# Velocity calibration for microseismic event location using surface data

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## Abstract

Because surface-based monitoring of hydraulic fracturing is not restricted by borehole geometry or the difficulties in maintaining subsurface equipment, it is becoming an increasingly common part of microseismic monitoring. The ability to determine an accurate velocity model for the monitored area directly affects the accuracy of microseismic event locations. However, velocity model calibration for location with surface instruments is difficult for several reasons: well log measurements are often inaccurate or incomplete, yielding intractable models; origin times of perforation shots are not always accurate; and the non-uniqueness of velocity models obtained by inversion becomes especially problematic when only perforation shots are used. In this paper, we propose a new approach to overcome these limitations. We establish an initial velocity model from well logging data, and then use the root mean square (RMS) error of double-difference arrival times as a proxy measure for the misfit between the well log velocity model and the true velocity structure of the medium. Double-difference RMS errors are reduced by using a very fast simulated annealing for model perturbance, and a sample set of double-difference RMS errors is then selected to determine an empirical threshold. This threshold value is set near the minimum RMS of the selected samples, and an appropriate number of travel times within the threshold range are chosen. The corresponding velocity models are then used to relocate the perforation-shot. We use the velocity model with the smallest relative location errors as the basis for microseismic location. Numerical analysis with exact input velocity models shows that although large differences exist between the calculated and true velocity models, perforation shots can still be located to their actual positions with the proposed technique; the location inaccuracy of the perforation is <2 m. Further tests on field data demonstrate the validity of this technique.

## Keywords

Velocity calibration Microseismic monitoring Double-difference RMS error Very fast simulated annealing Perforation-shot relocation## 1 Introduction

Hydraulic fracturing of low-permeability reservoirs generates many microseismic events due to pressure increase associated with fluid injection into treatment wells (Warpinski et al. 2005). Fracture development can be characterized by various microseismic monitoring techniques (Liang et al. 2015; Wang et al. 2013). Generally speaking, when the approximate locations of perforation shots can be resolved, we have the confidence to locate nearby microseismic events, and a usable velocity model plays an important role to achieve this goal (Usher et al. 2013). At present, because of the convenience of operation, surface observations are an effective technique when monitoring wells cannot be used. They are one of the main targets for improvement in future microseismic monitoring. Microseismic monitoring with surface observations requires a well-resolved velocity model, yet many factors can interfere with model calibration, as follows. (1) Well logs are influenced by many extraneous factors, such as pore pressure, stress accumulation, and mud invasion; in addition, seismic wave velocities around the reservoir can be altered by prior resource extraction, including mining. Consequently, velocity measurements from well logs are often unsuitable for microseismic event location (Grechka et al. 2011; Pei et al. 2009; Quirein et al. 2006; Zhang et al. 2013a, b). Moreover, log data may be incomplete, which naturally reduces the accuracy of the initial model. Methods based on searching for a local optimal solution (Pei et al. 2008; Tan et al. 2013) are not suitable for this task. (2) A particularly common problem in microseismic monitoring is a combination of little available source information (e.g., perforation shots), few receivers, and poor network coverage, resulting in a poorly constrained velocity model. (3) Perforations are often not precisely timed, so a velocity model cannot always be calibrated using perforation travel times alone. Although seismic tomography is widely used to image earth structure on local to global scales, the above limitations mean that we cannot expect the same high-quality results from microseismic monitoring data (Bardainne and Gaucher 2010). Several papers have proposed methods to construct reservoir velocity models for microseismic event location, most of which are based on the following steps: (1) A simple velocity model, using only a few parameters, is constructed from well logging data. (2) Known positions of perforation shots are iteratively relocated until a suitable velocity model is obtained. Pei et al. (2009) and Bardainne and Gaucher (2010) developed a fast simulated annealing algorithm to invert for a velocity model, which showed little dependence on initial values and outperformed the local optimal solution technique. However, the method still faced the problem that perforation shot origin times are generally inaccurate. Tan et al. (2013) proposed an inversion method based on time differences calculated from picked arrival times, which circumvented the issue of origin time inaccuracies. However, their method was still sensitive to the initial model. Anikiev et al. (2014) described a method in which the initial velocity of each layer was simultaneously increased or decreased using the accuracy of perforation shot relocations as an evaluation standard. They obtained a relatively accurate velocity model by inversion. However, their method still could not satisfy the precision requirements of microseismic event location.

This paper presents a new method to address the problem of velocity model calibration using surface data. A one-dimensional layered model is built, in which the difference between theoretical and expected models is characterized by the root mean square (RMS) errors of time double differences (DDrms) (Concha et al. 2010; Waldhauser and Ellsworth 2000; Zhang et al. 2009a, b; Zhang and Thurber 2003; Zhou et al. 2010). Using the relative differences of the first arrival times of multiple events, DDrms values are minimized using very fast simulated annealing (VFSA) (Pei et al. 2009). In order to obtain an optimal velocity model for perforation relocations, we select a subset of DDrms from the results of simulated annealing. A threshold is set near the minimum value, and velocity models with DDrms values between the threshold and the minimum are chosen for further analysis. These models are then used to relocate the perforation shots. We choose the velocity model with the smallest perforation shot location errors as the model for locating microseismic events.

This paper first introduces the principles of the method, and then conducts tests on synthetic data. We investigate the influences of velocity range constraints and picking errors on the proposed technique. Finally, the proposed technique is applied to data from a perforation shot at a gas shale reservoir as an example of velocity model calculation.

## 2 Travel time calculation

This study uses ray tracing to obtain travel times for microseismic events and perforation shots. Traditional two-point ray tracing algorithms mainly comprise shooting (e.g., Xu et al. 2004) and ray bending algorithms (e.g., Li et al. 2013). More recent works use wave front extension methods based on the eikonal equation and Huygens’ principle (e.g., Zhang et al. 2006a, b); the shortest path algorithm (Wang and Chang 2002; Zhang et al. 2006a, b; Zhao and Zhang 2014); and the LTI method (Zhang et al. 2009a, b), based on graph theory and Fermat’s principle. Compared with the above methods, ray tracing based on Snell’s law is not restricted by nodes and can provide accurate travel time and azimuth information (Zhang et al. 2013a, b). Traditional shooting methods were improved by Gao and Xu (1996), who proposed a new type of step-by-step iterative ray tracing algorithm that greatly improved computational efficiency. This method can also be used with a slightly more complicated velocity model than other techniques. In this paper, we expand the method to a 3D layered structure for calculating travel times.

### 2.1 Ray tracing in a layered medium

*Z*=

*z*

_{2}, where

*P*

_{1}is the launch point,

*P*

_{3}is the receiver,

*P*

_{2}is the intersection of

*P*

_{1}and

*P*

_{3}with the medium interface, and \(P_{3}^{\prime }\) is the end point of the test ray path.

*P*

_{1},

*P*

_{2}, \(P_{3}^{\prime }\) into the equation above, we have

*x*

_{3}, \(y_{3}^{\prime }\) =

*y*

_{3}, then

*c*= (\(x_{3}^{\prime }\) −

*x*

_{2})

^{2}+ (\(y_{3}^{\prime }\) −

*y*

_{2})

^{2}, then

*ɛ*= (\(z_{3}^{\prime }\) −

*z*

_{3}) < 0, or if

*b*> 1, then

*ɛ*= (\(z_{3}^{\prime }\) −

*z*

_{3}) > 0, then

*k*

_{ xy }= (

*y*

_{2}−

*y*

_{1})/(

*x*

_{2}−

*x*

_{1}) is the slope of the projection of the line segment onto the plane

*Z*=

*z*

_{2}. \(P_{2}^{\prime }\) is then obtained by Eqs. (6) and (7), and

*P*

_{2}is replaced by \(P_{2}^{\prime }\). The steps above are repeated until

*ɛ*is sufficiently small, which yields an estimate of \(P_{3}^{\prime }\).

### 2.2 Step-by-step iterative ray-tracing method

- (1)
The starting point

*P*_{0}and endpoint*P*_{ n }are connected with a straight line. The intersections of the line with each layer (denoted*P*_{1},*P*_{2},*P*_{3}, …*P*_{ n−1}) are calculated. - (2)
*Y*_{1}is taken as the first interface. A new intermediate refraction point \(P_{1}^{\prime }\) is calculated between*P*_{0}and*P*_{2}using the dichotomy method, and*P*_{1}is replaced by \(P_{1}^{\prime }\).*P*_{1},*P*_{2},*P*_{3}, …,*P*_{ n−1}can be obtained in the same way. - (3)
Repeat step (2) until

*t*−*t*′ <*ɛ*, where*t*is the travel time of the previous iteration and*t*′ is the current travel time. A series of intermediate points is obtained. The line that connects these points with the two endpoints is taken as the minimum travel time path.

## 3 Principle of velocity model perturbance

### 3.1 Very fast simulated annealing with DDrms

In field data, and particularly for surface observations, the location error of a perforation shot is always very large when one adopts a velocity model based on a priori well log data. Moreover, part of the data might be missing, which naturally affects the accuracy of the initial model. Therefore, methods of searching for a local optimal velocity model are not applicable. Simulated annealing (SA) is a search algorithm that seeks the global minimum of an objective function in a given model space. There is no need to solve large matrix equations, and constraints can be added easily.

- 1.
Velocity vector,

**V**= (**V**_{p1},**V**_{p2},**V**_{p3}, …,**V**_{pn })^{T}, where**V**_{pi }denotes the*P*-wave velocity of layer*i*. - 2.
Objective function,

*E*(**V**). Because perforation shot origin times are inaccurate, we use the RMS error of the time double-difference (DDrms) value, which can be computed from first arrival time differences. The procedure to compute the DDrms value is described below. - 3.
Initial temperature

*T*_{0}. The initial temperature must satisfy the requirement that all proposed models are acceptable solutions for the next iteration of the calculation. We choose a small positive number at first, then multiply by a constant value*b*> 1, until the probability of acceptance of each proposed model converges to unity. - 4.Temperature annealing parameter,
*T*_{k}, which for VFSA obeys the relationshipwhere$$T_{\text{k}} = T_{0} \exp ( - ck^{1/2N} )$$(8)*T*_{0}is temperature,*c*is a constant (for this application,*c*= 0.5 is a suitable value), and*N*is the total number of layers. - 5.
A random perturbation to the velocity vector, described below.

- 6.Termination criteria. In this application, we terminate the algorithm the first time one of the following three conditions is satisfied:
- (a)
Temperature

*T*_{k}is reduced to a certain value or close to zero. - (b)
DDrms value decreases below a predetermined threshold.

- (c)
DDrms value does not decrease after multiple iterations.

- (a)

#### 3.1.1 The objective function

We use the following procedure to construct an objective function based on DDrms values:

- 1.
Select the reference trace with the highest signal-to-noise ratio. This is denoted by the subscript M.

- 2.Compute the differences between the observed first arrival times of all traces and those of trace M:$$\Delta t^{\text{obs}} = \left[ {t_{1} - t_{k} , t_{2} - t_{k} , \ldots , t_{\text{M}} - t_{k} } \right].$$
- 3.
Generate an initial velocity model, \({\mathbf{V}}^{\prime } = \left[ {{\mathbf{V}}_{\text{p1}}^{\prime } , {\mathbf{V}}_{\text{p2}}^{\prime } , {\mathbf{V}}_{\text{p3}}^{\prime } , \ldots ,{\mathbf{V}}_{{{\text{p}}n}}^{\prime } } \right]\), from sonic log data.

- 4.
Calculate theoretical time differences for the reference trace, \(\Delta t^{\text{cal}} = \left[ {t_{1} - t_{k} , t_{2} - t_{k} , \ldots , t_{\text{M}} - t_{k} } \right]\), based on

**V**′. - 5.Determine the RMS error of the DDrms value using the equations$$\delta \Delta t = [\Delta t_{1}^{\text{obs}} - \Delta t_{ 1}^{\text{cal}} ,\Delta t_{ 2}^{\text{obs}} - \Delta t_{ 2}^{\text{cal}} , \ldots ,\Delta t_{n}^{\text{obs}} - \Delta t_{n}^{\text{cal}} ]$$(9)$$E({\mathbf{V}}) = \sqrt {\frac{ 1}{n}\sum\nolimits_{i = 1}^{n} {\delta \Delta t_{i}^{ 2} } }$$(10)

#### 3.1.2 Velocity perturbation vector

*i*, respectively, subject to the constraint

**V**

_{ i }∊ [\({\mathbf{V}}_{i}^{\hbox{min} }\), \({\mathbf{V}}_{i}^{\hbox{max} }\)];

*S*

_{fact}is a step-size factor that guarantees the DDrms value decreases stably; and

*x*∊ [−1,1] is a random number generated from the equation

*S*

_{fact}is approximately 0.1.

### 3.2 Selecting the optimal velocity model

**V**are preserved. In the simulated annealing algorithm,

**V**′ replaces

**V**if the condition

*E*(

**V**′) <

*E*(

**V**) is satisfied. On the other hand, if

*E*(

**V**′) >

*E*(

**V**), then

**V**is updated using the replacement probability.

*α*is an adjustment parameter. The number of velocity models in the model set is determined by

*α*; the more velocity models are preserved, the greater the likelihood of obtaining reliable results. However, computation time will increase accordingly.

## 4 Synthetic examples

*X*

_{s1}= 830 m,

*Y*

_{s1}= 840 m,

*Z*

_{s1}= −1180 m. This study uses a star-shaped array (6 lines, 96 geophones), as the aim is to place as many geophones as possible in a small area. The initial velocity values for each layer are 950, 1300, 1800, 2800, and 3300 m/s. The simulation requires 2133 s on a notebook computer with a 2.26-GHz Intel

^{®}processor. Observed values of first arrival time differences are determined from the true synthetic model using the ray tracing method described above (Fig. 5).

Synthetic velocity model parameters

Layer | Depth, m | Synthetic velocity model, m/s | Velocity constraint range ( |
---|---|---|---|

1 | 0–200 | 1200 | 600–1300 |

2 | 200–500 | 1600 | 1000–1800 |

3 | 500–700 | 2200 | 1600–2400 |

4 | 700–900 | 3200 | 2400–3600 |

5 | 900–1200 | 3800 | 3000–4200 |

### 4.1 Sensitivity to the constraints

*P*-wave velocities used in the model. Increasing this range will increase the solution space; if we use a simulated annealing algorithm with a larger velocity range and parameters that are otherwise unchanged, then source location accuracy and computational efficiency will both be reduced. If the range of velocity variations is too small, then we probably will not obtain a viable result, as shown in Fig. 7. It is therefore desirable to choose a reasonable range of velocity variations in each layer. Generally, the range of velocities in the objective layers is mainly determined from well logging data and local geology. Another constraint is the number of surface geophones, which determines the number of time double-differences available for the inversion. The same termination conditions are used for the SA algorithm, and adding surface geophones improves the convergence of the DDrms value, as shown in Fig. 8. However, the algorithm requires more computation time to converge. Less accurate travel time information is obtained when DDrms is reduced to a small value, and it can be the case that no acceptable velocity model is obtained at all. Figure 8 compares the velocity calibration results with different numbers of surface geophones; the termination conditions are the same for all simulations. Using the same line pattern as in Fig. 4, the number of geophones increases or decreases uniformly in each line.

The minimum DDrms value of Fig. 8a is 1.3e−6. The minimum DDrms value in Fig. 8b–d is less than 2.5e−6 s. As shown in Fig. 8, reliable location results are more likely when a large number of surface geophones are used. However, increasing the number of surface arrays arbitrarily may lead the reduction of DDrms value to be difficult. In this case, more computation time is needed; notably, increasing the number of rays also increases the forward calculation time.

### 4.2 The sensitivity to picking errors

Calibrating a velocity model for microseismic event location requires accurate information about perforation shots. In actual situations, seismic signals recorded by geophones are usually contaminated by noise, which may cause picking errors (Rodriguez et al. 2012; Song et al. 2010; Tan et al. 2014). Compared with borehole observations, the picking errors of *P*-wave arrivals are relatively large at the surface. This will affect the proposed technique. Therefore, to model our algorithm’s sensitivity to picking errors, we add a set of random picking errors to the synthetic arrival times at each receiver, ranging from 0 to 5 % of the calculated travel time. We use the same stop condition as in the numerical experiments above. The algorithm terminates when DDrms value reaches 7.84e−4 s. A threshold of 8.84e−4 s is set; 10 velocity models are selected to relocate the perforation shot, and an optimal velocity model is picked out by the method described above.

Figure 10 confirms that for microseismic events located far from the perforation, the optimal velocity model for the perforation shot introduces an inherent location error. Figure 11 suggests that increasing picking errors will increase location errors; for example, if picking errors reach 20 % of computed travel times, locations of microseismic events will be poorly constrained. The main reason for this is that the process of reducing the DDrms value is influenced by the picking errors; when picking errors are large, the DDrms value cannot be reduced to a sufficiently small value, and this reduces the chances to obtain a meaningful velocity model.

## 5 Field data experiments

Stratum velocity structure parameters

Layer | Depth, m | Starting velocity model, m/s | Velocity constraint range ( |
---|---|---|---|

1 | 0–200 | 2954.5 | 2650–3200 |

2 | 200–305 | 3214.5 | 3100–3500 |

3 | 305–418 | 3047.8 | 2800–3200 |

4 | 418–517 | 3348.6 | 3200–3500 |

5 | 517–645 | 2942.2 | 2600–3300 |

6 | 645–975 | 2690.1 | 2400–3000 |

7 | 975–1300 | 3408.2 | 3000–3800 |

## 6 Conclusion

- 1.
The proposed technique does not strongly depend on the initial velocity model, and can also overcome the problem of inaccurate perforation shot origin times. However, due to complex local geology and a lack of available information, the velocity model inversion is non-unique. Therefore, whether or not a velocity model is suitable for microseismic event location is determined based on the accuracy of perforation-shot relocation.

- 2.
Constraints on velocity structure have an effect on the proposed technique’s results. Reasonable velocity constraints should be imposed on each layer; if these are improperly selected, analyst errors will negatively affect the outcome. Second, the DDrms value should be reduced as much as possible; the smaller the DDrms value, the more reliable the optimal velocity model. When using the same simulated annealing termination condition as our examples, more surface geophones are needed.

- 3.
If the arrival times of a perforation shot include significant picking errors, it may be the case that DDrms values do not converge to a reasonable minimum. This can greatly influence the effectiveness of our technique, so it is imperative to minimize picking errors.

- 4.
When processing real field data, after velocity model calibration, a perforation shot could be located to its actual position. Thus, we believe nearby microseismic events can be located with confidence. However, due to many limitations of microseismic monitoring operations, we do not expect to obtain an accurate velocity model from only one perforation shot; this implies a certain risk for microseismic events located far from the perforation. Therefore, it may be necessary to introduce more complex velocity models and source information in our application.

## Notes

### Acknowledgments

This study is supported by the National Natural Science Foundation of China (No. 41074074).

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