Flow in Fractured Porous Media Modeled in Closed-Form: Augmentation of Prior Solution and Side-Stepping Inconvenient Branch Cut Locations

Carefully chosen complex variable formulations can solve flow in fractured porous media. Such a calculus approach is attractive, because the gridless method allows for fast, high-resolution model results. Previously developed complex potentials to describe flow in porous media with discrete heterogeneities such as natural fractures can be modified to expand the accuracy of the solution range. The prior solution became increasingly inaccurate for flows with fractures oriented at larger angles with respect to the far-field flow. The modified solution, presented here, based on complex analysis methods (CAM), removes the limitation of the earlier solution. Benefits of the CAM model are (1) infinite resolution, and (2) speed of use, as no gridding is required. Being gridless and meshless, the CAM model is computationally faster than integration methods based on solutions across discrete volumes. However, branch cut effects may occur in impractical locations due to mathematical singularities. This paper demonstrates how the augmented formulation corrects physically unfeasible refraction of streamlines across high-permeability bands (natural fractures) oriented at high angles with respect to a far-field flow. The current solution is an important repair. An application shows how a drained rock volume in hydraulically fractured hydrocarbon wells will be affected by the presence of natural fractures.

Fast Models of Hydrocarbon Migration Paths and Pressure Depletion Based on Complex Analysis Methods (CAM): Mini-Review and Verification

A recently developed code to model hydrocarbon migration and convective time of flight makes use of complex analysis methods (CAM) paired with Eulerian particle tracking. Because the method uses new algorithms that are uniquely developed by our research group, validation of the fast CAM solutions with independent methods is merited. Particle path solutions were compared with independent solutions methods (Eclipse). These prior and new benchmarks are briefly summarized here to further verify the results obtained with CAM codes. Pressure field solutions based on CAM are compared with independent embedded discrete fracture method (EDFM) solutions. The CAM method is particularly attractive because its grid-less nature offers fast computation times and unlimited resolution. The method is particularly well suited for solving a variety of practical field development problems. Examples are given for fast optimization of waterflood patterns. Another successful application area is the modeling of fluid withdrawal patterns in hydraulically fractured wells. Because no gridding is required, the CAM model can compute the evolution of the drained rock volume (DRV) for an unlimited (but finite) number of both hydraulic and natural fractures. Such computations of the DRV are based on the convective time of flight and show the fluid withdrawal zone in the reservoir. In contrast, pressure depletion models are based on the diffusive time of flight. In ultra-low permeability reservoirs, the pressure depletion zones do not correspond to the DRV, because the convective and diffusive displacement rates differ over an order of magnitude (diffusive time of flight being the fastest). Therefore, pressure depletion models vastly overestimate the drained volume in shale reservoirs, which is why fracture and well spacing decisions should be based on both pressure depletion and DRV models, not pressure only.

Impact on Drained Rock Volume (DRV) of Storativity and Enhanced Permeability in Naturally Fractured Reservoirs: Upscaled Field Case from Hydraulic Fracturing Test Site (HFTS), Wolfcamp Formation, Midland Basin, West Texas

Hydraulic fracturing for economic production from unconventional reservoirs is subject to many subsurface uncertainties. One such uncertainty is the impact of natural fractures in the vicinity of hydraulic fractures in the reservoir on flow and thus the actual drained rock volume (DRV). We delineate three fundamental processes by which natural fractures can impact flow. Two of these mechanisms are due to the possibility of natural fracture networks to possess (i) enhanced permeability and (ii) enhanced storativity. A systematic approach was used to model the effects of these two mechanisms on flow patterns and drained regions in the reservoir. A third mechanism by which natural fractures may impact reservoir flow is by the reactivation of natural fractures that become extensions of the hydraulic fracture network. The DRV for all three mechanisms can be modeled in flow simulations based on Complex Analysis Methods (CAM), which offer infinite resolution down to a micro-fracture scale, and is thus complementary to numerical simulation methods. In addition to synthetic models, reservoir and natural fracture data from the Hydraulic Fracturing Test Site (Wolfcamp Formation, Midland Basin) were used to determine the real-world impact of natural fractures on drainage patterns in the reservoir. The spatial location and variability in the DRV was more influenced by the natural fracture enhanced permeability than enhanced storativity (related to enhanced porosity). A Carman–Kozeny correlation was used to relate porosity and permeability in the natural fractures. Our study introduces a groundbreaking upscaling procedure for flows with a high number of natural fractures, by combining object-based and flow-based upscaling methods. A key insight is that channeling of flow through natural fractures left undrained areas in the matrix between the fractures. The flow models presented in this study can be implemented to make quick and informed decisions regarding where any undrained volume occurs, which can then be targeted for refracturing. With the method outlined in our study, one can determine the impact and influence of natural fracture sets on the actual drained volume and where the drainage is focused. The DRV analysis of naturally fractured reservoirs will help to better determine the optimum hydraulic fracture design and well spacing to achieve the most efficient recovery rates.