We present a model for explaining the seismicity rate of fluid-injection-induced earthquakes. It is assumed that pore-pressure diffusion is the main triggering mechanism, and that the criticality of the medium is randomly distributed. Based on these and other poroelastic and stochastic assumptions, we derive equations for the induced seismicity rate during and after fluid injection. On this basis, a method is proposed for estimating hydraulic diffusivity using only the observed seismicity rate and an estimate of the activated seismogenic volume. This approach differs from previous methods in that an accurate hypocentral parameter catalog is not required. Numerical investigation of the proposed method indicates that it is relatively robust with respect to the activated seismogenic volume dimension estimate. Application of the technique to two Hot Dry Rock experiments, Fenton Hill and Soultz, produces diffusivity values that are consistent with independent estimates from prior investigations.
 Seismic monitoring during borehole fluid injections is applied, e.g., to determine the extent of fracturing [e.g., Maxwell and Urbancic, 2003; Rutledge and Philips, 2003] or hydraulic properties [e.g., Shapiro et al., 1997, 2002, 2003; Audigane et al., 2002; Parotidis et al., 2004]. Accurate source location estimates are needed to apply these procedures which, in turn, requires deployment of an observation array sufficient to make accurate source location estimates. In many cases, such deployment is not experimentally feasible. Further, special software and processing are necessary; see, for example, Zoback and Harjes  for the KTB experiment and Philips et al.  and Urbancic et al.  for hydrocarbon and geothermal reservoir cases. This process is time-consuming and the corresponding investment is often not justifiable. Thus the need arises for a method, which could provide hydraulic estimations from seismicity data, for which exact locations of all registered events is not required.
 Here we present a method for estimating the in situ scalar hydraulic diffusivity, based on the seismicity rate of fluid-injection-induced microearthquakes. A hydraulic fracturing experiment is approximated by a point pore-pressure source causing pressure perturbations, which are assumed to propagate in accordance with the diffusion equation. Further poroelastic and statistical considerations allow us to derive equations for calculating the seismicity rate R(t), i.e., change of number of events Nev(t) with time t. Based on these equations the effective hydraulic diffusivity of rocks is estimated.
 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 · ).
 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).
 The units for rate are 1/s. So by multiplying the rate with a time interval Δt, we get the average number of events occurring during Δt. In the latter sense we will use the term rate below. For the estimation of the diffusivity D from the seismicity rate after equations (6) and (9). In addition to the observed rate, the spatial extent a of the seismically active area must be known. The factor F can be determined by fitting the calculated rate to that observed using the observed maximum event rate Rmax; and F = Rmax/ (after equation (6) for t → 0, Rb = F · ).
3. Numerical Model
Rothert and Shapiro  proposed a way of numerically simulating the triggering mechanism of pore-pressure diffusion. We use this type of modeling here for a point source with a constant magnitude of 5 MPa, and duration t0 = 3 · 105 s. With a FEM-algorithm (Finite Element Method) pore pressure is calculated according to the diffusion equation. For a 2D area a scalar diffusivity D = 0.4 m2/s is used in the model. Criticality C(r) follows a rectangular distribution. Events are triggered for each timestep in cells, where pore pressure exceeds criticality. Figure 2 (top) shows the simulated seismicity rate (bars). The calculated rates (curves) for diffusivities between 0.04 and 1.4 m2/s show that estimation accuracy within one order of magnitude is possible. Fluctuations of the modeled rate (see timestep 2 in Figure 2 (top)) can be explained by deviations of the statistical distribution of the criticality for the model from an exact rectangular distribution. Figure 2 (bottom) shows a sensitivity analysis for the parameter a, the radius of the seismically active region. The estimate is observed to be fairly robust: Even a 10% change of a would result in the same diffusivity estimation as above.
 Summarizing, the results of the numerical modeling are: The proposed analytical model for the seismicity rate is in agreement with numerical simulations; a scalar hydraulic diffusivity can be estimated from the seismicity rate.
4. Case Studies
 By the hydraulic fracturing test at the Hot Dry Rock geothermal energy site at Fenton Hill in New Mexico, USA, in 1983, at a depth of 3463 m water was injected for 61 h. More than 11000 seismic events were triggered and located with a relative uncertainty of 20–30 m [House, 1987]. The events occurred within 800 m of the 20 m long open-hole section. The surface injection pressure was about 48 MPa [Fehler et al., 1998]. Shapiro et al.  and Parotidis et al. , based on the assumption that pore-pressure diffusion is the main triggering mechanism but employing different methods, estimated the diffusivity for Fenton Hill between 0.14 and 0.17 m2/s. Figure 3 (top) shows the observed seismicity rate (bars) in Fenton Hill, and calculated rates for different diffusivity values (curves); for the seismically active area a = 620 m. The rate for D = 0.2 m2/s best fits the observed data, in agreement with previous estimations (see afore). Rates for diffusivities of 0.02 and 0.8 m2/s clearly do not fit the data, thus providing lower and upper limits for the diffusivity.
 On July 2000 about 23400 m3 of brine and water were injected at the European Hot Dry Rock (HDR) research site in Soultz, France. The goal of this experiment was to establish a geothermal reservoir at 4500–5000 m depth [EEIG “Heat Mining”, 2001]. Delepine et al.  estimated a scalar diffusivity for Soultz about 0.15 m2/s by also applying a method based on the assumption that pore-pressure diffusion is the main triggering mechanism. Figure 3 (bottom) shows that the calculated rate with D = 0.1 m2/s (with a = 640 m) fits best the observed rate, thus confirming the above estimation.
 The main advantage of the method proposed here is that it allows estimation of diffusivity for fluid injection experiments using only observed rates of seismicity and rough knowledge of the activated seismic volume. Thus no exact locations for all registered events are needed. As noted, use of some spatial information is required, namely the extent of the seismically active region, described here with a sphere of radius a; see equations (6) and (9). This implies that at least rough distance-estimates of the most remote events to the injection source must be determined. Such distance estimates can be easily found from P and S wave arrival times.
 The fluctuations of the observed seismicity rates relative to the calculated values (see Figure 3) can be explained by changing injection rates during the experiment, and by hydraulic and/or strength (criticality) heterogeneities, which deviate from a constant value and a rectangular probability distribution, correspondingly (i.e., the assumptions of the method presented here). These fluctuations are less for rates after the end of injection at time t0. Thus diffusivity estimates employing the method presented here are significantly improved by considering the seismicity rate after t0.
Nur and Booker  and Bosl and Nur  presented an equation analogous to equation (4) for describing the number of aftershocks, based on pore-pressure changes caused by the corresponding main shock. However, they did not introduce a criticality field for the medium and consequently no statistical distribution of it. Equations (1) and (2), and the left hand side of equations (3) and (4) are valid for general statistics of the criticality.
 The estimated diffusivity with the rate method corresponds to an effective value characterizing the whole seismically active region, including any hydraulic heterogeneities of the rock.
 The diffusivity estimate for the two Hot Dry Rock cases presented here agrees well with previous independent estimates, for which the method proposed by Shapiro et al. [1997, 1999] was used. Shapiro et al.  explain that the estimated diffusivity corresponds rather to the pre-hydrofracturing value of the rock. Although it cannot be excluded that the estimated diffusivity is influenced by permeability changes due to the fluid injection experiment, especially in the immediate vicinity of the borehole.
 The rate method presented here allows rough estimation (i.e., about an order of magnitude) of the scalar hydraulic diffusivity. The method is therefore suitable for preliminary estimations of diffusivity and particularly for application to cases without complete event location catalogs. For more precise diffusivity determinations, the earthquake locations and other methods are necessary (see Introduction).
 We present a model for rough estimation (within about an order of magnitude) of the in situ diffusivity, based on seismicity induced by fluid injections in boreholes. Exact locations of all registered events are not needed, only the spatial extent of the seismically active area, and the seismicity rate. The successful application of the method to two hydraulic fracturing experiments in Fenton Hill (USA) and Soultz (France), confirms the validity of the seismicity rate model presented here, at least for these cases, with the following main assumptions: the injection experiment can be approximated with a point boxcar pore-pressure source; pore-pressure diffusion is the main triggering mechanism; the pore-pressure values leading to earthquakes can be described by a criticality field, which approximately follows a rectangular probability distribution.
 This work was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft) under grant SH 55/3-1, and by SHELL International Exploration and Production B. V. Fenton Hill data was provided courtesy of M. Fehler, Geological Engineering Group, Los Alamos National Laboratory, USA. The Soultz data was acquired, processed and provided by the EEIG “Heat Mining”. The funding for the European HDR programme was provided by the European Commission (DG Research), ADEME (France), BMU (Germany), and the EEIG “Heat Mining” (France) and other national and private support.