Abstract: This paper describes the essential features of a numerical technique for the simulation of the coupled fluid flow and deformation in a 2D assembly of poroelastic blocks and transmissive fractures. The boundary element method (BEM) is applied to each block to reduce Navier and diffusion equations to a set of integral equations involving block boundary terms, whereas a Galerkin weighted-residuals finite element method (FEM) is applied to the fracture diffusion equations. In addition, fracture local equilibrium is rendered through spring-like equations relating the stresses to the relative displacements of the fracture walls. A time-marching process is implemented leading to an algebraic system where the right-hand side vector is built based on the collected solutions of the previous time steps. The technique requires the meshing of the fracture network only. The accuracy of the results is adequate even with relatively coarse meshes without the resort to small time steps at the beginning of the simulation. It furnishes outputs that focus only on the salient features of the response. The efficiency of the technique is demonstrated through the illustration of the results of three examples.

Abstract: In a fractured rock mass, variations in stress and fluid pressure induced by engineering activities can significantly affect the hydrogeological properties. A significant change in fracture transmis-sivities can also be experienced in the far-field. The simulation of this kind of change requires a Hydro-Mechanical (HM) coupled model. The purpose of this paper is to show how such a model can be used to analyse the evolution of deformation and pressure in a fracture subjected to fluid injection. A 2D BEM-EEM code is used to solve the non-linear system of equations that describe the dependency of transmissivity on local fracture closure. The results of a sensitivity analysis of the essential fracture parameters allow one to gain insight into the importance of the HM models in the framework of the hydrogeology of fractured rock masses. Results obtained from a system of two impervious blocks and a saturated fracture are reported, in order to show the possibilities offered by this technique.

Abstract: A boundary element method (BEM) solution for the problem of the fluid flow in a three-dimensional discrete fracture network (DFN) is proposed. A DFN is an assembling of polygons which resemble the fractures in a rock mass. The position, extension, orientation and transmissivity of each fracture of the network are excluded by specific statistical distributions. For a single problem, a significant number of DFNs has to be generated and the fluid flow has to be assessed in each of them in the context of a Monte-Carlo procedure. Even for relatively small domains, a DFN may include a large number of fractures. As a consequence, in order to solve the whole problem with standard finite-element method (FEM) codes, a big amount of core memory and large input data files are required. The main advantage of the proposed solution is mainly the minimization of the core memory. This is attained by handling the flow quantities in such a way the equation system of the overall network is never assembled. Only a relation per each fracture among nodal fluxes and heads of the traces (i.e., intersections among fractures) is defined and stored in a random access file. This relation is obtained by means of the application of the BEM to each single fracture of the network. Both constant and quadratic element representations are used in order to define the relevant nodal quantities. The use of constant elements allows to avoid the direct treatment of the points of flux discontinuity. No special care is applied to the discretization of the boundary of each fracture. The overall problem is solved by means of an iterative procedure, by retrieving the necessary coefficients from the random access file. The results are in acceptable agreement with the ones provided by a commercial FEM code. We remark that saving core memory without special care in the discretization of the DFN makes the solution competitive, especially when dealing with networks with a high number of fractures.

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Abstract: Discrete fracture network (DFN) models generally require solution of flow and transport equations in three-dimensional networks of either disc, polygonal, or pipe elements. Pipe network elements have significant advantages in computation for both flow and transport. However, there is a need to develop an efficient procedure for derivation of the properties of these pipes to ensure that they are hydraulically equivalent to the DFN network of polygonal elements. In this study a boundary element procedure for derivation of pipe properties is developed and demonstrated. The results show that the hydraulic behavior of pipe networks can be equivalent to that of polygonal-element DFN models.

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

Abstract: A numerical code which utilizes the boundary element method (BEM) for solving steady-state ground-water flow problems is illustrated. The paper concentrates on accuracy in studying the situations which are generally considered to involve some mathematical difficulty, such as zoned domains and flux discontinuities. A numerical BEM code is proposed, the main feature of which involves accurately calculating flux discontinuities using a structure particularly flexible in multizoned domains. Two examples are reported, the results of which are comparable with those available in the literature.

Abstract: In many rock formations the fluid flow takes place predominantly in fracture networks. Any model conceived for the prediction of fluid flow and transport in such formations should consider this evidence. Based on the statistical data extracted from observational supports about the geometrical features of the fractures and the results of hydraulic tests, synthetic realizations of the fracture network inside the rock mass can be produced. These objects are called DFNs (Discrete Fracture Networks) and can be utilized as predictive models, provided the fracture hydrological parameters are included. DFNs are gaining acceptance in the engineering practice, but their use is still limited, given the strong computational demand required when simulating the fluid flow. In some cases it is therefore reasonable to assume the Equivalent Porous Medium (EPM) conceptualization as a basis for the predictive model. An EPM encompasses the features of the fracture network leading also to a considerable simplification of the problem and reduction of computation. The equivalence is based on specific criteria but moves in any case from the information collected about the geometry of the fracture network and the hydraulic parameters of the fractures. The quality of a model based on EPM is less with respect to the quality of a DFNbased model; the loss of precision depends on the adequateness of the equivalence. Regular fracture networks can be assimilated to an EPM with a relative success. The concept of regularity is only qualitative and is explained in the following. Fracture networks can be defined as regular or irregular, having in mind that the same fracture network can be irregular when the flow domain is small, and regular when the domain is extended beyond a certain threshold. The use of EPM, when the fracture network is irregular, can instead bring to misleading results, therefore before any application, it is recommended to check whether the conditions for this regularity exist at the scale of the problem. Professionals sometimes prefer skipping the necessary steps required for the equivalence and rather testing the rock to infer the hydraulic parameters. Most of the time the hydraulic test results provide a picture which may differ from what is expected when having in mind a continuum-like material. In fact the tests usually give information about the preferential pathways and rarely provide directly the values of the hydraulic parameters, unless the fracture network inside the volume affected by perturbations is regular. This chapter concerns the hydraulic characterization of fractured rocks, with specific reference to what follows: the fluid flow in single fractures, the interpretation of hydraulic tests, the geometrical characterization and modeling of fracture networks. An appropriate model for the fluid flow and transport in fractured formations is based on these items. Finally, indications are also given about the best strategy to undertake in order to set up this model.