Hydraulic fracturing and its peculiarities
© Secchi and Schrefler; licensee Springer. 2014
Received: 15 November 2013
Accepted: 21 February 2014
Published: 16 May 2014
Simulation of pressure-induced fracture in two-dimensional (2D) and three-dimensional (3D) fully saturated porous media is presented together with some peculiar features.
A cohesive fracture model is adopted together with a discrete crack and without predetermined fracture path. The fracture is filled with interface elements which in the 2D case are quadrangular and triangular elements and in the 3D case are either tetrahedral or wedge elements. The Rankine criterion is used for fracture nucleation and advancement. In a 2D setting the fracture follows directly the direction normal to the maximum principal stress while in the 3D case the fracture follows the face of the element around the fracture tip closest to the normal direction of the maximum principal stress at the tip. The procedure requires continuous updating of the mesh around the crack tip to take into account the evolving geometry. The updated mesh is obtained by means of an efficient mesh generator based on Delaunay tessellation. The governing equations are written in the framework of porous media mechanics and are solved numerically in a fully coupled manner.
Numerical examples dealing with well injection (constant inflow) in a geological setting and hydraulic fracture in 2D and 3D concrete dams (increasing pressure) conclude the paper. A counter-example involving thermomecanically driven fracture, also a coupled problem, is included as well.
The examples highlight some peculiar features of hydraulic fracture propagation. In particular the adopted method is able to capture the hints of Self-Organized Criticality featured by hydraulic fracturing.
Fluid-driven fracture propagating in porous media is widely used in geomechanics to improve the permeability of reservoirs in oil and gas recovery or of geothermal wells. Another application of importance is related to the overtopping stability analysis of dams. In the case of reservoir engineering, water is forced under high pressure deep into the ground by injection into a well. The fluid, usually mixed with sand and some chemicals, penetrates in the reservoir rock, opening long cracks (fracking). Horizontal drilling together with hydraulic fracturing makes the extraction of tightly bound natural gas from shale formations economically feasible . In the field, it is unfortunately rather difficult to obtain direct information about the evolution of the crack in the ground, and very little data are known or accessible. Two types of measurements are mainly performed: monitoring of pressure fluctuations at the injection pump and registration of acoustic emissions at the surface . Fracking can also induce small earthquakes . Pressure-induced fracture propagation presents some peculiar features such as pressure peaks and stepwise advancement, which have been discovered only recently and need further investigation. It is recalled that differently from tensile experiments where the crack surfaces are stress free, in hydraulic fracturing, these surfaces are loaded by a pressure distribution resulting from the invading fluid or gas . Simulation is an extremely useful tool to obtain more insight into the problem. The paper addresses this issue.
Contributions to the mathematical modelling of fluid-driven fractures have been made continuously since the 1960s, beginning with Perkins and Kern . These authors made various simplifying assumptions, for instance, regarding fluid flow, fracture shape and velocity leakage from the fracture. For other analytical solutions in the frame of linear fracture mechanics, assuming the problem to be stationary, see [5–9]. They suffer the limits of an analytical approach, in particular the inability to represent an evolutionary problem in a domain with a real complexity. An analysis of solid and fluid behaviour near the crack tip can be found in [10, 11]. Boone and Ingraffea  present a numerical model in the context of linear fracture mechanics which allows for fluid leakage in the medium surrounding the fracture and assumes a moving crack depending on the applied loads and material properties. Tzschichholz and Herrmann  used a two-dimensional (2D) lattice model for constant injection rate and homogeneous and heterogeneous material which only breaks under tension. Carter et al.  show a fully three-dimensional (3D) hydraulic fracture model which neglects the fluid continuity equation in the medium surrounding the fracture. A discrete fracture approach with remeshing in an unstructured mesh and automatic mesh refinement is used by Schrefler et al. . An element threshold number (number of elements over the cohesive zone) was identified to obtain mesh-independent results. This approach has been extended to 3D situation in . Extended finite elements (XFEM) have been applied to hydraulic fracturing in a partially saturated porous medium by Réthoré et al.  in a 2D setting. In this case, a two-scale model has been developed for the fluid flow: in the cohesive crack, Darcy's equation is used for flow in a porous medium, and identities are derived that couple the local momentum and mass balances to the governing equations for the unsaturated medium at macroscopic level. As an example, the rupture of a saturated square plate (0.25 × 0.25 m) in plane strain conditions is investigated under a prescribed fixed vertical velocity v = 2.35 × 10-2 μm/s in the opposite direction at the top and bottom of the plate (tensile loading). The mesh used consists of 20 × 20 quadrilateral elements (12.5 × 12.5 mm each) with bilinear shape functions, and the time step size is 1 s. In the cracked region, the elements are further divided in four triangles. Mohammadnejad and Khoei  solve the same problem also with XFEM, using full two-phase flow throughout the region. Darcy flow is assumed within the crack. Finer meshes are used as above (smallest element size 4.5 × 4.5 mm) and much lower time steps (0.25 to 0.125 s). Cavitation is found in both papers, also due to the impervious boundary conditions chosen. Partition of unity finite elements (PUFEM) are used for 2D mode I crack propagation in saturated ionized porous media by Kraaijeveldt et al. . A pull test, a delamination test and an osmopolarity test are simulated with rather fine regular meshes (quadrangular elements with side length of the order of 2 mm and lower) and time step size down to 0.1 s. The time and space discretizations, including the element threshold number used for the solutions, are extremely important for catching the phenomena described next.
We address now the peculiar behaviour of hydraulic fracture propagation which has been observed only by a minority of the above-mentioned authors, but has been confirmed experimentally. Tzschichholz and Herrmann  have evidenced with their lattice model and constant injection rate a drop in pressure in time and oscillations on short time scales. These authors explain this by the fact that at the beginning high pressures are needed to push the fluid into the crack. The crack is enlarged and the pressure drops because the enlarged crack can now be opened much more easily than before. The pressure goes down although additional fluid has been added to the crack in the time step. If the pressure drops too much, the stresses at the crack tip fall below their cohesion value and the crack cannot grow at the next time step. By injecting more fluid into the crack, the pressure increases linearly in time until the cohesion forces can be overcome again. Using arguments from continuum mechanics, the authors show that the obtained value for pressure decline in the long term agrees acceptably with their numerical results. The short-term deviations are due the lattice model and the ensuing pressure drops. Oscillations are also obtained for the stored lattice energy. The breaking process is discontinuous in time with time intervals of quiescence where all beams on the crack surface are stressed below their cohesion thresholds and the acting pressure increases linearly in time. Tzschichholz and Herrmann  also find a temporal clustering of the breaking events, calling such a sequence bursts (avalanche behaviour). The bursts are unevenly distributed in time and occur relatively often for small times and become rarer later. There is resemblance between the obtained data and magnitude records of earthquakes or of acoustic emission records from laboratory experiments. We have shown with our porous media mechanics model in a 2D setting  that in the case of hydraulic fracturing the fracture advances stepwise. Two types of mesh refinement in space and refinement in time were used, but the stepwise advancement did not disappear. Such steps do not appear in other coupled solutions involving cohesive fracture, as e.g. the thermo-elastic one of  where the crack surfaces are stress free. The stepwise advancement and flow jumps were also found by Kraaijeveld  with a strong and a weak discontinuity model for flow. In , the stepwise advancement in mode I crack propagation is difficult to see because a continuous pressure profile across the crack is used. However, continuous pressure profile only works for sufficiently fine meshes. If the mesh is sufficiently fine, then the discretization can resolve the steep pressure gradients along the crack, but the advantage of PUFEM which allows keeping the mesh pretty rough all over the continuum is lost. Hence, dealing with the stepwise progression of the crack in this mode I model is only possible with a finer mesh than the one used (JM Huyghe, personal communication). This is why the authors state that the physical phenomenon challenges the numerical scheme. In mode II, as shown in [20, 21], this problem does not appear because a discontinuous pressure across the crack is accounted for. There it is not attempted to resolve the steep pressure gradient, but this gradient is reconstructed afterwards, using the Terzaghi analytical solution for pressure diffusion. This two-step procedure allows using a rough mesh and still handling a realistic pressure gradient. Stepwise crack advancement can clearly be observed in the crack length histories of Figure four of , while it does not appear in the solution for the same problem in . The cohesive fracture length for this problem is estimated with Barenblatt's expression (see Equation 22)  to be about 136 mm. Hence, there are about 10 elements over the cohesive zone in  and 30 elements in . The first value is probably below the element threshold number for this type of problem, even with XFEM (see also the large time steps used), while the second one is sufficient even for standard elements. While both use XFEM, the two-step procedure and the large time step size and coarse mesh in  hide the problem.
Finally, stepwise advancement and flow are also mentioned in , where PUFEM is used for 2D poroelastic media. Their method still suffers from mesh dependence because the crack propagates through one element each time step. Hence, their conclusions are not definite. However, Pizzocolo et al.  confirmed stepwise advancement experimentally with a test on a small hydrogel disk. The duration of the pause Δt between steps is found to be inversely related to the hydraulic permeability K according to Δt = Δx2 / KE with E Young's modulus and Δx length of the step. A possible explanation for the stepwise behaviour observed in [20, 21, 24] put forward in  is that an incompressible fluid consolidation comes into play which prevents tip advancement until the overpressures due to the last advancement have been dissipated, and the stress has been transferred again to the solid phase. This implies the existence of pressure peaks after each advancement stage. During the period of quiescence, the effective stress is below the breaking threshold. Consolidation as a possible explanation for the stepwise advancement needs further investigation in the case of fluid injection, because for some permeability values the tip pressure goes down to zero as shown below on an example. The existence of periods of quiescence is in line with the findings of . We will show that this phenomenon is not only relevant for small structures, where it has been observed experimentally, but also for large structures such as underground soil masses and dams. In that case, the fracture length is much larger, but the phenomenon is still there and the bursts can be felt at great distances compared to the crack length.
The first subsection presents the fracture model, the second subsection summarizes the governing equations and their numerical solution by means of the finite element method and the third subsection explains the adopted fracture advancement procedure and the required refinements necessary to obtain mesh-independent results.
The fracture model
τ0 being the maximum tangential stress (closed crack), δ τ the relative displacement parallel to the crack and δσ cr the limiting value opening for stress transmission. The unloading/loading from/to some opening δσ 1 < δσ cr follows the same behaviour as for mode I.
β being a suitable material parameter that defines the ratio between the shear and the normal critical components. For more details, see .
Governing equations and their discretization in space and time
where c is the cohesive traction on the process zone as defined above.
μ w is the dynamic viscosity and K f the bulk modulus of the fluid. The last term of (8) represents the leakage flux into the surrounding porous medium across the fracture borders and is of paramount importance in hydraulic fracturing techniques. This term can be represented by means of Darcy's law using the medium permeability and pressure gradient generated by the application of water pressure on the fracture lips. No particular simplifying hypotheses are hence necessary for this term. This equation can be directly assembled at the same stage as the mass balance Equation 11 for the saturated medium surrounding the crack, because both have the same structure: only the parameters have to be changed in the appropriate elements depending whether they belong to the fracture or to the surrounding medium.
where q w represents the water leakage flux along the fracture toward the surrounding medium of Equation 7. This term is defined along the entire fracture, i.e. the open part and the process zone. It is worth mentioning that the topology of the domains Ω and changes with the evolution of the fracture. In particular, the fracture path, the position of the process zone and the cohesive forces are unknown and must be regarded as products of the mechanical analysis.
The nodal displacement, velocity and pressure, , and , respectively, for the current step coincide with the unknowns at the end of the previous one, hence are known in the time marching scheme and coincide with the initial condition for the first time step. The system of algebraic equations is solved with a monolithic approach using an optimized non-symmetric sparse matrix algorithm. The number of unknowns is doubled with respect to the traditional trapezoidal method.
where is the cohesive traction rate and is different from zero only if the element has a side on the lips of the fracture Γ ′. Given that the liquid phase is continuous over the whole domain, leakage flux along the opened fracture lips is accounted for through the H matrix together with the flux along the crack. Finite elements are in fact present along the crack (not shown in Figure 1), which account only for the pressure field and have no mechanical stiffness. In the present formulation, non-linear terms arise through cohesive forces in the process zone and permeability along the fracture.
Fracture advancement and refinement strategy
Because of the continuous variation of the domain as a consequence of the propagation of the cracks, also the boundary Γ′ and the related mechanical conditions change. Along the formed crack edges and in the process zone, boundary conditions are the direct result of the field equations, while the mechanical parameters have to be updated. The following remeshing techniques account for all these changes [15, 32].
For the fracture nucleation and advancement, the Rankine criterion is used. More than one crack can open and fractures can also branch. Fracture forms and advances if the maximum principal stress exceeds in a point the fixed threshold. The fracture advancement procedure differs for 2D and 3D situations: in 2D, the fracture follows directly the direction normal to the maximum principal stress, while in 3D, the fracture follows the face of the element around the fracture tip which is closest to the normal direction of the maximum principal stress. In this last case, the fracture tip becomes a curve in space (front). The advancement of a fracture creates new nodes: in 2D, the resulting new elements for the filler at the front are triangles, while in 3D situation, they are tetrahedral. If an internal node along the process zone advances in a 3D setting, a new wedge element results in the filler .
Error measures defined in Equation 19 account at the same time for the cross effects among the different fields and the ones between space and time discretizations.
where θ < 1.0 is a safety factor. If the error is smaller than a defined value ηtoll,min, the step is increased using a rule similar to Equation 21.
As it stands, the refinements in space and time are carried out sequentially, starting with the space refinement, followed by the element threshold number and then the refinement in time. An eye is kept on the satisfaction of the discrete maximum principle  which states that it is not possible to refine in time below a certain limit depending on the material properties without also refining in space. A proper functional would be needed to link all the three refinements. A flow chart of the numerical procedure is given in the ‘Appendix’.
Results and discussion
Two different experiments are reported in . In the first, at a room temperature of 25°C, a load was applied 3 mm from the interface in the PMMA zone (Figure 3) to trigger the fracture process. The loading rate was very low and the resulting speed of crack propagation at the initial stages was also quite slow, so that quasistatic conditions can be assumed. The crack path was individuated, and stresses near the crack tip in the PMMA were measured using a shearing interferometer.
In the second experiment, the same operations were performed when the temperature of the aluminium was 60°C in steady state conditions. To reach these conditions, a cartridge heater (Q in Figure 3) was inserted into the aluminium part near the external vertical side. The variation in time of the PMMA temperature was checked before the fracture test, which was performed when steady state conditions were reached. The temperature of PMMA was recorded at the crack tip location, at 5 and 7 mm from the interface. Also in this case, the crack path was spotted. From the differences between the two situations, the authors gathered the thermal effects, which were independent of the magnitude of the applied mechanical load.
The initial condition is obtained under self-weight and the hydrostatic pressure due to water in the reservoir up to a level of 52 m. From this point, the water level increases until the overtopping level is reached (higher than the dam crest ). The increase of water level in the reservoir is specified in days according to the benchmark.
In all three examples of hydraulic fracturing, the fracture site is not easily accessible. However, the fact that the effects of the stepwise advancement can be felt also at distance as shown in two examples would make them possible to be monitored remotely. Field data and experimental evidence on reservoir rocks and large bodies are still missing. Data could possibly come from fracking sites or from some fracking-induced earthquakes .
A fully coupled model for pressure-induced cohesive fracture in a saturated porous medium and its solution by the finite element method has been shown. The model is of the discrete crack type and requires continuous updating of the mesh as the crack tip advances. This is achieved with powerful mesh generators. Three types of refinement are necessary to obtain mesh-independent results: a refinement over the domain of the Zienkiewicz-Zhu type, an element threshold number over the process zone and a refinement in time, here with DGT. The results show that in case of pressure induced fracture with pressure exchange and flow between the fracture and the surrounding medium the crack tip advances stepwise. This was found also by few other authors. Smooth diagrams are found on the contrary in a thermo-elastic fracture which is a coupled problem but with stress-free fracture surfaces. From a comparison of results obtained with different methods by other authors, it appears that in some situations a particular adopted method hides the problems discussed in this paper because the required refinements clash with the raison d'être of such methods like XFEM or PUFEM (adoption of rough meshes). This is the reason why ‘the physical phenomenon challenges the numerical scheme’  and why several authors dealing with hydraulic fracturing have not noticed the peculiar behaviour shown here. Also, two-step procedures may introduce some bias in the solution. Two different explanations are found in the literature for the discussed phenomena: one invokes pressure drop  and the other pressure peak  after crack advancement. The respective loading conditions are different, but the question deserves further scrutiny. Finally, the stepwise advancement may be relevant for earthquake engineering, see e.g. the resemblance between the obtained data of  and magnitude records of earthquakes. In many earthquake-prone regions, there is plenty of water available at the level where the rupture takes place [38, 39]. The problem solved in  has been solved again with XFEM and finer mesh in  and the steps in the fracture advancement featured in  disappeared. This implies that XFEM yields a smooth solution for a phenomenon which in nature is not smooth: as shown in  hydraulic fracturing exhibits avalanche behaviour and hints of Self-Organized Criticality.
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