scholarly journals An Analytical Solution of Partially Penetrating Hydraulic Fractures in a Box-Shaped Reservoir

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
He Zhang ◽  
Xiaodong Wang ◽  
Lei Wang
Keyword(s):  
Analytical Solution ◽  
Transient Behavior ◽  
Hydraulic Fractures ◽  
Pressure Derivative ◽  
Laplace Domain ◽  
Fracture Length ◽  
Dirac Function ◽  
Diffusivity Equation ◽  
Fracture Height ◽  
First Time

This paper presents a new method to give an analytical solution in Laplace domain directly that is used to describe pressure transient behavior of partially penetrating hydraulic fractures in a box-shaped reservoir with closed boundaries. The basic building block of the method is to solve diffusivity equation with the integration of Dirac function over the distance that is presented for the first time. Different from the traditional method of using the source solution and Green’s function presented by Gringarten and Ramey, this paper uses Laplace transform and Fourier transform to solve the diffusivity equation and the analytical solution obtained is accurate and simple. The effects of parameters including fracture height, fracture length, the position of the fracture, and reservoir width on the pressure and pressure derivative are fully investigated. The advantage of the analytical solution is easy to incorporate storage coefficient and skin factor. It can also reduce the amount of computation and compute efficiently and quickly.

An Analytical Solution of Two-Dimensional Flow and Deformation Coupling Due to a Point Source Within a Finite Poroelastic Media

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Peichao Li ◽  
Detang Lu
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Point Source ◽  
Transient Behavior ◽  
Finite Domain ◽  
Two Dimensional ◽  
Pressure Derivative ◽  
Poroelastic Media ◽  
Two Dimensional Flow ◽  
Dependent Flow

An analytical solution is derived for the time-dependent flow and deformation coupling of a saturated isotropic homogeneous incompressible poroelastic media within a two-dimensional (2D) finite domain due to a point source at some arbitrary position. In this study, the pore pressure field is assumed to conform to the second type of boundary conditions. Boundary conditions of the displacement field are chosen with care to match the appropriate finite sine and cosine transforms and simplify the resulting solution. It is found that the analytical solution is always independent of the Poisson’s ratio. The detailed solutions are given for the case of a periodic point source with zero pressure derivatives on the boundaries and for an imposed pressure derivative on the lower edge in the absence of a source. The presented analytical solutions are highly applicable for calibrating numerical codes, and meanwhile they can be used to further investigate the transient behavior of flow and deformation coupling induced by fluid withdrawal within a 2D finite poroelastic media.

scholarly journals Semi-Analytical Solution to Assess CO2 Leakage in the Subsurface through Abandoned Wells

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2452
Author(s):  
Tian Qiao ◽  
Hussein Hoteit ◽  
Marwan Fahs
Keyword(s):  
Analytical Solution ◽  
Storage Site ◽  
Co2 Leakage ◽  
Transient Pressure ◽  
Flow Paths ◽  
Abandoned Wells ◽  
Diffusivity Equation ◽  
Carbon Dioxide Co2 ◽  
Good Agreement

Geological carbon storage is an effective method capable of reducing carbon dioxide (CO2) emissions at significant scales. Subsurface reservoirs with sealing caprocks can provide long-term containment for the injected fluid. Nevertheless, CO2 leakage is a major concern. The presence of abandoned wells penetrating the reservoir caprock may cause leakage flow-paths for CO2 to the overburden. Assessment of time-varying leaky wells is a need. In this paper, we propose a new semi-analytical approach based on pressure-transient analysis to model the behavior of CO2 leakage and corresponding pressure distribution within the storage site and the overburden. Current methods assume instantaneous leakage of CO2 occurring with injection, which is not realistic. In this work, we employ the superposition in time and space to solve the diffusivity equation in 2D radial flow to approximate the transient pressure in the reservoirs. Fluid and rock compressibilities are taken into consideration, which allow calculating the breakthrough time and the leakage rate of CO2 to the overburden accurately. We use numerical simulations to verify the proposed time-dependent semi-analytical solution. The results show good agreement in both pressure and leakage rates. Sensitivity analysis is then conducted to assess different CO2 leakage scenarios to the overburden. The developed semi-analytical solution provides a new simple and practical approach to assess the potential of CO2 leakage outside the storage site. This approach is an alternative to numerical methods when detailed simulations are not feasible. Furthermore, the proposed solution can also be used to verify numerical codes, which often exhibit numerical artifacts.

scholarly journals The Effect of Perforation Spacing on the Variation of Stress Shadow

Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 4040
Author(s):  
Weige Han ◽  
Zhendong Cui ◽  
Zhengguo Zhu
Keyword(s):  
Compressive Stress ◽  
Reservoir Rock ◽  
Hydraulic Fractures ◽  
Extended Finite Element ◽  
Gas Reservoir ◽  
Fracture Length ◽  
Shadow Effect ◽  
Stress Shadow ◽  
Initiation Pressure ◽  
Small Perforation

When the shale gas reservoir is fractured, stress shadows can cause reorientation of hydraulic fractures and affect the complexity. To reveal the variation of stress shadow with perforation spacing, the numerical model between different perforation spacing was simulated by the extended finite element method (XFEM). The variation of stress shadows was analyzed from the stress of two perforation centers, the fracture path, and the ratio of fracture length to spacing. The simulations showed that the reservoir rock at the two perforation centers is always in a state of compressive stress, and the smaller the perforation spacing, the higher the maximum compressive stress. Moreover, the compressive stress value can directly reflect the size of the stress shadow effect, which changes with the fracture propagation. When the fracture length extends to 2.5 times the perforation spacing, the stress shadow effect is the strongest. In addition, small perforation spacing leads to backward-spreading of hydraulic fractures, and the smaller the perforation spacing, the greater the deflection degree of hydraulic fractures. Additionally, the deflection angle of the fracture decreases with the expansion of the fracture. Furthermore, the perforation spacing has an important influence on the initiation pressure, and the smaller the perforation spacing, the greater the initiation pressure. At the same time, there is also a perforation spacing which minimizes the initiation pressure. However, when the perforation spacing increases to a certain value (the result of this work is about 14 m), the initiation pressure will not change. This study will be useful in guiding the design of programs in simultaneous fracturing.

Utilizing Microseismic to Monitor Fracture Geometry in a Horizontal Well in a Tight Sandstone Formation

2021 ◽  
Author(s):  
Abu M. Sani ◽  
Hatim S. AlQasim ◽  
Rayan A. Alidi
Keyword(s):  
Horizontal Well ◽  
Fracture Propagation ◽  
Horizontal Stress ◽  
Height Growth ◽  
Tight Sandstone ◽  
Fracture Geometry ◽  
Fracture Length ◽  
Sandstone Formation ◽  
Fracture Height ◽  
Maximum Horizontal Stress

Abstract This paper presents the use of real-time microseismic (MS) monitoring to understand hydraulic fracturing of a horizontal well drilled in the minimum stress direction within a high-temperature high-pressure (HTHP) tight sandstone formation. The well achieved a reservoir contact of more than 3,500 ft. Careful planning of the monitoring well and treatment well setup enabled capture of high quality MS events resulting in useful information on the regional maximum horizontal stress and offers an understanding of the fracture geometry with respect to clusters and stage spacing in relation to fracture propagation and growth. The maximum horizontal stress based on MS events was found to be different from the expected value with fracture azimuth off by more than 25 degree among the stages. Transverse fracture propagation was observed with overlapping MS events across stages. Upward fracture height growth was dominant in tighter stages. MS fracture length and height in excess of 500 ft and 100 ft, respectively, were created for most of the stages resulting in stimulated volumes that are high. Bigger fracture jobs yielded longer fracture length and were more confined in height growth. MS events fracture lengths and heights were found to be on average 1.36 and 1.30 times, respectively, to those of pressure-match.

Analytical Buckling Loads of Columns Weakened Simultaneously with Transverse Cracks and Partial Delamination

2019 ◽  
Vol 19 (03) ◽  
pp. 1950027 ◽  
Author(s):  
Igor Planinc ◽  
Simon Schnabl
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Parametric Study ◽  
Transverse Crack ◽  
Transverse Cracks ◽  
Buckling Loads ◽  
Crack Location ◽  
Benchmark Solution ◽  
First Time

This paper focuses on development of a new mathematical model and its analytical solution for buckling analysis of elastic columns weakened simultaneously with transverse open cracks and partial longitudinal delamination. Consequently, the analytical solution for buckling loads is derived for the first time. The critical buckling loads are calculated using the proposed analytical model. A parametric study is performed to investigate the effects of transverse crack location and magnitude, length and degree of partial longitudinal delamination, and different boundary conditions on critical buckling loads of weakened columns. It is shown that the critical buckling loads of weakened columns can be greatly affected by all the analyzed parameters. Finally, the presented results can be used as a benchmark solution.

scholarly journals Apparent Toughness Anisotropy Induced by “Roughness” of In-Situ Stress: A Mechanism That Hinders Vertical Growth of Hydraulic Fractures and Its Simplified Modeling

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2148-2162 ◽  
Author(s):  
Pengcheng Fu ◽  
Jixiang Huang ◽  
Randolph R. Settgast ◽  
Joseph P. Morris ◽  
Frederick J. Ryerson
Keyword(s):  
Hydraulic Fracture ◽  
High Stress ◽  
In Situ Stress ◽  
Height Growth ◽  
Hydraulic Fractures ◽  
Simple Relationship ◽  
Vertical Growth ◽  
Stress Profiles ◽  
Fracture Height

Summary The height growth of a hydraulic fracture is known to be affected by many factors that are related to the layered structure of sedimentary rocks. Although these factors are often used to qualitatively explain why hydraulic fractures usually have well–bounded height growth, most of them cannot be directly and quantitatively characterized for a given reservoir to enable a priori prediction of fracture–height growth. In this work, we study the role of the “roughness” of in–situ–stress profiles, in particular alternating low and high stress among rock layers, in determining the tendency of a hydraulic fracture to propagate horizontally vs. vertically. We found that a hydraulic fracture propagates horizontally in low–stress layers ahead of neighboring high–stress layers. Under such a configuration, a fracture–mechanics principle dictates that the net pressure required for horizontal growth of high–stress layers within the current fracture height is significantly lower than that required for additional vertical growth across rock layers. Without explicit consideration of the stress–roughness profile, the system behaves as if the rock is tougher against vertical propagation than it is against horizontal fracture propagation. We developed a simple relationship between the apparent differential rock toughness and characteristics of the stress roughness that induce equivalent overall fracture shapes. This relationship enables existing hydraulic–fracture models to represent the effects of rough in–situ stress on fracture growth without directly representing the fine–resolution rough–stress profiles.

A new approach to evaluate the performance of partially propped hydraulic fractures

2013 ◽  
Vol 53 (1) ◽  
pp. 355 ◽  
Author(s):  
Luiz Bortolan Neto ◽  
Aditya Khanna ◽  
Andrei Kotousov
Keyword(s):  
Sensitivity Study ◽  
Economic Benefits ◽  
Productivity Index ◽  
Hydraulic Fractures ◽  
Fracture Geometry ◽  
Fracture Conductivity ◽  
New Approach ◽  
Fracture Length ◽  
Compressive Stresses ◽  
Initial Fracture

A new approach for evaluating the performance of hydraulic fractures that are partially packed with proppant (propping agent) particles is presented. The residual opening of the partially propped fracture is determined as a function of the initial fracture geometry, the propped length of the fracture, the compressive rock stresses, the elastic properties of the rock, and the compressibility of the proppant pack. A mathematical model for fluid flow towards the fracture is developed, which incorporates the effects of the residual opening profile of the fracture and the high conductivity of the unpropped fracture length. The residual opening profile of the fracture is calculated for a particular case where the proppant pack is nearly rigid and there is no closure of the fracture faces due to the confining (compressive) stresses. A sensitivity study is performed to demonstrate the dependence of the well productivity index on the propped length of the fracture, the proppant pack permeability, and the dimensionless fracture conductivity. The sensitivity study suggests that the residual opening of a fracture has a significant impact on production, and that partially propped fractures can be more productive than fully propped fractures. Application of this new approach can lead to economic benefits.

Investigation of Rupture and Slip Mechanisms of Hydraulic Fractures in Multiple-Layered Formations

SPE Journal ◽  
2019 ◽  
Vol 24 (05) ◽  
pp. 2292-2307 ◽  
Author(s):  
Jizhou Tang ◽  
Kan Wu ◽  
Lihua Zuo ◽  
Lizhi Xiao ◽  
Sijie Sun ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Hydraulic Fracture ◽  
Fracture Propagation ◽  
Propagation Model ◽  
Rock Deformation ◽  
Hydraulic Fractures ◽  
Fracture Geometry ◽  
Fracture Width ◽  
Hydraulic Fracture Propagation ◽  
Fracture Height

Summary Weak bedding planes (BPs) that exist in many tight oil formations and shale–gas formations might strongly affect fracture–height growth during hydraulic–fracturing treatment. Few of the hydraulic–fracture–propagation models developed for unconventional reservoirs are capable of quantitatively estimating the fracture–height containment or predicting the fracture geometry under the influence of multiple BPs. In this paper, we introduce a coupled 3D hydraulic–fracture–propagation model considering the effects of BPs. In this model, a fully 3D displacement–discontinuity method (3D DDM) is used to model the rock deformation. The advantage of this approach is that it addresses both the mechanical interaction between hydraulic fractures and weak BPs in 3D space and the physical mechanism of slippage along weak BPs. Fluid flow governed by a finite–difference methodology considers the flow in both vertical fractures and opening BPs. An iterative algorithm is used to couple fluid flow and rock deformation. Comparison between the developed model and the Perkins–Kern–Nordgren (PKN) model showed good agreement. I–shaped fracture geometry and crossing–shaped fracture geometry were analyzed in this paper. From numerical investigations, we found that BPs cannot be opened if the difference between overburden stress and minimum horizontal stress is large and only shear displacements exist along the BPs, which damage the planes and thus greatly amplify their hydraulic conductivity. Moreover, sensitivity studies investigate the impact on fracture propagation of parameters such as pumping rate (PR), fluid viscosity, and Young's modulus (YM). We investigated the fracture width near the junction between a vertical fracture and the BPs, the latter including the tensile opening of BPs and shear–displacement discontinuities (SDDs) along them. SDDs along BPs increase at the beginning and then decrease at a distance from the junction. The width near the junctions, the opening of BPs, and SDDs along the planes are directly proportional to PR. Because viscosity increases, the width at a junction increases as do the SDDs. YM greatly influences the opening of BPs at a junction and the SDDs along the BPs. This model estimates the fracture–width distribution and the SDDs along the BPs near junctions between the fracture tip and BPs and enables the assessment of the PR required to ensure that the fracture width at junctions and along intersected BPs is sufficient for proppant transport.

Controlling the Height of Multiple Hydraulic Fractures in Layered Media

SPE Journal ◽  
2016 ◽  
Vol 21 (01) ◽  
pp. 256-263 ◽  
Author(s):  
Aditya Khanna ◽  
Andrei Kotousov
Keyword(s):  
Fluid Pressure ◽  
Pressure Control ◽  
Shielding Effect ◽  
Hydraulic Fractures ◽  
Fracturing Fluid ◽  
Fracture Spacing ◽  
Gas Recovery ◽  
Fracture Opening ◽  
Fracture Height ◽  
Fracture Height Containment

Summary Fracture-height containment is desirable in hydraulic-fracturing treatments because it can result in better efficiency of oil or gas recovery and have less impact on the environment. Several mechanisms of the containment of a single hydraulic fracture were investigated in the past, and the outcomes of these studies are now well-documented in the open literature. However, the effectiveness of these mechanisms in the case of multiple closely spaced hydraulic fractures has not received much attention. The latter situation typically arises in the case of multiple transverse fractures emanating from a single horizontal wellbore. In this paper, we develop a mathematical model that one can use to assess the fracture-interaction phenomenon as well as the effect of the modulus contrast between adjacent rock layers. We consider the situation in which one must contain the hydraulic fractures entirely in the pay zone and investigate fracturing-fluid-pressure control as a possible mechanism of height containment. It is demonstrated that when the fracture spacing becomes comparable with the fracture height, the interaction between the fractures produces a shielding effect. In this case, the fracturing-fluid pressure that ensures fracture containment is greater in comparison with the case of a single isolated fracture. However, the fracture opening is also smaller in the case of closely spaced fractures. The dependence of the fracturing-fluid pressure and fracture opening on the fracture spacing needs to be taken into consideration during the selection of fracture spacing for a particular treatment.

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