# Numerical Simulation of Deformation Band Occurrence and the Associated Stress Field during the Growth of a Fault-Propagation Fold

^{*}

## Abstract

**:**

## 1. Introduction

_{1}− σ

_{3}) and P is the effective mean stress (here taken as a biaxial parameter, P = (σ

_{1}+ 2σ

_{3})/3). In a Q-P plot (Figure 1), a direct analysis can be made between the stress path and the failure envelopes, whose sizes can vary, depending on the porosity or the grain radius of the rock [13,14]. Interestingly, the type of deformation bands can also be related to the band orientation where the stress path intersects the failure envelope; for low Q/P values (0.35 < Q/P < 0.7), the bands are perpendicular or sub-perpendicular to the maximum stress direction and are called Pure Compaction Bands (PCBs) [15]. For higher Q/P values (>0.7), the bands are oblique to the maximum stress direction, with a lower α angle (Figure 1B): Such structures are called Shear Enhanced Compaction Bands (SECBs) and can be observed with conjugated orientations [15]. The conjugated angle (2α Figure 1B) can change within the SECB domain [16] and may present a shear component that offsets preexistent structures. The Q/P threshold value separating the two types of bands (i.e., PCBs and SECBs) is not fixed and can depend on the nature of the materials. In the following, we will take a value of Q/P = 0.7 (Figure 1), which is consistent with the average observation provided by geomechanical experiments conducted on different sandstones [14,17,18,19,20,21,22,23].

## 2. Methods

#### 2.1. Numerical Method

_{n}representing the normal stress, φ is the internal friction and c is the cohesion of the material.

#### 2.2. Model Details

_{i}. The sedimentary rocks above the décollement have a bulk friction angle (Coulomb criterion) φ

_{B}, and the friction on the décollement and the thrust is φ

_{d}(Figure 2). A low value for φ

_{d}was chosen (φ

_{d}= 4°), corresponding to classical values of friction in evaporites. The thrust retains a dip angle of α = 23°, which is the approximate value observed in the Boixols thrust, with an evolving height of the fault tip (h

_{r}) throughout the fold growth. The density of these rocks is ρ = 2300 kg/m

^{3}. The material below the décollement is the same as that above it. A tectonic force Q parallel to the décollement is applied on the hinterland side of the wedge, above the décollement (Figure 2, red arrows). Note that its magnitude is an outcome of the numerical mechanical analysis. The topographic surface is free of any stress. On the other prototype boundaries (base, extreme left or right sides of the models), displacements are constrained, as indicated in Figure 2. Adding an internal topography allows one to use the Critical Coulomb Wedge (CCW) theory, which guarantees that the compression Q will result in activation of the thrust in the central region. The models are size calibrated compared with the stress gradient evolving with depth, taken at the fold front, away from the deformed zone, and following the relation σ

_{v}= ρ·g·z, where σ

_{v}represents the vertical stress, g is the standard gravity (g = 9.8 m.s

^{−2}), z is the depth in meters, and ρ indicates the rock density, as previously mentioned. This means that for a depth of 2000 m, which is the depth between the Maastrichtian and Trias décollement in the Sant-Corneli area [69,70], the principal minimum stress (σ

_{3}= σ

_{v}) is approximately 45 MPa. The models are geometrically constrained by several parameters that are evolving with the growth of the fold, as detailed in Table 1.

_{1}) and minimum (σ

_{3}) stresses obtained from the numerical results were used to calculate the effective mean stress (P) and the differential stress (Q), and then the Q/P ratios.

## 3. Results

#### 3.1. Q/P Ratios Distribution and Stress Path

#### 3.2. Failure Envelopes

#### 3.3. Band Timing and Distribution

## 4. Discussion

#### 4.1. Chronology of Band Occurrence

#### 4.2. Deformation Band Occurrence in Heterogeneous Granular Materials

^{−3/2}, where α is considered a constant mostly depending on grain elastic properties (Fig. 7). Our study shows that the Aren formation cannot follow the same empirical relation base for measurements on siliciclastic materials. Indeed, the stress path for the kinematic we proposed is insufficient to reach the empirical envelope (“Env 1”). Another study [76] tested the empirical relationship between P*, φ, and R for carbonate rocks: a slope of −0.7 was found to match the data for carbonate rocks (Figure 7), which is different from the siliciclastic one (−1.5).

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Q-P plot with definition of the stress path (adapted from Figure 17 in [24]; Burial in dark blue and extensive regime in light blue, followed by the compressive regime in red). The stress path crosscut the envelopes, leading to the deformation bands’ set up. The type of band depends on the crosscut area; Shear Enhanced Compaction Bands (SECBs) will form in Q/P ratios between 0.7 and 1.2 (grey area); Pure Compaction Bands (PCBs) are set up between 0.7 > Q/P > 0.35 (red area); finally, for lower values of Q/P ratios (below 0.35), no localized structures will form as the deformation is homogeneous. (

**B**) Examples of band structures potentially set up depending on the Q/P ratios, where α represents the angle between the maximum principal stress and the band orientation.

**Figure 2.**Schematic base model geometry at an intermediary stage. H

_{i}: initial pre tectonic thickness; L

_{psi}: length of the fold top; L

_{i}: horizontal length at the base of the fold; A: amplitude of the front limb of the fold; h

_{r}: height of the fault tip; α: angle of the thrust; β: angle of the fold front limb; S: shortening; e: thickness of the syn-tectonic layer; φ

_{B}: internal friction of the bulk material; φ

_{d}: internal friction of the decollement (d, green line); φ

_{S}: internal friction of the syn-tectonic material. Green symbols correspond to the movements allowed at the border of the section.

**Figure 3.**Schematic representation of the fold evolution geometry for each step and localization of the different key points used for the local stress estimations represented by the markers.

**Figure 4.**Q/P ratio estimated by the Limit Analysis Method (LAM) model for each step. White color is for a Q/P ratio lower than 0.35; red is for a Q/P ratio between 0.35 and 0.7; black is for ratios higher than 0.7. Shadow markers show the point measurements (Point I is not shown in this figure).

**Figure 5.**Stress path evolution for the different key points (See Figure 3 for point positions) in a Q-P plot from the numerical results. (

**A**) Point I; (

**B**) Point II; (

**C**) Point III; (

**D**) Point IV; (

**E**) Point V; (

**F**) Point VI. Stage numbers are indicated next to the points. Three yield envelopes are represented. “Env 1” represents the empirical value of the calcarenite formation with P* = 116 MPa (see text for details); “Env 2” corresponds to the Boise II sandstone with P* = 42 MPa [14]; and “Env 3” corresponds to the theoretical envelope of the calcarenite proposed by Robert et al. [24], with P* = 30 MPa.

**Figure 6.**Deformation band distribution with the numerical stress results in comparison to the yield cap of the theoretical Aren calcarenite (P* = 30 MPa) for the steps 2, 6, 10, 12, and 14. Pure Compaction Bands (PCBs) are represented in red and Shear Enhanced Compaction Bands (SECBs) are represented in grey. The blue part on the top of the fold represents the extensional domain (maximum stress is sub-vertical). In the background, light red and grey colors represent the Q/P ratio obtained in Figure 4.

**Figure 7.**Diagram of critical effective pressure (P*) vs. the product of porosity and the grain radius. Results of geomechanical experiments and slope trend are plotted for siliciclastic materials [13,14], porous carbonate rocks [76], and an Otter Sherwood Sandstone (OSS, [77]). Theoretical results for the Aren calcarenite of Aren and Orcau sites are also plotted, see text for calculus details.

**Table 1.**Parameter table for each step of the fold evolution (See parameter details in Figure 2’s caption).

Exp. | H_{i} (km) | L_{psi} (km) | L_{i} (km) | A (km) | h_{r} (km) | α (°) | β (°) | S (km) | e (m) | φ_{B} (°) | φ_{d} (°) | φ_{s} (°) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Step 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 30 | 4 | 30 |

Step 2 | 2 | 3.1 | 3.2 | 0.1 | 0.15 | 23 | 75 | 0.1 | 70 | 30 | 4 | 30 |

Step 3 | 2 | 2.7 | 3.3 | 0.25 | 0.25 | 23 | 75 | 0.3 | 125 | 30 | 4 | 30 |

Step 4 | 2 | 2.6 | 3.6 | 0.5 | 0.45 | 23 | 75 | 0.6 | 250 | 30 | 4 | 30 |

Step 5 | 2 | 2.4 | 3.7 | 0.6 | 0.55 | 23 | 75 | 0.9 | 325 | 30 | 4 | 30 |

Step 6 | 2 | 2.1 | 3.9 | 0.9 | 0.75 | 23 | 75 | 1.2 | 450 | 30 | 4 | 30 |

Step 7 | 2 | 1.9 | 4.1 | 1 | 0.85 | 23 | 75 | 1.4 | 500 | 30 | 4 | 30 |

Step 8 | 2 | 1.8 | 4.3 | 1.2 | 1.05 | 23 | 75 | 1.8 | 540 | 30 | 4 | 30 |

Step 9 | 2 | 1.6 | 4.4 | 1.4 | 1.15 | 23 | 75 | 2.1 | 700 | 30 | 4 | 30 |

Step 10 | 2 | 1.3 | 4.5 | 1.5 | 1.35 | 23 | 75 | 2.3 | 750 | 30 | 4 | 30 |

Step 11 | 2 | 1.15 | 4.8 | 1.7 | 1.5 | 23 | 75 | 2.6 | 850 | 30 | 4 | 30 |

Step 12 | 2 | 1 | 5 | 2 | 1.7 | 23 | 75 | 2.8 | 1000 | 30 | 4 | 30 |

Step 13 | 2 | 0.5 | 5.1 | 2.2 | 1.9 | 23 | 80 | 3 | 1080 | 30 | 4 | 30 |

Step 14 | 2 | 0 | 5.2 | 2.3 | 2 | 23 | 85 | 3.1 | 1130 | 30 | 4 | 30 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Robert, R.; Souloumiac, P.; Robion, P.; David, C. Numerical Simulation of Deformation Band Occurrence and the Associated Stress Field during the Growth of a Fault-Propagation Fold. *Geosciences* **2019**, *9*, 257.
https://doi.org/10.3390/geosciences9060257

**AMA Style**

Robert R, Souloumiac P, Robion P, David C. Numerical Simulation of Deformation Band Occurrence and the Associated Stress Field during the Growth of a Fault-Propagation Fold. *Geosciences*. 2019; 9(6):257.
https://doi.org/10.3390/geosciences9060257

**Chicago/Turabian Style**

Robert, Romain, Pauline Souloumiac, Philippe Robion, and Christian David. 2019. "Numerical Simulation of Deformation Band Occurrence and the Associated Stress Field during the Growth of a Fault-Propagation Fold" *Geosciences* 9, no. 6: 257.
https://doi.org/10.3390/geosciences9060257