The concept of radius of investigation is fundamental to well test analysis and is routinely used to design well tests and to understand the reservoir volume investigated. The radius of investigation can also be useful in identifying new well locations and planning, designing and optimizing hydraulic fractures in unconventional reservoirs. It has additional implications in estimating reserves and understanding stimulated reservoir volumes. There are many definitions of radius of investigation in the literature and Kuchuk (2009) summarized them recently. Although these definitions vary in detail, they all relate to the propagation of a pressure disturbance or impose thresholds on detectable pressure or rate changes. In this article we will focus on the definition proposed by Lee (1982). Lee defines the radius of investigation as the propagation distance of the “peak” pressure disturbance for an impulse source or sink. For simplified flow geometries and homogeneous reservoir conditions, the radius of investigation can be calculated analytically. However, such analytic solutions are severely limited for heterogeneous and fractured reservoirs, particularly for unconventional reservoirs with multistage hydraulic fractures. Generalization of the Concept How can we generalize the concept of radius of investigation to heterogeneous reservoir conditions including unconventional reservoirs with horizontal wells and multistage hydraulic fractures? For such general situations, it will be more appropriately called “the depth of investigation” rather than the radius of investigation. The simplest, not necessarily the most desirable, approach will be to use a numerical reservoir simulator. For example, we can simulate a constant rate drawdown test and observe the pressure response at every grid block in the simulation model. It is as if, we have distributed sensors throughout the reservoir. We can now compute the time derivative of the pressure at each grid block and note the time when the derivative reaches a maximum. We can then simply contour this “peak” arrival time at every grid block. Note that because the constant rate test corresponds to a step function (from 0 to Q), its derivative is an impulse function. Thus, by contouring the arrival time of the maximum of the pressure derivative, we are actually looking at the arrival time of the maximum of an impulse response as defined by Lee (1982). How well does the approach work? Fig. 1a shows the evolution of the radius of investigation for homogeneous radial flow using Lee’s analytic solution. Fig. 1b shows the radius of investigation obtained from numerical simulation. We have superimposed the analytic solution (black lines) on the results from the numerical simulation. We do see a close correspondence, although the numerical results have difficulties resolving the pressure transients away from the well. In spite of its limitations, the numerical approach is very general and can be applied to arbitrary reservoir and well conditions. The computation time and expenses, however, make the numerical simulation approach unfeasible for routine applications.

We present a method for history matching and uncertainty quantification for channelized reservoir models using Level Set Method and Markov Chain Monte Carlo. Our objective is to efficiently sample realizations of the channelized permeability fields conditioned to the production data and facies observation at the wells. In our approach, the channel field boundary is first described by a level set function, e.g., a signed distance function or any other indicator function. By solving the level set equation (motion in a prescribed direction), we are able to gradually move the channel boundaries and evolve the channelized reservoir properties. Our approach allows representing facies via a parameterization of the velocity field that deforms the interface. Thus facies can be parameterized in the space of smooth velocity fields. The dimension reduction can be achieved for covariance-based velocity fields by re-parameterizing with SVD techniques. After parameterization, Markov Chain Monte Carlo method is utilized to perturb the coefficients of principal components of velocity field to update channel reservoir model matching production history. One advantage of this approach is that it is easy to condition the channel model to the facies observations at well locations by constraining the random velocity field to zero at well locations. To speed up the computation and improve the acceptance rate of the MCMC algorithm, we employ two stage methods where coarse-scale simulations are used to screen out the undesired proposals. The MCMC algorithms naturally provide multiple realizations of the permeability field conditioned to well and production data and thus, allow for uncertainty assessment in the forecasting. We demonstrate the effectiveness of the level set MCMC algorithm using both 2D and 3D examples involving waterflood history matching.

Gas flow in shale gas reservoirs occurs primarily from ultra low permeability shale rocks through a complex network of natural and induced hydraulic fractures. Consequently, fracture parameters (conductivity and half length), fracture location and distribution are the dominant factors influencing well drainage volumes and shale gas well performance. Stimulated reservoir volume or SRV, estimated from microseismic event clouds or rate/pressure transient analysis, describes a measurement of overall reservoir volume impacted by fracture treatments. With SRV as well as the dynamic production/pressure response, reservoir simulation models can be calibrated to actual well performance in shale gas reservoirs leading to improved understanding, forecasting and future well placement.In this paper, we first introduce a novel approach for computing well drainage volume for shale gas wells with multistage fractures and fracture clusters. Next, we calibrate the shale gas reservoir model by matching the drainage volume with the SRV within specified confidence limits. The matching of the SRV is done in addition to the traditional history matching of production/pressure response and further constrains the estimation of fracture parameters. An evolutionary algorithm with design of experiments is used for the assisted history matching. Sensitivities to various parameters such as fracture conductivity, fracture half lengths and rock compaction have also been investigated. The proposed approach has been applied to a generic shale gas well designed after a real field case. The results clearly indicate the benefits of including SRV during history matching, leading to improved fracture/matrix parameter estimation and performance forecasting. Our proposed approach provides an important tool that can be used to optimize well placement, fracture treatments and improve the economics of shale gas plays.