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Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms

Ya Deng1, Rui Guo1, Zhongyuan Tian1, Cong Xiao2*, Haiying Han1 and Wenhao Tan2

Abstract:

 

Multi-stage  fracturing  horizontal  well currently  has been proved to be the most effective method  to produce shale gas. This method  can activate  the natural fractures  system  defined  as stimulated  reservoir  volume  (SRV),  the remaining region similarly is defined as un-stimulated  reservoir volume (USRV). At present, no type curves have been developed for hydraulic fractured shale gas reservoirs in which the SRV zone has triple-porosity  dual-depletion  flow behavior and the USRV zone has double porosity flow behavior. In this paper, the SRV zone and USRV zone respectively  are simplified as cubic triple-porosity  and slab dual porosity media. We have established  a new productivity model for multi-fractured horizontal well shale gas with Comprehensive  consideration  of desorption, diffusion, viscous flow, stress sensitivity and dual-depletion  mechanism in matrix. The rate transient responses are inverted into real time space with stehfest numerical inversion  algorithm. Type curves are plotted, and different flow regimes in shale gas reservoirs are identified. Effects of relevant parameters are analyzed as well. The whole flow period can be divided into 8 regimes: bilinear flow in SRV;  pseudo  elliptic  flow;  dual  inter-porosity  flow;  transitional  flow;  linear  flow  in  USRV;  inter-porosity  flow  and boundary-dominated flow. The stress sensitivity  basically  has negative influence  on the whole productivity  period .The less the value of Langmuir volume and the lager the value of Langmuir pressure, the more lately the inter-porosity  flow and boundary-dominated flow occurs. It in concluded that the USRV zone has positive influence on production and could not be ignored.

1. INTRODUCTION

 

 

Shale gas are typical unconventional  reservoir due to its ultra-low   permeability   and  porosity.   Generally   speaking, there is no natural productive capacity for those kinds of reservoirs.  Multi-stage  fracturing  horizontal  well  currently has  been  proved  to  be  the  most  effective  way  to  produce shale gas, and this method can not only create several high- conductivity  hydraulic  fractures,  but also activate  and connect the existing  natural  fractures  so as to form large  spacious  network  system  [1].  The  zone  containing  the  main high-conductivity hydraulic fractures and large spacious network  system  both of which  can  effectively  improve  the wells performance  is defined  as SRV  (stimulated  reservoir volume), and the remaining zone which hardly influenced by the treatment of hydraulic fracturing  is similarly  defined  as USRV (un-stimulated reservoir volume) [2-4].

At present, a number of scholars have done large amount of researches about transient rate analysis for shale gas, some analytical  and  semi-analytical   solutions  are  developed  as well. Shale gas reservoir  is the classical naturally  fractured reservoir  (NFR)  which  contains  complex  natural  fractures and ultra-low  permeability.

In terms of this kinds of reservoirs, Barenblatt (1960), Warren and Root (1963) originally proposed  the  dual-porosity  model  which  assumed  pseudo steady  state  fluid  transfer   between   matrix  and  fractures [5, 6], and then Kazemi (1969),

de Swaan (1976) and Ozkan et al. (1987) developed some other dual-porosity models for shale gas reservoirs to enrich the former productivity model, these models assumed unsteady-state  (transient) flow condition  between  matrix  and  fractures  [7-9].  However,  all  of these dual-porosity  models  neglected  the diffusion  and adsorption phenomenon in shale gas reservoirs.

 

Some  scholars  investigated  amount  of field production data and found that these dual-porosity  models may not be true in actual reservoirs.  An improvement  to overcome  this drawback  is  to  considerate  two  different  fracture  systems with different properties.  This system is so-called triple porosity system. Al-Ghamdi and Ershaghi (1996) initiatively proposed  the  dual  fracture  triple-porosity  model  for  radial flow [10], and then Liu et al. (2003), Wu et al. (2004) and Dreier (2004) enriched the triple-porosity model [11-13], but unfortunately these model still did not considerate the impact of adsorption  and diffusion. However,  the linear flow stage are apparently identified in some real productivity curves, especially for these fractured reservoirs, therefore, the linear flow models for shale gas are proposed by some scholars. El-Banbi (1998) proposed a linear dual- porosity model in linear fractured reservoirs [14], and originally derived the solutions in Laplace space, but the impact of desorption,  diffusion  and  USRV  zone  on the production  is ignored;  Hasan and Al-Ahmadi (2011) proposed a triple-porosity linear flow model with consideration  of the impact of shale gas desorption and diffusion [15], however, the impact of USRV zone was neglected; Xu et al. (2012) analyzed the effect of USRV zone on shale gas production, at the same time, the impact of desorption  diffusion  is considered  as well [16]; Zhao et al. (2013) proposed triple-porosity  spherical flow model for the fractured  infinite  shale gas reservoirs  which  considered  the impact of diffusion and desorption [17], however, they considered artificial fractures as infinite conductivity.

In terms of these naturally fractured reservoirs,  the phenomenon of stress sensitivity is readily observed. Samaniego VF et al. (1980), Raghavan  et al. (2004) employed  simulation and experiment  methods  to make some research  about the impact of stress sensitivity on the conventional reservoirs [18, 19]. Pedrosa et al. (1986) first applied the mathematical method  to study the stress sensitivity  for homogeneous  and dual-porosity  reservoirs  [20];  Wang  (2013)  develop  a dual porosity  spherical  flowing  model with consideration  of the stress sensitivity  in micro-fractures  for shale gas reservoirs [21].

 

Ezulike Daniel Obinna and Dehghanpour  Hassan (2014) first  proposed  the  triple-porosity  dual  inter-porosity  linear flow model [22], that is to say, the gas simultaneously  depletes from matrix into micro-fracture and macro-fracture, however,  the desorption,  diffusion  and  stress  sensitivity  in fracture are ignored.

 

In view of this, shale gas transfer in the reservoir  is the result of mutual effects of various percolation  mechanisms, and  therefore,  it is necessary  to  comprehensively  consider the impact of various mechanisms  in order to obtain important dynamic parameters for shale gas reservoirs. This paper simplifies  the SRV zone and USRV zone as triple-porosity cubic model and dual-porosity slab model respectively, comprehensively  taking  various  mechanisms  into  account, such  as  adsorption  and  diffusion  in  shale  matrix,  viscous flow and stress sensitivity  in fractures,  besides, we assume the gas simultaneously  depletes from the matrix into micro- fractures  and  macro-fractures  in SRV  zone  and  this  is the essence  of this  paper.  Laplace  transformation,  perturbation method are employed to solve this new model. The transient rate responses are inverted into real time space with stehfest numerical inversion algorithm [23]. Type curves are plotted, and different flow regimes in shale gas reservoirs are identified. The effects of relevant parameters are analyzed as well. Besides, this model also compares with numerical simulation and exhibits good agreements.

2. PRODUCTIVITY  MODEL

 

 

2.1. Physical Model

 

The  schematic  illustration  in  Fig.  (1a)  shows  a  multi-stage fracturing horizontal well. Multi-stage fracturing shale gas  reservoir  is  divided  into  SRV  zone  and  USRV  zone, SRV  zone  and  USRV  zone  is  simplified  as  cubic  triple-porosity  model  and dual-porosity  slab model.  A horizontal well located  in the center of a rectangular  closed formation producing  at constant wellbore pressure. The other assump tions are as follows:

 

(1) The  initial  pressure  distribution  in  the reservoir  is uniform which equals to Pi, the SRV zone contains  micro-fractures, macro-fractures and matrix, the USRV zone contains micro-fractures  and matrix, the fractures in different zone have different properties.

 

(2) The  macro-fracture   is  perpendicular   to  the  horizontal well and evenly  distributed  along  the wellbore,  the microfractures  are perpendicular  to the macro-fractures  as well, the length of reservoir and horizontal wellbore are equivalent, the length of micro-fracture  and the width of reservoir respectively equals to yf and ye.

 

(3) Macro-fractures have finite conductivity and are assumed to be penetrated fully, considering stress sensitivity in macro-fractures.

 

(4) Flowing  is sequential  from one medium  to another  medium. In the SRV zone, only fluid flow from macro- fractures to wellbore is considered; The shale gas simultaneously deplete from matrix into micro-fractures and macro-fractures with pseudo-steady state inter-porosity flow; the fluid flow between micro-fractures  and macro- fractures  is unsteady  state  flowing.  In the USRV  zone, the fluid flowing from matrix to the fractures is pseudo- steady state inter-porosity  flow; the connection  between SRV  zone  and  USRV  zone  via  the  macro-fractures  in SRV and micro-fractures in USRV.

 

(5) Slightly compressible shale gas and compressibility coefficient is constant;

 

(6) Shale gas desorption and diffusion respectively meets the Langmuir  isotherm  equation  and  the first  law  of diffusion;

 

(7) The impact of gravity and capillary pressure is ignored.

 

This  paper  considers  the  simultaneous  depletion  from matrix into micro-fractures  and macro-fractures.  To analyze this  flow  process  conveniently,  the matrix  in SRV  zone  is artificially divided into two distinct segments which have different permeability and porosity ratio respectively. The schematic illustration in Fig. (1b) shows the depletion process  from  matrix  to  micro-fractures  and  macro-fractures  in SRV zone.

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

2.2. Mathematical Model

 

Based on the mass balance principle, the governing equations respectively in micro-fractures, matrix, and macro- fractures in SRV zone and USRV zone with consideration of adsorption,  diffusion, viscous flow and stress sensitivity are as follows:

 

SRV Zone: Macro-fracture

Micro-fracture

The matrix in SRV zone are divided into two segments, the segment denoted as matrix-1 which permeability and porosity equal to km1 and   ϕm1 depletes into macro-fractures and another segment denoted as matrix-2 which permeability and  porosity  equal  to  km2   and    ϕm2   depletes  into  micro- fractures. Therefore, the governing equations of these two segments are as follows:

 

Matrix-1:

Matrix-2:

USRV Zone

Fracture

Matrix:

initial condition

Inner boundary condition

Inner boundary condition

Inner zone micro-fracture

Outer boundary condition

To  simplify  these  equations,  some  dimensionless  variables  are defined  (seen  from  appendix  A)  and  substituting these dimensionless variables into equation 4-15, the dimensionless governing equations are as follows:

 

SRV Zone

 

Macro-fracture:

Micro-fracture

Matrix-1

Matrix-2:

USRV Zone:

Fracture

Matrix:

Inner boundary condition

Interface condition

Inner Zone fracture

Outer boundary condition

It’s not difficult to find that the macro-fracture  equation contains  strong  nonlinear,  the  perturbation  technology  and the Presoda transformation are applied to linearize this equation [20], the formula is as follows:

According to the theory conducted by Wang (2013) [24] and Presoda (1986), performing  a parameter  perturbation  in ζD by defining the following series:

Finally,  the final  governing  equation  of SRV  zone  andvUSRV zone can be changed into the following profiles. SRV Zone

 

Macro-fracture

Micro-fracture

Matrix-1

Matrix-2:

USRV Zone:

Fracture

Matrix:

Inner boundary condition

Interface condition

Inner Zone fracture

Outer boundary condition

2.3. Model Solution

 

The Laplace transformation  is used to solve equation 25-35,  the details of this process are listed in appendix B. The final solution of these equations  in Laplace space is as fol- lows:

where

However,  to  analyze  the  impact  of  relative  parameters and identify the shale gas flowing regimes, the transient rate responses are inverted into real time space with stehfest numerical inversion algorithm.

3. NUMERICAL SIMULATION VALIDATION

 

 

Numerical  models  were used to validate  this new analytical solution proposed in the paper. This validation  process  is operated  in  the  commercial  software-CMG.  We  assume  that the rate of water  is ignored  and gas is the only phase  existing  in  the  reservoir.  The  basic  parameters  are listed in the Table 1. The radial coordinate is converted into x-y  coordinate,  and  the value  of  the permeability  of  these two directions is different based on the flowing regimes. In SRV zone, the vertical permeability of some grids which are located near the wellbore is assumed to be very high so as to ensure the facts that the gas can flow into the wellbore. It is difficult to simulated the adsorption used CMG, for simplicity, the adsorbed  gas is not included.  We focus on the tendency of curves.

For gas case, as seen in Fig. (2), the simulation data from CMG are converted into the formation of dimensionless time and dimensionless  rate for the convenience of analysis. It is concluded that the curve is in good agreement with the analytical solution at the stage 1   which represents  the flowing regime  in fractures  in SRV  zone  and  stage  3 which  represents the flowing regime in fractures in USRV zone, while it deviates  a lot at the stage 2 which  represents  the flowing regime in matrix in SRV zone and stage 4 which represents the flowing regime in matrix in USRV zone. The main reason for the deviation is that adsorption is neglected in simulation  model.  The deviation  also could  be caused  by using the inappropriate average pressure to calculate the pseudo pressure.  At  the  very  early  times,  the  flow  is  fracture- dominated  flowing regime in SRV zone, the adsorptive gas has not desorbed from the surface of matrix, so the predictive results are similar with those from the numerical simulation  (as  seen  in stage 1 and 3). However,  at  the  middle times and late times, the flow is matrix-dominated and boundary-dominated flowing regime, and the situation is different. The adsorptive gas can desorb from the surface of matrix and depletes into fractures,  this process can increase the shale gas production (as seen in stage 2 and 4).

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Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

4. TYPE CURVES AND DISCUSSIONS

 

 

In this paper, we make some assumptions that SRV zone is cubic triple-porosity  model and USRV zone is slab dualporosity  model.  Some  characteristic   parameters   of  these kinds of shale gas reservoirs include: ωf, ωf1, ωm1, ωm2, ωf2, ωm3. However, if these parameters are equal to some special value, the new model proposed in this paper can be changed into some other familiar models. For example,   ωf1 is equal to 0, the model of SRV zone can be dual porosity slab model; when the ωm3  equals to 0, USRV zone can be simplified  as homogeneous model;  ωm1 = 0, the gas only depletes from the matrix   to  micro-fractures   in  SRV   zone   (Hasan   A.  Al- Ahmadi,  2011),  in  another  words,  the  new  model  can  be similar  with  the  triple-linear  model  proposed  by  El-Banbi (1988) as well;  ωm2 = 0, SRV zone can be simplified as dual- porosity model (Xu et al, 2013). In short, this new model has universal application to different formations when these characteristic  parameters  are equal to different  values.  It is worth mentioning that the stress sensitivity in the macro- fractures is considered, this consideration extremely matches the real production  condition  in  fractured  reservoirs,  especially for the ultra-low permeability shale gas reservoirs .The mechanism  of the stress sensitivity  will be analyzed  in the following sections.

 

Combined  with  stehfest  numerical  inversion  algorithm, the type curves concerning dimensionless  rate derivative qD' with respect to tD  and dimensionless  rate qwith respect to tD  are plotted under different condition of formation properties, eight flow regimes can be easily observed by analyzing the following type curves Seen from Fig. (3):

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

Regime I: bilinear flow regime. The value of the slope of the rate derivative  curve is equal to -1/4. This flow regime occurs  between  the  micro-fracture  and  macro-fractures  in SRV  zone,  because  of  the  existence  of  micro-fractures  in SRV zone, it is difficult to find the single linear flow in the macro-fractures  in the SRV zone, especially, when the permeability of macro-fractures is big.

 

Regime II: pseudo elliptic flow regime. The value of the slope of the rate derivative  curve equals to -1/3, this is the transitional flow stage from the bilinear flow regime between the micro-fracture  and macro-fractures  to linear flow in micro-fractures in SRV zone.

 

Regime III: linear flow regime. The value of the slope of the rate derivative  curve is equal to -1/2. This is the linear flow which occurs in the micro-fractures of SRV zone.

 

Regime  IV: pseudo-steady  state  inter-porosity  regime. Two  consecutive  incomplete  concave  dips  denoted  with  a circle can be ambiguously  observed,  these two sections  respectively  present two different inter-porosity  flow regimes which simultaneously  occur from matrix  to micro-fractures and macro-fractures in SRV zone;

 

Regime  V:  transition  flow  regime.  The  pressure  wave reaches to the interface between  SRV  zone and USRV  zone, USRV zone supply to SRV zone, the decline ratio of rate decreases as well. As a result, the rate derivative  curves elevate and the transition flow regime occurs between these two zones.

 

Regime VI: linear flow regime. The value of the slope of the rate derivative curve is equal to -1/2. This regime mainly occurs in the fractures  in USRV zone so that the shale gas stored  in the USRV  zone can deplete  from USRV  zone to SRV zone via interface.

 

Regime  VII: pseudo-steady  state  inter-porosity  flowing regime. The shape of rate derivative curve likes a "concave" which is typical feature of pseudo-steady state inter-porosity. This regime mainly occurs between  matrix and fractures  in USRV zone;

 

Regime VIII: Boundary-dominated  flow regime. At this time, the boundary  has influence on dynamic production  of well, the rate and rate derivative curve decrease rapidly.

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

The paper considers the impact of stress sensitivity in the macro-fractures  in SRV zone. Fig. (4) shows that stress sensitivity has great negative  influence on horizontal  well performance  during  the  whole  productivity  period.  However, with the extension  of production  time, the negative  impact on rate becomes less and less, especially  during the boundary-dominated  flow stage and the negative  impact basically vanishes. The permeability  of macro-fractures  in SRV zone consecutively decreases with the reduction of pressure, when the pressure wave has reached to the outer boundary, the permeability of the macro-fractures is so small that the influence of stress sensitivity  can be ignored.  Besides,  with the increasing of stress sensitivity coefficient (βD= 0, βD= 0.8, βD=1.5), dimensionless rate curve (Fig. 4a) and dimensionless rate derivative curve (Fig. 4b) simultaneously  descend. By analyzing the mechanisms, the more the pressure reduces in  macro-fractures,  the  more  severely  the  macro-fractures close, as a result, the permeability  of macro-fracture  rapidly decreases.

 

Two most important  characteristic  parameters  are Langmuir pressure  and Langmuir  volume  which  reflect the feature of adsorption.  PL  and  VL  primarily  affect  the stage of pseudo-steady  state inter-porosity  between  matrix  and fracture in SRV zone, linear flow in USRV zone and outer boundary-dominated flow. The larger the value of PL (PL perspective  is equals to 2MPa, 5MPa and 10MPa) or the smaller the value of VL (VL perspective is equal to 5 sm3/m3, 10 sm3/m3, 15 sm3/m3), the less the shale gas rate during the whole process of productivity  period, and the later the stage of pseudo-steady state inter-porosity between matrix and fractures in SRV zone occurs, and linear flow in USRV zone and outer boundary-dominated  flow occurs.

 

From  the  analysis  of  micro-mechanisms,  the  larger  the value  of Langmuir  volume  is and  the more  the content  of adsorbed  gas are. When  the pressure  is smaller  than Langmuir  pressure,  the  adsorbed  gas  starts  to  desorb  from  the surface of matrix and spread into fractures so that the decline ratio of pressure and rate slows down. At the same time, due to inter-porosity flowing between matrix and fracture, the negative  of  outer  boundary  will  delate  lately,  in  another word, the dimensionless  rate derivative curve wholly moves right (Fig. 5a, d). In short, due to the gas desorption and diffusion,  the decline  ratio of pressure  becomes  slow and the rate is less affected by outer boundary. When shale gas desorption reaches a certain level, the amount of desorption is insufficient to cover the impact of outer boundary, dimensionless rate derivative curve moves downward (Fig. 5b, c).

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

Fig. (5). Type curve under different value of Langmuir parameters.  (a) qD vs tD under different Langmuir volume; (b) q’D vs tD under different Langmuir volume; (c) qD vs tD under different Langmuir pressure; (d) q’D vs tD under different Langmuir pressure

The paper assumes that gas simultaneously depletes from matrix  into  micro-fractures and macro-fractures in  SRV zone. The matrix  is divided  into two segments,  where,  the segment denoted as matrix-1 which permeability  and porosity  equal to km1  and  ϕm1  depletes  into  macro-fractures  and another segment denoted as matrix-2 which permeability and porosity equal to km2  and ϕm2 depletes into micro-fractures. We  introduce  the  pore  volume  ratio  denoted  as χ  which equals to ϕm1/(ϕm1m2), and χ=0, χ=1, 0<χ<1 respectively represents: (1) the matrix only depletes into micro-fractures; (2) the matrix only depletes into macro-fractures;  (3) matrix simultaneously depletes from matrix into micro-fractures and macro-fractures.  The  type  curves  (Fig.  6)  under  different value of χ, km1 and km2 are plotted. Some conclusions can be obtained via the analysis of curves: (1) km1> km2, the larger the value of χ  is, the higher the rate of shale gas is (Fig. 6a); (2) km1 <= km2, on the one hand, during  the early  flowing stage, the larger the value of χ is, the lower the rate of shale gas is, on the other hand, during the rest of flowing period, the larger the value of χ is, the higher the rate of shale gas is (Fig. 6b). From  the analysis  of micro-mechanisms,  km1 <= km2  < kFi, as a result, the inter-porosity  flowing capacity between matrix-2 and micro-fractures  is stronger than that be- tween matrix-1 and macro-fractures,  therefore, the larger the pore volume ratio of matrix-1 which depletes into macro- fractures is, the less the total amount of shale gas from reservoir during the same productivity period is. In conclusion, as for the actual shale gas reservoir, we can acidize the matrix to increase  its permeability  so that the shale  gas stored  in matrix can directly flow into the macro-fractures,  this treatment can significantly  improve the horizontal well performance.

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

This  paper  assumes  that  USRV  zone  is  dual-porosity slab model including matrix and fracture. The dimensionless rate type curves  (Fig.  7) are plotted  under  different  conditions in USRV zone, such as different matrix permeability, different fracture permeability  and different width of reservoir. Observing  from  all of the following  four figures,  the rate rapidly  declines  without  consideration  of USRV  zone. Generally speaking, the width of the whole reservoir is much larger than that of SRV zone; therefore,  the flow occurs in the USRV zone can last longer than that in SRV zone (Fig.7a). Fig. (7b) represents dynamic principle of rate under different permeability  of matrix, the larger the value of matrix permeability is, the more the amount of gas deplete from matrix into fractures  in USRV zone is, the later the boundary-dominated occurs. Fig. (7c) represents the dynamic principle  of  rate  under  different  permeability   of  fracture  in USRV zone, during the early productivity  period, the larger the value of fracture permeability is, the more the gas rate is, however,  the sooner the pressure wave reaches to the outer boundary, what’s worse, the flowing regimes of matrix and micro-fractures  in the these two zones are hardly  observed (seen from the blue circle in Fig. 7c). Fig. (7d) reflects the impact of the width of reservoir,  the wider the reservoir  is, the more apparent the flowing regimes we can observe from the type curves. Through the above analysis, we can recognize that it is not reasonable to neglect the impact of USRV zone,  especially  for  these  kinds  of  reservoirs  which  have great formation properties.

Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

Fig. (7). Impact of USRV zone : (a) SRV zone and all zone; (b) different matrix permeability in USRV; (c) different fracture permeability in USRV; (d) different width of reservoir.

CONCLUSION

 

 

New  analytical   solution  for  shale  gas  reservoir   with multi-stage  fracturing  horizontal  well  has  been  developed which  considers  the USRV  zone as a dual porosity  system and the SRV zone as a triple-porosity system. The solution is more general for type curve analysis  both in homogeneous and   naturally   fractured   reservoirs.   Numerical   simulation model was used to validate the analytical solutions  and obtains an excellent agreement.

 

Analyzing the type curves, shale gas flow is divided into eight  regimes:  bilinear  flow  of macro-fractures  and  micro- fractures  in SRV  zone; pseudo-elliptic  flow; linear  flow of micro-fracture   in   SRV   zone;   dual-pseudo   steady   inter-porosity flow; transition flow between SRV and USRV zone; linear  flow  of fracture  in outer  zone;  pseudo-steady  state inter-porosity flow; outer boundary-dominated  flow.

The impact of adsorption,  stress sensitivity  in shale gas reservoirs  must not be ignored;  otherwise,  the performance of horizontal  well  cannot  be predicted  precisely.  It is also concluded that the dual-porosity behavior of USRV zone has a positive  effect  on production,  the larger  the value of the permeability  of matrix in USRV zone is, the more apparent the positive effect is. The stress sensitivity  has negative  influence on production during the whole productivity period.

 

This paper initially introduces the triple-porosity dual- depletion  model in SRV zone for shale gas. It is concluded that  the USRV  zone  has  positive  influence  on production. Reservoirs could be acidized so as to increase the permeability of matrix in SRV zone and optimize the performance  of horizontal wells for shale gas reservoirs.

CONFLICT OF INTEREST

 

 

The authors confirm that this article content has no conflict of interest.

 

 

ACKNOWLEDGEMENTS

 

 

The authors acknowledge a fund from the National Natural  Science  Foundation  (NNSF)  of  China  (No.  51204193) and supports  from the MOE Key  Laboratory  of Petroleum Engineering. And this paper is supported by the fund NNSF of China (No. 51204193).

 

Notes: the program can provide if it is necessary.

 

 

Cong Xiao: Address correspondence to this author at the College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China;

Tel: 18810907235; E-mail: 987558984@qq.com

 

 

NOMENCLATURE

APPENDIX A: DIMENSIONLESS  DEFINITION

APPENDIX B: SOLUTION OF EQUATIONS

 

 

Applying the Laplace Transform, the governing equations and boundary conditions are changed into:

 

SRV Zone

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Received: November 05, 2014

Revised: January 10, 2015

Accepted: January 21, 2015

 

© Deng et al.; Licensee Bentham Open.

 

This is  an  open access article licensed under the  terms of  the  Creative Commons Attribution Non-Commercial License (http://creativecommons.org/- licenses/by-nc/3.0/) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.

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Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale
Productivity Model for Shale Gas Reservoir with Comprehensive Consideration of Multi-mechanisms, Deng, Guo, Tian, Xiao, Han, Tan, Allaboutshale

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