A continuous fracture front tracking algorithm with multi layer tip elements (MuLTipEl) for a plane strain hydraulic fracture

TL;DR

This paper develops MuLTipEl, a fixed-mesh front-tracking algorithm for a plane-strain hydraulic fracture in layered media. It introduces a fictitious tip stress to keep a partially filled tip element closed while allowing the fracture front to evolve smoothly within the element, enabling accurate front tracking on coarse meshes. The method extends to viscosity and leak-off through an apparent-toughness framework and incorporates layering by computing local apparent toughness, stress differences, and leak-off as functions of the front position. Benchmarking across homogeneous and layered scenarios demonstrates mesh-independent accuracy and efficient handling of many thin layers, with significant potential for upscaling high-resolution rock-property data in hydraulic fracturing simulations.

Abstract

The problem of a plane strain hydraulic fracture propagating in a layered formation is considered. Fracture toughness, in-situ stress, and leak-off coefficient are assumed to vary by layer, while the elastic properties are kept constant throughout the domain for simplicity. The purpose of this study is to develop a numerical algorithm based on a fixed mesh approach, which is capable to solve the above problem accurately using elements which can even be larger than the layer size. In order to do this, the concept of fictitious tip stress is first introduced for determining the fracture front location. In this technique, an additional stress is applied to the tip element with the purpose to suppress opening and to mimic width corresponding to the actual fracture front location. A theoretical basis for this concept has been established and it is further calibrated for piece-wise constant elements. Once the ability to track the crack front location is developed, the effect of layers is included by vary properties as a function of front location. Several numerical examples benchmarking the numerical solution, as well as highlighting capabilities of the algorithm to tackle multiple thin layers accurately are presented.

Figures (9)

• Figure 1: Schematics of a plane strain hydraulic fracture propagating in a layered formation, in which stress, toughness, and leak-off vary with depth.
• Figure 2: Illustration of the fictitious stress concept. Panel $(a)$ schematically shows the exact solution with fracture front located within an element. Panel $(b)$ shows the fictitious solution, in which the fracture front is located at the edge of an element. The dashed red line shows the exact solution for comparison.
• Figure 3: Panel $(a)$: Variation of the dimensionless fictitious stress due to toughness versus fill ratio to the power of $3/2$ (black solid line) for $N=50$ fracture elements. The red dashed line shows the linear fit (\ref{['sigmaKfit']}). Panel $(b)$: Comparison between the exact (black lines) and numerically computed fracture width for $f=0$ (red line) and $f=1$ (magenta line) for the case of $N=5$ fully open elements and $\Delta\sigma=0$. Panel $(c)$: Variation of the dimensionless fictitious stress due to stress barrier, normalized by $f^{1/2}$, versus fill ratio to the power of $3/2$ (black solid line) for $N=50$ fracture elements. The red dashed line shows the linear fit (\ref{['sigmaSfit']}). Panel $(d)$: Comparison between the exact (black lines) and numerically computed fracture width for $f=0.2$ (red line) and $f=1$ (magenta line) for the case of $N=5$ fully open elements and $K'=0$.
• Figure 4: Schematics of the upward and downward fracture propagation. The central "main" element is highlighted by the bold black perimeter and is surrounded by the elements above and below. Layers are shown by the blue lines. Fracture front is shown by the red line.
• Figure 5: An example variation of properties within the element and its neighbours. The first three tracks show variation of toughness, stress, and leak-off versus depth. The second three tracks show the apparent toughness, stress difference, and average leak-off for the upwards fracture propagation. The last three tracks show the apparent toughness, stress difference, and average leak-off for the downwards fracture propagation.