New developments in the application of configurational mechanics to crack propagation

The numerical description of discrete cracks and their propagation remains one of the main difficulties in the modeling of quasi-brittle materials such as rock or concrete. An emerging powerful approach is the use of Configurational Mechanics concepts, in such a way that crack trajectory really corresponds to a structural energy minimum and is not predetermined by the initial mesh lines. In the implementation developed, discrete cracks, represented by zero-thickness interface elements, are reoriented on the basis of configurational or material forces, calculated in a FEM context by an integration over the elements of the Eshelby energy-momentum tensor. The strategy is illustrated with an application example for which the fracture path is known a priori, and the initial mesh layout is chosen such that the lines zig-zag significantly with respect to it. The results show that the procedure implemented works successfully, that is, mesh lines do succeed in reorienting themselves during configurational iterations, so that the developing crack progressively matches the known physical trajectory

Zero-thickness interface elements in petroleum geomechanics : sand production and hydraulic fracture

Aplicat embargamet des de la data de defensa fins el dia 20 de setembre de 2019This thesis describes the extension of the Finite Element Method with zero-thickness interface elements (FEM+z) to 3D, large and complex problems in geomaterials, with special interest in petroleum geomechanics. This general objective has led to specific developments and applications such as the 3D code implementations and parallelization, and the specific petroleum geomechanics studies, both the macroscale (hydraulic fracture) and microscale (sand production). The extension to 3D of the hydro-mechanical formulation of double node zero-thickness interface elements proposed earlier, has been developed and implemented in the computer code, with satisfactory results in the verification examples. From the theoretical viewpoint, the formulation is generalized via the definition of \quotes{transport} matrices for both mechanical and hydraulic formulations, so that the two levels of the formulation can be separated: the nodal variables of the interface element, and the mid-plane variables. The formulation described is successfully validated with benchmarking examples based on analytical expressions of a hydraulic fracture. The parallelization of the code DRAC is achieved through the implementation of public domain library PETSc. The new code structure is conceived to perform a correct subdivision of tasks associated to each processor. For this purpose, a domain decomposition strategy has been implemented, which is crucial for an efficient matrix generation and assembly. The results obtained show a good degree of parallelization, demonstrated with a cube benchmark test. The applications to hydraulic fracture have served a dual purpose. First, the examples of a single fracture have been used to validate the proposed formulation, since it has been possible to compare the results with the predictions of analytical expressions such as GDK or PKN, and to other numerical results from the literature. Second, the examples of multiple interacting fractures have shown the capabilities to analyze large and complex cases. The studies performed have shown a number of relevant aspects of multiple fracturing such as the effect of geometry (distance between injections) and the effect of in situ stresses. Finally the thesis is devoted to the micromechanical analysis of sand production, including the generation and testing of micromechanical models based on the use of zero-thickness interface elements. Micromechanical (mesoscopic level) analysis with FEM+z has been successfully used to model the mechanical behavior of rock materials, using a similar approach as used previously for other heterogeneous materials. The rock grains are modeled as a group of continuum elastic elements and the cement (or matrix) is modeled with zero-thickness interfaces. This kind of modeling has been successfully used in uniaxial and triaxial compression test simulations. These numerical tests have been used for calibration purposes, comparing the macroscopic results obtained with the existing laboratory data. Due to the availability of experimental data, the simulation of sand production has been focused on the modeling of the hollow cylinder test. The simulations have been divided into two parts. First, using a prototype material, the sensitivity of the method to geometric and microstructural variations has been analysed, and the effect of the perforation size is clearly observed. The second part deals with the analysis of a real case of rock sanding, that includes micromechanical tests for parameter calibration and the simulation of sand production. The results, despite the intrinsic variability of the samples, have shown a satisfactory agreement with average experimental results, both in terms of the rock sanding initiation and of sand production.En aquesta tesi es presenta l'extensió del mètode d'elements finits amb elements junta d'espessor zero (FEM+z) a casos en 3D, per geometries grans i amb fenòmens complexos de fracturació per geomaterials, amb especial interès en la geomecànica del petroli. Aquest objectiu general ha donat lloc al desenvolupament i la generació aplicacions específiques, com ara les implementacions del codi a 3D i la paral·lelització, i els estudis específics de geomecànica del petroli, tant a macroescala (fractura hidràulica) com a microescala (producció de sorra). L'extensió a 3D de la formulació hidromecànica d'elements junta de doble nus proposada anteriorment, s'ha desenvolupat i implementat en el codi de càlcul, amb resultats satisfactoris en els exemples de verificació. Des del punt de vista teòric, la formulació s'ha generalitzat mitjançant la definició de matrius de transport per les dues formulacions, la mecànica i la hidràulica, de manera que es poden separar els dos nivells de la formulació: les variables nodals de l'element junta i les variables del pla mig. La paral·lelització del codi DRAC s'ha assolit mitjançant la implementació de la llibreria en codi lliure PETSc. La nova estructura del codi s'ha concebut per tal de realitzar una subdivisió correcta de les tasques associades a cada processador. Així, s'ha implementat una estratègia de descomposició de dominis, que és fonamental per a la generació i assemblatge de matrius de manera eficient. Els resultats obtinguts mostren un bon grau de paral·lelització. Les aplicacions de fractura hidràulica han tingut un doble propòsit. En primer lloc, s'han utilitzat els exemples d'una sola fractura per tal de validar la formulació proposada, ja que s'han pogut comparar els resultats amb les prediccions d'expressions analítiques com GDK o PKN i altres resultats numèrics de la literatura. En segon lloc, els exemples d'interacció entre fractures múltiples han demostrat la capacitat d'analitzar casos grans i complexos. Els estudis realitzats han mostrat una sèrie d'aspectes rellevants de la fractura múltiple, com l'efecte de la geometria (distància entre les injeccions) i l'efecte de les tensions in situ. Per finalitzar, la tesi presenta l'anàlisi micromecànic de la producció de sorra, que inclou la generació i la verificació dels models micromecànics basats en l'ús d'elements junta d'espessor zero. L'anàlisi micromecànic (nivell mesoscòpic) fet amb FEM+z s'ha utilitzat amb èxit per modelitzar el comportament mecànic de les roques. Els grans de roca es modelen com un grup d'elements elàstics de continu i el ciment (o matriu) es modelitza amb les juntes d'espessor zero. Aquest tipus de modelització s'ha utilitzat amb èxit en les simulacions d'assaigs a compressió uniaxial i triaxial. Les simulacions numèriques han servit per calibrar els paràmetres del model, comparant els resultats macroscòpics obtinguts amb les dades del laboratori existents. Degut a la disponibilitat de dades experimentals, la simulació de la producció de sorra s'ha centrat en la modelització de l'assaig de cilindres buits. Les simulacions s'han dividit en dues parts. En primer lloc, mitjançant l'ús d'un material prototip, s'ha analitzat la sensibilitat del mètode a les variacions geomètriques i microestructurals, on s'observa clarament l'efecte de mida de la perforació. La segona part ha tractat l'anàlisi d'un cas real de producció de sorra, que inclou assaigs micromecànics per la cal·libració dels paràmetres i la simulació de la producció de sorra. Els resultats, tot i la variabilitat intrínseca de les mostres, han demostrat un semblança satisfactòria amb els resultats experimentals, tant pel que fa a la iniciació com a la producció de sorra.Postprint (published version

New developments in the application of configurational mechanics to crack propagation

The numerical description of discrete cracks and their propagation remains one of the main difficulties in the modeling of quasi-brittle materials such as rock or concrete. An emerging powerful approach is the use of Configurational Mechanics concepts, in such a way that crack trajectory really corresponds to a structural energy minimum and is not predetermined by the initial mesh lines. In theimplementationdeveloped, discrete cracks, represented by zero-thickness interface elements, are reoriented on thebasis of configurational or material forces, calculated in a FEM context by an integration over the elements of the Eshelby energy-momentum tensor. The strategy is illustrated with an application example for which the fracture path is known a priori, and the initial mesh layout is chosen such that the lines zig-zag significantly with respect to it. The results show that the procedure implemented works successfully, that is, mesh lines do succeed in reorienting themselves during configurational iterations, so that the developing crack progressively matches the known physical trajectory.Postprint (published version

Estudio Microestructural del Rock Sanding

Castellà: El estudio de la producción de arena (rock sanding) en pozos petrolíferos ha sido objeto importante de investigación durante los últimos años debido al auge de las nuevas técnicas de producción: perforación con cañones (perforating guns). La producción de arena es un fenómeno de disgregación del medio que se genera en formaciones débilmente cohesionadas, como arenas poco cementadas. Este fenómeno está gobernado por varios factores, de los que destacan dos principalmente: el estado tensional del medio y la presión del flujo de extracción. Los estudios realizados hasta el momento han puesto de manifiesto que el primer factor es el responsable de la fisuración del medio, y el segundo el responsable del transporte. El objeto de esta tesina ha sido el de iniciar el estudio de dicha problemática desde un punto de vista numérico microestructural, utilizando el método de los elementos finitos con elementos junta. Para ello se ha discretizado una sección transversal de una perforación generada mediante cañones de perforación. Estas excavaciones se caracterizan por su pequeño diámetro, 2 cm, y por no estar revestidas. La discretización (bidimensional) del medio rocoso se ha llevado a cabo mediante dos zonas bien diferenciadas: una corona circular que rodea el hueco de perforación, en la que se representan explícitamente granos de arena distribuidos de forma aleatoria mediante elementos finitos triangulares de comportamiento elástico lineal. El contacto entre granos se simula mediante elementos junta sin espesor, cuya ley constitutiva es no lineal con reblandecimiento por trabajo de fractura, formulada en términos de tensiones y desplazamientos relativos. Una malla de elementos continuos, que rodea la zona anterior, con una extensión suficiente como para garantizar la transmisión de las tensiones de confinamiento, a la vez que permite disminuir el número de elementos de la discretización. La simulación numérica ha consistido en imponer inicialmente la presión de confinamiento en los bordes de la malla y posteriormente disminuir gradualmente la presión en el interior de la perforación. La disgregación del medio, objeto de la modelización de “rock sanding”, ha sido reducida a una extracción de las partículas de arena (granos), cuando los elementos junta que las mantienen en contacto con otros granos, alcanzan un grado elevado de deterioro. A fin de poner de manifiesto la potencialidad del procedimiento se han realizado tres tipos de análisis. Los principales resultados obtenidos son: Estudio de la producción de arena. Se han obtenido interesantes resultados cualitativos de la evolución de la fisuración alrededor de los granos cercanos al hueco de la perforación. Este hecho ha servido de referencia para los posteriores casos. La curva presión-producción de arena aún no ha sido posible de obtener debido al alto coste computacional del caso de estudio. Estudio del efecto de tamaño. Se ha realizado un estudio comparativo del mismo problema utilizando diferente tamaño de perforación. Los resultados obtenidos han permitido capturar el efecto de tamaño: las perforaciones más pequeñas resultan menos dañadas a igual condición. Estudio de un estado no hidrostático de tensiones iniciales. Se ha aplicado un estado inicial de carga con diferentes valores de las componentes horizontal y vertical de tensiones. El resultado obtenido durante la descarga ha evidenciado el hecho comprobado experimentalmente de que se produce una rotura preferencial en las direcciones perpendiculares a la tensión máxima aplicada. Los resultados preliminares obtenidos hasta el momento permiten reproducir de forma cualitativa algunos aspectos importantes del comportamiento real observado, lo cual pone de manifiesto la potencialidad de esta metodología microestructural para profundizar en la investigación futura de “rock sanding”

Coupled h-m fracture interaction using fem with zero-thickness interface elements

Intensive hydraulic fracturing is a procedure employed for low permeability reservoir stimulation. This technique consists of generating a sequence of regularly spaced parallel fractures (multi-stage fracturing). The generation of a fracture involves the modification of the local stress state, and therefore, in the case of multi-stage fracturing, the propagation of a certain fracture can be affected by the injection sequence, as it has been observed with microseismicity monitoring [1]. This paper describes a study of this technique by means of the Finite Element Method with zero-thickness interface elements for the geo-mechanical modelling of discontinuities [2]. The technique consists in inserting interface elements in between standard elements to allow jumps in the displacement solution fields. For the mechanical problem, their kinematic constitutive variables are relative displacements, and the corresponding static variables are stress tractions. The relationship between variables is controlled via a fracture-based constitutive law with elasto-plastic structure [3]. Concerning the hydraulic problem, the interface formulation includes both the longitudinal flow (with a longitudinal conductivity parameter strongly dependent on the fracture aperture), as well as and the transversal flow across the element [4]. Previous work by the authors focused on the validation of the method, the analysis a single fracture plane problem [5, 6]. In this case the method is extended to allow free propagation of fractures in any direction, by means of inserting interface elements between all continuum elements. The results presented in this paper analyse the effect of material properties, in particular fracture characterization, in the propagation and the effect of different major to minor principal horizontal stress ratio, on the trajectory and interaction of the fractures

Micromechanical analysis of damage and fracture in sandstone rock specimens, using zero-thickness interface elements

This numerical study deals with the analysis of rock specimens from a micromechanical point of view. The analysis is based on the Finite Element Method (FEM) with fracture-based zero-thickness interface elements, and numerically generated micromechanical geometries. In previous studies this approach has been used very successfully to represent the mechanical behaviour of concrete and other quasi-brittle materials under a variety of loading scenarios. The current study, motivated by longer-term research on rock sanding in oil wells, is focused on the capability of the model to represent the type of failure observed near the walls of tunnels or holes bored in a medium subject to initial compressive stresses. In this context, the microstructure of sandstone rock is generated via Voronoi tessellation. Interface elements are inserted along all polygon contacts, and, for some calculations also within the polygons themselves. The two main loading scenarios explored are direct uniaxial compression, and uniaxial extension after hydrostatic loading. Besides a general good capability to represent the desired behaviour, the results show that direct uniaxial compression requires intra-granular cracks in the model in order to reach failure. However, if the same final loading state is reached via initial hydrostatic loading followed by uniaxial extension, (as typical in tunnel or borehole walls), then failure kinematics changes and failure may take place with the inter-granular interfaces exclusively (i.e. no need to consider intra-granular cracks to represent failure)

XFEM formulation for discontinuities in fractured rock masses

The paper describes some aspects of the application of XFEM to represent Geomechanical discontinuities, including the choice of additional nodal variables and the appearance and remedies to the oscillations that may take place depending on the mesh layout. An example of application to recover the stresses along a discontinuity line emanating from a tunnel cross-section is presented together with the comparison to an analytical solution. The formulation is developed in terms of the “overhang” displacement variables on the other side of the discontinuity (instead of more traditional displacement jump variables), and the oscillations associated to nodes too close to the discontinuity are solved by moving those nodes onto the discontinuity (instead of moving them away as seems more common in current practice)

Zero-thickness interface elements with h-m coupling, formulation and applications in geomechanics

Previous developments on zero-thickness interface elements for coupled Hydro-Mechanics problems are briefly described, followed by some new geomechanical applications, particularly to hydraulic fracture in rock. The results are compared satisfactorily to approximate formulas and previously published numerical results. Once verified, the model is applied to new cases to show the capabilities of the approach

Avoiding fracture instability in wedge splitting tests by means of numerical simulations

In this paper, unstable fracture propagation obtained in a in-house performed experimentalWedge Splitting Test (WST) is simulated by means of the FEM and fracture- based zero-thickness interface elements. In order to obtain a specimen geometry suitable for a stable WST without modifying the remaining significant parameters of the test (machine stiffness and control parameter), additional simulations were performed varying the length of the specimen notch, until a load-COD (Crack Opening Displacement) curve without snap-back was obtained. Finally, a new experimental WST with the modified geometry was carried out leading to a stable load-COD curve. In the simulations, elastic continuum elements were used to represent the rock, the steel loading plates and the test- ing machine compliance via an “equivalent spring”, whereas interface elements were used for the notch and along the potential crack path. The interface elements representing the notch were equipped with linear elastic constitutive law, with very low elastic stiffness Kn and Kt so that they do not oppose any significant resistance to opening. For the inter- face elements along the fracture path, an elastoplastic constitutive model with fracture energy-based evolution laws was used.Postprint (published version