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Fluid Injection in Deformable Geological Formations pp 553–680Cite as

Cracks in Saturated Porous Media: Desiccation Cracks, Hydraulic Fracturing, and Microseismicity

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Abstract

Desiccation of fluid-infiltrated porous media is a highly non-homogeneous process that in most cases is bound to generate cracks. Under sufficiently slow crack propagation in an elastic fluid-saturated porous medium, the tip of a steadily running crack is drained. On the other hand, the tip of steadily running cracks that propagate fast enough in regard to the local diffusivity and length scale is not drained. In fact, the pressure there is negative and large. During soil and gel desiccation, the suction increases in time close to the evaporation surface. The developments of cracks due to desiccation, on one side, and hydraulic fracturing, on another side, are both considered in fluid-saturated porous solids. While elements of fracture mechanics of brittle elastic solids with and without cohesive zone are exposed, focus is on the enhancement of geothermal reservoirs via hydraulic fracturing, rather than on petroleum engineering. The microseismic effects resulting from fluid injection into, or extraction from, rock formations during waste water disposal, geothermal operations, and carbon dioxide sequestration are reviewed. Simplified models defining the motions of the free surface resulting from these operations are developed using the Mindlin solution for a half-space and illustrated.

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Notes

  1. 1.

    As the main focus in Sects. 8.1 and 8.2 of this chapter is on soils, emphasis is laid on the Terzaghi’s effective stress. The Biot’s effective stress for unsaturated rocks is obtained by introducing the Biot’s coefficient \(\kappa \), namely along Loret and Khalili (2000):

    figure a
  2. 2.

    The relations below are worth recording:

    $$\frac{\cos \phi }{1+\sin \phi } =\frac{1-\sin \phi }{\cos \phi } =\tan \left( \frac{\pi }{4}-\frac{\phi }{2}\right) , \quad \frac{1-\sin \phi }{1+\sin \phi }=\tan ^2\left( \frac{\pi }{4}-\frac{\phi }{2}\right) .$$
  3. 3.

    In fact, the Coulomb criterion applies in arbitrary directions, that is, on arbitrary planes of normal direction \(\mathbf{n}=(n_1,n_2,n_3)\). Component representation is easier when the working axes are the principal directions of the stress (or equivalently effective stress). The effective traction \( {\varvec{\sigma }}'\cdot \mathbf{n}=\sigma '\,\mathbf{n} + \tau \,\mathbf{n}^\perp \) is uniquely decomposed along the normal \(\mathbf{n}\) and along an orthogonal direction \(\mathbf{n}^\perp \) that is defined by this relation (\(\mathbf{n}\) and \(\mathbf{n}^\perp \) are unit vectors). Then, \(\sigma '=\mathbf{n}\cdot {\varvec{\sigma }}'\cdot \mathbf{n}\), \(\tau =\mathbf{n}^\perp \cdot {\varvec{\sigma }}'\cdot \mathbf{n}\), \(( {\varvec{\sigma }}'\cdot \mathbf{n})^2=(\sigma ')^2+\tau ^2\), and componentwise,

    figure b

    The shear stress \(\tau \) is by definition insensitive to isotropic variations of the stress.

  4. 4.

    The failure pressure is equal to \((1+\sin \phi )/(1+(2\,\kappa -1)\,\sin \phi )\,u_{ts}\ge u_{ts}\) according to the Coulomb criterion.

  5. 5.

    For a curve \(\mathbf{U}(\theta )\) parameterized by the scalar \(\theta \), the radius of curvature \(\rho (\theta )\) is equal to \(|\mathbf{U}'|^3/|\mathbf{U}'\wedge \mathbf{U}'' |\) with \((\cdot )'=d(\cdot )/d\theta \). For an ellipse \(\mathbf{U}=(r_{ix}\,\cos \theta ,r_{iy}\,\sin \theta )\), the radius of curvature \(\rho (\theta )\) is equal to \((r_{ix}^3\,\sin ^3\theta +r_{iy}^3\,\cos ^3\theta ))/(r_{ix}\,r_{iy})\), that is, to \(r_{iy}^2/r_{ix}\) at \(\theta =0^{\circ }\).

  6. 6.

    This analysis was an exercise taught to the students of the Ecole Nationale des Techniques Avancées (School for Ocean Engineering), Paris, France, as I was a teaching assistant of Professor André Zaoui.

  7. 7.
    figure c
  8. 8.

    The integrals \(\displaystyle \int _0^s\,\frac{du}{\sqrt{\ell ^2-u^2}}=\arcsin \Big (\frac{s}{\ell }\Big )\) and \(\displaystyle \int _s^{\ell }\,\frac{du}{\sqrt{u^2-s^2}}=\mathrm{arccosh}\Bigg (\frac{\ell }{s}\Bigg )\) for \(s<\ell \) are worth recording.

  9. 9.

    The inversion of the order of integration in the second line of (8.5.12) can be checked by drawing two sketches in the axes \((\xi ,s)\) showing that the domains of integration are identical to the first line.

  10. 10.

    The basic tool of an analytical treatment of fracture mechanics is the complex analysis. A sufficient introduction is found in, e.g. Kanninen and Popelar (1985). The starting point is to define two complex potentials \(\phi \) and \(\psi \) of the complex variable \(z=x+i\,y\) that are analytical except on the crack line \(x>0\), \(y=0\). The displacement components \(u_x\) and \(u_y\),

    figure j

    and stress components,

    figure k

    derive from these potentials. Actually, attention is restricted to the x-axis and to a close neighborhood of the crack tip. A solution with a vanishing shear stress is obtained by setting \(\psi (z)=\phi (z)-z\,\phi '(z)\) and requiring \(\phi \) to be a real function. The second equation (8.5.37) becomes elusive. To leading order, \(\phi (z)\) is sought as a monomial\(\frac{A}{2\,\alpha }\)\(\,z^\alpha \) with A real. Equation (8.5.36) then provides the displacement discontinuity (aperture) across the crack axis \(z=|x|\,e^{\pm i\,\pi }\), and the first Equation (8.5.36) yields the pressure.

  11. 11.

    The earthquakes of the largest magnitude are expected to take place along the boundaries of the continental plates. Earthquakes along intra-continental faults like the San Andreas Fault in California and the North Anatolian Fault in Turkey do not reach magnitudes larger than 8, but they are dangerous because they concern densely populated areas and their hypocenter is located at shallow depth. Late 1811 and early 1812, several earthquakes struck the New Madrid area, at the boundary between Arkansas and Missouri, with a magnitude estimated to 7.5–7.7.

  12. 12.

    The Gutenberg–Richter rule states that the decimal logarithm of the number of events N per unit time with magnitude larger than M is, in a given volume, an affine function of the latter, \(\log ( N\ \mathrm{of\ magnitude}\, > M ) = a - b\,M\). The parameter a depends on the volume of interest, while the parameter b indicates the influence of the magnitude: values larger than 1 are associated with a large number of small events.

  13. 13.

    The seismographic network of the Lamont Doherty Earth Observatory (LDEO) records earthquakes of magnitude 2 and above over the Marcellus Shale of the Appalachian Basin. The UK network for the North Sea has threshold 2 and did not detect events from the Sleipner field. A magnitude 1 event has been recorded from the In Salah test field.

  14. 14.

    Indeed, consider the diffusion equation \(\partial u/\partial t \ -\ D\ \partial ^2 u/\partial x^2=0\) for the unknown function \(u=u(x,t)\), with x space, t time, and \(D>0\) [m\(^2\)/s] diffusivity. Scaling the space by the length L implies the scaled time \(t\,D/L^2\). The diffusion front reaches the position \(x=L\) at a characteristic time \(t=L^2/D\). Conversely, at time t, the diffusion front is located at the characteristic position \(x=\sqrt{D\,t}\). A number of field examples are documented in Shapiro (2015).

  15. 15.

    Their analysis of fluid diffusion is curious: their diffusivity defined by their equations 10 and 11 would tend to infinity for a soil. In contrast, the hydraulic diffusivity of the rock, Equation (9.5.8) in Loret and Simões (2016), tends to the value \(2\,G\,(1-\nu )/(1-2\,\nu )\times k_\mathrm{in}/\eta _\mathrm{w}\), Eq. (7.5.17) in LS, with \(G\) shear modulus, \(\nu \) drained Poisson’s ratio, \(k_\mathrm{in}\) permeability, \(\eta _\mathrm{w}\) dynamic viscosity of water that applies to a porous medium with incompressible water and solid grains. For a soil, the undrained Poisson’s ratio tends to 1/2, Eq. (7.4.44) in LS.

  16. 16.

    Quite generally, the diffusions of temperature and fluid are endowed with distinct time scales, and a thermal recorder and a pressure recorder placed inside the rock formation would take note of their respective information at different times. These time scales depend on the constitutive parameters. Often, the fluid flow is faster and the thermal signal is late. However, the thermal signal can trigger a pore pressure raise only if the two time scales are not too far apart. Therefore, an attempt to explain the events after shut-in as induced by thermomechanical coupling associated with this slower thermal diffusion is not convincing.

  17. 17.

    It might be worth to provide the rationale behind this procedure which has been extensively used in the modeling of heterogeneous engineering materials and living tissues. Indeed, the inclusion \(\Omega \) is cut from the medium, subjected to the eigenstrain, replaced at its original position and glued to the medium. The strain which is imposed to the inclusion is viewed as an eigenstrain by requesting (8.7.49). The formalism used here considers linear elasticity so that superimposing the initial geostatic stresses is not an issue. In the presence of nonlinear constitutive responses, the inclusion analysis should be viewed as incremental.

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Loret, B. (2019). Cracks in Saturated Porous Media: Desiccation Cracks, Hydraulic Fracturing, and Microseismicity. In: Fluid Injection in Deformable Geological Formations. Springer, Cham. https://doi.org/10.1007/978-3-319-94217-9_8

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