Reduced order model for simultaneous growth of multiple closely-spaced radial hydraulic fractures
Introduction
Reduced order models (ROMs) have a great potential for enabling optimization and uncertainty quantification for hydraulic fracturing. However, ascertaining the essential ingredients necessary for a reasonably accurate and suitable efficient ROM for simulating systems of multiple, simultaneously-growing hydraulic fractures remains a challenging and open problem.
Hydraulic fracturing (HF) is a well stimulation technique used in many industrial applications include mining, waste disposal, and enhanced geothermal systems [1], [2], [3] The most well-known application is its use for increasing the rate at which oil and gas can be extracted from wells. In this application, pressurized fluid drives growth of cracks through the reservoir rock, carrying granular proppant that is left behind in the created fractures. The resulting high conductivity pathways promote an increased flow of hydrocarbons from the reservoir formation towards the well (as described in further detail by e.g. [4]). Both vertical and horizontal wells are stimulated in this way, with vertical well simulation comprising most cases over the 70 year history of hydraulic fracturing and horizontal well fracturing comprising the essential advance for unlocking unconventional (low-permeability) resources in the past two to three decades [5]. Essentially all horizontal wells in unconventional reservoirs (such as shale gas and oil) are treated by hydraulic fracturing, and the most common approach is to stimulate in a sequential manner from the “toe” to the “heel” of the well (see description in e.g. [6]). Within each of these sequential “stages”, multiple clusters of perforations comprise the reservoir entry points, with the intention that injected fluid is reasonably uniformly distributed among these possible entry points, thereby uniformly stimulating the reservoir rock. Although such a multistage technique has enabled tremendous cost savings, analysis of production logs over several basins tends to show that between 20 to 40 percent of perforation clusters do not contribute to production [7], indicating current simulation strategies are highly non-optimal. One contributing factor is the non-uniformity of reservoir properties, including the in-situ stresses along the well e.g. [8], [9]. Another factor is almost certainly the widely recognized phenomenon known of “stress shadowing” (see e.g. field evidence in [10]). Stress shadowing refers to suppression of some HFs as a result of the compressive stresses exerted on them by other, nearby HFs (e.g. [11], [12], [13]). One result is that the ideal case of uniform hydraulic fracture growth (Fig. 1(a)) is probably never achieved. Instead, some hydraulic fractures are suppressed due to the presence of locally elevated compressive stress (Fig. 1(b) as previously discussed by e.g. [14], see also. [11], [12], [15], [16], [17], [18]).
While there are certainly demonstrations showing use of hydraulic fracture simulators to identify approaches that improve uniformity of stimulation (see e.g. [6], [19]), optimization is challenging because of the simulations' computational intensity. Overcoming this challenge has opened a growing area of interest in generating reduced order models for hydraulic fractures, for example following formalisms that involve order reduction via an empirical search for eigenfunction bases that can be used to capture system behavior over some subdivision of the time domain [20], [21], [22], [23], [24]. Here we follow a different approach, but the goal is the same, namely, to obtain a reduced order model that provides a useful approximation to the full model, and with the key feature being capturing interaction of simultaneously growing hydraulic fractures.
While there are several possible threads in the literature that aim generally at simulating and optimizing multistage completions, here we will briefly introduce the background most relevant to the current contribution. The Implicit Level Set Algorithm, or “ILSA” [25] was extended by [19] for multiple parallel-planar HFs, including full 3D elastic coupling between the simultaneously propagating fractures (“ILSA II”). This simulator has been used to demonstrate that the stress shadow effect can be reduced with appropriate placement of interior HFs close to the outer HFs to inhibit their growth relative to the other fractures in the array.
Although ILSA II is a fully coupled benchmark simulator (to use terminology commonly contrasted with ROMs, we also can call this a “large scale” or “high-fidelity” model), implementing state of the art approaches to enable accurate calculations on very coarse meshes, the model can require several days, and sometimes over one week, to compute a multi-fracture result at typical reservoir length and time scales (note timing is for single node calculations, ∼2.5 GHz processor speed). Hence, optimization of HF design, which can require hundreds or thousands of model runs, is not practical with this or other models with run times on the order of tens of hours to days. Similarly, uncertain quantification, which also can require thousands or model evaluations, is not typically possible. A first step is, therefore, addressing the need for rapid, even if approximate, simulation. Such ROM simulators can be used to do broad explorations of high dimensional parametric spaces, identifying combinations of parameters, which can be examined in detail by a few, fully-coupled simulations.
We previously demonstrated the feasibility and basic concept of a new HF simulator, called “C2Frac”, which very rapidly estimates the growth of an array of HFs [26]. In this prototype model, the HFs are restricted to radial, planar growth – as in the current version presented here – but under the additional limitation that fractures remain small in radius compared to their separation. The method uses semi-analytical HF solutions (after [27]), coupling a far field approximation of the interaction stress via an overall energy balance. In this way, the model predicts each HF's aperture (t), net pressure (t), radius (t), and inflow rate (t) for different choices of uniform or non-uniform spacing among N HFs. The validating shows good agreement between C2Frac and ILSA II benchmarks, however, because of the use of a far-field approximation of the interaction stress between the HFs, the C2Frac estimates diverge from fully coupled benchmark solutions when the fracture radii become similar to the fracture spacing. Additionally, because the prototype model does not account for near field stress interaction, it does not capture some of the complex behaviors predicted by fully coupled simulations when the fracture spacing is non-uniform. In particular, the previous model cannot capture when the interior fractures switch from being suppressed to accepting the majority of the fluid, as observed in fully-coupled simulations by [19]. Simulating this phase is essential for obtaining accurate predictions, but it can only be captured when the impacts of near field stress interaction between very closely spaced fractures are appropriately modeled.
The necessary model improvements are here enabled by developing a new algorithm leading to numerical simulations approximating the benchmark solutions for all times, regardless of fracture radius and spacing, while running times faster than the fully coupled benchmark simulator. In this paper, the new model, called “C3Frac”, is developed and validated. We begin by presenting the governing equations. We then introduce a new approach to approximation of the interaction stress from each fracture based on a uniformly pressurized crack with equal volume and radius to the actual HF. Next, we describe an interaction stress coupled elasticity function, which preserves volume balance by ensuring the elasticity solution is consistent with the inlet flow rate boundary condition. Then, the system of governing equations is completed by requiring that the fluid is partitioned among the multiple entry points so as to maintain equality of the wellbore pressure predicted for each fracture while also conserving the fluid injected into the wellbore. These final conditions are required by both the fully coupled and approximate simulator. In the case of the fully coupled simulator the wellbore pressure is predicted by carefully simulating fluid flow at all locations within the fracture so as to obtain an accurate estimate of the pressure at the fracture inlet (wellbore). In contrast, the approximate simulator approximates the fluid flow in a manner preserving the main contribution to viscous energy dissipation and then predicts the inlet pressure for each fracture using global energy balance.
After presenting the model, we next show how well it approximates the fully coupled simulations. Following this validating, numerical experiments illustrate cases for uniform and non-uniform spacing designs to indicate how spacing effects the hydraulic fracture growth. Thus, we utilize the new C3Frac model to search for optimized HF scenarios in terms of created fracture surface area, providing examples of optimized designs for different stage lengths, inflow rates, and pumping times. The work concludes with a demonstration of the benefits of optimization and the potential for non-uniform fracture spacing to promote multiple methods for promoting multiple HF growth.
Section snippets
Governing equations
In a typical HF treatment of an oil or gas well, one or more fractures is/are created by injection of fluid. The fracture is initiated within a rock formation that contains the hydrocarbons (the reservoir), and propagates perpendicularly to the orientation of the minimum in situ confining stress . Here the HFs are considered to grow transversely to a horizontal well, as illustrated by Fig. 1. This model accounts for the growth of N fractures within a single stage and, for now, neglects the
Interaction stress approximation
The main challenge and interest of the problem is due to HF interaction. In general, the interaction stresses need to be computed based on the details of the pressure distribution inside each HF (as in e.g. [19]). However, such an approach is not compatible with the desire for rapid, approximate computation. So, for this model, we propose an approximation of the interaction stress using the uniformly-pressurized crack solution of [49], whereby the normal component of stress performed by
Validating and overall behavior of the solution
We validate and illustrate the use of the model considering cases with 5 HFs. The fractures are placed symmetrically relative to the middle fracture. Hence the “outer” fractures, 1 and 5, are identical. So also the “inner” fractures, 2 and 4, are identical. Fracture 3 always occupies the center of the array and will henceforth be called the “middle” fracture. The validating is comprised of comparison of the C3Frac approximations (ROM) to fully coupled large-scale (“high fidelity”) simulations
Parametric study
A few examples illustrate the optimization enabled by the rapid computation times associated with C3Frac. The metric by which we evaluate the performance of a given configuration is taken as the total surface area of all the fractures in the array until time t, which we represent by A(). It is useful to normalize by (t), the total fracture area of N non-interacting fractures each taking the same total volume of fluid and growing exactly uniformly according to the relevant analytical
Conclusion
A new approximate ROM simulator, C3Frac, rapidly predicts how mechanical interaction among simultaneously growing radial hydraulic fractures effects their growth. This approximate simulation method is based on preserving global volume and energy balance and the elastically-determined crack opening while approximating the fluid flow via a functional form preserving the pressure gradient near the inlet and tip and approximating the interaction stresses based on the analytical solution for
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2021, Computers and GeotechnicsMulti-fracture interactions during two-phase flow of oil and water in deformable tight sandstone oil reservoirs
2020, Journal of Rock Mechanics and Geotechnical EngineeringCitation Excerpt :In view of negative effects, interactions of multi-fractures constrain the initiation and propagation of some fractures during hydraulic fracturing. The fracture aperture is reduced by interactions, particularly for fractures in the interior of one cluster, leading to increasing perforation friction and entry resistance of subsequent injected fluid flowing into the fracture and further decreasing proppant conductivity (Weng, 1993; Bunger et al., 2013b; Vahab and Khalili, 2018; Zhou et al., 2018; Cheng and Bunger, 2019). Meanwhile, the unexpected non-uniform flow rates partitioned among perforations are formed (Zhao et al., 2016; Cheng and Bunger, 2019).
Applications of the fast multipole fully coupled poroelastic displacement discontinuity method to hydraulic fracturing problems
2019, Journal of Computational PhysicsCitation Excerpt :Naturally, this limits its application in hydraulic fracturing of elastic, rather than the realistic poroelastic media, and fractured media. The issues associated with BEM can be addressed by approximating the solutions using reduced order models [55,11], or by using the concept of back stress [15,16]. Another approach is to expedite the dense matrix-vector multiplications by the fast multipole method (FMM).