Backflow from a model fracture network: an asymptotic investigation

2019 ◽  
Vol 864 ◽  
pp. 899-924 ◽  
Author(s):  
Asaf Dana ◽  
Gunnar G. Peng ◽  
Howard A. Stone ◽  
Herbert E. Huppert ◽  
Guy Z. Ramon
Keyword(s):  
Fracture Network ◽  
Relaxation Processes ◽  
Late Time ◽  
Model Structure ◽  
Time Dynamics ◽  
Fracture Length ◽  
Drainage Time ◽  
Asymptotic Investigation ◽  
Self Similar ◽  
By Products

We develop a model for predicting the flow resulting from the relaxation of pre-strained, fluid-filled, elastic network structures. This model may be useful for understanding relaxation processes in various systems, e.g. deformable microfluidic systems or by-products from hydraulic fracturing operations. The analysis is aimed at elucidating features that may provide insight on the rate of fluid drainage from fracturing operations. The model structure is a bifurcating network made of fractures with uniform length and elastic modulus, which allows for general self-similar branching and variation in fracture length and rigidity between fractures along the flow path. A late-time $t^{-1/3}$ power law is attained and the physical behaviour can be classified into four distinct regimes that describe the late-time dynamics based on the location of the bulk of the fluid volume (which shifts away from the outlet as branching is increased) and pressure drop (which shifts away from the outlet as rigidity is increased upstream) along the network. We develop asymptotic solutions for each of the regimes, predicting the late-time flux and evolution of the pressure distribution. The effects of the various parameters on the outlet flux and the network’s drainage efficiency are investigated and show that added branching and a decrease in rigidity upstream tend to increase drainage time.

Dynamics of viscous backflow from a model fracture network

2017 ◽  
Vol 836 ◽  
pp. 828-849 ◽  
Author(s):  
Asaf Dana ◽  
Zhong Zheng ◽  
Gunnar G. Peng ◽  
Howard A. Stone ◽  
Herbert E. Huppert ◽  
...  
Keyword(s):  
Oil And Gas ◽  
Produced Water ◽  
Numerical Solutions ◽  
Fracture Network ◽  
Maximum Pressure ◽  
Single Fracture ◽  
Relaxation Processes ◽  
Late Time ◽  
Network Systems ◽  
Time Required

Hydraulic fracturing for production of oil and gas from shale formations releases fluid waste, by-products that must be managed carefully to avoid significant harm to human health and the environment. These fluids are presumed to result from a variety of fracture relaxation processes, and are commonly referred to as ‘flowback’ and ‘produced water’, depending primarily on the time scale of their appearance. Here, a model is presented for investigating the dynamics of backflows caused by the elastic relaxation of a pre-strained medium, namely a single fracture and two model fracture network systems: a single bifurcated channel and its generalization for $n$ bifurcated fracture generations. Early- and late-time asymptotic solutions are obtained for the model problems and agree well with numerical solutions. In the late-time period, the fracture apertures and backflow rates exhibit a time dependence of $t^{-1/3}$ and $t^{-4/3}$, respectively. In addition, the pressure distributions collapse to universal curves when scaled by the maximum pressure in the system, which we calculate as a function of $n$. The pressure gradient along the network is steepest near the outlet while the bulk of the network serves as a ‘reservoir’. Fracture networks with larger $n$ are less efficient at evicting fluids, manifested through a longer time required for a given fractional reduction of the initial volume. The developed framework may be useful for informing engineering design and environmental regulations.

scholarly journals Averaging generalized scalar field cosmologies II: locally rotationally symmetric Bianchi I and flat Friedmann–Lemaître–Robertson–Walker models

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Genly Leon ◽  
Sebastián Cuéllar ◽  
Esteban González ◽  
Samuel Lepe ◽  
Claudio Michea ◽  
...  
Keyword(s):  
Scalar Field ◽  
Nonlinear Dynamical Systems ◽  
Perturbation Parameter ◽  
Time Dependent ◽  
Late Time ◽  
De Sitter ◽  
Time Dynamics ◽  
Locally Rotationally Symmetric ◽  
Bianchi I ◽  
Rotationally Symmetric

AbstractScalar field cosmologies with a generalized harmonic potential and a matter fluid with a barotropic equation of state (EoS) with barotropic index $$\gamma $$ γ for the locally rotationally symmetric (LRS) Bianchi I and flat Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. Methods from the theory of averaging of nonlinear dynamical systems are used to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, the simplest time-averaged system determines the future asymptotic behavior. Depending on the values of $$\gamma $$ γ , the late-time attractors of physical interests are flat quintessence dominated FLRW universe and Einstein-de Sitter solution. With this approach, the oscillations entering the system through the Klein–Gordon (KG) equation can be controlled and smoothed out as the Hubble parameter H – acting as time-dependent perturbation parameter – tends monotonically to zero. Numerical simulations are presented as evidence of such behavior.

scholarly journals A New Approach for Automatic Optimization of Complex Fracture Network in Shale Reservoirs

Lithosphere ◽  
2021 ◽  
Vol 2021 (Special 1) ◽  
Author(s):  
Haibo Wang ◽  
Tong Zhou ◽  
Fengxia Li
Keyword(s):  
Shale Gas ◽  
Flow Model ◽  
Optimization Method ◽  
Fracture Network ◽  
Gas Reservoirs ◽  
Fracture Length ◽  
Complex Fracture ◽  
Shale Gas Reservoirs ◽  
Automatic Optimization ◽  
Complex Fracture Network

Abstract Shale gas reservoirs have gradually become the main source for oil and gas production. The automatic optimization technology of complex fracture network in fractured horizontal wells is the key technology to realize the efficient development of shale gas reservoirs. In this paper, based on the flow model of shale gas reservoirs, the porosity/permeability of the matrix system and natural fracture system is characterized. The fracture network morphology is finely characterized by the fracture network expansion calculation method, and the flow model was proposed and solved. On this basis, the influence of matrix permeability, matrix porosity, fracture permeability, fracture porosity, and fracture length on the production of shale gas reservoirs is studied. The optimal design of fracture length and fracture location was carried, and the automatic optimization method of complex fracture network parameters based on simultaneous perturbation stochastic approximation (SPSA) was proposed. The method was applied in a shale gas reservoir, and the results showed that the proposed automatic optimization method of the complex fracture network in shale gas reservoirs can automatically optimize the parameters such as fracture location and fracture length and obtain the optimal fracture network distribution matching with geological conditions.

Best Practices in DFIT Interpretation: Comparative Analysis of 62 DFITs from Nine Different Shale Plays

2022 ◽  
Author(s):  
Mark Mcclure ◽  
Garrett Fowler ◽  
Matteo Picone
Keyword(s):  
Pressure Drop ◽  
Pore Pressure ◽  
Time Pressure ◽  
Early Time ◽  
Horizontal Wells ◽  
Late Time ◽  
Fracture Length ◽  
Well Spacing ◽  
Net Pressure ◽  
The Difference

Abstract In URTeC-123-2019, a group of operators and service companies presented a step-by-step procedure for interpretation of diagnostic fracture injection tests (DFITs). The procedure has now been applied on a wide variety of data across North and South America. This paper statistically summarizes results from 62 of these DFITs, contributed by ten operators spanning nine different shale plays. URTeC-123-2019 made several novel claims, which are tested and validated in this paper. We find that: (1) a ‘compliance method’ closure signature is apparent in the significant majority of DFITs; (2) in horizontal wells, early time pressure drop due to near-wellbore/midfield tortuosity is substantial and varies greatly, from 500 to 6000+ psi; (3) in vertical wells, early-time pressure drop is far weaker; this supports the interpretation that early- time pressure drop in horizontal wells is caused by near-wellbore/midfield tortuosity from transverse fracture propagation; (4) the (not recommended) tangent method of estimating closure yields Shmin estimates that are 100-1000+ psi lower than the estimate from the (recommended) compliance method; the implied net pressure values are 2.5x higher on average and up to 5-6x higher; (5) as predicted by theory, the difference between the tangent and compliance stress and net pressure estimates increases in formations with greater difference between Shmin and pore pressure; (6) the h-function and G-function methods allow permeability to be estimated from truncated data that never reaches late-time impulse flow; comparison shows that they give results that are close to the permeability estimates from impulse linear flow; (7) false radial flow signatures occur in the significant majority of gas shale DFITs, and are rare in oil shale DFITs; (8) if false radial signatures are used to estimate permeability, they tend to overestimate permeability, often by 100x or more; (9) the holistic-method permeability correlation overestimates permeability by 10-1000x; (10) in tests that do not reach late-time impulse transients, it is reasonable to make an approximate pore pressure estimate by extrapolating the pressure from the peak in t*dP/dt using a scaling of t^(-1/2) in oil shales and t^(3/4) in gas shales. The findings have direct practical implications for operators. Accurate permeability estimates are needed for calculating effective fracture length and for optimizing well spacing and frac design. Accurate stress estimation is fundamental to hydraulic fracture design and other geomechanics applications.

scholarly journals Universal late-time dynamics in isolated one-dimensional statistical systems with topological excitations

2020 ◽  
Vol 101 (10) ◽  
Author(s):  
Alvise Bastianello ◽  
Alessio Chiocchetta ◽  
Leticia F. Cugliandolo ◽  
Andrea Gambassi
Keyword(s):  
Late Time ◽  
One Dimensional ◽  
Topological Excitations ◽  
Time Dynamics ◽  
Statistical Systems

ANALYTICAL SOLUTIONS FOR PERMEABILITY OF A THREE-DIMENSIONAL FRACTAL-LIKE TREE NETWORK MODEL WITH FRACTURES HAVING VARIABLE WIDTHS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050013
Author(s):  
RICHENG LIU ◽  
LIYUAN YU ◽  
YANG GAO ◽  
MING HE ◽  
YUJING JIANG
Keyword(s):  
Network Model ◽  
Analytical Solutions ◽  
Constant Width ◽  
Fracture Network ◽  
Length Ratio ◽  
Width Ratio ◽  
Fracture Aperture ◽  
Fracture Width ◽  
Fracture Length ◽  
Aperture Ratio

This study proposed analytical solutions for permeability of a fractal-like tree network model with fractures having variable widths, which has not been reported before, if any. This model is more realistic with natural fracture networks than the traditional constant width fracture network models. The results show that considering fracture width variations decreases the permeability. Taking the fracture width ratio that equals to 0.6 and the total number of branching levels that equals to 30 as an example, the permeability decreases by more than three orders of magnitude with respect to that of a constant width fracture network model. The fracture length ratio plays a more significant role in permeability when it is larger than 0.8 than that is less than 0.8. The permeability is more sensitive to the fracture aperture ratio that is less than 0.8. When the total number of branching levels is large (i.e. 30), the permeability changes significantly (i.e. more than three orders of magnitude); whereas when the total number of branching levels is small (i.e. 5), the permeability varies in a small range (i.e. less than one order of magnitude). When taking into account the relationships among fracture length ratio, fracture aperture ratio and fracture width ratio, the parameters can be easily obtained and analytical solutions for permeability can also be easily derived. The empirical function for predicting critical hydraulic gradient is proposed, which can be used to estimate whether the fluid flow is within the linear flow regime and whether the proposed analytical solutions are applicable in the present study.

scholarly journals A new method for probing the late-time dynamics in the Lorentzian type IIB matrix model

2017 ◽  
Vol 2017 (8) ◽  
Author(s):  
Takehiro Azuma ◽  
Yuta Ito ◽  
Jun Nishimura ◽  
Asato Tsuchiya
Keyword(s):  
Matrix Model ◽  
New Method ◽  
Late Time ◽  
Time Dynamics

scholarly journals Dynamics of the cerebral blood flow response to brief neural activity in human visual cortex

2019 ◽  
Vol 40 (9) ◽  
pp. 1823-1837 ◽  
Author(s):  
Jung Hwan Kim ◽  
Amanda J Taylor ◽  
Danny JJ Wang ◽  
Xiaowei Zou ◽  
David Ress
Keyword(s):  
Blood Flow ◽  
Cerebral Blood Flow ◽  
Visual Cortex ◽  
Temporal Dynamics ◽  
Oxygen Metabolism ◽  
Late Time ◽  
Neural Activation ◽  
Hemodynamic Response Function ◽  
Time Dynamics ◽  
Human Visual Cortex

The blood oxygen-level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal depends on an interplay of cerebral blood flow (CBF), oxygen metabolism, and cerebral blood volume. Despite wide usage of BOLD fMRI, it is not clear how these physiological components create the BOLD signal. Here, baseline CBF and its dynamics evoked by a brief stimulus (2 s) in human visual cortex were measured at 3T. We found a stereotypical CBF response: immediate increase, rising to a peak a few second after the stimulus, followed by a significant undershoot. The BOLD hemodynamic response function (HRF) was also measured in the same session. Strong correlations between HRF and CBF peak responses indicate that the flow responses evoked by neural activation in nearby gray matter drive the early HRF. Remarkably, peak CBF and HRF were also strongly modulated by baseline perfusion. The CBF undershoot was reliable and significantly correlated with the HRF undershoot. However, late-time dynamics of the HRF and CBF suggest that oxygen metabolism can also contribute to the HRF undershoot. Combined measurement of the CBF and HRF for brief neural activation is a useful tool to understand the temporal dynamics of neurovascular and neurometabolic coupling.

scholarly journals A general buoyancy–drag model for the evolution of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities

2003 ◽  
Vol 21 (3) ◽  
pp. 347-353 ◽  
Author(s):  
YAIR SREBRO ◽  
YONI ELBAZ ◽  
OREN SADOT ◽  
LIOR ARAZI ◽  
DOV SHVARTS
Keyword(s):  
Temporal Evolution ◽  
Random Noise ◽  
Single Mode ◽  
Time Dependent ◽  
Late Time ◽  
Random Perturbations ◽  
Linear Stage ◽  
Drag Model ◽  
Characteristic Wavelength ◽  
Self Similar

The growth of a single-mode perturbation is described by a buoyancy–drag equation, which describes all instability stages (linear, nonlinear and asymptotic) at time-dependent Atwood number and acceleration profile. The evolution of a multimode spectrum of perturbations from a short wavelength random noise is described using a single characteristic wavelength. The temporal evolution of this wavelength allows the description of both the linear stage and the late time self-similar behavior. Model results are compared to full two-dimensional numerical simulations and shock-tube experiments of random perturbations, studying the various stages of the evolution. Extensions to the model for more complicated flows are suggested.

Sign in / Sign up

Export Citation Format

Share Document