Coalbed methane (CBM) is one of the most important gas resources developed recently in China. The permeability of coalbed in China is generally very low. The productivity of low permeability coalbed is poor,which makes the production rate of water and gas to be very slow. In order to improve the productivity of wells in coalbed,the hydraulic fracturing technology is used in most of the CBM wells in China. A fracture through the wellbore is formed after the hydraulic fractures. The fracture can R_eDuce the flow resistance near the wellbore. Assessing the characters of the fracture and the coalbed from well test data is an especial key step for a reasonable producing process.

A great number of studies have been published in the literature on hydraulic fractuR_eD wells. However,most of the published well testing models are analytical models,and only a fewnumerical models have been presented. The analytical models are based on many assumptions. There are many methods to develop fracture well test models. Gringarten et al.^[1] developed a well test model for the vertical fractuR_eD well in a normal reservoir by using the source and Green’s functions. They assumed that the fracture was a uniform flux fracture or an infinite conductivity fracture. Cinco-Ley and Meng^[2] and Cinco-Ley and Samaniego^[3] introduced a more general well test model for the fractuR_eD well with a finite conductivity vertical fracture. Lee and Brockenbrough^[4] developed a fractuR_eD well model by assuming the trilinear flow near a finite conductivity fracture. Riley et al.^[5] presented exact analytic solutions for an elliptical finite conductivity fracture. Liu and Yang^[6] and Liu and Liu^[7] developed an elliptical flow model for a finite conductivity fractuR_eD well. Anbarci and Ertekin^[8] developed a uniform-flux vertical fracture model in consideration of the unsteady-state sorption phenomena in coalbed. He and Hang^[9] found a model to assess a new finite conductivity fractuR_eD well by considering the deformation of media in coalbed. Liu et al.^[10] developed a numerical model for the multiwell in coalbed. The above results of CBM wells are all derived from the pseudo-steady state desorption model^[11] or unsteady state desorption model^[12] by using the Langmuir adsorption theory^[13]. In 2010,Ouyang and Liu^[14] developed a new model for a fractuR_eD well in coalbed with an infinite conductivity fracture by considering the steady state desorption. The advantage of the numerical approach is that it is based on fewer assumptions than analytic solutions,and hence has greater generality. In this paper,based on the above published works,a new numerical well test model for a fractuR_eD well with a finite conductivity vertical fracture is presented.

The main aspects consideR_eD in the model are as follows:

(i) The desorption phenomenon^[15] is one of the key features between normal gas and coalbed methane in the producing process,and the effects of the steady and unsteady terms of desorption are dealt within the model.

(ii) The hydraulic fracture is vertical.

(iii) More stuffs make lower conductivity fractures as finite vertical fractures.

(iv) The asymmetry of the fracture is around the axis of the well.

Since the governing equation and the inner boundary condition in the coalbed are very complex,finite element method (FEM) is used to solve the new mathematical model. The effects of the desorption factor,size,and permeability of the damage region,fracture length, asymmetry coefficient on the type curve are all discussed in detail in this paper. In order to verify the new model,a set of field well test data is analyzed.

a) The media is homogenous in the coalbed.

b) The fluid in the coalbed is compressible and Newtonian with constant viscosity.

c) The fluid flow in the coalbed is laminar,and obeys the Darcy theorem. The model considers the wellbore boundary and the flux of wellbore boundary.

d) The manufactuR_eD fracture is vertical with certain permeability and width. The value of permeability in the fracture is greatly larger than that of the coalbed. There is a pressure difference in the fracture. The wellbore is not uniform,but the pressure at each point of the wellbore is assumed to be equal.

e) The desorption values of the CBM are linear with pressure,and are independent of time during the test processing.

f) The saturation of water within the coalbed is very low,or the distribution of saturation of water is stable and almost irR_eDucible. Only coal gas is consideR_eD in the coalbed. Water affects the permeability of gas there.

g) The changes of gravity and temperature are neglected,and other physical and chemical effects are also ignoR_eD.

The governing equation is

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The initial condition is

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The inner boundary condition is

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The infinite outer boundary is

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The constant outer pressure boundary is

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The closed outer boundary is

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In the above equations,pD is the dimensionless pressure,TD is the dimensionless time, (xD,yD) is the dimensionless position in the coalbed,pCD is the dimensionless critical desorption pressure^[16],α1D is the dimensionless steady desorption coefficient,α2D is the dimensionless unsteady desorption coefficient,Kr is the ratio of fracture permeability to coalbed permeability, LiD is the dimensionless inner boundary length of the ith element,pwD is the dimensionless wellbore pressure,piD is the dimensionless pressure of the ith element,􀀀in indicates the inner boundary,􀀀out indicates the outer boundary, indicates the research region,and C_fD is the dimensionless combined fracture storage.

To solve the above mathematical model,the Galerkin finite element method with the weighted residual method is used,and the selected interpolation function Ni is used as the weight function,where

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The weak solution formation is

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The finite element discrete equation is

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The simultaneous equations (10)-(12) are assembled by unit equations so as to solve the system equations. Then,the pressure value at the grid node (x,y) and the (n + 1) instant and the pressure derivative on the fracture can be obtained immediately.

In order to calculate the finite elements,it is necessary to discrete the research region in the coalbed by using the triangular unstructuR_eD grids within the round out-boundary.

In order to ensure the stability of the finite element solutions,the maximum spectral radius of the transformation matrix needs to be less than or equal to 1. The transformation matrix is associated with the grid,time step,and other calculated parameters. Therefore,the maximum spectral radius of the transformation matrix must be calculated at each time step,different grids,or different parameters. If the maximum spectral radius of the transformation matrix is greater than 1,the time step can be modified. The error formula of the linear interpolation for the triangular element^[17] is

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Through the analysis of the error formula,we can see that large obtuse triangle is not allowed in order to ensure the convergence of the finite element solutions. The grid is generated by Gambit (see Fig. 1(a)),which can make sure that the triangle is approximate to the equilateral triangles except some small acute triangles in the fracture (see Fig. 1(b)). Moreover,the grid size is requiR_eD to be large enough so as to make sure that the solutions are not sensitive to the grid.

Fig. 1 UnstructuReD grid chart of research region in coalbed .

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The following parameters are taken: the dimensionless fracture storage C_fD is 0.001,the dimensionless fracture conductivity is 2,the dimensionless fracture width is 0.002,and the radius of the closed outer boundary is 100. The calculation results are shown in Fig. 2. The well test type curve is generally divided into 4 parts based on the characteristics in different stages. The first part is the wellbore control section line with the slope 1. The second partindicating the finite conductivity fracture feature is the straight line with the slope 0.25. The third part is a horizontal line of 0.5 with a radial fluid flow around the wellbore. The fourth part is the straight line with the slope 1,showing the effect of the closed outer boundary.

Fig. 2 Log-log type curves for fractuReD well with finite conductivity fracture .

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The pressure distribution in the bed is shown in Fig. 3. The flow is the radius flow in the far field from the well bore. However,the flow forms “eye shape” nearby the fracture of the well bore,and forms ellipse shape at the place a little bit far from the well. It can be thought that the propagation speed of the pressure in the fracture is faster than that in any other region owing to its much bigger permeability.

Fig. 3 Distributions of pressure in bed .

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Fracture storage is defined as the storage capacity of a fracture. We choose the fracture storage C_fD to be 0.000 1,0.001,0.01,and 0.1,respectively,and the fracture conductivity to be 2. The results are shown in Fig. 4. From Fig. 4,it can be seen that the smaller C_fD is,the shorter the storage time lasts and thus the longer the time of the linear flow in the fracture lasts. If C_fD becomes larger,the fracture features are not much obvious. When C_fD is large enough (see Curve 4 in Fig. 4),the features of the linear flow are coveR_eD by the wellbore storage.

Fig. 4 Log-log type curves with different CfD .

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The fracture conductivity is defined as the conductive capacity of the fracture. First,the fracture conductivities for different fractures affected on the calculating results are developed. We choose F_CD to be 0.1,1,10,and 100,respectively,C_fD to be 0.001,and the radius of the closed outer boundary to be 100. The results are shown in Fig. 5. From Fig. 5,we can see that larger fracture conductivity corresponds to lower ratio of dimensionless pressure. It means that the greater the fracture conductivity is,the lower the flow resistance of the well bottom is,and the more convenient the fracture conductivity is to produce. When the fracture conductivity is very small (see the curve corresponding to F_CD = 0.1 in Fig. 5),the fracture feature is not distinct and there is no linear section on the curve. Therefore,after reaching the storage section,the radius flow immediately comes in the wellbore. Moreover,the conductivity capacity of the fracture is very great. See the curve of F_CD = 100 in Fig. 5 for example,the fracture feature section with the slope nearly 0.5 is on the characteristic curve. However,on the distribution of pressure (see Fig. 6),there is elliptic flow in the fracture region. The pressure difference does not exist in the fracture. It is pointed out that the features mentioned above are similar to infinite conductivity. It indicates that infinite conductivity is a special case of finite conductivity. It is reasonable and natural.

Fig. 5 Type curves with different conductivity capacities .

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Fig. 6 Distributions of pressure field with different FCD .

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Considering the effect of the steady desorption coefficient of the coalbed,i.e.,α1D,we take α1D to be 0,−10⁻⁴,and −2.0 × 10⁻⁴,respectively,the unsteady desorption coefficient α2D to be 0,the fracture storage of the well bore C_fD to be 0.001,and the radius of boundary R_eD to be 100. The results are shown in Fig. 7. The early section of the type curve is not affected by desorption but by the storage of the wellbore and fracture. The steady desorption strongly affects the late section of the type curve. Desorption can relieve pressure. The stronger the

Fig. 7 Type curves for fractuReD well with dif- ferent α1D .

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We examine unsteady desorption,i.e.,α2D,effects. We take

Fig. 8 Type curves for fractuReD well with dif- ferent α2D .

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As the pressure decreases in the bed unsteadily,the desorption speed of the coalbed increases, and thus the pressure decrease in the coalbed is relieved. However,because of the effect of the closed outer boundary,the pressure in the coalbed must continue to decrease,and the relevant speed of the unsteady desorption varies faster. When the desorption speed contents the production value,the pressure keeps to be constant and the pressure derivative decreases. With the effect of the unsteady desorption,the pressure in the coalbed can reach stability without any change. It is also similar to a constant pressure boundary. On the other hand,on the pressure distribution,desorption can relieve propagation of the pressure wave. When the desorption speed contents production,the propagation of the pressure wave will cease.

Normally,the fractures are symmetric around the wellbore in sand oil reservoir,but the hydraulic fractures are asymmetric in coalbed. The asymmetric fracture effects on the curve are analyzed now. We take the fracture length ratios for both sides of the wellbore to be 1 : 1, 0.8 : 1.2,0.6 : 1.4,0.4 : 1.6,and 0.2 : 1.8,respectively,the fracture storage C_fD to be 0.001, and the radius of the closed out boundary R_eD to be 100. From Fig. 9 and Fig. 10,it can be seen that the more asymmetric the fractures are,the greater the dimensionless pressure is. The phenomenon can be explained by that the flow resistance in the bottom of the well is so great that it is not convenient for the fluid flow.

Fig. 9 Type curves for asymmetry fracture with different fracture length ratios for both sides of wellbore.

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Fig. 10 Type curves for asymmetry fracture with different fracture length ratios for both sides of wellbore (part am-plified) .

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Figure 11 shows the pressure distributions around the wellbore with different fracture length ratios for both sides of the wellbore. The speed of the pressure propagation is different for both sides because of the fracture asymmetry (see Fig. 11). The results concluded that faster pressure propagation speed corresponds to longer fracture and lower flow resistance.

Fig. 11 Pressure distributions for asymmetry fracture with different fracture length ratios for both sides of wellbore .

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In order to verify the numerical mathematical model,a set of field well test data is used to match the type curve generated by the new model. The field well test is executed at one CBM well in the X-CBM reservoir of China in March 2011. The depth of the CBM well is 860m. The injection rate is 3.2m3/d. The thickness of the coalbed is 5.6m. The test lasts 36 hours. The results are shown in Fig. 12. From Fig. 12,we can see that the pressure changes along with the time. The solid line is the theoretical result which is calculated by the model introduced in this paper.

Fig. 12 Well test history sketch of pressure .

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By matching the real test data with the well test type curve similar to Fig. 13,the parameters are obtained from an advanced well test analyzing method. The results show that the perme- ability of the coalbed is 0.126mD,The fracture conductivity is 0.193mD·m. The half-length of the fracture is 1.53m. The fracture asymmetry equals 1.0. These results are accepted by CBM reservoir engineers.

Fig. 13 Well test data matching theoretical type curve of new model .

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A numerical model of well tests for the well with a finite conductivity vertical hydraulic fracture in coalbed is developed in this paper by considering the fracture features of the finiteconductivity,the storage coefficient of the wellbore,the conductivity capacity of the fracture, the desorption of the coalbed,and the symmetry of the fracture. FEM is used to solve the model. The conclusions can be made as follows:

(i) There is an obvious fracture feature in the initial theoretical characteristic curve. It indicates a straight line with a slope 0.25. The pressure distribution in the coalbed indicates that the fluid flow is a radial flow far from the wellbore. However,the flow forms as an elliptic flow nearby the fracture in the early time.

(ii) Smaller combined fracture storage coefficient C_fD corresponds to shorter storage period and longer time for the linear flow in the fracture. When the storage coefficient C_fD becomes larger,the fracture features are not distinct. Therefore,the behavior of linear flow may disap- pear when the storage capacity is large enough.

(iii) The greater the fracture conductivity is,the smaller the flow resistance of well is,and thus the more effective the pressure is on the producing rate. The fracture feature is consistent to the infinite conductivity when the conductivity capacity of the fracture is strong enough, i.e.,infinite conductivity is a special case of finite conductivity. Moreover,the fracture feature is not evident when the conductivity capacity is weak. It is difficult to see that the elliptic flow in the pressure field and its effect on the pressure is very poor.

(iv) Desorption of CBM is able to relieve pressure drop. The steady desorption which can R_eDuce pressure decay is related to the values of the steady desorption coefficient and the region,and it is not related to the values of the pressure drop. However,the ability to relieve the pressure drop is increased with the increase in the amplitude of the pressure drop for unsteady desorption. When the desorption speed contents production,the propagation of the pressure wave will cease. It seems to reach a constant pressure boundary.

(v) The fracture asymmetry goes against the flow to in the wellbore. When the fracture asymmetry is serious,the flow resistance becomes greater.