# A Stochastic Analysis of the Non-Newtonian Hydraulic Behaviour of Rough Fractures

Alessandro Lenci, Yves Méheust, Mario Putti, Vittorio Di Federico

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Alessandro Lenci, Yves Méheust, Mario Putti, Vittorio Di Federico. A Stochastic Analysis of the Non-Newtonian Hydraulic Behaviour of Rough Fractures. Interpore 2021, May 2021, Online, United States. 2021, Book of abstracts. 10.1029/1888g11008464. insu-03257831

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# A STOCHASTIC ANALYSIS OF THE NON-NEWTONIAN HYDRAULIC BEHAVIOUR OF ROUGH FRACTURES

Alessandro Lenci<fnref n="1" />,<fnref n="2" />, Yves Méheust<fnref n="2" />, Mario Putti<fnref n="3" />, Vittorio Di Federico<fnref n="1" />

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alessandro.lenci@unibo.it

Session: MS 21 – (MS21) Non-linear effects in flow and transport through porous media

1

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2

## FLOW OF SHEAR-THINNING FLUIDS IN GEOLOGICAL FRACTURES

The **hydraulic behavior of geological formations** is mainly governed by **fractures connectivity** and **permeability**, whose heterogeneity strongly affects flow and transport, with fluid rheology playing an important role often oversimplified.

* Typically, unconventional and deep geothermal reservoirs present both **low-porosity** and **low-permeability**

* Activities in gas shale or hot rock requires **hydraulic stimulation** to enhance productivity and become cost effective

* Fluid viscosity is strictly linked with the activity effectiveness (e.g., proppant transport or stimulation fluid recovery).

Fluids involved in subsurface industrial activities present a **shear-thinning behaviour** at **continuum scale**, due to their complex make-up.

* muds

* foams

* water-based suspensions

**Subsurface industrial activities:**

* Enhanced Oil Recovery (EOL)

* Enhanced Geothermal Systems (EGS)

* Carbon sequestration

* Lost circulation of drilling fluids

## NUMERICAL MODELING

### 2.1 Synthetic fracture generator

A synthetic aperture field is estimated by mating two **isotropic self-affine surfaces** $h_i(x)$

$$w(x)=h_{1}(x)-h_{2}(x)+\langle w \rangle$$

A rough surface can be generated as a 2D white noises and introducing spatial correlation: multiplying the modulus of the Fourier transform by the modulus of the wave numbers

$$|k| = (k_x^2 + k_y^2)^{1/2} \text{ to the power } -1-H$$

$$|h(k)| \rightarrow |k|^{-1-H} |h(k)|$$

### 2.2 Fluid Rheology: Ellis model

The Ellis model is a **three-parameter generalized Newtonian rheology**

$$\eta = \frac{\mu_0}{1 + \left( \frac{\tau_{xz}}{\tau_{1/2}} \right)^{\frac{1}{n} - 1}}$$

$\mu_0$ low-shear rate viscosity ($\eta \rightarrow \mu_0$ for $\dot{\gamma} \rightarrow 0$)

$n$ shear-thinning index

$\tau_{1/2}$ characteristic shear stress: $\eta(\tau_{1/2})=\mu_0/2$

<figure><img src="image_5.png" /></figure>

### 2.3 Generalized Non-linear Reynolds Equation

* Lubrication theory holds ($\nabla w \ll 1$ and $Re \ll 1$)

$$-\nabla \cdot \left[ \frac{w(\mathbf{x})^3}{12\mu_0} + \frac{n}{(2n+1)} \left( \frac{1}{2^{1+n}\mu_0^n \tau_{1/2}^{1-n}} \right)^{\frac{1}{n}} w(\mathbf{x})^{\frac{2n+1}{n}} |\nabla P|^{\frac{1}{n}-1} \right] \nabla P = 0$$

Numerical modeling via finite volume method:

* The non-linear system of equations is solved via **inexact Newton-Krylov** method

* **Variable-fill-in Cholesky preconditioned conjugate gradient** → linear problem

* A **parameters continuation strategy** is adopted to handle strongly non-linear cases

3

## STOCHASTIC ANALYSIS

### Parameter variability

<figure><img src="image_6.png" /></figure>

### Fracture Geometry

<table><tr><td>σ<sub>w</sub>/⟨w⟩</td><td>0.8</td></tr><tr><td>⟨w⟩</td><td>10<sup>-3</sup> m</td></tr><tr><td>H</td><td>0.8</td></tr><tr><td>L<sub>c</sub></td><td>0.1 m</td></tr></table>

### Fluid Rheology

<table><thead><tr><td></td><td>n (-)</td><td>μ<sub>0</sub> (Pa·s)</td><td>τ<sub>1/2</sub> (Pa)</td></tr></thead><tbody><tr><td>Silicon Oil</td><td><strong>SO</strong></td><td><strong>0.61</strong></td><td>9.75</td><td>9.23</td></tr><tr><td>Xanthan Gum</td><td><strong>XG</strong></td><td><strong>0.36</strong></td><td>4.42</td><td>0.199</td></tr></tbody></table>

A **Monte Carlo Framework** is adopted to study the impact of:

* Pressure gradient ($\overline{\nabla P}$) → Flow regime

* Shear-thinning behaviour (n) → Fluid rheology

* Fracture dimension (L) → Scale

For each combination of the parameters (12 in total), the flow is solved considering a set of **1000 fracture realizations**.

Eventually, ensemble statistics are then analysed for:

* Velocity components and magnitude ($u_x, u_y, u$)

* Apparent Transmissivity (T)

4

### 4.1 Conclusion

* Velocity components displays narrow PDFs with nearly exponential decay

* High $\overline{\nabla P}$ and low n increments shear-thinning behaviour, favouring flow localization

* Qols PDFs of the smaller fracture length present higher dispersion

* Shear-thinning behaviour enhances fracture transmissivity, leading to non-darcian flow regime $\langle u \rangle \propto \overline{\nabla P}^n$

### 4.2 Perspectives

Flow localization impacts the effectiveness of any subsurface activities

* **High velocities** magnitude PDFs → **channeling metrics**

* **Low velocities** magnitude PDFs → **anomalous transport**

Mèheust, Y., Schmittbuhl, J. (2000) Flow enhancement of rough fracture. *Geophys. Res. Lett.*, 27(18), 2989-2992. doi: 10.1029/1888gl1008464

Lenci, A., Mèheust, Y., Putti, M., Di Federico, V., (2021) Shear-thinning hydraulic behaviour of rough fractures: a stochastic analysis via Monte Carlo simulations, *Water Resour. Res.* (under submission)

GEO SCIENCES
Rennes

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UNIVERSITÀ
DEGLI STUDI
DI PADOVA

Abstract ID 298

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ALMA MATER STUDIORUM
UNIVERSITÀ DI BOLOGNA

<footnote n="1">¹ Università di Bologna, Department of Civil, Chemical, Environmental and Materials Engineering, 40126 Bologna, Italy</footnote>

<footnote n="2">² Université de Rennes 1, CNRS, Géosciences Rennes UMR 6118, 35042 Rennes, France</footnote>

<footnote n="3">³ Università di Padova, Department of Mathematics, 35131 Padua, Italy</footnote>