Abstract

Discrete Element Methods (DEM) explicitly model the mechanics of the discontinuities of naturally fractured rock masses. However, due to the large number of degrees of freedom in DEM simulations and the requirement of small times steps, the application of DEM simulations to reservoir-scale problems and long-term fluid injection problems is often computationally prohibitive.

In order to reduce the computational costs associated with full-scale DEM simulations, an up-scaling method is presented in which Representative Elementary Volume (REV) DEM simulations are used to calibrate the parameters of a Continuum Damage Mechanics (CDM) constitutive model that is then used for Finite Element Analysis (FEA). The CDM model empirically captures the effect of the degradation of the rock integrity due to the yielding and sliding of natural fractures in the rock mass.

Up-scaling is achieved through homogenization, in which the spatially averaged stress-strain behavior of various DEM RVE simulations is computed. Subsequently, a CDM constitutive relationship is fitted using the Levenberg-Marquardt Algorithm (LMA) and the homogenized DEM simulation data. The CDM model is then used in FEA reservoir scale simulations. The CDM model is implemented in ABAQUS^{TM} and DEM simulations were conducted using UDEC^{TM}. The up-scaling methodology is demonstrated through a case study on a naturally fractured carbonate reservoir in which the up-scaled CDM model compares well with a direct numerical simulation with the DEM model but requires an order of magnitude less computational time.

Distinct Element Method (DEM) models are used commonly in geomechanics to explicitly model the mechanics of Naturally Fractured Rock (NFR) masses (citation not found: jing_review_2003). NFR is often modeled as a multiscale material due to the vastly different length scales involved in the deformation process (citation not found: zhou_flow_2003). At the fracture scale (10^{-3} m), the physics is dominated by brittle fracture propagation and fracture-to-fracture contact force interaction, while one is normally interested in the reservoir scale (10^{3} m) response as a result of the spatial extension of the fractures. Because these scales of interest span approximately six orders of magnitude, multiscale methods are required to assess the overall response as modeling with fracture scale resolution at the reservoir scale becomes computationally prohibitive.

DEM models, unlike standard continuum models, consider the fractures within the rock mass as a Discrete Fracture Network (DFN), which explicitly defines the geometry of the fracture network. The physics of block interaction is then governed by the motion, contact forces and traction-separation laws between the rock blocks and the fractures (citation not found: Cundall_1979). Because NFR behavior is complex, even sophisticated phenomenological constitutive relationships may be inadequate to describe the complete rock mass behavior. The DEM approach aims to address this continuum behavioral deficiency by only requiring constitutive relations for the block interactions (citation not found: Cundall_2001).

That being said, the main issue with DEM models is primarily the computational demands. Due to the large number of degrees of freedom in the models and the requirement for very small time steps — because of the constant need for contact detection between blocks — running reservoir scale models is computationally prohibitive. The intent of this article is to develop a framework that incorporates the response of the DEM models while harnessing the computational speed of the continuum models. Up-scaling is accomplished in this paper by ‘calibrating’ a continuum model with DEM virtual experimental data using a combination of a heuristic optimization algorithm and an iterative least squares regression algorithm.

The general goal of up-scaling is to formulate simplified coarse-scale governing equations that approximate the fine-scale behavior of a material (citation not found: Geers_2010). In the case of the DEM simulations in this investigation, the aim of up-scaling is to identify the parameters of a continuum model that best mimics the response of the DEM model.

Multiscale methods that can be considered often fall into one of two classes: hierarchical or concurrent (citation not found: Gracie_2011). In concurrent multiscale models, different scales are used in different regions of the domain; the solution of the coupled model proceeds by solving both scales simultaneously. This approach is very expensive since the time step of the whole simulation is controlled by the fine-scale model; however, the solution is often more accurate. In hierarchical multiscale methods, the constitutive behavior at the coarser scale is determined by exercising a finer scale RVE. The finer scale models vary from relatively simple models, as in micromechanics, to complex nonlinear models. This approach is much more efficient, but can be less accurate. Up-scaling in this investigation can be considered to be a hierarchical multiscale method using computational homogenization.

Many multiscale homogenization techniques have been developed and proposed in the past, (citation not found: Aanonsen_2006) (citation not found: Temizer_2009) (citation not found: Loehnert_2005), but none have addressed the problem of up-scaling DEM simulations of NFR to continuum damage mechanics models using parameter estimation techniques.