Simple models of the hydrofracture process

Hydrofracturing to recover natural gas and oil relies on the creation of a fracture network with pressurized water. We analyze the creation of the network in two ways. First, we assemble a collection of analytical estimates for pressure-driven crack motion in simple geometries, including crack speed as a function of length, energy dissipated by fluid viscosity and used to break rock, and the conditions under which a second crack will initiate while a first is running. We develop a pseudo-three-dimensional numerical model that couples fluid motion with solid mechanics and can generate branching crack structures not specified in advance. One of our main conclusions is that the typical spacing between fractures must be on the order of a meter, and this conclusion arises in two separate ways. First, it arises from analysis of gas production rates, given the diffusion constants for gas in the rock. Second, it arises from the number of fractures that should be generated given the scale of the affected region and the amounts of water pumped into the rock.

Figure 1

Schematic geometry of horizontal well.

Figure 2

Histogram showing the number of wells with different effective spacings $d,$ estimated from production history in Barnett Shale.

Figure 3

Geometry first employed by Perkins and Kern [4] to study fluid-driven fractures.

Figure 4

Geometry of lubrication approximation.

Figure 5

Structure of lattice model of hydrofracture. The lattice spacing is $a$. Mass points are located at points $u_{\vec{R}}$. When the distance between mass points exceeds a critical value (the increase is small compared to the lattice spacing) bonds break, channels open up between them, and their width is $w_{ij}^{x,y}$. Channels are indicated by solid lines.

Figure 6

(a) Crack geometry. There is initially a horizontal channel and a vertical seed crack 1m high. The system is 5 m wide with stress-free boundaries. The dot indicates the fluid injection point. (b) Pressure versus time, theory, and simulation. The simulation was carried out with $H=10$ m, $Q=18.5$ bbl/min, and $Y/(1-{\nu }^2)=50$ GPa.

Figure 7

Comparison of simulation results for pressure with the analytical expression Eq. (7) and a test of the relation between $P$ and $w$ given in Eq. (6).

Figure 8

(a) Configuration with two seed cracks, each 1 m long, on opposing sides of a well. The dot shows the location of water injection. (b) Pressure versus time showing pressure drops corresponding to initiation of the two cracks.

Figure 9

Two cracks are placed at perpendicular distance $d$ from each other. The leftmost one begins to grow. If the rightmost one were very far away it would begin to grow at time $t_{\infty }$, but because of shielding from the left it takes longer. The solid line shows the estimate Eq. (44) for $L_0=2$ m. The solid squares show results of numerical simulations.

Figure 10

Fracture network created by two minutes of pumping water at the rate of 37.5 barrels per minute into a region of size $250\times 20$ m. Blue regions are cracks filled with water. The vertical tube represents an initial long crack into which water is injected at one point. The initial condition also has two horizontal seed cracks of 4 m in length, centered on the vertical crack. A net compressive stress of 10 MPa has been applied at three different angles in relation to the horizontal axis.

Figure 11

Total area of channel network created by systematic variation of the magnitude and angle of stress anisotropy in hydraulic fracture simulation. The area is given by multiplying the total length of the channels (in meters) by a height of 10 m. The fractures are created by pumping 37.5 barrels per minute for two minutes.