Inference of permeability distribution from injection-induced discrete microseismic events with kernel density estimation and ensemble Kalman filter

Authors


Corresponding author: B. Jafarpour, Viterbi School of Engineering, University of Southern California, 925 Bloom Walk, HED 313, Los Angeles, CA 90089, USA. (behnam.jafarpour@usc.edu)

Abstract

[1] Hydraulic stimulation of subsurface rocks is performed in developing geothermal and hydrocarbon reservoirs to create permeable zones and enhance flow and transport in low-permeability formations. Borehole fluid injection often induces measurable microearthquakes (MEQs). While the nature and source of the processes that lead to triggering of these events is yet to be fully understood, a major hypothesis has linked these events to an increase in pore pressure that decreases the effective compressional stress and causes sliding along preexisting cracks. Based on this hypothesis, the distribution of the resulting microseismicity clouds can be viewed as monitoring data that carry important information about the spatial distribution of hydraulic rock properties. However, integration of fluid-induced microseismicity events into prior rock permeability distributions is complicated by the discrete nature of the MEQ events, which is not amenable to well-established inversion methods. We use kernel density estimation to first interpret the MEQ data events as continuous seismicity density measurements and, subsequently, assimilate them to estimate rock permeability distribution. We apply the ensemble Kalman filter (EnKF) for microseimic data integration where we update a prior ensemble of permeability distributions to obtain a new set of calibrated models for prediction. The EnKF offers several advantages for this application, including the ensemble formulation for uncertainty assessment, convenient gradient-free implementation, and the flexibility to incorporate various failure mechanisms and additional data types. Using several numerical experiments, we illustrate the suitability of the proposed approach for characterization of reservoir hydraulic properties from discrete MEQ monitoring measurements.

1. Introduction

[2] The production of geothermal energy and unconventional resources from tight and low-permeability reservoirs is achieved by hydraulic stimulation of the rock through borehole injection to create permeable zones, a process that involves fracture initiation and/or activation of discontinuities such as faults and joints due to pore pressure and in situ stress perturbations. Hydraulic stimulation of rock is typically accompanied by multiple microseismic events [Zoback and Harjes, 1997; Fehler et al., 1998; Audigane et al., 2002; Rutledge and Phillips, 2002], which are believed to be associated with rock failure. While the true nature and source of such events remains to be fully understood [Trifu, 2002], alternative mechanisms for triggering microearthquake (MEQ) events have been proposed in the literature. Among the existing hypotheses, pore pressure relaxation is widely studied in the literature [Nur and Booker, 1972; Fletcher and Sykes, 1977; Ohtake, 1974; Pearson, 1981; Talwani and Acree, 1985; Zoback and Harjes, 1997; Shapiro, 2008].

[3] The pore pressure hypothesis postulates that a rise in fluid pressure in a reservoir increases the pressure in the connected pore space of the rock, thereby increasing the pore pressure and decreasing the effective normal compressional stress on the rock surfaces. In critical locations of the rock the fall in the compressional stress can result in sliding along some of the preexisting cracks. The pore pressure relaxation hypothesis is supported by several observations [Shapiro et al., 2005a, 2005b]. The spatiotemporal distribution of MEQ events has been observed to have signatures of a diffusion-like process (including a forward triggering front and back front of seismicity waves [Audigane et al., 2002; Shapiro et al., 2003; Parotidis et al., 2004]) consistent with the diffusive nature of pore pressure distribution. Other observations supporting the above hypothesis are related to the ellipsoid-shaped seismicity clouds after normalization of the event coordinates by their occurrence time [Shapiro et al., 2003] and the spatial density of MEQ events [Shapiro et al., 2005b].

[4] If pore pressure diffusion can be used to explain the spatiotemporal signatures of microseismic event clouds, the MEQ events are expected to reveal important information about the distribution of hydraulic diffusivity or permeability in reservoirs. This concept has been exploited in the development of seismicity-based reservoir characterization (SBRC) methods of Shapiro et al. [1997, 2002, 2005a, 2005b] and Rothert and Shapiro [2003] where hydraulic rock properties are estimated from analysis and integration of injection-induced MEQ monitoring measurements. The SBRC analysis involves solving the parabolic equation of pore pressure diffusion in the rock mass and comparing the distribution of pore pressure with the rock criticality distribution to identify locations that undergo failure. In this context, rock criticality at a given location refers to the minimum pore pressure required to trigger a seismic event.

[5] The microseismic signals contain information about the triggering source locations and have been used to understand the hydraulic fracturing process [e.g., Foulger et al., 2004; Vandamme et al., 1994; Warpinski et al., 1999]. Detection and interpretation of microseismic events is useful for estimating the reservoir permeability, the stimulated zone and fracture growth, as well as the geometry of the geological structures and the in situ stress state [Warpinski et al., 2001; Pine and Batchelor, 1984]. While the SBRC approach is a useful framework for estimating large-scale reservoir properties, it can be substantially improved by developing more sophisticated inversion algorithms that offer a number of important estimation properties (features). First, the process can be automated in an iterative or sequential manner. Second, the inversion can be implemented to account for the uncertainties in the prior models or the observed microseismic events. Third, and more importantly, through a stochastic inversion approach, a mechanism can be included for quantitative assessment of the quality of the solution obtained and for rigorous characterization of solution uncertainty. The need for such a stochastic inversion approach is imperative in light of the significant uncertainties that exist in describing the spatial distribution of rock physical properties. As we will discuss soon, an outstanding challenge in applying state-of-the-art inversion method for MEQ data integration is the discrete nature of these events, which does not fit into most of the conventional estimation methods that are designed for assimilating continuous data.

[6] Adopting the pore pressure relaxation hypothesis in this study, the initiation of MEQ events can be associated with pore pressure and stress values that exceed the rock criticality. Hence, the distribution of MEQ observations in the reservoir can be correlated with pore pressure distribution, which is in turn related to hydraulic properties of the reservoir rock. Therefore, the MEQ events are viewed as a new source of monitoring measurements that, after interpretation into prior descriptive models, are expected to reveal important information about the distribution of rock flow properties.

[7] In this view, the clouds of microseismic observations in the reservoir consist of discrete events that contain information about the approximate seismic source locations where rock failure has occurred. Since most established model calibration algorithms are designed to integrate continuous measurements, inverting the discrete microseismic events calls for development of inversion methods to handle discrete data types. In some cases, however, it may be possible to equivalently interpret discrete data sets as continuous measurements (through a simple conversion) that can be readily processed using well-established inversion techniques. One way to model the MEQ data is to consider the density of these discrete events. This interpretation leads to combining (counting) the discrete microseismic events at each location in the reservoir and interpreting the results as the distribution of “seismicity density”. This conceptual framework is followed in this paper by taking advantage of kernel density estimation (KDE) methods [Scott, 1992; Wand and Jones, 1995].

[8] The KDE methods are common for smoothing data and estimating nonparametric probability density functions [Scott, 1992; Wand and Jones, 1995]. We use this approach to convert discrete MEQ events into a map of seismicity density as a continuous representation of the data. Detailed description of this method is provided in section 2. The continuous representation of MEQ data can then be used with a data integration technique to estimate the relevant reservoir properties, in this case permeability distribution. The data integration of our choice in this paper is the ensemble Kalmn filter (EnKF) [Evensen, 1994, 2007]. We evaluate the feasibility of using the EnKF for estimating permeability distributions from the KDE-based continuous representation of microseismic measurements.

[9] The EnKF has been widely established as a practical data integration method for large-scale nonlinear dynamical systems and has been received favorably by the scientific and research community in a range of applications including hydrology [Reichle et al., 2002], meteorology and oceanography [Evensen and van Leeuwen, 1996; Houtekamer and Mitchell, 1998; Madsen and Canizares, 1999], groundwater model calibration [Chen and Zhang, 2006; Franssen and Kinzelbach, 2008; Nowak, 2009; Schöniger et al., 2011], and oil reservoir characterization [Nævdal et al., 2005; Wen and Chen, 2005; Jafarpour and McLaughlin, 2009; Aanonsen et al., 2009; Jafarpour and Tarrahi, 2011]. Evensen [2009] reviews the EnKF formulation and its wide range of applications. Ehrendorfer [2007] presents a review of important issues that are encountered in implementing the EnKF. Despite the existing limitations in operational implementation of the EnKF for more complex (non-Gaussian) and challenging large-scale problems, this approach has become popular as a promising approximate nonlinear estimation method in several applications. In this paper, we propose applying the EnKF for MEQ data integration and evaluate its performance using several numerical experiments. Throughout the paper, all EnKF updates are applied to the natural logarithm of permeability.

[10] The remainder of the paper is outlined as follows. We begin with section 2, which covers an overview of the proposed framework followed by a description of each step of the formulation and a concise presentation of the experimental setup used to test the described methodology. Next, we present and discuss the results of applying the proposed approach to a series of two-dimensional and three-dimensional experiments in section 3. We close the paper with section 4, which includes general remarks about the presented formulation, its advantages, limitations and possible future extensions.

2. Methodology and Experimental Setup

[11] The proposed EnKF-based inversion framework begins by generating an ensemble of N prior realizations of the permeability model based on prior information (e.g., using geostatistical simulation methods). These realizations are used to predict the pore pressure distributions in the reservoir. Comparison of the pore pressure predictions with a spatially uncorrelated random rock criticality map is used to establish the triggering mechanism and to predict microseismic events. These predictions are then converted into continuous seismicity densities, using the KDE method, and used in the EnKF update equation. The main steps involved in the implementation of the proposed method are schematically shown in Figure 1 and summarized as follows.

Figure 1.

Overall workflow of the forward model that relates permeability to microseismicity density observations.

[12] 1. Convert discrete microseismic data into quantified continuous seismicity density maps using KDE method (section 2.2).

[13] 2. Generate an ensemble of rock permeability models from available prior information (e.g., using geostatistical simulation techniques). Repeat steps 3–5 until all measurements are processed.

[14] 3. For the prediction step, using numerical simulation of the parabolic pressure diffusion equation, forecast the pore pressure distribution for each member of the most recently updated ensemble realizations (section 2.1).

[15] 4. Predict the microseismic events for each realization by comparing the local pore pressure to the random local rock criticality values and convert the results into seismicity density maps using the KDE approach (section 2.2).

[16] 5. Use the EnKF analysis equation with the seismicity density observations from (1) to update the ensemble of reservoir permeability models (section 2.3).

[17] The details of each of these steps are discussed next.

2.1. Forward Model: Pore Pressure Prediction

[18] The Frenkel-Biot equations [Biot, 1962] in a homogeneous isotropic saturated poroelastic medium, identifies three waves (P, S and a dissipative slow wave named Frankel-Biot) propagating from a source to an observation point. Field evidence suggests that evolution of MEQ events is a relatively slow process that is likely to be associated, at least in part, with the Frankel-Biot wave. The pore pressure variation in the slow wave can be described by a simple diffusion equation for a homogeneous isotropic porous medium. Recent studies [see Shapiro, 2008, and references therein] have proposed a similar diffusion equation for describing the spatiotemporal evolution of pore pressure relaxation in heterogeneous anisotropic porous media.

[19] We follow the same approach in this paper. We also assume the rock failure occurs at locations in the reservoir where the pore pressure exceeds rock criticality. To implement this failure criterion with the pore pressure relaxation assumption, we numerically solve the diffusive pore pressure equation for a heterogeneous reservoir. This forward model relates rock hydraulic conductivity distribution to pore pressure distribution, which is directly related to the rock failure mechanism and the distribution of microseismic events (data). For single phase flow with slightly compressible fluid, the pore pressure equation is expressed as

display math

where u denotes the location in space, t is time, φ is porosity, c is total compressibility, and μ represents viscosity. In equation (1), the spatial distribution of permeability in space is denoted by math formula while the spatiotemporal distribution of pore pressure is represented with math formula. We solve the diffusion equation in (1) for each permeability realization math formula using a finite difference-based commercial fluid flow simulator [Eclipse, 2010]. Under the assumptions described above and heterogeneity and anisotropy of hydraulic permeability, equation (1) can be derived from the Frenkel-Biot equations in the low-frequency range [Rice and Cleary, 1976; Van der Kamp and Gale, 1983; Biot, 1956].

[20] To generate the corresponding microseismicity clouds for each permeability model realization, we apply the failure criterion used by Shapiro et al. [2005a] using the predicted pore pressure distributions to generate the spatiotemporal distribution of seismicity events. An important property of the EnKF inversion is that the forward and observation models can be quite general with varying level of complexity. However, the update equation is designed for continuous random variables (parameters and observations). As a result, the updates can be applied under various forms of event triggering mechanisms and failure criteria as long as the random variables representing the states, parameters and measurements are continuous. We consider the pore pressure as the MEQ triggering mechanism and investigate the feasibility of estimating permeability distribution from microseismic observations using the EnKF.

2.2. Observation Model and Kernel Density Estimation

[21] To infer the permeability distribution from MEQ observations, we use the ensemble of permeability models to predict an ensemble of microseismic clouds to be used in the EnKF update. Denoting rock critical stress as math formula and the pore pressure distribution at location ui and time tj as math formula, we assume that a microseismic event (failure) is triggered at location ue and time te if

display math

From equation (2), at each time step t the comparison between predicted pore pressure and criticality at different locations in the reservoir identifies the distribution of seismicity clouds. Figure 1 illustrates this procedure schematically for a two-dimensional homogenous permeability model and a given random criticality distribution. Note that following [Shapiro, 2008], we have represented the rock criticality at each grid block as an uncorrelated random variable. This assumption can be easily relaxed and a spatially correlated random field (possibly correlated with other rock physical properties) can be considered as a criticality distribution. In section 2.3, we use this procedure with a synthetic true permeability map to generate simulated microseismic data sets. In experiment 1 of section 3, the sample microseismicity cloud shown in Figure 2c is used as the synthetic observations corresponding to the shown permeability model (which will be considered as unknown to be estimated).

Figure 2.

Microseismicity cloud generation in a two-dimensional model with homogeneous permeability (20 mD): (a) snapshots of diffusive pore pressure distributions at different time steps math formula, (b) spatially uncorrelated Gaussian (white noise) rock criticality math formula, and (c) the cloud of microseismic events generated by comparing rock criticality with pore pressure distributions at different times steps.

[22] In practice, the discrete microseismic events identify the location of the passive seismic sources and are often generated through seismic source inversion methods. The raw seismic data (collected either from surface or borehole geophones) are inverted to map the location of seismic sources and characterize the associated uncertainty. In this paper, however, we skip the seismic source inversion part and assume that, after seismic data analysis, the map of observed source (event) locations is available.

[23] The available seismic observations, however, are of discrete nature since they only identify the seismic status (active or inactive) of a cell in the reservoir model. The discrete nature of MEQ events introduces a difficulty in implementing inversion methods that are designed for continuous problems. For gradient-based methods, the discrete form of MEQ observations complicates the calculation of their gradients with respect to unknown parameters. On the other hand, while the EnKF does not require gradient information explicitly, by construction it is formulated for estimation of continuous variables and observations. To address this issue, we interpret the MEQ events as continuous measurements using the kernel density estimation method. KDE is often used for nonparametric approximation of continuous probability density functions (PDFs). The general idea is to convert the discrete MEQ data (and their predictions) into a smooth and continuous seismicity density map. For this purpose, at each time step, we replace each MEQ event/source with a Gaussian kernel function centered at the event location. By adding up the kernels, we construct a continuous function over the model grid that represents the spatial density of the MEQ events. The procedure for implementing the KDE method is illustrated in Figure 3 for a one-dimensional example. We note that the procedure in Figure 3 can be easily applied to three-dimensional problems. Mathematically, the continuous seismicity density map can be written as

display math

where math formula is a Gaussian kernel, math formula is the number of MEQ events at each time step, math formula denotes the location coordinate of the MEQ events (center of the individual Gaussian kernels) and math formula is the covariance matrix of the Gaussian kernel. The continuous map math formula represents the seismicity density at all locations in the reservoir and constitutes the observations for the EnKF update.

Figure 3.

Schematic illustration of converting discrete microseismic event measurements to continuous seismicity density observations in one dimension using kernel density estimation method. The crosses on the x axis show the reconstruction of the discrete microseismic events, while the short symmetric curves display the corresponding Gaussian kernels used to represent each event as a density function with maximum value at the location of the discrete events. The red line shows the density of the microseismic events in space as a continuous observation to be used in the EnKF.

[24] The Gaussian kernel has the form

display math

in which the covariance matrix can be specified either globally for all events locally (or separately) for individual events. We generate the covariance matrices for each MEQ event by randomizing the covariance matrix of the kernel function. The covariance matrix for the kernel determines the shape, size and orientation of the Gaussian ellipsoid centered at the microseismic event location. In this paper, we select an isotropic Gaussian kernel for quantification of the microseismic events and the uncertainty in the MEQ locations. We note that, in practice, the values of the bandwidth used for the kernel functions are identified from the uncertainty in locating the seismic sources from the raw surface or borehole seismic data. In this paper, a simple sensitivity analysis revealed that selecting a bandwidth parameter as large as twice the dimension of each grid block leads to reasonable results. An alternative approach to the KDE method for quantification of microseismicity is the spatial bin-counting process in which coarser grid blocks (bins) are assigned seismicity density values proportional to the number of events recorded within them. This approach results in a coarse-scale seismicity density map that reduces the number of measurements. One advantage of the KDE is that it also provides a convenient procedure to account for the spatial uncertainty in the location of the events.

2.3. Estimation With Ensemble Kalman Filter

[25] The classical Kalman filter [Kalman, 1960] is a sequential state estimation method for characterization of the first and second statistical moments of the states posterior distribution. Hence, the filter fully characterizes the posterior distribution of linear state-space systems that are characterized with jointly Gaussian distributions [Kalman, 1960; Gelb, 1974]. The implementation of the filter involves two steps: (1) a forecast step, in which a linear state propagation model is used to predict the mean and covariance of the states at the next time step; and, (2) an analysis step that updates the mean and covariance of the states using the dynamic observations and the forecast states mean and covariance. These two steps are repeated sequentially until all observations are assimilated.

[26] For nonlinear dynamical systems, the EnKF provides a practical approximation of the Kalman filter that has been successfully applied to many applications ranging from hydrology, meteorology and oceanography to groundwater and oil reservoir model calibration [see, e.g., Evensen, 2007, 2009; Aanonsen et al., 2009, and references therein]. The sequential formulation of the EnKF distinguishes a forecast (or prior) PDF for the states (augmented vector of permeability and continuous seismicity response xt) math formula, conditioned on all measurements math formula taken through time t − 1, and an updated (or posterior) density math formula conditioned on all measurements math formula (continuous seismicity response maps) taken through time t. To compute the cross covariance between predicted observations and parameters, the original state vector is augmented with uncertain model parameters (permeability distribution) and predicted measurements [Evensen, 2007]. This state augmentation approach can be used to update states and parameters simultaneously. Alternatively, one can only update the uncertain parameters and derive the updated states by solving the flow equations with the updated parameters. This is the approach taken in this paper. The measurements math formula consist of seismicity density map math formula, defined in equation (3) that represent microseismic measurements in space at time t.

[27] Since the general multivariate PDFs and their statistical moments are difficult to characterize, the EnKF uses a Monte Carlo approximation approach by sampling an initial set of realizations from the high-dimensional prior PDF of the uncertain properties to form an ensemble of reservoir states (and/or parameters). These property maps are then used to generate an ensemble of state and measurement predictions that can be used to compute a sample (prior) covariance matrix for the EnKF update step as described below.

[28] The forecast step in the EnKF can be written as

display math

where ·|t represents conditioning on observations up to time t; math formula is a vector of known (nonrandom) time-dependent boundary conditions and controls (such as injection rate); and math formula is a vector of random variables that accounts for modeling errors. The function math formula represents the state propagation equation from time t − 1 to time t. The notations j and Ne are used to indicate the realization index and total number of realizations, respectively. In our application, equation (5) represents the solution of the parabolic diffusion equation that describes the time evolution of pore pressure distribution for each individual realization j of the ensemble permeability. At time steps when MEQ data are available, the EnKF analysis equation is used to update the permeability realizations using the gain matrix and the misfit between predicted and observed seismicity density maps for each realization. At the update step we use an augmented state vector consisting of spatially distributed permeability realizations (parameters to estimate) and realizations of the predicted continuous seismicity density map. After each update we apply a confirmation step [Wen and Chen, 2005] by forecasting the future states and predictions from the initial time step with the updated parameters. We repeat the sequence of prediction and update steps until all measurements are integrated.

[29] For a model with Nb grid blocks, each log permeability realization math formula and its corresponding microseismicity density response math formula are vectors of size Nb × 1. In this paper, the log permeability models are jointly Gaussian random fields that are generated using the sgsim [Deutsch and Journel, 1998] geostatistical simulation technique. The augmented state vector for this case is of the form

display math

The EnKF analysis equation that is used to update each log permeability realization can be expressed as

display math

where K is the Kalman gain matrix and the subscripts u and f denote updated and forecast quantities while the superscript e indicates ensemble calculated statistics. The notations math formula and math formula represent the states sample covariance and observation covariance matrices, respectively. The measurement matrix math formula, where math formula and math formula are zero and identity matrices of the specified dimensions, respectively, acts as a selection operator that extracts the predicted measurement components from the augmented state vector. The notation math formula is used to represent the jth realization of the perturbed observations. The states sample covariance math formula can be computed from the ensemble of state vectors

display math

where math formula is used to denote the ensemble mean of the forecast states (that is, the log permeability distribution from the previous step and the corresponding microseismic response forecasts). In the EnKF implementations, the covariance matrix in equation (8) need not be constructed explicitly and the update can be applied using its low-rank representation though a compact SVD implementation. The covariance matrix in equation (8) contains the covariance information about the log permeability field as well as the cross covariance information between the log permeability and (microseismic) measurements. It is the latter cross covariance that allows the estimation of uncertain log permeability distributions from microseismic observations. This relation bears similarity with the use of covariance and cross covariance in the kriging/simulation [Vargas-Guzman and Yeh, 1999; Deutsch and Journel, 1998] and cokriging/cosimulation [Kitanidis and Vomvoris, 1983; Yeh et al., 1995; Graham and McLaughlin, 1989; Hanna and Yeh, 1997; Deutsch and Journel, 1998] methods, respectively. Note that in equation (6), the term math formula is the misfit between the jth perturbed observation and prediction, which in this case represents the observed and predicted continuous map of seismicity density. Several remarks regarding the update equation for our problem will follow.

[30] In addition to nonlinearity in the forward flow model, a complexity of the measurement model in our application is the nonlinear failure criterion (i.e., hard truncation) that is used to convert the continuous pore pressure distribution to discrete microseismic events. The Gaussian kernel that we apply to convert the MEQ predictions to continuous maps of seismic density makes the data more amenable to processing with the EnKF. However, the relationship between the magnitude of pore pressure and the resulting seismicity map remains complex.

[31] The modeling and calibration process employed in this paper does not consider the changes in rock properties as a result of the pressure increase due to bulk injection. To incorporate such effects, a model is needed to describe the dynamic changes in the rock permeability as a function of its geomechanical properties and reservoir states (i.e., stress regime, failure mode, etc.). While such effects are not included in the forward model of this study, the EnKF-based workflow, with slight modifications, remains applicable to more sophisticated forward models (e.g., coupled flow and geomechanical models).

[32] In our EnKF implementation, to perturb the observations, we add an uncorrelated realization from a Gaussian random noise, with a specified observation covariance matrix math formula, to the value of the observed quantities. We assume that the observation error is proportional to the value of the observed quantity and compute the diagonal elements of the observation error matrix as

display math

where math formula is the observation variance at the kth grid block (the kth diagonal entry for the observation covariance matrix), math formula and math formula are the minimum and maximum variances specified for the observations, respectively. The notation math formula represents the observed seismicity density at location k while math formula, math formula represent the minimum and maximum observed values of the seismicity density, respectively. The realization j of the perturbed observation at location k, can then be written as

display math

In the experiments of this paper, we assume an uncorrelated Gaussian observation error with zero mean and standard deviation obtained from equation (9). We note that other methods for generating the perturbed observations may also be considered. In particular, given the large dimension and the spatial correlation that may exist between the observation errors, one could assimilate the resulting observations in a low-rank subspace defined by the left singular vectors of the ensemble observations perturbations matrix in a similar way to Keepert [2004].

[33] To generate the ensemble of permeability realizations, we used a variogram-based geostatistical simulation method with specified variogram parameters. The sgsim algorithm [Deutsch and Journel, 1998] was used to implement the geostatistical simulations. In real applications, the number of realizations is typically determined through a trade-off between available computational resources and the desired statistical accuracy in computing the required sample statistics. For large-scale problems where the number of realizations are limited, practical considerations such as localization or local analysis [Hamill et al., 2001] have been proposed to avoid inaccurate updates due to spurious (nonphysical) correlations and to reduce the possibility of an ensemble collapse. In the examples that follow, we implement the EnKF algorithm with Ne = 100 and do not apply any localizations. The initial realizations of the heterogeneous log permeability are conditioned on the log permeability value at the well location in the center.

2.4. Description of Experimental Setup

[34] In section 3, we present two sets of experiments covering a two-dimensional (2-D) and a three-dimensional (3-D) reservoir model. For the 2-D example, we consider the estimation of a heterogeneous permeability model and show that the distribution of the MEQ events can be used to infer the spatial distribution of the permeability field. Our second experiment is based on a 3-D reservoir configuration with a heterogeneous permeability model. In these experiments, one water injection well is located at the center of the field and the boundaries are closed to flow (no-flow boundary conditions). The injection-induced MEQ events for this injection well are used to estimate the permeability in the reservoir. The 2-D examples consist of 100 × 100 discretized models, leading to Nb = 10,000 grid blocks. In this paper, we have assumed that an interpretation of the microseismic data (through seismic source inversion) in some preprocessing step provides a spatial map of the seismic event locations. Therefore, at each update step, a vector of 10,000 observations of seismicity density values is assimilated. In the 3-D example, the reservoir is discretized into a 50 × 50 × 30 (Nb = 75,000) grid configuration. Also in this case, one injection well is located at the center of the domain, which is perforated throughout the entire thickness of the formation. The source locations of the MEQ events throughout the 3-D domain are used to estimate the heterogeneous permeability distribution.

3. Results and Discussion

[35] We present and discuss the results of applying our methodology to the experiments described above. We present the results in terms of the estimated property maps and the ensemble statistics prior to and after data integration. As is common in ensemble data assimilation, we use the evolution of log permeability estimation root-mean-square error (RMSE) and the ensemble spread (Sp) as performance measures. These measures are computed using the following equations:

display math
display math

where n is the number of parameters (same as number of grid blocks here), Ne is the number of realizations and mi,j is the ith parameter of realization j. We plot the ensemble spread as a percentage of the initial ensemble spread.

3.1. Experiment 1: Two-Dimensional Heterogeneous Example

[36] As our first experiment we consider estimation of a simple two-dimensional heterogeneous permeability field. The field setup for this example is shown in Figure 2. The estimation results for this experiment are shown in Figure 4. Figure 4a shows the true permeability, while Figure 4b displays the corresponding MEQ event locations. Notice the MEQ events distribution in Figure 4b contains signatures of low- and high-permeability trends. Due to rapid pressure diffusion to high-permeability areas, these regions experience early and more concentrated MEQ events while low-permeability areas do not experience any microseismicity due to slower pressure propagation from the injection point to these regions. Figure 4c illustrates, from left to right, the initial ensemble mean, standard deviation, and a sample permeability realization maps. The corresponding updated maps at the final step of data integration are shown in Figure 4d. The estimated permeability maps tend to identify the major high- and low-permeability regions in the reservoir. It is evident from these maps that the EnKF can infer information about the permeability distribution by integrating the data about the distribution of the MEQ event locations. The permeability estimation RMSE and ensemble spread for this example are also shown in Figures 4e and 4f, respectively. These plots indicate decreasing trends in these performance measures with time, indicating increased confidence in the updated ensemble after integrating the MEQ data. Similar results were obtained for different reference log permeability models and also when the criticality map was assumed to be a spatially correlated random field (not shown).

Figure 4.

The EnKF estimation results for experiment 1 with a two-dimensional heterogeneous permeability model: (a) the true log permeability model, (b) the corresponding discrete microseismic events, (c) initial log permeability ensemble mean (left), standard deviation (middle), and an individual realization (right), (d) final log permeability ensemble mean (left), standard deviation (middle), and individual realization (right) after seven update steps, and time evolution of (e) the log permeability RMSE and (f) normalized ensemble spread.

3.2. Experiment 2: Three-Dimensional Heterogeneous Example

[37] The second experiment involves the application of the proposed formulation to a 3-D problem. For this example, Figure 5 shows the experimental setup with the true permeability (Figure 5a), the random rock criticality (Figure 5b), the corresponding MEQ observations (Figure 5c), and the pore pressure distribution in the reservoir as a function of time (Figure 5d). As expected, the distribution of MEQ events indicates positive correlations with the permeability and pressure distributions. It is this correlation that is exploited by the EnKF update to reconstruct the trend in the permeability map. It is also important to note that the overall field pore pressure increases with time, implying that more seismic activities are expected at later times.

Figure 5.

Experimental setup for experiment 2 with three-dimensional permeability model: (a) true log permeability distribution, (b) spatially uncorrelated criticality distribution, (c) simulated cloud of microseismic event distribution, and (d) snapshots of diffusive pore pressure distribution in time.

[38] Figure 6 presents a summary of the estimation results. In Figure 6a, the true permeability field is replotted for convenient comparison with the estimated permeabilities. Figures 6b and 6c show the RMSE and ensemble spread for this experiment, respectively. Again, the decreasing behavior in these performance metrics is indicative of an improving estimation result in time. Figure 6d displays, from left to right, the ensemble mean for permeability at the initial and after the first, fourth, and sixth update steps, respectively. The initial permeability ensemble mean appears to be almost homogeneous while at later time steps this mean field begins to reveal the correct permeability trend in the field. This is also observed in Figures 6e and 6f, where similar update results are shown for two different sample permeability fields (i.e., the time evolution of two example permeability realizations). The updated variance maps for the permeability ensemble are depicted in Figure 6g. The variability in the initial ensemble is gradually reduced with each update step as more MEQ data are integrated, while the permeability estimates recover the main trends in the field.

Figure 6.

EnKF estimation results for experiment 3 with a three-dimensional heterogeneous permeability model: (a) the true log permeability model, time evolution of (b) the log permeability RMSE and (c) normalized ensemble spread, (d) estimated mean of the log permeability ensemble at different data integration steps, (e and f) estimated maps of two individual log permeability realizations after selected update steps, and (g) time evolution of the corresponding ensemble log permeability standard deviation maps.

[39] The results of the above experiments (and many others with similar well setup but different reference permeability or spatially correlated criticality maps that were not included) indicate that the EnKF can be used to successfully infer permeability distributions from continuous interpretations (through KDE) of the discrete MEQ monitoring measurements. This outcome has important implications for characterization of subsurface reservoirs from MEQ events as an emerging monitoring technology in several important energy and environmental applications. While simple and easy to implement, the EnKF proves to be an effective model calibration tool for nonlinear problems where the optimality requirements of the original Kalman filter update equation, namely jointly Gaussian states and measurements and linear state-space model assumptions, are not strictly met.

[40] While the examples illustrated in this paper clearly show the feasibility of applying the EnKF to constrain permeability distributions based on microseismic event locations, we did not consider the seismic analysis step that is required to provide the MEQ sources locations. In addition to event locations, other information about the seismic source may be extracted from the raw seismic data (e.g., the magnitude of events) and be used to further constrain rock property distributions. In particular, when a coupled flow and geomechanical model is used to describe the behavior of the reservoir and rock failure, one can expect to correlate rock mechanical properties to observed microseismic events, and hence apply the proposed method for joint geomechanical and hydraulic parameter estimation. An important aspect that was not considered in this study is the presence, initiation and propagation of fractures in the rock during the hydraulic fracturing process. In general, microseismic events can carry important information about the location and geometrical attributes of the fractures, which can be exploited for fracture model calibration purposes.

4. Conclusion

[41] Seismicity-based reservoir characterization is a promising approach for monitoring and improving reservoir performance in a number of important energy and environmental applications. We formulated an EnKF-based model calibration approach to integrate discrete MEQ events into the description of reservoir permeability models. Since the EnKF is a continuous estimation approach, we introduced a new interpretation of the MEQ event locations as a continuous seismicity density map that is amenable to assimilation with the EnKF. A main advantage of the EnKF to previously introduced SBRC methods is that it is a stochastic inversion that provides an ensemble of solutions to facilitate uncertainty assessment. Other important advantages of the EnKF are the ability to systematically incorporate uncertainty in models and observations, and its generality for application under any forward model, failure criteria, and MEQ event triggering mechanisms. In addition, the simple and versatile implementation of the EnKF allows for estimation of different types of parameters from various data types.

[42] In this paper, we employed a previously used failure criterion for triggering MEQ event, based on the pore pressure relaxation hypothesis, and focused on developing a framework for automatic and robust integration of MEQ-type discrete data sets using the EnKF. However, in principle, more sophisticated failure criteria can be used with the EnKF formulation. An important property of the EnKF is that its sequential update scheme provides different representations of unknown parameters after each update. By construction, the EnKF is designed to update time-varying states of a system. In several emerging applications where field disturbances can be significant enough to change the rock physical properties (parameters) with time, the EnKF-type sequential filtering techniques shall prove quite useful for estimation of dynamically varying parameters.

[43] While the hypothesis and theoretical model for pore pressure relaxation provide a sound modeling framework, it uses several simplifying assumptions, such as homogeneous and isotropic medium and simplified random statistical distribution of rock criticality. The criticality distribution may also be modeled as a spatially correlated property, possibly correlated with other rock physical properties such as permeability distribution, which can only be known with significant uncertainty, or be considered as an additional unknown to be estimated. Also, detailed modeling and analysis of the geomechanical effects on the triggering mechanisms can lead to more sophisticated, quantitative and geomechanically supported failure criteria. With a more detailed modeling of the coupled flow and geomechanical processes comes an opportunity to use the MEQ data to also learn about geomechanical properties of the reservoir. While we only considered permeability estimation from injection-induced MEQ data, work is underway to use this data type to estimate other parameters such as geomechanical rock properties (e.g., tensile strength or Young's modulus) as well as rock criticality distribution. In addition, the proposed KDE approach for transforming the discrete MEQ data in this paper inevitably introduces some error into the estimation results. A more natural estimation approach for integration of MEQ data is one that does not convert the discrete events into continuous measurements. Developing discrete data integration algorithms can eliminate the discrete data quantification step and potentially lead to additional improvements in the estimation results. Here, we adopted a continuum approach and did not include fracture systems in the estimation. In general, one may need to characterize fractures and fracture networks as part of the model calibration process. Further refinements of the workflow may also be possible by including seismic modeling as one of the components in the inversion framework that can help better characterize the MEQ events both in terms of their distribution and intensity. Analyses of the raw microseismic data can lead to additional information about the induced fractures and their properties.

Acknowledgments

[44] This project was supported by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy under cooperative agreement DE-PS36-08GO18194. This support does not constitute an endorsement by the U.S. Department of Energy of the views expressed in this publication.

Ancillary