Nonlocal Control of Fracture Propagation in Numerical Simulations

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 201346, “Are We Overstimulating Our Laterals? Evaluating Completion Design Practices Based on Field Offset Well-Pressure Measurements,” by Puneet Seth, SPE, The University of Texas at Austin, and Brendan Elliott, SPE, and Trevor Ingle, SPE, Devon Energy, et al., prepared for the 2020 SPE Annual Technical Conference and Exhibition, originally scheduled to be held in Denver, Colorado, 5–7 October. The paper has not been peer reviewed. Increased injection volumes coupled with a suboptimal completion design can lead to overstimulation at current well-spacing densities. In the complete paper, the authors analyze offset well-pressure measurements in the Permian Basin to evaluate if a fracturing job is overstimulated. Additionally, numerical modeling studies are performed to evaluate the extent of overstimulation in different scenarios and provide recommendations to maximize the capital efficiency of a fracturing job. In their analysis, the authors focus on the scenario in which fracturing hits occur when child-well fractures intersect with the parent well. Field Data Analysis Pumping for the full designed volume and time (typically 90 minutes) according to well-stimulation procedures is currently common in the industry. Often, the observation of hydraulic interactions is not coupled with a decision to alter or change the stimulation. The authors analyzed the offset well-pressure response monitored with a surface pressure gauge in multiple parent wells in the Permian Basin during stimulation in nearby child wells. The child wells were stimulated after roughly 1 year of production from the parent wells. The focus of this study was to identify fracture-driven interactions—specifically the timing of intersection of the child-well fractures with the offset parent wells, which are recorded as massive hydraulic pressure responses. The results of this analysis for different well pairs are presented in the complete paper. To better understand the factors that affect fracture propagation from the child wells toward the parent wells, fracture arrival times, and capital efficiency of a fracturing job, a series of numerical simulations was performed with a fully coupled hydraulic fracturing simulator. Simulation Results Numerical simulations were performed using an integrated hydraulic fracturing and reservoir simulator developed at The University of Texas at Austin. This simulator solves for flow and geomechanics in the reservoir, fracture, and wellbore domains in a tightly coupled manner. Hydraulic fractures are modeled as compliant discontinuities in the reservoir rather than high-permeability gridblocks. This is important in order to capture the stress alterations around a propagating fracture accurately. Effect of Parent-Well Production (Depleted Region). For this study, two scenarios were analyzed. In the first case, fracture propagation from a child well stimulated near a recently fractured unproduced parent well (no depletion) was considered. In this case, the fracture from the child well propagates away from the parent well because of elevated stresses near the parent well. In the second case, a child well is stimulated near a parent well that has been producing for 300 days before child-well stimulation. In this scenario, the child-well fracture propagates toward the parent well because of a depleted region that develops near the parent well (because of production) and relaxes the reservoir stresses around the parent well. This causes the child-well fracture to grow preferentially toward the parent well (toward the low-stress region). In fact, in this scenario, as the fracture reaches the depleted reservoir region, its growth accelerates toward the parent well and intersects with the parent well. Even minor depletion can induce asymmetric growth of infill child-well fractures toward the parent well.

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A Γ-convergence result for fluid-filled fracture propagation

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2020016 ◽

2020 ◽

Vol 54 (3) ◽

pp. 1003-1023

Author(s):

Annika Bach ◽

Liesel Sommer

Keyword(s):

Asymptotic Analysis ◽

Numerical Simulations ◽

Discontinuous Galerkin ◽

Phase Field ◽

Field Model ◽

Phase Field Model ◽

Fracture Propagation ◽

Convergence Result ◽

Elastic Media ◽

Γ Convergence

In this paper we provide a rigorous asymptotic analysis of a phase-field model used to simulate pressure-driven fracture propagation in poro-elastic media. More precisely, assuming a given pressure p ∈ W 1,∞ (Ω) we show that functionals of the form $$ E(\vec{u})={\int }_{\mathrm{\Omega }} e(\vec{u}):\mathbb{C}e(\vec{u})+p\nabla \cdot \vec{u}+\left\langle \nabla p,\vec{u}\right\rangle\enspace \mathrm{d}x+{\mathcal{H}}^{n-1}({J}_{\vec{u}}),\enspace \vec{u}\in \mathrm{G}{SBD}(\mathrm{\Omega })\cap {L}^1(\mathrm{\Omega };{\mathbb{R}}^n) $$ can be approximated in terms of Γ-convergence by a sequence of phase-field functionals, which are suitable for numerical simulations. The Γ-convergence result is complemented by a numerical example where the phase-field model is implemented using a Discontinuous Galerkin Discretization.

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Direct Numerical Simulations of Gas–Liquid Multiphase Flows

10.1017/cbo9780511975264 ◽

2001 ◽

Cited By ~ 151

Author(s):

Grétar Tryggvason ◽

Ruben Scardovelli ◽

Stéphane Zaleski

Keyword(s):

Numerical Simulations ◽

Multiphase Flows ◽

Direct Numerical Simulations

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Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

Astronomy and Astrophysics ◽

10.1051/0004-6361/202038023 ◽

2020 ◽

Vol 640 ◽

pp. A53

Author(s):

L. Löhnert ◽

S. Krätschmer ◽

A. G. Peeters

Keyword(s):

Growth Rate ◽

Numerical Simulations ◽

Accretion Disks ◽

Gravitational Instability ◽

Turbulent Viscosity ◽

Radial Distance ◽

Maximum Growth ◽

Wave Number ◽

Radiative Cooling ◽

Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

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