On the residual opening of hydraulic fractures

Abstract

Hydraulic stimulation technologies are widely applied across resource and power generation industries to increase the productivity of oil/gas or hot water reservoirs. These technologies utilise pressurised water, which is applied inside the well to initiate and drive fractures as well as to open a network of existing natural fractures. To prevent the opened fractures from complete closure during production stage, small particles (proppants) are normally injected with the pressurised fluid. These particles are subjected to confining stresses when the fluid pressure is removed, which leads to a partial closure of the stimulated fractures. The residual fracture openings are the main outcome of such hydraulic stimulations as these openings significantly affect the permeability of the reservoirs and, subsequently, the well productivity. Past research was largely focused on the assessment of conditions and characteristics of fluid driven fractures as well as proppant placement techniques. Surprisingly, not much work was devoted to the assessment of the residual fracture profiles. In this work we develop a simplified non-linear mathematical model of residual closure of a plane crack filled with deformable particles and subjected to a remote compressive stress. It is demonstrated that the closure profile is significantly influenced by the distribution and compressibility of the particles, which are often ignored in the current evaluations of well productivity.

Appendix: Validation of the computational approach

Cox and Rose (1996) work focused on the modelling of the non-linear behaviour of composite patching repair of fatigue cracks when a notch of length $b$ is present. Despite having a different nature, its mathematical formulation is very similar to the problem considered above. The analytical results presented by these authors were, therefore, adopted as the benchmark for validating the mathematical model described in Sect. 3. Their solution utilises elastic/perfectly-plastic springs to model the crack bridging patch. Therefore, changes in the formulation presented earlier are needed in order to make the comparisons. The required changes specifically concern the boundary conditions and $\sigma _n^{\prime } \left( x \right)$, being given by:

$$\begin{aligned}&\sigma _{yy} \left( {x,y} \right) \!=\!\sigma ^{\infty }\!-\!\sigma _n^{\prime } \left( x \right) , b\le \left| x \right| \le a, y\!=\!0;\end{aligned}$$

(38a)

$$\begin{aligned}&\sigma _n^{\prime } \left( x \right) =\bar{{E}}k\delta \left( x \right) , \delta \left( x \right) <\delta _p ;\end{aligned}$$

(38b)

$$\begin{aligned}&\sigma _n^{\prime } \left( x \right) =\sigma _p =\bar{{E}}k\delta _p ,\delta \left( x \right) \ge \delta _p ; \end{aligned}$$

(38c)

where $k$ is a constant characterising the spring stiffness in the linear range, $\sigma _p$ is the yield stress, and $\delta _p$ is a characteristic crack opening beyond which the spring response changes from being elastic to being perfectly plastic, see Cox and Rose (1996). Additionally, the initial stress intensity factor $K_0$ and the initial crack opening $\delta _0$ set at zero.

Fig. 6

Normalised stress intensity factor $K_N$ development for both elastic and elastic/perfectly-plastic cases. Situations (a) with zero notch length $\left( {B=0} \right)$ and (b) with moderate notch length $\left( {B=2} \right)$ are shown

The formulae presented by Cox and Rose (1996) for predicting the stress intensity factor $K$ and the crack opening $\delta \left( x \right)$ are, respectively, as follow:

$$\begin{aligned}&K=\sigma ^{\infty }\sqrt{\pi a}-2\sqrt{\frac{a}{\pi }} \int \limits _b^a \frac{\sigma _n^{\prime } \left( x \right) }{\sqrt{a^{2}-x^{2}}}\text{ d }x,\end{aligned}$$

(39)

$$\begin{aligned}&\delta \left( x \right) =4\frac{\sigma ^{\infty }}{\bar{{E}}}\sqrt{a^{2}-x^{2}}\nonumber \\&\qquad \qquad -\frac{4}{\pi \bar{{E}}} \int \limits _b^a \text{ ln } \left| \frac{\sqrt{a^{2}-\xi ^{2}}+\sqrt{a^{2}-x^{2}}}{\sqrt{a^{2}-\xi ^{2}}-\sqrt{a^{2}-x^{2}}} \right| \sigma _n^{\prime } \left( \xi \right) \text{ d }\xi . \nonumber \\ \end{aligned}$$

(40)

These formulae can be rewritten in a numerical fashion and solved numerically.

A concise description of the employed algebraic manipulations and obtained results can be achieved by the adoption of the following normalisations:

$$\begin{aligned} A&= \frac{4ka}{\pi },\end{aligned}$$

(41a)

$$\begin{aligned} B&= \frac{4kb}{\pi },\end{aligned}$$

(41b)

$$\begin{aligned} K_N&= \frac{K\sqrt{k}}{\sigma _p }. \end{aligned}$$

(41c)

$A$ being the normalised crack length, $B$ the normalised notch length and $K_N$ the normalised stress intensity factor.

The development of the normalised stress intensity factor $K_N$ for various normalised fracture lengths $A$ as predicted by the above developed formulation, for a varying $N$, is compared against Cox and Rose (1996) approach (Eqs. 39–40). Such comparison is presented in Fig. 6, which provides a succinct overview of the solutions obtained.

The comparison presented in Fig. 6 demonstrates a good agreement between the numerical approach derived above and the results presented by Cox and Rose (1996). The difference totally disappears with an increase of the number of integration points in the numerical solution.