The equivalent permeability, keq of stratified fractured porous rocks and its anisotropy is important for hydrocarbon reservoir engineering, groundwater hydrology, and subsurface contaminant transport. However, it is difficult to constrain this tensor property as it is strongly influenced by infrequent large fractures. Boreholes miss them and their directional sampling bias affects the collected geostatistical data. Samples taken at any scale smaller than that of interest truncate distributions and this bias leads to an incorrect characterization and property upscaling.

To better understand this sampling problem, we have investigated a collection of outcrop-data based Discrete Fracture and Matrix (DFM) models with mechanically constrained fracture aperture distributions, trying to establish a useful Representative Elementary Volume (REV). Finite-element analysis and flow-based upscaling have been used to determine keq eigenvalues and anisotropy.

While our results indicate a convergence towards a scale-invariant keq REV with increasing sample size, keq magnitude can have multi-modal distributions. REV size relates to the length of dilated fracture segments as opposed to overall fracture length. Tensor orientation and degree of anisotropy also converge with sample size. However, the REV for keq anisotropy is larger than that for keq magnitude. Across scales, tensor orientation varies spatially, reflecting inhomogeneity of the fracture patterns. Inhomogeneity is particularly pronounced where the ambient stress selectively activates late- as opposed to early (through-going) fractures. While we cannot detect any increase of keq with sample size as postulated in some earlier studies, our results highlight a strong keq anisotropy that influences scale dependence.

Is the permeability of naturally fractured rocks scale dependent

The influence of far-field stresses on fracture apertures in a fragmented rock layer is investigated using finite element analysis of a three-dimensional mechanical model. The model implements realistic boundary conditions, interactions between the fragmented layer and neighboring plastic rock layers, and frictional interfaces between the rock blocks. Stress-strain analysis is conducted to obtain stress variations within the fragmented rock layer and the block displacements and rotations. The fracture apertures are calculated using the local stress states instead of the far-field stresses simply being projected on the fractures. It is observed that fracture apertures can vary for the fracture segments over the individual blocks. Ensemble permeability is calculated by running a single-phase flow analysis considering the obtained fracture apertures for fracture segments. The influence of the rock block displacements, rotations and deformations, difference between the mechanical properties of the rock layers, and the orientation of the horizontal stresses is investigated on the ensemble permeability. It is demonstrated that the compressibility of the neighboring layers and block rotations and deformations have significant influence on the permeability of the fragmented rock layer. These effects, which may be ignored in simpler aperture calculation models, can result in considerable inaccuracies in the estimation of fracture apertures and ensemble permeability. Hence, such methods may only be used as indicative tools.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2017JB015365

Stress Influence on Fracture Aperture and Permeability of Fragmented Rocks

A Tracing Algorithm for Flow Diagnostics on Fully Unstructured Grids With Multipoint Flux Approximation

The role of natural fracture activation in hydraulic fracturing for deep unconventional geo-energy reservoir stimulation

Numerical investigation of fracture-rock matrix ensemble saturation functions and their dependence on wettability

Caroline Milliotte and Stephan K. Matthai show how a former structurally complex oil reservoir that is evaluated as a geological CO2 storage complex, can be modelled at a very high level of geological detail using a new workflow involving unstructured finite element meshes. The resulting simulation model includes discrete representations of complex intersecting faults.

From seismic interpretation to reservoir model: an integrated study accounting for the structural complexity of the Vienna Basin using an unstructured reservoir grid

The grid-block scale ensemble relative permeability, kri of fractured porous rock with appreciable matrix permeability is of decisive interest to reservoir simulation and the prediction of production, injector-producer water breakthrough, and ultimate recovery. While the dynamic behaviour of naturally fractured reservoirs (NFR) already provides many clues about (pseudo) kri on the inter-well length scale, such data are difficult to interpret because, in the subsurface, the exact fracture geometry is unknown. Here we present numerical simulation results from discrete fracture and matrix (DFM) unstructured grid hybrid FEM-FVM simulation models, predicting the shape of fracture-matrix kri curves. In contrast to earlier work (Matthai et al. 2007, Nick and Matthai, 2011), we also simulate capillary fracture matrix transfer (CFMT) and without relying the frequently made simplifying assumption that fracture saturation reflects fracture-matrix capillary pressure equilibrium. We also use a novel discretization of saturation which permits jump discontinuities to develop across the fracture-matrix interface. This increased physical realism permits – for the first time - to test the Matthai and Nick (2009) semi-analytical model of the flow rate dependence of relative permeability, ensuing from CFMT. The sensitivity analysis presented here constrains the CMFT-related flow rate dependence of kri and illustrates how it manifests itself in two geometries of layer-restricted well-developed fracture patterns mapped in the field. In a companion paper (Lang et al.), also investigate the dependence of kri on fracture aperture as computed using discrete element analysis for plausible states of in situ stress. Our results indicate that fracture-matrix ensemble relative permeability is matched – for fast flow rates – by the semi-analytic model of Matthai and Nick (2009). For slow rates the strong impact of CFMT leads to significantly different behaviour requiring a more elaborate treatment.

Numerical prediction of relative permeability in water-wet naturally fractured reservoir rocks

Abstract Discrete fracture and matrix (DFM) homogenization, simultaneously capturing reservoir layers and contained fractures, is an alternative to discrete fracture network (DFN) upscaling. Here, the DFM approach was applied to a fractured carbonate reservoir. Honouring geostatistical data from well logs, near-well multilayer reservoir models were constructed and analysed. Fracture aperture variations were modelled with a new semi-analytical model including a special treatment of layer-restricted fractures. Important results concern both pre-processing of stochastically generated DFMs for finite-element meshing, and the ensemble permeability values obtained by numerical homogenization of single v. multilayer models, respectively. Upscaling by volume averaging of vertically stacked single-layer DFMs results only in a fraction of the equivalent horizontal permeability that is obtained by homogenization of the multilayer models. Inspection of the flow patterns shows that this discrepancy arises because many fractures contact each other at layer boundaries fostering cross-flow. This effect is further enhanced where fractures intersect multiple layers. Compared to earlier DFN models for this reservoir, the DFM-derived fracture and matrix ensemble permeabilities are up to four times higher, highlighting how important it is to include the rock matrix into equivalent permeability calculations.

Well-data-based discrete fracture and matrix modelling and flow-based upscaling of multilayer carbonate reservoir horizons

The cross-disciplinary, multi-tool modelling and simulation approach we pursue on the Amellago Canyon platform carbonate outcrop (High Atlas, Morocco) aims at a rigorous characterization of stratigraphy, petrology, structure and flow, trying to capture a maximum of geological features of layered carbonate NFRs so that DFM simulation can reveal what level of detail has to be resolved for predictive simulation of reservoir behaviour / determination of equivalent properties by means of flow-based upscaling. This question of model convergence towards a sufficient level of realism is addressed by comparison of results obtained from a hierarchy of thin vertical slab models: Model (1) a coarse model with faults and fracture corridors only; Model (2) an intermediate model adding multi-layer fractures including the WE striking set with stylolitic overprint, and Model (3) a fine model also including different sets of bed-confined fractures. These DFM models are unique because they are largely deterministic: fractures are interpreted from the LiDAR survey data, including texture-mapped high-resolution photographic images. Bed confined fractures seem to contribute significantly to the equivalent permeability only when they intersect more continuous fractures, like in the fracture corridors or multi-layer fractures spanning low permeability layers.

3D Deterministic Multi-Level Fracture and Matrix Flow Model of a Layered Platform Carbonate Reservoir Analog (Morocco)

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Caroline Milliotte

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From seismic interpretation to reservoir model: an integrated study accounting for the structural complexity of the Vienna Basin using an unstructured reservoir grid

Numerical prediction of relative permeability in water-wet naturally fractured reservoir rocks

Well-data-based discrete fracture and matrix modelling and flow-based upscaling of multilayer carbonate reservoir horizons

3D Deterministic Multi-Level Fracture and Matrix Flow Model of a Layered Platform Carbonate Reservoir Analog (Morocco)