The term criticality is used to describe the pore-pressure value that must be reached in order to trigger an earthquake. We assume a rectangular probability density function f(C) for the criticality C(r) of the medium at any point with position vector r, with the minimum and maximum criticality values Cmin and Cmax, respectively; then f(C) = 1/dC, with dC = Cmax − Cmin.
 We consider a point source that causes pore-pressure perturbations p(r,t) which propagate according to the diffusion equation ∂p/∂t = D∇2p, with t the time, and D the scalar hydraulic diffusivity; this equation is valid for an irrotational fluid displacement field [Wang, 2000], which we consider as an approximation for pore-pressure perturbations due to borehole fluid injections. The permeability k is related to diffusivity by D = k/(μS), where S is the uniaxial specific storage coefficient, and μ is the fluid viscosity. Generally, for the Earth's crust D is between 10−4 and 10 m2/s [Talwani and Acree, 1984; Kuempel, 1991; Wang, 2000; Scholz, 2002]. As long as pore pressure rises at a given point, an earthquake can be triggered. Thus the minimum monotonous majorant g(r,t) of pore pressure is the decisive parameter for triggering seismicity (Figure 1). The absolute value of the position vector equals r = , with x, y, z the Cartesian coordinates, and the point source placed at the origin of the coordinate system. The medium is subdivided into N cells. Then the probability that an event occurs in a cell will be given by the following value of the probability distribution function of the criticality:
For the whole volume V of the area of interest the cumulative number of triggered events Nev(t) is defined as:
Using δV to represent the volume in each cell, we can rewrite equation (2):
and the seismicity rate R(t) for the whole volume V (i.e., a sphere with radius a) is (in spherical coordinates):
Firstly, we consider time t only during fluid injection with duration t0. For a step function point pore-pressure source of strength q, pore pressure pb (r,t) rises monotonically during t0; the subscript b denotes before t0. Thus for t ≤ t0 it is gb(r,t) = pb (r,t), which in 3D is:
with erfc the complementary error function erfc(x) = 1 − erf(x) [e.g., see Carslaw and Jaeger, 1959]. By substituting the solution of the diffusion equation, equation (5), in equation (4), differentiating and integrating we receive for the seismicity rate Rb(t) up to t0 in 3D:
with a the radius of the sphere comprising the volume of the seismically active region, and F = q/(dC · δV · ).
Figure 1. The majorant function g(r,t) (bottom) results from the pore pressure function p(r,t) (middle), shown here for a boxcar pore-pressure source (top) of duration t0 = 2·105 s. The shown pressure distribution p(r,t), for r = 450 m, rises up to time t1, where its maximum is reached. Thus after t1, the minimum monotonous majorant g(r,t) of pressure p(r,t) is constant and equal to the maximum pore pressure.
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 Secondly, we consider the time t after the end of injection at t0, where the pore pressure pa(r,t) continues to increase as a function of time and distance from the injection point (Figure 1). At points where pressure reaches its maximum, no further seismic events can be triggered. Thus a zone of seismic quiescence is developing with r < rbf(t). Here rbf(t) is the back front of seismicity that Parotidis et al.  described with the following equation:
Then using equations (4) and (7) we describe the rate Ra(t) after t0 in 3D according to
with u1 = −a/ · exp (−a2/(4Dt)); u2 = · erf(a/); u3, u4 equal −u1, −u2, correspondingly, but a is substituted by rbf. The v terms equal the u terms, but t is substituted by (t − t0).