^{1}

^{1}

^{2}

^{3}

^{1}

^{1}

^{2}

^{3}

Crustal-scale fluid flow can be regarded as a bimodal transport mechanism. At low hydraulic head gradients, fluid flow through rock porosity is slow and can be described as diffusional. Structures such as hydraulic breccias and hydrothermal veins both form when fluid velocities and pressures are high, which can be achieved by localized fluid transport in space and time, via hydrofractures. Hydrofracture propagation and simultaneous fluid flow can be regarded as a “ballistic” transport mechanism, which is activated when transport by diffusion alone is insufficient to release the local fluid overpressure. The activation of a ballistic system locally reduces the driving force, through allowing the escape of fluid. We use a numerical model to investigate the properties of the two transport modes in general and the transition between them in particular. We developed a numerical model in order to study patterns that result from bimodal transport. When hydrofractures are activated due to low permeability relative to fluid flux, many hydrofractures form that do not extend through the whole system. These abundant hydrofractures follow a power-law size distribution. A Hurst factor of ~0.9 indicates that the system self-organizes. The abundant small-scale hydrofractures organize the formation of large-scale hydrofractures that ascend through the whole system and drain fluids in large bursts. As the relative contribution of porous flow increases, escaping fluid bursts become less frequent, but more regular in time and larger in volume. We propose that metamorphic rocks with abundant veins, such as in the Kodiak accretionary prism (Alaska) and Otago schists (New Zealand), represent regions with abundant hydrofractures near the fluid source, while hydrothermal breccias are formed by the large fluid bursts that can ascend the crust to shallower levels.

Fluid flow through rocks and sediments plays a crucial role in many geological and geomechanical processes. They are responsible for a variety of geological phenomena, for example, the formation of veins and hydraulic breccias. Fluids carry dissolved components and heat that advect together with the fluid and are a critical control on many geological processes (e.g., [

Veins and breccias provide abundant evidence in the geological record, that natural fluid flow is not always steady state. Veins are dilatant structures, typically fractures, in which minerals precipitated from fluids [

Breccias are rock masses composed of broken rock fragments or clasts. Breccias can form by deposition of transported clasts (namely sedimentary breccias [^{2} hydrothermal breccia with clasts sizes from tens of microns to hundreds of meters, with mixing of clasts that were originally >km apart vertically [^{3}.

Examples of hydrothermal breccias. (a) Teufelsgrund mine, Black Forest, Germany (47°50

The permeability of rocks is a primary and critical geological parameter because migrating fluids in the Earth’s crust play a major role in mass and heat transfer and for crustal rheology. Permeability has been regarded as a dynamic parameter, as it changes in response to stress, fluid production, and geochemical reactions [^{2}) and fluid viscosity (

Permeability varies by >16 orders of magnitude in geological materials (from

The driving pressure is the difference between the equilibrium fluid pressure and the actual fluid pressure [

Transport of clasts over longer distances, as well as structures, indicates that fluid flow rates may be as high as m/s [^{-2} to 10^{-1} m/s have been estimated based on analyses of breccia fragments [

Intermittent flow is predicted to occur when the matrix permeability of a rock is insufficient to accommodate fluid flow, which leads to an increase in fluid pressure and opening of fractures [

Despite the tendency for crustal permeability to decrease with time, long-lived (10^{3}-10^{6} years) hydrothermal systems exist [

Many transport systems show intermittent behaviour in experiments [

A large number of studies modelled hydrofracture formation and dynamic fluid flow, using, e.g., finite element [

Steady Darcy flow and intermittent fracture flow both seem to be important mechanisms, and these two end-member fluid flow modes have been investigated extensively. There is abundant evidence that such transport systems are neither purely diffusive nor purely intermittent, but surprisingly, little work has been published on the transition between the two regimes (e.g., [

We assume a rigid matrix model in which porosity (^{-1}) of the fluid. We can express this with:

^{3} times the porosity) of fluid at

Note that this equation is similar to Fick’s second law for diffusion, with the pressure diffusion coefficient ^{2}/s).

Following Fick’s second law, pressure gradients can dissipate in a diffusional fashion with an effective diffusivity ^{2}/s). The compressibility of fluids varies with pressure and temperature [^{-5} to 10^{-1} m^{2}/s for

Symbols.

Symbol | Description | SI unit |
---|---|---|

Flux | ||

Permeability | m^{2} | |

Fluid viscosity | Pa·s | |

Compressibility | Pa^{-1} | |

Pressure diffusion coefficient | m^{2}/s | |

Porosity | ||

Solid pressure | Pa | |

Effective lithostatic pressure | Pa | |

Volume | m^{3} | |

Solid density | kg/m^{3} | |

Fluid density | kg/m^{3} | |

Gravitational acceleration | m/s^{2} | |

Time | s | |

Depth | m | |

Model variable: Average P over cluster elements | ||

Model variable: Element size | ||

Subscript indicates model parameters | ||

Hurst exponent |

In order to simulate bimodal fluid transport by the two mechanisms, Darcian flow and transport through fractures, we use a square, 2-dimensional orthogonal grid of ^{4} Pa/m with depth ^{3} and a gravitational acceleration, ^{2}). Fluid flow is implicitly modelled by tracking the evolution of

Flow diagram of the numerical simulations. The model consists of a rectangular grid of

The basic loop in the simulation is as follows:

One element in the bottom row is randomly selected and the pressure in that element is increased by

After the pressure increase, the possible initiation of a hydrofracture is assessed for every element in the model. A fracture is initiated when the pressure exceeds the effective lithostatic pressure ^{3}, and

If a fracture is initiated, a propagation loop is started. As only one hydrofracture can exist at any one time in the simulation, all elements with the label “broken” form one connected cluster. First step of the fracture propagation subloop is to equalise the pressure to

none of the elements in the cluster reaches the failure criterion, or

the cluster reaches the surface. In the latter case, the pressure in all elements within the cluster is set to zero, which implies that fluid pressure is reduced to hydrostatic and any excess fluid is released at the surface. Once fracture propagation is finished, all elements within the cluster are reset to “unbroken” (implying instantaneous closing or healing of the fracture) and the time and size (number of broken elements) of the cluster is recorded.

Finally, once pressures are increased in all elements in the bottom row (step 2) and any resulting hydrofractures are dealt with (step 3), Darcian flow is simulated using an explicit, forward finite difference scheme. Pressures in the top row are set to zero.

In the current model, fractures completely close or heal after one calculation step. Healing is thus effectively instantaneous relative to the diffusional flow process. Microstructures of crack-seal veins do show that cracks can and do commonly seal faster than fluid pressure builds up. We therefore chose to model the healing as effectively instantaneous, which obviates the need to add a healing-rate parameter. It should, however, be borne in mind that this is an end member and that reducing the healing rate changes the patterns of fractures [

Equation (

Scaling of the model values (subscript ^{4} Pa/m. From this, we obtain for the scaling between

We assume a fixed fluid flux ^{2} (assuming the model is 1 m thick in the third dimension) and is added to the fluid residing in the ^{3} pore space of the element. This results in a pressure increase

Assuming a porosity

In the model, the pressure at the base is raised by

For the given ^{-7} to 0.005 corresponds to a permeability range of

Simulation parameters and movie names (in the supplementary material (available

Permeability | Timesteps [ | Movie name | |||
---|---|---|---|---|---|

0 | 0 | 0 | 1_D0.avi | ||

0 | 0 | 0 | 10_D0.avi | ||

0 | 0 | 0 | 100_D0.avi | ||

10^{-7} | 10^{-23} | 100_D2e-9.avi | |||

10^{-4} | 10^{-20} | 100_D2e-6.avi | |||

10^{-3} | 10^{-19} | 100_D2e-5.avi | |||

100_D6e-5.avi | |||||

10^{-4} | 100_D1e-4.avi |

Frequency distributions of hydrofracture sizes have been evaluated. Simulation visualisation and calculation of the time-averaged vertical pressure profile have been done with ImageJ [

Where

Figure ^{-4}. All the fluid is transported by hydrofractures when the diffusion coefficient is

Snapshots of simulation results, in the form of maps of the pressure field. Movies of the model can be found in the supplementary material (available

At

The average pressure in the system (Figure

Mean pressure in the whole model as a function of time. No and little diffusion produces an intermittent and irregular behaviour. Higher diffusion coefficients result in more periodical pressure fluctuations.

Whereas Figure

(a) Time-integrated vertical pressure profile. Red line indicates theoretical graph for pure diffusion. (b) Effective diffusion coefficient with depth.

Figure

The nature of the pressure fluctuations is illustrated with the rescaled-range analysis (

Self-similarity analysis of pressure fluctuations with rescaled range analysis. (a) Rescaled range (

Similar models and experiments [

Absolute frequency of hydrofracture size distributions (a) for different element sizes (

The resulting best-fit values for

Data and power-law fits of data shown in Figure

Volume of fractures reaching the surface (%) | Volume of fractures not reaching the surface (%) | No. of events reaching the surface | No. of events reaching the surface (%) | No. of events not reaching the surface | No. of events not reaching the surface (%) | Power-law fit for fractures not reaching the surface | |
---|---|---|---|---|---|---|---|

14.81 | 85.19 | 10 | 0.52 | 1,906 | 99.48 | ||

2.78 | 97.22 | 34 | 0.19 | 18,197 | 99.81 | ||

1.87 | 98.13 | 401 | 0.16 | 253,050 | 99.84 | ||

1.87 | 98.13 | 393 | 0.15 | 251,940 | 99.84 | ||

1.88 | 98.12 | 459 | 0.17 | 265,630 | 99.83 | ||

2.93 | 97.06 | 447 | 0.25 | 179,091 | 99.75 | ||

85.14 | 14.86 | 159 | 18.73 | 690 | 81.27 | ||

97.99 | 2.00 | 15 | 75 | 5 | 25 |

Hydrofractures that reach the surface are invariably large and their sizes do not follow the aforementioned power-law trends. Fractures of about 10% of the model area are the most frequent. As the hydrofractures that reach the surface are always large, their number relative to the ones that do not reach the surface is small. The fraction of hydrofractures that reach the surface equates the change that one hydrofracture reaches the surface. This fraction is approximately constant at about 0.15% at low

With increasing

Hydrofractures that do not reach the top of the model (Figure

The many hydrofractures that do not reach the surface are restricted to the bottom of the model, just above the level where fluid is produced. The expression in the geological record would be abundant crack-seal veins. Abundant veins are common in metamorphic rocks in accretionary prisms, such as the Kodiak accretionary complex, Alaska, USA, [

Hydrofractures that do reach the top of the model (Figure

The simulations shown here may give some indication on the character of escaping fluid batches. In case of high fluid fluxes (high fluid production rate and/or short duration of the fluid production), the system is expected to be close to the pure hydrofracturing end member. Pressure diffusion would be too low to significantly modify pressure buildup. Fluid escape events would be frequent, but escaping volumes relatively small. The maximum duration of the formation of Jurassic hydrothermal ore deposits in the Black Forest (Figure

The 10 km^{2} Hidden Valley hydrothermal megabreccia (Figure

A numerical model is presented to explore the effect of the relative contributions of Darcian porous flow and flow through hydrofractures on crustal-scale fluid flow. This is achieved through varying the fluid pressure diffusivity, a function of permeability, while keeping the hydrofracture initiation and fluid flux constant

When hydrofracture transport dominates, the system self-organizes. Abundant hydrofractures form at the base of the model and their size-frequency distributions are power laws. The Hurst parameter of ~0.9, calculated from the mean pressure variations over time, supports the development of a self-organized critical state. The hydrofracture size distributions show “dragon-king”-like large hydrofractures that deviate from the power-law distribution. These large hydrofractures actually drain the fluid from the system

With the increasing contribution of Darcian porous flow, pressure fluctuations become larger in magnitude and more regular and cyclical. The transitional regime to Darcian flow is thus characterised by fewer, but larger fluid expulsion events

The observed fluid transport behaviour may explain the abundance of crack-seal veins in metamorphic rocks in, for example, accretionary complexes, as well as the development of hydrothermal hydraulic breccia deposits at shallower crustal levels

All the models and data in this article are based on the equations published here and parameters described in the main text and Tables

The authors declare that there is no conflict of interest regarding the publication of this paper.

EGR acknowledges funding by the Spanish Ministry of Science, Innovation and Universities (“Ramón y Cajal” fellowship RYC2018-026335-I and research project PGC2018-093903-B-C22).

Movies of pressure field in the model box over time, for the simulations indicated in Table