Fluid flow through a single fracture is traditionally described by the cubic law, which is derived from the Navier-Stokes equation for the flow of an incompressible fluid between two smooth-parallel plates. Thus, the permeability of a single fracture depends only on the so-called hydraulic aperture which differs from the mechanical aperture (separation between the two fracture wall surfaces). This difference is mainly related to the roughness of the fracture walls, which has been evaluated in previous works by including a friction factor in the permeability equation or directly deriving the hydraulic aperture. However, these methodologies may lack adequate precision to provide valid results. This work presents a complete protocol for fracture surface mapping, roughness evaluation, fracture modeling, fluid flow simulation, and permeability estimation of individual fracture (open or sheared joint/pressure solution seam). The methodology includes laboratory-based high-resolution structure from motion (SfM) photogrammetry of fracture surfaces, power spectral density (PSD) surface evaluation, synthetic fracture modeling, and fluid flow simulation using the Lattice-Boltzmann method. This work evaluates the respective controls on permeability exerted by the fracture displacement (perpendicular and parallel to the fracture walls), surface roughness, and surface pair mismatch. The results may contribute to defining a more accurate equation of hydraulic aperture and permeability of single fractures, which represents a pillar for the modeling and upscaling of the hydraulic properties of a geofluid reservoir.
Fractures exert a vital contribution on determining the migration and storage for geofluids, such as groundwater, and hydrocarbons. Thus, the analysis and modeling of fractures are imperative for characterizing reservoirs and simulating their behavior during the production stage. Fluid flow through fractures is traditionally described by the cubic law, derived from the Navier-Stokes equation for the flow of an incompressible fluid between two smooth-parallel plates [
Several authors have studied the effect of roughness of the walls on fracture permeability working with various materials, such as glass [
Another widely used methodology derives the hydraulic aperture from the mechanical aperture,
Considering the previous arguments, the main objective of this work is to find empirical equations that describe the effect of fracture roughness on permeability at different apertures. In order to reach this goal, some problems which should be overcome are (i) to develop a protocol for mapping the fracture surface, (ii) to evaluate the fracture roughness as a function of the wavelength of the asperities, and (iii) to validate the relationships using a significant number of samples, roughness values, and aperture scenarios.
Various approaches have been reported in the literature for mapping the surface of fractures and faults in the field or laboratory involving the use of Lidar [
Evaluation of fracture roughness is achieved by implementing the power spectral density (PSD), which provides a more objective description based on the frequency distribution of the asperities in the Fourier domain. This approach has been successfully applied by previous authors for describing the roughness of fractures (e.g., [
A key benefit of incorporating computer-generated synthetic fractures is the capability to work with a large amount of fracture data to perform direct fluid flow simulations, such as (i) the finite difference method (e.g., [
In this study, the permeability values of single isolated fractures (synthetic and natural) were calculated via LBM, using the PALABOS open source library [
The selected study area, the Roman Valley Quarry (Figure
(a) Structural map of the Roman Valley Quarry (modified after Volatili et al. [
The Roman Valley quarry has been heavily studied by previous authors focusing on the structural, sedimentological, and diagenetic properties [
Characteristics of lithofacies exposed in the Roman Valley Quarry.
Lithofacies | Thickness | Bitumen distribution | |||
---|---|---|---|---|---|
Au: | Alternation of medium- to coarse-grained bioclastic grainstones (Au1) and medium-grained bioclastic grainstones (Au2). | 40 to 60 m | ~27.5 | 83.13 (V) |
Abundant in both matrix and fractures near faults |
B: | Medium-grained grainstones. | 10-to 15-m | ~26.4 | 444.82 (V) |
Abundant in both matrix and fractures near to faults |
C: | Alternations of two echinoid plates and spines rich facies: fine-grained bioclastic grainstones (C1) and fine- to very fine-grained bioclastic packstones (C2). Argillaceous to marly beds (<3 cm thick) are common. | 10 to 15 m | ~10.9 | ~0.30 (V) |
Absent in matrix and oil stain in fractures |
E: | Alternation of two planktonic foraminifera facies: marly wackestones (E1) and marly mudstones (E2). | 60-65-m | ~28.8 | ~0.085 (V) |
Absent in both matrix and fractures |
Notes: lithofacies description from Rustichelli et al. [
Considering the significance of these fractures, this work focused on investigating both cases of open mode and sheared fractures with a small sliding/tearing mode displacement, in the order of millimeters, allowing the assumtion of a negligible wall wearing. For this last case, the mismatch between opposite walls was also computed due to its importance as a mechanism for maintaining fracture openings even at reservoir depths.
In this work, we present a multiphase integrated methodology for characterizing fracture surfaces and their effect on permeability. This approach combines fracture surface scanning using structure from motion photogrammetry, a statistical and spectral description of individual natural fracture surfaces, modeling of synthetic fractures, and computational calculation of permeability by fluid flow simulation.
During the summer of 2018, a suite of oriented hand samples was collected from the study site comprising three (i.e., Au, B, and C) of the four major lithofacies of the Bolognano Formation present in the quarry (Figure
Sample location sites. (a) Sampling locations inside the Roman Valley Quarry. (b) Sample site 1 from lithofacies (b) with scale card showing centimeter increments. (c) Sample site 2 from lithofacies Au. (d) Sample site 3 from lithofacies (c). Rock hammer for scale, 22 cm in length.
The workflow for mapping surface topography involves the following key stages: (1) fracture surface image set acquisition and (2) image alignment and three-dimensional digital rock model creation using SfM.
SfM photo scanning was performed at the University of Camerino photogrammetry laboratory (Figure
Photogrammetry setup and three-dimensional SfM procedure. (a) Photo light box used in the photogrammetry lab. (b) The collected sample placed on the rotating stage with unique photo-targets generated by Agisoft PhotoScan. (c) Sparse point cloud generated during the photo alignment phase of the SfM procedure. (d) Fully rendered photo-realistic 3D model showing camera positions.
Fracture surface models were aligned and processed using Agisoft PhotoScan Pro (
Agisoft-generated coded targets were placed inside the scene to aid in the imagery processing; these coded targets are automatically recognized by the software and help build connection points between the image sets (Figure
As a measure to define the error of the model, we follow the methodology established by Corradetti et al. [
Trimmed 3D point clouds were exported from Agisoft as “.xyz” text files. Then, a rectangular subregion of each fracture surface of interest was extracted from the point cloud and processed to remove undesirable trend and eventual noise (Figure
Fracture surface processing. (a) Original exported fracture surface containing an artificial trend. (b) Resulting image after removing the artificial trend and assigning a new reference grid according to a millimeter scale.
A complete description of the fracture roughness is given by the specification of two functions: the probability density function (depending on the media and standard deviation) for heights and the PSD [
Illustration of a complete description of surface roughness: (a) in terms of statistical height distribution, probability density and (b) in terms of frequencies distribution, Fourier power spectrum (modified from Brown [
The Fourier power spectrum,
From a physical aspect, the fractal dimension shows the proportion of high-frequency to low-frequency sinusoid components (roughness). High
Since a limited number of natural fracture surfaces were available, additional synthetic fracture surfaces were used to strengthen the statistical significance of the results. Following the procedure described by Ogilvie et al. [
The individual fracture surfaces (natural and synthetic) were used to model dilation associated with (Figure
Mechanisms considered for possible fracture aperture generation in the study area. (a) Opening mode displacement, E, (joint and/or opened pressure solution seam) and (b) sliding/tearing mode displacement, S, (sheared joint and/or sheared pressure solution seam).
Lattice-Boltzmann simulations were performed using the open-source computational fluid dynamics software PALABOS [
This procedure consists of imposing a single-phase fluid flow through a 3D porous media maintaining a fixed pressure gradient between the inlet and outlet opposing faces of the model; the rest of the faces were padded (Figure
Examples of lattice velocity field volumes with the corresponding streamlines. (a) Fracture with opening mode displacement equals to 1 mm. (b) Fracture with sliding/tearing mode displacement equals to 50 mm. Both fractures have a considerable roughness (
The simulation ended once the imposed steady-state condition was reached (standard deviation of the average
The mismatch between the opposite fracture walls is of extreme importance since this factor may keep fractures open even at reservoir depths. Since the mismatch was not imposed during fracture modeling, it was measured after the generation of the synthetic fractures. The mismatch was evaluated only for the sliding/tearing mode fractures, whereas it was unnecessary in the case of opening mode fractures since the aperture is constant. For the evaluation of the mismatch value, the methodology of the power spectral density ratio (PSDr), introduced by Ogilvie et al. [
The results of this calculation can be represented in a graph where the parameters associated with the mismatch and the degree of mismatch between the surfaces at different wavelengths can be obtained (Figure
A typical PSD ratio graph used for defining the mismatching parameters.
Following the definition of Ogilvie et al. [ Minimum mismatch length (ML_min): wavelength at which the fractures start to match, indicated by the wavelength where the PSD ratio values fall below its maximum value (PSDr_max) Maximum mismatch length (ML_max): wavelength at which the fracture opposing surfaces reach the maximum matching, thus the minimum value of PSD ratio (PSDr_min)
The calculation of these parameters was made using a MATLAB code. In this case, (ML_min) is considered as the only reliable indicator of the mismatch since (ML_max) often falls outside the scale of the study (Figure
The results of this work consist of an analysis of surface topography performed on fracture samples from three lithofacies (Au, B, and C), and the computed permeability in function of the fracture properties, including fractal dimension, opening and sliding/tearing displacement, and minimum mismatch length.
In Table
Results of the surface analysis.
Field description | Surface Analysis | ||||||
---|---|---|---|---|---|---|---|
ID | Orientation | Set | Lithofacies | JRC | SD | ||
F-1 | 200/72 | PS2a | Au | 10 | 1.22 | 1.89 | 1.91 |
F-2 | 285/85 | PS2b | Au | 12 | 2.85 | 1.85 | 1.93 |
F-3 | 195/80 | PS2a | B | 9 | 1.49 | 1.90 | 1.78 |
F-4 | 210/V | PS2a | C-2 | 8 | 1.97 | 1.90 | 1.67 |
F-5 | 200/80 | PS2a | C-1 | 10 | 1.66 | 1.85 | 1.96 |
F-6 | 120/85 | PS2b | C-1 | 11 | 0.82 | 1.96 | 1.95 |
Notes: orientation noted as dip direction/dip angle. SD: standard deviation of asperity height.
The results of the work are presented in Figures
(a) Single-fracture permeability versus opening mode displacement; results indicate a positive power-law relationship (nearly cubic). (b) Hydraulic aperture (computed with the equation (
Single fracture permeability versus sliding/tearing displacement. (a) Permeability along the shear direction. (b) Anisotropy permeability ratio,
Permeability as a function of fractal dimension. (a) Permeability versus open mode displacement. Note the inversely proportional control of the fractal dimension on permeability. (b) Permeability versus sliding/tearing displacement. Note the proportional control of the fractal dimension on permeability. Axes are in logarithmic scale, and the dashed lines correspond to the best-fitting power laws.
Porosity versus sliding/tearing displacement. Error bars indicate the standard error.
Permeability components—(a) parallel to shear,
(a) Minimum mismatch length versus sliding/tearing displacement. (b) Permeability versus minimum mismatch length. Permeability is directly proportional to the minimum mismatch length, which is related to the shear displacement.
In the case of opening mode displacement, the results indicate that the permeability increased proportionally to the mechanical aperture following a positive power-law relationship (Figure
With respect to the case of sliding/tearing displacement, the results indicate that the permeability component parallel to the displacement (
The fracture roughness, expressed in terms of fractal dimension (
It is expected that this positive relationship between displacement and permeability should stabilize at a certain point, as the permeability and porosity cannot increase indefinitely. This behavior is observed when the porosity is evaluated at higher displacement values (Figure
The fracture permeability is related to the porosity following a power law (Figure
The minimum mismatch length, ML_min, was evaluated as a function of the displacement (Figure
The present work evaluates the effect of fracture surface features such as roughness, aperture, and mismatch on permeability using fracture surface scanning by SfM photogrammetry, numerical modeling, and lattice-Boltzmann fluid flow simulation.
The results of this study demonstrate the versatility of the SfM procedure as an analytical tool which can be applied at a wide range of scales including millimeter-scale features such as fracture surfaces. The controlled conditions in the photogrammetry laboratory allowed a highly detailed scan and extraction of the micro surface topography of samples sized
This methodology proved to be highly efficient in expressing the fracture roughness allowing a more accurate and representative measure with respect to the relative hydraulic roughness [
Another important aspect of these results is the reproducibility of synthetic fractures with similar characteristics. This step was important to increase the data volumes leading to a greater statistical significance of the results and validity of the inferred relationships. The lattice-Boltzmann procedure also played a key role in this study as it allows the estimation of permeability values for controlled scenarios with different imposed properties (i.e., roughness, opening mode, and sliding/tearing displacement). This permits evaluation of the relationship between permeability, porosity, mismatch, and other imposed properties. The computed permeability may present some inaccuracy in low-resolution models as previously reported by Zambrano et al. [
Two situations were considered to explain the presence of open fractures: (i) dilation due to opening mode displacement (joint or opened pressure solution seam) and (ii) dilation due to mismatch caused by shearing and sliding/tearing displacement.
In the first case, the results followed the expectation and confirmed previous interpretations: (i) permeability tends to increase with opening following a nonlinear relationship, (ii) a higher fractal dimension (greater roughness) correlates to lower permeability, and (iii) the effect of roughness is less significant at greater opening values. It is expected that higher roughness (higher frequencies of asperities) may expose a wider area in contact with the migrating fluid, diminishing its velocity due to friction. Evidently, at higher opening values, this effect should be less evident because it is the specific area (area/volume) which has a control on the permeability, as has been previously reported by Zambrano et al. [
The second case (sliding/tearing displacement) creates an aperture due to the mismatch between the opposite walls of the fracture. We found significant differences between these two cases concerning the effect of the roughness of the permeability. In fact, the effect of roughness on permeability is inverse. Given the same displacement, fractures with higher roughness values permit the creation of larger voids and therefore enhance the fluid flow. So, the effect of friction exerted by the roughness on fluid flow has a secondary role in the case of mismatch due to sliding/tearing displacement. The continuous dilatancy of the fracture due to sliding/tearing displacement should cease at a certain value depending on the asperity frequencies present in the fracture. Nevertheless, it is difficult to verify this behavior for real fractures, where a 20 mm sliding/tearing displacement likely leads to fracture wall wearing and the generation of cataclastic material [
The permeability anisotropy in fractures with sliding/tearing displacement is significant and dependent on the roughness (fractal dimension). For low displacement (0.5 mm), the anisotropy can reach values up to 2.6 for fractures with a fractal dimension of 2.0. For the same displacement, fractures with high roughness (
The mismatch itself has a positive control on the permeability. The importance of this result is that the mismatch could also be produced by diagenetic processes (e.g., cementation, dissolution) and shearing. Zambrano et al. [
The results agree with the macroscale observations of previous authors in the study area, where both opened/sheared pressure solution seams and fault-related joints present the greatest values of aperture and the most important bitumen impregnation [
The relationship between permeability and porosity for rough fractures clearly deviates from the ideal smooth parallel plate case (for the studied scenarios). Fracture permeability is lower for the same porosity range (<0.2%) in comparison to the theoretical values. Instead, the power-law slope is higher, indicating a more important control of porosity as it was expected. The equation itself may be useful to estimate the permeability of fractures if the fracture porosity is known.
After their formation, both closing and opening mode fractures are often subjected to a shear process, and even with a small imperceptible sliding/tearing displacement, they cannot be modeled as simple opening mode fracture. At reservoir depth, preexisting fractures (joints and pressure solution seams) that are favorably oriented to be sheared (accordingly to the orientations of the stress field which affected the area) may be characterized by a mismatch between the fracture walls enhancing the fracture opening. Therefore, the findings of this work may have a significant impact on fracture modeling workflows for subresolution faults (e.g., [
We presented a new multifaceted approach to characterize surface fracture roughness by SfM photogrammetry, numerical modeling, and computational fluid dynamics simulation. This methodology provides a better quantification of surface parameters that are not possible to obtain using former surface roughness measurement and analysis tools.
In addition, this study illustrates the crucial relationships between permeability and other fracture properties, such as roughness, porosity, opening mode-sliding/tearing displacement, and mismatch. The obtained relationships pointed out the following statements:
In joints (opening mode fractures) and/or opened pressure solution seams, the roughness tends to reduce the permeability. Thus, the permeability is inversely proportional to the fractal dimension In sheared joints and/or pressure solution seams (assuming an insignificant surface wearing), the sliding/tearing mode displacement may cause mismatch and therefore enhance the porosity and permeability. The validity of this behavior may depend on the point that displacement starts to produce cataclastic material. Small shear displacements and mismatch may be extremely important to guarantee storage and migration of geofluid at depth thanks to asperities-supported aperture. Permeability anisotropy is very significant for small shear displacements, characterized by higher values of permeability component perpendicular to the shear displacement Porosity exerts a more important control on permeability in rough fractures (higher power-law slope). The empiric relationship may result in greater utility for estimating the fracture permeability if the fracture porosity is known
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This research was supported by the FAR Project 2014 “Characterization and modeling of natural reservoirs of geofluids in fractured carbonate rocks,” funded by the University of Camerino, Principal investigator Emanuele Tondi, and the Reservoir Characterization Project (