We study a phenomenon occurring by hydraulic fracturing stimulations, where events are triggered during injection and also after the end of injection, sometimes over hours or even days. We assume that pore-pressure diffusion is the main triggering mechanism. Based on the theory of linear poroelasticity, an equation is derived, that describes the distance from injection point at which seismicity is terminated at a given time after the end of fluid injection. We call this distance the back front of induced seismicity. Using numerical modeling and data from three case studies we show that the back front is observed in reality. The existence of the back front is an important phenomenon supporting the idea of the pore-pressure diffusion nature of fluid injection triggered seismicity. Moreover, it provides an additional means of rock characterization allowing the estimation of the scalar hydraulic diffusivity of the seismically active area.
 Fluid injections in boreholes are carried out, e.g., for oil extraction, hydraulic fracturing or Hot Dry Rock experiments. Thereby, seismicity is triggered both during and after the end of injection. In some cases seismicity has been monitored for hours and even days after the end of injection, e.g., in Cotton Valley, USA [Urbancic et al., 1999; Rutledge and Philips, 2003], Fenton Hill, USA [Fehler et al., 1998], and Soultz, France [Dyer et al., 1994].
 The question is what could be the responsible triggering mechanism for this time-dependent seismicity evolution. To give an answer we approach the phenomenon of fluid injection triggered seismicity from the poroelasticity point of view. We assume that pore-pressure diffusion is the main triggering mechanism. Shapiro et al. [1997, 2000, 2002, 2003] already developed a methodology for analyzing earthquake data, based on the pore-pressure diffusion equation, allowing hydraulic characterization of the seismically active region.
 This concept is extended to the phenomenon of seismicity triggered by fluid injections after the end of injection: Based on poroelasticity theory equations are derived, that describe the spatio-temporal evolution of seismicity after injection end in 2D and 3D, when the main triggering mechanism is pore-pressure diffusion. A 2D numerical model is defined for verifying the theoretical results and determining applicability through sensitivity studies. For three case studies of hydraulic fracturing and Hot Dry Rock experiments, the concept is successfully applied for explaining the back front of induced seismicity.
 The time-dependent interaction of fluid flow and rock deformation is described by the theory of linear poroelasticity, which is based on Biot's equations [Biot, 1962]. For an irrotational displacement field in an infinite domain, assumed to be approximately valid for fluid injections, the spatio-temporal distribution of pore pressure p is according to the diffusion equation:
D is hydraulic diffusivity, t time. In Equation (1) the medium is considered homogeneous and isotropic regarding elastic and hydraulic properties. The permeability is related to the diffusivity by D = k/(μ S), where S is the uniaxial specific storage coefficient. D is generally assumed to be between 10−4 m2/s and 10 m2/s for the Earth's crust [Talwani and Acree, 1984; Kuempel, 1991; Wang, 2000; Scholz, 2002].
Shapiro et al. [1997, 2002] developed a method based on the diffusion equation, initially for describing pore-pressure perturbations caused by fluid injections into a borehole. They solved Equation (1) for a point pore-pressure source in a homogeneous isotropic saturated poroelastic medium, and estimated the distance r of the propagating pore-pressure front from the source, i.e., the injection point, with . This is the equation of a parabola in an r-t plot. It describes a characteristic signature for seismicity triggered by pore-pressure diffusion, called the parabolic signature.
3. The Back Front Signature
 Now we concentrate on fluid injection triggered events after the end of injection at time t0, assuming that pore-pressure diffusion is the main triggering mechanism. A fluid injection borehole-experiment is approximated by a point source of magnitude q, and duration t0, i.e., a boxcar function of duration t0 (Figure 1, top). The 2D solution of Equation (1) for time t ≤ t0 gives the pore-pressure distribution pb(r, t) during injection; r is the distance from the injection point source:
with D the hydraulic diffusivity, and the so-called Ei-function (e.g., see Carslaw and Jaeger, , p. 262). For t > t0 the following equation is applied for pore-pressure pa(r, t) after the end of injection:
Figure 1 shows the pore-pressure distribution for a point close to and one far from the injection point. For small r pore pressure rises immediately with start of injection and drops promptly after t0. For larger r, pore pressure begins to increase after the beginning of injection until time t1, where the maximum pressure value is reached. Time t1 is reached later than t0 and increases with distance r. So, for constant diffusivity D, t1 depends solely on the distance r from the injection point. Assuming that events may be triggered only for increasing pore-pressure values, would result in no later seismic activity where t1(r) is reached. Depending on distance and time, this end of seismic activity should correspond to the solution of the equation where the time derivative of pore pressure after t0, Equation (3), equals zero:
For given diffusivity D, and injection duration t0, the solutions for t in Equation (4) give for every distance r the times t1(r), where the maximums of pore pressure have been reached. These solutions can be represented in an r-t plot by solving Equation (4) for r(t), resulting in the 2D back front equation:
Equations (2)–(5) are valid for a 2D case. By considering the 3D equations for pore-pressure distribution the equation for the 3D back front can be derived:
The following numerical modeling investigates the consequence of diffusive pore-pressure distribution after injection end in the spatio-temporal pattern of the triggered events.
4. Numerical Model
 A numerical model for simulating pore-pressure diffusion triggered seismicity has been presented by Rothert and Shapiro . Here a 2D numerical model is defined. With a FEM program (Finite Element Method) pore-pressure distribution is calculated after the diffusion equation, Equation (1), for a boxcar point source q = 16 MPa. Diffusivity is defined with D = 0.1 m2/s. The point source is placed in the center of the area of interest with 500 × 500 m2. The result of the FEM-calculations is pore pressures for different timesteps. Prior to triggering events a randomly distributed criticality field is defined. It gives for each cell the value of pore pressure that must be reached to trigger an earthquake. Therefore for each timestep the criticality field is compared to the pore-pressure solutions. The next step is to produce the r-t plot from the above results; distance r of each event is calculated with reference to the injection point, and t corresponds to occurrence time of each event (Figure 2, top).
 The numerical model shows that a boxcar pore-pressure point source, in a poroelastic homogeneous and isotropic medium, triggers events after the end of injection following a spatio-temporal pattern, which is completely described by the back front, i.e., equation (5) for 2D and equation (6) for 3D. The input parameters for the calculation of the back front are duration of injection t0, and diffusivity D. Their influence is investigated in the following sensitivity analysis.
 The first parameter to analyze is the duration of the pore-pressure perturbation t0. For the above model, back fronts are calculated for t0 = 35 h and t0 = 45 h. That is a variation of more than 10% relative to the correct modeling input value of t0 = 40 h. Figure 2, middle, shows the results. When by real field experiments pore pressure is measured then t0 is exactly known.
 The second parameter to examine is diffusivity. The following values are considered, D = 0.02 m2/s, and D = 0.5 m2/s. That means the correct value D = 0.1 m2/s is reduced/increased by a factor of 5. The plot in Figure 2, bottom, shows that the determination of the back front allows a confident estimation of diffusivity with better than one order of magnitude accuracy.
5. Case Studies
 For three case studies the back front is calculated for the seismic events triggered after the end of injection at time t0. In earlier publications all three were independently studied, thus providing a possibility for comparison.
 A hydraulic fracture stimulation was carried out in May 1997 in the Carthage Cotton Valley gas field in Texas, USA. Within the study area the top of the Cotton Valley formation is about 2600 m deep and 325 m thick. Seismicity was monitored for three completion depth intervals. The analyzed stage-3 completion interval is in the upper Cotton Valley formation. About 994 events were localized during the injection and up to 10 h after the beginning of injection [Urbancic et al., 1999]. The location error ellipsoids major axes have a median length of 16 m. The bottom-hole pressure varied between 33 MPa and 42 MPa [Rutledge and Philips, 2003]. Figure 3, top, shows the r-t plot with the back front for this first case study, for the same diffusivity estimated by Rothert et al. .
 At Fenton Hill, a Hot Dry Rock geothermal energy site in New Mexico, USA, 11366 events were located using P- and S-arrival times during a hydraulic injection in December 1983 at a depth of 3463 m. Events occurred within 800 m of the 20 m long open-hole injection interval in the wellbore [Fehler et al., 1998]. Because of too few working stations, no locations were determined for 30–40 h after the initiation of injection. The relative location uncertainty is about 20 to 30 m [House, 1987]. The surface injection pressure was approximately 48 MPa. Previous estimations with D = 0.14 m2/s for the Fenton Hill data are found in Shapiro et al. . The same value is used for the back front calculation in Figure 3, middle.
 The third case study is the Hot Dry Rock project in Soultz, France, where the geothermal anomaly of the Rhine Graben is used to produce geothermal energy [Dyer et al., 1994]. Two wells were drilled to depths of 3600 m and 3800 m. Different hydraulic fracture phases were performed for stimulating the fracture system in the granite. During the experiment of 1 to 22 September 1993, 25000 m3 of water were injected, and more than 9300 microseismic events were localized. Shapiro et al.  analyzed the data and estimated a diffusivity of 0.05 m2/s. Figure 3, bottom, shows the corresponding r-t plot.
 The assumptions for the proposed concept are that a fluid injection experiment can be approximated with a point pore-pressure source, and the resulting pore-pressure changes propagate according to the diffusion equation. This point source is approximated with a boxcar function of duration t0, corresponding to injection duration. The duration time t0 of the pore-pressure source is the time of pore-pressure perturbation and not of flow rate. When pressures are not measured, a back front can still be calculated, by estimating t0. The sensitivity analysis showed that different t0-times result in a time-shift of the back front, thus allowing a rough estimation of t0, even when no relative data are available.
 Decisive for the application of the back front concept is the sensitivity of the method with respect to the hydraulic diffusivity D. This allows the use of the methodology for estimating independently a scalar diffusivity from the events triggered after the end of injection. Thus monitoring seismicity after injection end is mandatory for estimating diffusivity, by applying the back front method.
 Three case studies of fluid injections are analyzed. The corresponding r-t plots (Figure 3) show that the majority of data could be approximated with a back front. But there are still a few events not fitted by the back front. The assumptions for the definition of the back front (i.e., boxcar point source and a homogeneous isotropic rock) are for the r-t plot of the numerical model in Figure 2, top, strictly satisfied; but in reality deviations exist. Especially the existence of diffusivity heterogeneities is a cause for triggered events not fitted by the back front. These case studies were independently studied for the same triggering mechanism of pore-pressure diffusion, with a different methodology proposed by Shapiro et al. [1997, 2000, 2002, 2003], resulting in diffusivity estimations. The application of the back front signature confirmed the previous estimations.
 We present a new method for analyzing seismicity triggered by fluid injection experiments. Especially we concentrate on the events triggered after the end of injection. Equations for the 2D and 3D cases are derived, that predict the spatio-temporal evolution of these events, when pore-pressure diffusion is the main triggering mechanism. These equations allow the definition of a back front that fits the seismic data after the end of injection. The successful application of the concept on three case studies and the conformity with previous independently estimated results, confirms the applicability potential of the here presented methodology, for determining the main triggering mechanism and characterizing hydraulically the seismically active region.
 This work was funded by the German Science Foundation (Deutsche Forschungsgemeinschaft) under grant SH 55/3-1, and by SHELL International Exploration and Production B.V. Cotton Valley data was provided courtesy of T. Urbancic, Engineering Seismology Group Inc., Canada. Fenton Hill data was provided courtesy of M. Fehler, Geological Engineering Group, Los Alamos National Laboratory, USA. The data set for Soultz was provided courtesy of SOCOMINE, France.