In order to better understand the interaction between pore-fluid overpressure and failure patterns in rocks we consider a porous elasto-plastic medium in which a laterally localized overpressure line source is imposed at depth below the free surface. We solve numerically the fluid filtration equation coupled to the gravitational force balance and poro-elasto-plastic rheology equations. Systematic numerical simulations, varying initial stress, intrinsic material properties and geometry, show the existence of five distinct failure patterns caused by either shear banding or tensile fracturing. The value of the critical pore-fluid overpressure pc at the onset of failure is derived from an analytical solution that is in excellent agreement with numerical simulations. Finally, we construct a phase-diagram that predicts the domains of the different failure patterns and pc at the onset of failure.
 The effect of the homogeneous pore-pressure increase on the strength of crustal rocks and failure modes has been studied by many authors [e.g., Terzaghi, 1943; Skempton, 1961; Paterson and Wong, 2005]. Their results show that, provided that the rocks contain a connected system of pores, failure is controlled by the Terzaghi's effective stress defined as
where σij is the total stress; p is the pore fluid pressure, and δij is the Kronecker delta (by convention, compressive stress is positive).
 However, many geological systems, such as magmatic dykes, mud volcanoes, hydrothermal vents, or fluid in faults, show evidence that pore pressure increase might be localized, instead of being homogeneously distributed [Jamtveit et al., 2004]. Localized pore-pressure variations couple pore-fluid diffusion to rock deformation through the seepage force generated by pressure gradients [Rice and Cleary, 1976]. The seepage force introduces localized perturbation of the effective stress field and may promote various failure patterns. The effect of seepage forces caused by laterally homogeneous pore-pressure increase on failure patterns was recently studied experimentally by Mourgues and Cobbold . In the present study, we explore both numerically and analytically how an essentially two-dimensional, i.e. localized both at depth and laterally, increase in pore-pressure affects failure patterns in porous elasto-plastic rocks. In section 2, we discuss the effect of localized pore pressure increase on tensile and shear failure. Section 3 is devoted to the characterization of the various failure patterns using finite element and finite difference simulations that solve the gravitational force balance equation and the fluid filtration equation in a poro-elasto-plastic medium. In section 4, we predict the fluid pressure at the onset of failure using new analytical solution. Finally we discuss the geological implications in section 5.
2. Effect of Pore Pressure on Rock Failure
 In nature, rock failure occurs in two different modes: shear bands and tensile fractures. Laboratory triaxial experiments show that the Mohr-Coulomb criterion (equation 2) provides an accurate prediction for shear failure [Paterson and Wong, 2005]:
where τ = is the stress deviator, σ′m = − p is the mean effective stress, C is the rock cohesion and ϕ is the internal friction angle.
 On the other hand, Griffith's theory provides a theoretical criterion for tensile failure of a fluid-filled crack [Murrell, 1964]:
where σT is the tensile strength of the rock. This criterion has also been verified experimentally [Jaeger, 1963].
 In Figure 1, we show how a homogeneous or localized increase of pore-fluid pressure influences rock failure. There, lm is the Mohr-Coulomb envelope (equation 2) and kl is the tensile cut-off limit (equation 3). The Mohr circle indicates the initial state of stress, with zero pore-fluid overpressure. As the pore-fluid pressure increases homogeneously by an amount p, the radius of the Mohr circle remains constant and the circle is displaced to the left until it touches the failure envelope (blue curve). Depending on the location where the circle touches the failure envelope, the formation of shear bands or tensile fractures takes place. In both cases, when pore-fluid pressure increase is homogeneous, the orientation and onset of failure patterns can be predicted [Paterson and Wong, 2005]. Shear bands form at an angle of to the direction of maximum compressive stress; tensile fractures develop perpendicularly to the direction of maximum tensile stress.
 We explore a more complex scenario, where the pore-fluid increase is localized into a narrow source, so that seepage forces modify locally the stress-state. As shown on Figures 1c and 1d, the radius of the initial Mohr circle does not remain constant, as for the homogeneous pore-fluid pressure increase case. For a localized fluid pressure increase equal to p, the radius of the Mohr circle is changed by an amount of βτp, and the center of the circle is displaced to the left by an amount of βσp. The two dimensionless parameters βτ and βσare derived below both by numerical and analytical means and given by (equations 14–15).
3. Numerical Model and Numerical Results
 We consider a 2D porous medium embedded in a box of length L and height h ≪ L (Figure 2a). At the bottom of this box, we define a pore-fluid over-pressure source of width w ≪ h. We consider plane-strain deformation in a material with constant and homogeneous intrinsic properties. We solve numerically the fluid filtration equation and the force balance equation, using a poro-elasto-plastic rheology relationship between the stress and the strain rates. At initial conditions the deformation of the material and the pore-fluid overpressure are equal to zero everywhere in the system. The initial mechanical state is chosen below the failure limits everywhere in the system. The initial vertical stress σV (along y axis) is equal to the weight of the overburden. The initial horizontal stress σH is proportional to the vertical stress:
where ρ is the total density of the rock including pore fluid, g is the gravitational acceleration, −y is the depth and A is a constant coefficient. The initial shear stress is zero everywhere. The top boundary (y = 0) has a free surface condition, with zero overpressure (pf = 0). The lateral and bottom walls are fixed, with free-slip condition (including the pore-fluid source), and impermeable (excluding pore-fluid source). The boundary conditions on lateral walls represent far-field fluid pressure and mechanical state undisturbed by the localized fluid pressure at the center of the model, which is achieved by using large enough horizontal extent of the models, L ≫ max (h, w). Perturbations of stress and displacement are negligibly small at the lateral boundaries, thus either fixed stress or fixed displacement lateral boundary conditions lead to the similar numerical results. At time t = 0, the fluid pressure is slowly increased everywhere on the source segment (pf = p(t)), until failure nucleates and propagates. The process of fluid overpressure build up at a small segment of lower boundary is unspecified but assumed slow compared to the characteristic time for establishing a steady-state distribution of the fluid pressure. This quasi-static slowly driven evolution of the pressure field in a domain with constant permeability is governed by the Laplace equation
 The gravitational force balance equation formulated for the total stress is given by
 The initial state of the stress defined in (equation 4) fulfills this relationship.
where the superscripts pe and pl denote the poro-elastic and the plastic components, respectively. The poro-elastic constitutive relation can be written as:
where α is the Biot-Willis poro-elastic coupling constant [Paterson and Wong, 2005], v is the drained Poisson's ratio, and G is the shear modulus. The plastic strain rates are given by
 Here, we chose the yield function in the form f = max (ftension, fshear), where ftension and fshear are yield functions for failure in tension and in shear, respectively, defined as:
 The parameter λ in (equation 9) is the non-negative multiplier of the plastic loading [Vermeer, 1990], and q is the plastic flow function, defined as follows for tensile (associated flow rule) and shear failure (non-associated flow rule), respectively:
where υ is the dilation angle (υ < ϕ). Note that the total stress is used in (equations 6, 8), whereas the Terzaghi's effective stress (equation 1) applies in the failure equations (9)–(11). Substitution of stresses (equation 8) into the force balance equation (6) renders gradient of the fluid pressure, commonly referred as seepage forces, as a cause of the solid deformation.
 Solving this set of equations, we aim to predict the localization and quasi-static propagation of plastic deformations into either shear bands or tensile fractures. The term tensile fracture is used here to describe the inelastic material response in the process zone area that accompanies fracture onset and propagation [Ingraffea, 1987].
 In order to check the independence of the simulation results on the numerical method, we have developed two codes (finite element and finite difference). Extensive numerical comparisons indicate that the results converge to the same values when increasing the grid resolution. The poro-elastic response of codes was tested using a new analytical solution (see auxiliary materials). The plastic response of the code was tested by stretching or squeezing of lateral walls. The numerical results were consistent with both numerical [Poliakov et al., 1993] and laboratory experiments of rock deformation [Paterson and Wong, 2005].
 In the case when the lateral walls are fixed, but the localized pore-fluid overpressure increases, the simulations show that the rock starts swelling poro-elastically (Figure 2b). When the pore-fluid pressure exceeds a critical threshold value pc at the injection source, the homogeneous deformation evolves into a pattern where either a tensile fracture or highly localized shear bands nucleate and propagate in a quasi-static manner (Figure 2c).
 Solving equations (5)–(11), the model selects the failure mode (shear or tensile) and the propagation direction. Systematic numerical simulations show the existence of five distinct failure patterns when the pore-fluid pressure exceeds pc (Figure 3, see auxiliary materials for animations). Deformation patterns I (normal faulting) and II (reverse faulting) form by shear failure in compressive (σV < σH) and in extensional initial stress states (σV > σH), respectively. Patterns III (vertical fracturing) and IV (horizontal fracturing) are caused by tensile failure in compressive and extensional initial stress states, respectively. The nucleation of failure for patterns I–IV is located at the fluid overpressure source. It is located at the free surface for pattern V, which is also called soil-piping mode in hydrology [Jones, 1971]. After nucleation at the free surface as tensile fracture in response to swelling caused by the fluid pressure build up, pattern V develops by downwards propagation of a tensile failure.
4. Analytical Solution and Failure Pattern Phase Diagram
 We have derived an analytical solution for pre-failure stress distribution caused by seepage forces sharing the same set of governing parameters as our numerical setup but a different geometry of the outer free surface boundary (see auxiliary materials). We report an excellent agreement between numerical and analytical predictions of the maximum pre-failure pore-fluid pressure. In order to calculate this critical pore-fluid pressure pc, we consider the stress and failure conditions at the fluid source (patterns I–IV in Figure 3) and at the free surface (pattern V in Figure 3).
 It follows from the analytical solution that the center, σm, and the radius, τ, of the Mohr circle do not vary along the fluid source segment. They are related by the following expressions to the initial (and far field or “global”) stresses and to the fluid overpressure at the localized source (auxiliary materials):
where two parameters βσ and βτ control the shift of Mohr circle and radius change, respectively (Figure 1):
 Using τ and, according to the Terzaghi's law (equation 1), σ′m = σm − p in the local (i.e. evaluated at the potential failure point) failure criteria allows predictions of failure pattern and initiation criteria as a function of the “global” and undisturbed by the localized fluid pressure rise far-field stresses σV and σH. Equations 12–13 can be interpreted as a generalized form of the Terzaghi's law expressed in terms of the far-field stresses for the case of “local” fluid pressure not necessarily equal to the far-field fluid pressure. According to (equation 15), during the fluid pressure increase, the radius of the Mohr-circle decreases when is positive (patterns I and III) and increases when it is negative (patterns II and IV). In the case when ln () ≪ 1, the radius of the Mohr circle does not vary. If the rock is incompressible (ν = 0.5) or if the Biot-Willis coupling constant is set to zero, then equations (14)–(15) recovers the expected Terzaghi's limit (βσ = 1 and βτ = 0) Indeed, the case when the fluid pressure gradients are not coupled to solid deformation must be in agreement with the classical effective stress law well supported by experiments with a homogeneous fluid pressure distribution [Garg and Nur, 1973; Paterson and Wong, 2005].
 Thus, after evaluating initial stresses at depth y = −h using (equation 4) and substitution of τ and σ′m from (equations 12–13) using (equations 14–15) into the shear and tensile failure conditions (equations 2–3), the critical pore-fluid pressure pc is calculated. Similarly, using the analytical solution and equation (4) at the free surface we obtain σH = 0 and the tensile failure condition: 2βτpc = σT for failure pattern V.
 Based on the above calculations, we obtain a generalized expression for failure criteria for all failure patterns as a linear combination of initial stresses evaluated at appropriate depth:
 By rearranging, we obtain a unified expression for the critical pore-fluid pressure pc:
where kb, kτ, kσ, and kf are constant coefficients (see Table 1) for the various failure patterns shown on Figure 3. Using these coefficients and equation (17), the phase-diagram for the different failure patterns can be calculated. The minimum value of pc for patterns I–V defines the pore-fluid overpressure at failure nucleation (Figure 4, red color). If pc ≤ 0, the rock is at failure without fluid overpressure (Figure 4, white color). In Figure 4, the vertical axis corresponds to the vertical stress σV and the horizontal axis to the stress difference (σV − σH), both axes being normalized by C, the cohesion of the rock. Any initial stress state in the model corresponds to a point on the diagram. The contours in the colored regions plot the dimensionless pressure pc* = pc at failure onset. If the value of the localized pore-fluid pressure is smaller than pc*, then the system is stable. However, if it is equal or larger, then the porous material fails with a predictable pattern-onset, that depends on the position in the phase diagram.
Table 1. Critical Pore-Pressure (Equation 17) at the Onset of Failure, Corresponding to the Various Failure Patterns (I–V) Shown in Figure 3a
The coefficients also define the domains in the phase diagram of Figure 4.
2(βτ + sin(ϕ) βσ)
2(βτ − sin(ϕ) βσ)
2(βτ + βσ)
2(βτ − βσ)
 White thick lines on Figure 4 define the topology of transition boundaries between different failure patterns (pic = pjc where i and j are patterns I–V in (equation 17) and Table 1, i ≠ j). Their equations can be calculated using equation (17) and the parameters given in Table 1. If a point in the failure diagram lies on one of these transition lines, the yield functions for the two corresponding domains are both equal to zero, implying that both failure modes could occur.
 We present an analytical and numerical analysis of the effect of a localized pore-fluid pressure source on the failure pattern of crustal rocks. The main results are the following: (1) Depending on the initial conditions, the geometry, and the material properties, five different patterns of failure can be characterized, either with tensile or shear mode. (2) The critical fluid pressure at the onset of failure could also be determined for all failure patterns and an analytical solution for pc is given in equation (17) and Table 1.
 These results can be used in many geological applications, including the formation of hydrothermal vent structures triggered by sill intrusion [Jamtveit et al., 2004], the aftershocks activities caused by motions of fluids inside faults [Miller et al., 2004], or the tremors caused by sediments dehydration in subduction zones [Shelly et al., 2006]. Finally, our simulations did not allow studying any transient effects in the fluid pressure during fracture propagation. It has also been shown that fluid lubrication [Brodsky and Kanamori, 2001] could have strong effect on the dynamics of rupture propagation. We are currently neglecting these additional effects, which could be integrated in an extended version of our model.
 We would like to thank Dr. R. Mourgues for the constructive review. This work was financed by PGP (Physics of Geological Processes) a Center of Excellence at the University of Oslo.