Simulation of the hydraulic fracture process in two dimensions using a discrete element method

We introduce a discrete element simulation for the hydraulic fracture process in a petroleum well which takes into account the elastic behavior of the rock and the Mohr-Coulomb fracture criterium. The rock is modeled as an array of Voronoi polygons joined by elastic beams, which are submitted to tectonical stresses and the hydrostatic pressure of the fracturing fluid. The fluid pressure is treated like that of a hydraulic column. The simulation reproduces well the time and dimensions of real fracture processes. We also include an analysis of the fracturing fluid loss due to the permeability of the rock which is useful in an efficiency analysis of the treatment. The model is a first step for future applications in the petroleum industry.

Figure 1

A typical pressure behavior for a microhydraulic fracturing test of the in situ stress.

Figure 2

A typical set of Voronoi polygons and points. If a line is drawn from one Voronoi point to another, this line will be perpendicular to the common side of the two corresponding Voronoi polygons.

Figure 3

Here we show the action of the granular repulsive force. The vertical line goes between the two points that define the intersection of the polygons. The force is perpendicular to this line, acts on the middle point of the line, and is proportional to the overlapping area between the polygons (in gray).

Figure 4

(Top) The fracture obtained in the $20\times 20$ polygon array without tectonical stresses. (Bottom) The fracture alone.

Figure 5

Computation of the fractal dimension for the fracture perimeter in Fig. 4. The fractal dimension is the exponent of the relation between the minimum number of covering boxes $\left(N\right)$ and the size of each box $\left(s\right)$.

Figure 6

Fractal dimension versus the size of the squared array of polygons employed. The two values reported for an array of $100\times 100$ squares in Ref. 11 (1.44 and 1.39) are shown as a horizontal line for comparison.

Figure 7

(Top) The evolution of the fluid pressure, (center) the fracture area, and (bottom) the perimeter in absence of external stresses for the fracture of Fig. 4.

Figure 8

Three different fractures for tectonical stresses of $34.5\mathrm{MPa}$ along the $y$ axis and the values of (top) 34.5, (center) 69, and (bottom) $103.5\mathrm{MPa}$ for the stress along the $x$ axis.

Figure 9

(Top) The evolution of the fluid pressure, (center) the fracture area, and (bottom) the perimeter in the presence of external stresses for the fracture of Fig. 8 (bottom).

Figure 10

Pressure (in MPa) at any point of our array of polygons for a particular step of time.

Figure 11

Fracturing fluid pressure as a function of time for a simulation on an array of $25\times 25$ polygons with a constant injection rate $q=0.0069{\mathrm{m}}^3∕\mathrm{s}$ and the parameters of Table . Vertical lines divide the process into three stages.

Figure 12

(Top) Rate of lost fluid, (center) fracture fluid, and (bottom) fracture volume at the final stage (from $1600\mathrm{s}$) in log-scale showing a possible power law. $t_0=1600\mathrm{s}$ and $V_0$ is the volume at this time. The curves are from the same simulation as Fig. 11.