵波速度和SV波速度,在不同尺度裂缝岩样中的差异更明显.在含数量多的小尺度裂缝的岩样中纵波各向异性和横波各向异性最高,而含少量的大尺度的裂缝的岩样中的纵波各向异性和横波各向异性较低.实验测量结果与Hudson理论模型预测结果进行了对比分析,结果发现Hudson理论考虑到了裂缝尺度对纵波速度和纵波各向异性的影响,但是忽略了其对横波速度和横波各向异性的影响.     

Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations

In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions.     

Model structure and uncertainty for stochastic non-point source modelling applications

Abstract

The uncertainty associated with a rainfall–runoff and non-point source loading (NPS) model can be attributed to both the parameterization and model structure. An interesting implication of the areal nature of NPS models is the direct relationship between model structure (i.e. sub-watershed size) and sample size for the parameterization of spatial data. The approach of this research is to find structural limitations in scale for the use of the conceptual NPS model, then examine the scales at which suitable stochastic depictions of key parameter sets can be generated. The overlapping regions are optimal (and possibly the only suitable regions) for conducting meaningful stochastic analysis with a given NPS model. Previous work has sought to find optimal scales for deterministic analysis (where, in fact, calibration can be adjusted to compensate for sub-optimal scale selection); however, analysis of stochastic suitability and uncertainty associated with both the conceptual model and the parameter set, as presented here, is novel; as is the strategy of delineating a watershed based on the uncertainty distribution. The results of this paper demonstrate a narrow range of acceptable model structure for stochastic analysis in the chosen NPS model. In the case examined, the uncertainties associated with parameterization and parameter sensitivity are shown to be outweighed in significance by those resulting from structural and conceptual decisions.

Citation Parker, G. T. Rennie, C. D. & Droste, R. L. (2011) Model structure and uncertainty for stochastic non-point source modelling applications. Hydrol. Sci. J. 56(5), 870–882.     

Deriving reservoir operational strategies considering water quantity and quality objectives by stochastic fuzzy neural networks

By taking advantage of the close relationship between quality and quantity of water, we investigated the potential improvements of the in-reservoir water quality through the optimization of reservoir operational strategies. However, the few available techniques for optimization of reservoir operational strategies present some limitations, such as restrictions on the number of state/decision variables, the impossibility considering stochastic characteristics and difficulties for considering simulation/prediction models. One technique which presents great potential for overcoming some of these limitations is applied here and investigated for the first time in such complex system. The method, named stochastic fuzzy neural network (SFNN), can be defined as a fuzzy neural network (FNN) model stochastically trained by a genetic algorithm (GA) based model to yield a quasi optimal solution. The term “stochastically trained” refers to the introduction of a new loop within the training process which accounts for the stochastic variable of the system and its probabilities of occurrence. The SFNN was successfully applied to the optimization of the monthly operational strategies considering maximum water utilization and improvements on water quality simultaneous. Results showed the potential improvements on the water quality through means of hydraulic control.     

Stochastic coupled simulation of strong motion and tsunami for the 2011 Tohoku,Japan earthquake

This study conducts coupled simulation of strong motion and tsunami using stochastically generated earthquake source models. It is focused upon the 2011 Tohoku, Japan earthquake. The ground motion time-histories are simulated using the multiple-event stochastic finite-fault method, which takes into account multiple local rupture processes in strong motion generation areas. For tsunami simulation, multiple realizations of wave profiles are generated by evaluating nonlinear shallow water equations with run-up. Key objectives of this research are: (i) to investigate the sensitivity of strong motion and tsunami hazard parameters to asperities and strong motion generation areas, and (ii) to quantify the spatial variability and dependency of strong motion and tsunami predictions due to common earthquake sources. The investigations provide valuable insights in understanding the temporal and spatial impact of cascading earthquake hazards. Importantly, the study also develops an integrated strong motion and tsunami simulator, which is capable of capturing earthquake source uncertainty. Such an advanced numerical tool is necessary for assessing the performance of buildings and infrastructure that are subjected to cascading earthquake–tsunami hazards.