Productivity Prediction Research of Fractured Horizontal Wells for Low Permeability Gas Reservoirs

In order to effectually resolve the problems as embarrassing solving and cumbersome calculation in design and simulation of hydraulic fracturing in coal petrography, the productivity prediction analysis software of horizontal wells has been developed in this study based on the object-oriented visual programming environment. The analysis software can determine the rock mechanics parameters, the distribution of crustal stress and the coalbed methane production, which can greatly improve the work efficiency of the engineering staff. The software has better applicability and can provide a foundation for the analysis of CBM productivity prediction. The results show that: simulation analysis is of high precision, can satisfy the actual engineering needs. Fracture number and half-length have a greater impact on fracturing horizontal well production; moreover, crack width almost has no influence on production.


INTRODUCTION
The permeability of coal seams in China is low with mostly less than 50 mD.Almost all coalbed methane wells with the gas production above 1000 m 3 /d are reformed by hydraulic fracturing during the past 20 years of coalbed methane exploration and development process (Archer and Roland, 2001).Hydraulic fracturing can effectively ameliorate flow channel, improve coal rock diverting capacity and increase productivity, which has become the effective measure of CBM exploration.Moreover, it has caused universal attention and special research of domestic and foreign experts and scholars.The cracks of hydraulic fracturing are closely related to the strata structure, rock mechanics property and ground stress (Clarkson et al., 2011;Keim et al., 2011).Fracturing crack simulation analysis belongs to an important subject of hydraulic fracturing technology research field.Meanwhile, the fracturing effect is directly related to the production capacity of coalbed methane well (Maricic et al., 2008).Economic and effective fracturing should connect mutual-disconnected cracks in the original fractures, broaden the original cracks and effectively accelerate the pressure decrease and transmission during the process of discharge and mining, so as to achieve the purpose of efficient production.In case of hydraulic fracturing operation failure, it will cause the destruction of coal seam pressure system, the abandon of coalbed methane well and significant economic loss.Therefore, hydraulic fracturing crack simulation analysis has become the key problem on coal rock fracture design, construction and production forecast.
This study analyzes CBM horizontal well production and studies sensitive factors.The results can not only provide the fundamental basis for resource evaluation and fracturing stimulation plans, but also have important practical significances for exploration and development of coalbed methane wells.

THEORETICAL BASES
Mechanical parameters: Coal seam is quite different from conventional sandstone reservoir with complex physical and mechanical properties, which has many crack and pore systems of a high degree of uncertainty.Therefore, coal reservoir physical and mechanical parameters on hydraulic fracturing of coal rock have always been one of the hottest issues in the international engineering research.It can not only provide references for productivity prediction after hydraulic fracturing, but also be the most important information on the destruction rule of coal rock strain and stress.
• Elasticity modulus: Elastic modulus is an important performance parameter in engineering, can be considered as measure index of rock to produce elastic deformation.The bigger the value is, the greater the stress to have elastic deformation: • Shear modulus: It stands for the rock's ability to resist shear strain.The greater the rock, the more difficult it is to produce shear deformation: (3) • Bulk modulus: It can be defined as the pressure required producing unit relative volume contraction: • The far field pore pressure value is constant • The stratum is linear porous elastomer • The lateral strain is zero: where, H = The vertical depth, m g = The gravity acceleration, m/s 2 • Fluid is single-phase and incompressible Newtonian fluid; the flow of the whole system is an isothermal process.• Fractured horizontal well only goes through one reservoir with the same thickness.• Cracks distribute symmetrically by horizontal wellbore, the fracture height is the thickness of reservoir.
Assume that the production length of fractured horizontal well is L, which can be divided into M parts with ∆L of each section.The half-length of crack is x f which can be divided into R parts (number from crack tip to wellbore) with ∆x f of each section.Fracture section parameter diagram is shown in Fig. 1.The flows in each infinitesimal horizontal section and infinitesimal rectangular crack are infinite diversion with uniform flow distribution.Each micro horizontal well section or micro crack can be equivalent to a vertical well.Equivalent borehole diameter of infinitesimal crack: Equivalent borehole diameter of infinitesimal horizontal well section: According to the theory of potential superimposition, the potential of any point W(x, y, z) can be written as below: where, r wf, jk is the equivalent borehole diameter of the k th infinitesimal section of the jth crack, m; r we, i stands for the equivalent borehole diameter of the i th infinitesimal section of horizontal well, m; h is the thickness of coal seam, m; β stands for coefficient of permeability anisotropy; r w is the wellbore diameter, m; z w, i is the distance between the i th infinitesimal section and coal seam bottom, m; N is the crack number; q w, i stands for production per unit length of the i th infinitesimal section, m 3 /d/m; q f, jk is the flow rate per unit length of the k th infinitesimal section of the j th crack, m 3 /d/m; C is undetermined constant, MPa.
Taking special points along the equivalent vertical sidewall and the supply boundary, the relations between bottom-hole pressure and flow rate are: where, p wr, i is casing pressure of the i th infinitesimal section along the horizontal well, MPa; µ is fluid viscosity, mPa.s; k stands for reservoir permeability, mD; r e is reservoir supply radius, m; r wa, wi is the distance between the centre of the i th infinitesimal section and the a th infinitesimal section along the wellbore, m; p fr, jk is casing pressure of the k th infinitesimal section of the j th crack, MPa; r fjk, wi is the distance between the centre of the i th infinitesimal well section and the centre of the k th infinitesimal section of the j th crack, m; r wi, fjk is the distance between the centre of the k th infinitesimal section of the j th crack and the centre of the i th infinitesimal well section, m; r fst.fjk stands for the distance between the centre of the k th infinitesimal section of the j th crack and the centre of the t th infinitesimal section of the s th crack, m.
When the fracture fluids go towards the wellbore, the flow can be treated as one dimensional linear flow.The pressure drop of the k th infinitesimal section of the j th crack can be written as: where, k f = Fracture permeability, mD w f = For fracture width, m q ft, jk = The section flow rate of the k th infinitesimal section of the j th crack, m 3 /d Based on the momentum and mass conservation principle, the pressure drop of the i th infinitesimal section of the fractured horizontal well is: Where, ∆p w, i is the pressure drop of the i th infinitesimal section of fractured horizontal well, MPa; ρ i stands for mixed fluid density of the i th infinitesimal section along the horizontal well, kg/m 3 ; q wt, i is the section flow rate of the i th infinitesimal section of horizontal well, m 3 /d; D is the wellbore diameter, m; θ i is the hole deviation angle of the i th infinitesimal section of horizontal well, ͦ ; f t,i is the friction coefficient between the i th infinitesimal section of horizontal well and pipe wall; g is gravitational acceleration, m/s 2 .
According to the pressure continuous principle, the pressure of fluid flow at the wellbore in the reservoir should be equal to the pressure at the wellbore in the borehole, furthermore, the pressure of fluid flow at crack wall in the reservoir should be equal to the pressure of fluid flow at crack wall in the crack.Therefore, we can obtain CBM horizontal well production prediction model.

ANALYSIS OF COMPUTING RESULTS
We take L7 well of Shengli oil field to illustrate the application of the software.The block has medium coal rank with buried depth of 420 m.Three-dimensional wellbore trajectory is shown in Fig. 2 with the horizontal length of 470 m.The basic parameters of stratum and L7 well are written in Table 1; moreover, the basic parameters of cracks are shown in Table 2.The following studies the effects of fracture parameters on fractured horizontal well production.
Fracture number: Taking the values in Table 1 and 2 as the basic parameters, the results are shown in Table 3 when well bottom pressure difference of 2.8 MPa.With the increasing of fracture number, coalbed methane production of fractured horizontal well improves gradually.Crack number has a great influence on production.Fracture width: With the change of well bottom flowing pressure, fractured horizontal well production value list is shown in Table 5.
modulus, MPa ρ = For rock density, kg/m 3 ∆t s = Transverse wave offset time, µs/m ∆t p = Longitudinal wave offset time, µs/m v = For Poisson's ratio G = Shear elasticity, MPa K = Bulk modulus, MPa Layered ground stress model: Three assumptions are used in the model, namely:

Fig. 1 :
Fig. 1: Diagram of fracture segmentation parameters σ v , σ h , σ H = The vertical stress, horizontal minimum and maximum principal stresses respectively, MPa T, T 0 = The original temperature and formation temperature in production process, K α = The linear expansion coefficient, 1/K p = Formation pore pressure, MPa σ gHmax = The maximum horizontal constitution stress, MPa Seepage near-wellbore and productivity prediction model: It has the following assumptions:

FigFig. 3 :
Fig. 2: Wellbore trajectory graph Fracture half-length: Coalbed methane production increases with crack half-length increasing.Based on production well bottom hole flowing pressure of 2.8 MPa, Production and fracture half-length relationship table is shown in Table 4. Coalbed methane production increases from 1154.916 m 3 /d to 1467.309 m 3 /d when crack half-length increases from 50 m to 65 m with a raise of 27.05%.The production increases from 2051.768 m 3 /d to 2325.629 m 3 /d when crack halflength increases from 95 m to 110 m with a raise of 13.35%.Fracture half-length has a great influence on fractured well production.

FractureCONCLUSION•
1531.483 m 3 /d to 1545.745 m 3 /d when crack width increases from 15 mm to 20 mm with a raise of 0.93%.Above all, crack width has no obvious effect on improving fractured horizontal well production.According to the example of L7 well, the results

Table 1 :
Basic parameters of stratum and L7 well

Table 3 :
Calculation result list of production and fracture number

Table 4 :
Calculation result list of production and fracture half-length

Table 5 :
Calculation result list of production and fracture width