Modeling well performance for fractured horizontal gas wells

Journal of Natural Gas Science and Engineering 18 (2014) 180e193

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Modeling well performance for fractured horizontal gas wells Jiajing Lin*, Ding Zhu Texas A&M University, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 September 2012 Received in revised form 19 February 2014 Accepted 22 February 2014 Available online

In tight gas reservoirs, horizontal wells have been used to increase reservoir recovery and hydraulic fracturing has been applied to further extend the contact with the reservoir. In the past, many models, analytical or numerical, were developed to describe the flow behavior in horizontal wells with fractures. Source solution is one of analytical/semi-analytical approaches. The source method was advanced from point sources to volumetric source, and pressure change inside fractures was considered in the volumetric source method. We have developed a method that can predict horizontal well performance, and the model can also be applied to fractured horizontal wells. The method solves the problem by superposing a series of slab sources under transient or pseudo-steady state flow conditions. The principle of the method comprises the calculation of semi-analytical response of a rectilinear reservoir with closed outer boundaries. The slab source approach assigns sources a geometry dimension, similar to the volumetric source method; but has the solution similar to the point source method by neglecting the effect of the flow inside the source. When solving the source problem the pressure/flow effect inside source is considered sequentially by superposition principle over multiple sources. The pressure response is integrated over time to provide continuous pressure behavior. Flow effect inside fractures can be studied by dividing the fracture into several segments, and each can be treated as a slab source. The method is validated by comparison with the results of analytical solutions and other commercial software of horizontal wells with uniform flux and infinite conductivity, and fractured wells with uniform flux, finite or infinite conductivity. For multiple fractures in a horizontal well, the method was also compared with some published field data. The method provides an effective tool for horizontal well design and well stimulation design for gas reservoirs. In this paper, we present the details of model development. We use a case study to illustrate how the model can help to optimize wellbore and fracture design by comparing production performances of vertical well, slanted well, horizontal well, and fractured vertical and horizontal wells. The method in this paper is more accurate compared with conventional point-source solution, and can handle the transaction from transient flow to pseudo-steady state flow smoothly. Ó 2014 Elsevier B.V. All rights reserved.

Keywords: Horizontal wells Hydraulic fracture Tight gas formation Semi-analytical solution

1. Introduction Over the past decades, point source integrated over a line and/or a surface has been mostly used in solving single-phase flow problems in porous media when fluid movement is from a complex fractured well system. Horizontal well models with point source solution have been presented in many literatures. Gringarten and Ramey (1973) provided the tables of instantaneous Green’s and source functions which can be used in combination with Newman’s product method to generate solutions for different reservoir flow problems. Many

* Corresponding author. 901G RICH, 3116 TAMU, College Station, TX 77845, United States. Tel.: þ1 979 845 2920, þ1 979 571 2406 (mobile). E-mail address: [email protected] (J. Lin). http://dx.doi.org/10.1016/j.jngse.2014.02.011 1875-5100/Ó 2014 Elsevier B.V. All rights reserved.

studies of horizontal well problems used this approach (Clonts and Ramey, 1986; Daviau et al., 1988; Babu and Odeh, 1989). The analytical models are developed under assumptions about boundary conditions. Steady-state models assumed a constant pressure at the drainage boundary (Butler, 1994; Furui et al., 2003), pseudo-steadystate models assumed no flow crossing the boundary with either constant pressure gradient or constant flow rate (Babu and Odeh, 1988, 1989), and transient flow models uses an infinite acting drainage domain (Goode, 1987; Ozkan, 1988; Ozkan et al., 1995a,1995b; Goode and Kuchuk, 1991; Economides and Frick, 1994). For low permeability formation, transient flow period of a horizontal well may be significantly longer than for conventional formations. Valko and Amini (2007) developed a method with distributed volume sources to simulate fractured horizontal wells in a box-shaped reservoir. A source term was added to the diffusivity

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

equation to calculate the pressure distribution. Then the production rate from a fracture is computed. Different from the other point source methods, the volume source approach is able to describe the pressure behavior inside sources and its influence to the flow field. Meyer and Bazan developed approximate analytical solution of complex linear flow for multiple finite-conductivity transverse fractures (Meyer et al., 2010). Fractured horizontal well performance is also studied by using reservoir simulation tools. Even though the numerical simulation is a more powerful tool to handle complex reservoirs, analytical/semi-analytical models by source approaches offer flexibility to count for the detail structure of wellbore, completion and fracture geometry; and thus, are invaluable tools in well performance prediction. In this paper, we present a different approach to the problem of unsteady state flow of a compressible fluid in a rectilinear reservoir. The model is based the solution of slab sources. It can be used to calculate well performance for horizontal gas wells with or without fractures. Fractures can be longitudinal or transverse, single or multiple, and fractures can be infinite conductivity or uniform influx. Using the slab source approach, we assigned the sources (horizontal wells or fractures) a geometry dimension, and the effect of pressure behavior inside sources is considered by superposition principles. This method is relatively easy to apply because flow rate could be calculated directly from pressure difference between initial reservoir pressure and pressure in fracture, which is the same as wellbore flow pressure for an infinite conductivity fracture. 2. Semi-analytical slab source model The slab source method solves the flow problem in a parallelepiped porous medium with a slab source, s, placed in the domain, as shown in Fig. 1. The porous medium is assumed to be an anisotropic reservoir. Following the same approach as the conventional point source solution to apply Newman’s principle, the three-dimensional pressure response of the system to an instantaneous source can be obtained as the production of the solutions of three one-dimensional problems from each principal direction. The diffusivity equation in dimensionless format of a singlephase incompressible fluid is written as

V2 pD ¼

vpD vtD

(1)

For an anisotropic medium, we can write the diffusivity in three directional as 2

2

2

v pD v p v p vpD þ 2þ 2 ¼ vtD vx2D vyD vzD

(2)

The dimensionless variables in the above equations are defined in Appendix A. Because the diffusivity equation is in the same format as the heat conduction problems, we can directly apply the sink/source technique to solve the flow in porous media. The solution from this technique applies to different state in the flow period, both transient flow and stabilized flow. The boundary

181

condition of the reservoir can be constant pressure, no-flow or mixed boundary, which makes the model practical to a wide range of flow problems in petroleum engineering. The procedure of obtaining the solution is to obtain onedimensional solution of the slab problem, applying Newman’s product method based on instantaneous source function in an infinite reservoir to get three-dimensional solution, and then integrates the three-dimensional solution over time to get a continuous source function. Modifying the point source domain by placing a pair of parallel plates in the domain, as shown in Fig. 2, we began our model with one-dimensional instantaneous infinite slab source in an infinite slab reservoir. Green’s functions (Carslaw and Jaeger, 1959) for different boundary conditions in infinite slab reservoirs with a system scheme in Fig. 2 are shown in Table 1. To apply the instantaneous Green’s function in the slab source model, Newman’s method has been applied which states that at certain types of initial and boundary conditions, the solution of a three-dimensional problem is equal to the product of the solutions of three one-dimensional problems. We start with an instantaneous slab source in an infinite one-dimensional reservoir (Fig. 2), overlay three of such sources in x, y, and z direction to make a threedimensional instantaneous slab source in a box-shaped reservoir. To obtain the solution of the new system, we multiply the three solutions of the original one-dimensional problem to an instantaneous solution for the three-dimensional system. Integrate over the well trajectory or the fracture length and height to get the instantaneous slab source solution for the performance of the well, and then integrate over the time to get the three-dimension continuous slab source solution to solve practical reservoir problems. The procedure is summarized in Fig. 3. The solution as instantaneous source depends on the locations of the slab source and the box shape reservoir. To apply this method for horizontal wells with or without fractures, we define the source term (the location and the dimensions of the source) and the main domain according to each individual physical system. For instance, the pressure drop as a results of a constant production, q, at a position (x0, y0, z0) in an anisotropic box-shaped reservoir measured at a position (x, y, z) is readily calculate by

vpD ¼ sx sy sz vxD vyD vdzD vtD

(3)

where, Sx, Sy and Sz are the slab source functions in each direction, as shown in Table 1. For example, for the no-flow boundary, the sx, sy, and sz is

sx ¼

"  # N xf npxf 4a X 1 npxo npx n2 p2 kx s sin exp  1þ cos cos pxf n ¼1 n a a 2a a a2

sy ¼

" N yf mpyf 4b X 1 mpyo mpy sin cos cos 1þ pyf m ¼ 1 m b b 2b b !# 2 2 m p ky s  exp  b2

"  # N zf lpzf 4h X 1 lpzo lp z l2 p2 kz s sin exp  sz ¼ cos cos 1þ pzf l h b 2h h h2 l¼1

Fig. 1. Scheme of the slab source model.

After obtain the instantaneous slab source solution under defined boundary conditions, we integrate the instantaneous point source over a time interval to attain the continuous slab source solution. The pressure drop at point (x, y, z) as a result of the continuous production or injection at position (x0, y0, z0) in an anisotropic box-shaped reservoir then is

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0

X0

0

Xe

X0

w

0

Xe

X0

Xe

w

w

Fig. 2. Instantaneous Green’s function.

Table 1 Instantaneous Green’s function in 1D infinite slab reservoir (Carslaw and Jaeger, 1959). Boundary conditions

Instantaneous Green’s functions  PN 1 npx n2 p2 kx s 4 sin 2xef sin npxexw sin nxpex p n ¼ 1 n exp  ax2e " #  xf 4xe PN 1 exp  n2 p2 kx s sin npxf cos npxw cos npx 1 þ 2 n¼1 n pxf xe xe xe 2xe ax

Constant pressure at x ¼ 0 and x ¼ xe No-flow at x ¼ 0 and x ¼ xe

e

No-flow at x ¼ 0 Constant pressure at x ¼ xe

vpD ¼ vxD vyD vzD

Zt

8

p





ð2nþ1Þ2 p2 kx s 4ax2e

ð2nþ1Þpx pxw px cos ð2nþ1Þ sin 4xe f cos ð2nþ1Þ xe xe

(4)

0

For the slab source representing a fracture, the solution of the continuous slab source can be written as,

pD ¼

1 n ¼ 1 2nþ1 exp

2.1. Uniform flux horizontal well

  sx sy sz ds



PN

1 L3 ðx2  x1 Þðy2  y1 Þðz2  z1 Þ

 Z t Zz2 Zy2 Zx2

For a horizontal well located in a box shape reservoir, the well can be simply treated as integrating the slab source over the length of the wellbore. The cross-section area of the source, which is shown in Fig. 4, is equivalent to 2 As ¼ rw

  sx sy sz dxdydzds

0 z1 y1 x1

(6)

For a horizontal well, we assume that anisotropy is only in vertical and horizontal direction (kx ¼ ky ¼ kh).

(5) 2.2. Horizontal well with a transverse fracture where (z2  z1), (y2  y1) and (x2  x1) are the height, length and width of the fracture. The detailed derivation of the above equations is shown in Appendix A and Appendix B.

Newman Method

x

Fig. 5 shows the scheme of a transverse fracture case. We assume that the fracture is dominating the total production to the

Integrate over time

for 3D solution

y

z

s

y

z

x

x Integration over

Well Trajectory or Fracture Geometry

Fracture

s

Continuous

wellbore

Fig. 3. Slab source solution approach.

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

183

h l

As b a Fig. 6. Superposition for one transverse fracture. Fig. 4. Slab source solution for horizontal well.

well and the flow to the wellbore is neglected. If the fracture is infinitely conductive or with uniform flux, the fracture can be treated as a slab source integrate along both the fracture length and height directions. So far the slab source solution has neglected the flow inside fracture. To add the effect of flow inside fracture, we divided the original fracture source to multiple smaller sources, as shown in Fig. 6. Multiple sources can also consider the heterogeneity of the reservoir. Depending on the permeability and porosity distribution of the reservoir, each source could be assigned with different properties. We first define the inner boundary condition at the interface of the source and the domain (for example, wellbore and reservoir). Then we divide the fracture into multiple segments. The segments are then connected to each other by superposition in the space. By using this technique, a set of linear equation is generated and solved to predict the fractured horizontal well performance. Fig. 6 shows an example for four sources fracture. We first allow source 1 to exist in the reservoir and let it generate a flow rate of q1 at the location. The flow results in corresponding pressure changes at locations of sources 2, 3, and 4. Then if we only let source 2 exists the pressure also changes at all source locations. We can apply this procedure to all 4 sources in the system. To illustrate the influence of the source location, we use dimensional format of the equations in this section. By the superposition principle, the total pressure drop and production rate at each source location as a result of each segment produces at a constant rate is calculated by

effect on the object segment. The pressure drop measured at segment i as a result of the production at segment j is evaluated by multiplying qj with Fi,j(x, y, z, t) as shown in Eq. (7). For the entire fracture (N segments), we obtain a set of linear equation shown as,

p  pi ðx; y; z; tÞ ¼ qj Fi;j ðx; y; z; tÞ

2.3. Horizontal well with multiple fractures

(7)

where Fi,j(x, y, z, t) is the coefficient of single slab source solution of segment j relative to the segment i in the domain. This coefficient is presented in Appendix B (Eq. B6 for a constant production rate condition). The subscripts x, y, z indicate the center locations of fracture, and i, j are fracture index. i is the index for object segment and j the index of segments other than object segment which has

Fracture h

well b a Fig. 5. Slab source solution for one transverse fracture.

q1 F1;1 þ q2 F1;2 þ / þ qN F1;N ¼ Dp1 q1 F2;1 þ q2 F2;2 þ / þ qN F2;N ¼ Dp2 « q1 FN;1 þ q2 FN;2 þ / þ qN FN;N ¼ DpN

(8)

where qj is the flow rate into segment j and Dpj is the pressure drop for segment j due to the flow into the fracture. The total production form the entire fracture is calculated by

XN

q j¼1 j

¼ qtotal

(9)

where qtotal is the total production over every segment, or the maximum flow rate if the well constraint is constant production rate. By using the above method, we can calculate the fractured horizontal well performance in uniform flux boundary condition for each segment, infinite or finite conductivity. When flow rate inside the fracture is high, finite conductivity should be used to take the pressure drop in the fracture into account. On the other hand, if the flow rate is relatively low and pressure drop in the fracture is insignificant, infinite conductivity could be applied. The fractured well is predicted under constant wellbore-pressure constraint.

As a simple example, we consider two transverse fractures which are shown in Fig. 7. For transverse fracture intercepting a horizontal well, we represent the fractures as infinitely conductive or with uniform flux under the assumption that the fractures are dominating the total production to the well. Each fracture can be treated as an individual source and their effects to other fractures are included through superposed pressure drawdown. Eq. 7 can be directly used in this case. In the equation i denotes the location that observes the pressure change, and j denotes the fracture that causes the pressure change. If considering pressure drop in the wellbore between fractures

pwf ;i  pwf ;i1 ¼ Dpfrac;i

(10)

To calculate the well performance, we first place fracture 1 in the system, which causes a flow rate of q1 at the location of the fracture 1. The flow results in corresponding pressure changes at both locations of the fracture 1 and the fracture 2. Then if we only let fracture 2 exist, the pressure also changes at both locations. Since the total pressure drawdown at each fracture should be the sum of the pressure drops caused by all the fractures in the system. By the superposition principle, we have

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Fig. 7. Superposition for two transverse fractures.

pre  pwf ;1 ¼ Dp1;1 þ Dp1;2

(11)

and

pre  pwf ;2 ¼ Dp2;1 þ Dp2;2

(12)

If we assume there is no pressure drop along the wellbore, then

pwf ;1 ¼ pwf ;2

(13)

3. Validation of the model The solution presented above is validated by the solutions of different methods. For horizontal wells, we compared the solution with an analytical solution presented by Babu and Odeh (1988), and for fracture cases, we validated the slab source model with the volumetric source model (Valko and Amini, 2007) and a commercial software, Ecrin-Kappa. We validated the multiple source fracture case with the single segment fracture model. The multiple fracture result was compared with published filed data (Meyer et al., 2010).

3.1. Uniform flux horizontal well Babu and Odeh (1988) presented a method to obtain the performance of a horizontal well. They developed a line source solution to represent a horizontal well. The model is under pseudosteady state condition. The input data is given in Table 2. Comparing the slab source model with Babu and Odeh’s model, Fig. 8 shows the results of the two solutions for the example of uniform flux horizontal well. The result of the slab source model is close to Babu and Odeh’s model. Since Babu and Odeh’s model assumed pseudo-steady state, the reservoir pressure is an average pressure over the reservoir volume, but the slab model is a transient model and pressure declines from the initial reservoir pressure. We used the material balance to approximate the average reservoir pressure decline when using Babu and Odeh’s model, which causes inaccurate results. 3.2. Fully penetrating transverse fracture intercepting a horizontal well The input data used in validating the fracture model is shown in Table 3. The result of a fully-penetrating transverse fracture with

Table 2 Input data for horizontal cell validation. Parameter

Value

Unit

Reservoir length (assumed), b Reservoir width (assumed), a Reservoir thickness, h Horizontal wellbore length, L Porosity, f Reservoir initial pressure, pi Reservoir temperature, T Bottomhole pressure, pwf Gas specific gravity, gg Water specific gravity, gw 8 < H2 S Gas component CO2 : N2

4000 2000 200 3000 9.0% 2335 146 1886 0.836 1.005 0% 1% 1.4%

ft ft ft ft

Reveal data Horizontal permeability, kH Vertical permeability, kv Gas viscosity, mg

0.25 0.1 0.0156

psi  F psi

md md cp

Fig. 8. Comparison of the Slab Source model with Babu and Odeh’s model.

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

185

Table 3 Input data for single transverse fracture validation. Parameter

Value

Unit

Reservoir length (assumed), b Reservoir width (assumed), a Reservoir thickness, h Porosity, f Reservoir initial pressure, pi Reservoir temperature, T Bottomhole pressure, pwf Gas specific gravity, gg Horizontal permeability, kH Vertical permeability, kv Gas viscosity, mg Fracture length, Lf Fracture height, hf Fracture width, wf

4000 2000 200 0.09 2335 146 1885 0.836 0.5 0.25 0.0156 1000 100 0.033

ft ft ft ft psi  F psi md md cp ft ft ft

uniform flux is compared with the solution by the distributed volumetric source (Valko and Amini, 2007) and the commercial software Ecrin-Kappa. In this case, as demonstrated in Fig. 9, the three methods showed a good agreement. It should be noticed that the slab source model neglected the pressure drop caused by the Non-Darcy flow inside fracture. To compare with the distributed volumetric source model, this pressure drop is not included in the distributed volumetric methods either. When flow rate is high, this assumption is not valid (Miskimins et al., 2005) and could result in some errors in rate estimation. From the comparisons with other models at the appropriate conditions, we validated of the slab model for horizontal wells with an excellent agreement for uniform flux and infinite conductivity solution. For the case of fractured wells, the results are in very good agreement (within 1% difference). 3.3. One transverse fracture with multiple sources by superposition The transverse fracture is divided into 4 or 9 segments in this study. The wellbore is placed in the middle of the fracture. Superposition method has been applied to the problem. If we assume that the pressure for each segment is equal to the wellbore pressure, then the production from each segment will be equal, so does the total flow rate from the cases with different segment number. In addition, the total flow rate should be equal to the rate from nonsegment fracture model if neglecting the pressure drop inside the fracture. We use this to validate the superposition operation. The input data is given in Table 3. From the results (Fig. 10), the production rates for 1, 4 and 9 sources are the same, which confirms

Fig. 9. Comparison of the Slab Source model, DVS and simulation by Ecrin.

Fig. 10. Confirmation of superposition procedure.

the superposition method we used in the study. More segment number can be used to count for conversion in the fracture. The effect of heterogeneity in the reservoirs can also be studied by this method by assigning different permeability to different segments. 3.4. Multiple fractures along a horizontal well To validate the multiple fracture calculation, we compared the model result with a published field production data. The history match is based on a horizontal well in Marcellus shale (Meyer et al., 2010). The horizontal well was completed with a six stage fracture treatment over a lateral length of 2100 ft. The production data was matched with the single phase, multiple transverse fractures in horizontal wellbores, and the result is shown in Fig. 11. The reservoir and fracture properties are showed in Table 4. A petrophysical analysis showed an average permeability of 366 nD over a formation thickness of 162 ft that included the Upper and Lower Marcellus. The calculated result is on the same order of the field observation. Because fractured well performance depends on numerous parameters including reservoir properties and fracture geometries which both contain uncertainty and it is not surprised that a perfect match was not obtained. From the results (Fig. 11), the higher production rate at early time from the slab model could

Fig. 11. History match of a gas well in Marcellus Shale.

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J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193 Table 4 Marcellus shale-reservoir and fracture properties. Formation

Marcellus

Depth (ft) Thickness (ft) Drainage Area (acres) Reservoir permeability (nD)

7876 162 80 390

Wellbore/Fracture Wellbore Radius (ft) Lateral Length (ft) Number of stages Propped length (ft) Conductivity, (mD-ft)

0.36 2100 6 320 3.9

caused by the assumptions of ideal condition of fractures and neglected Non-Darcy choking effect inside fracture.

Fig. 12. Effect of wellbore length on production rate for kh ¼ 0.01 md, kv ¼ 0.005 md.

4. Results and discussion Examples of horizontal well with different wellbore length, a horizontal well with single transverse fracture and multiple transverse fractures intercepts horizontal well have been studied by the slab model. The input data is shown in Table 5. The results are presented in this section. 4.1. Horizontal well To produce tight gas reservoirs, horizontal wells are more efficient compared with vertical wells. The length of the horizontal laterals relative to reservoir drainage dimension will be a key parameter for well production. In general, since mobility dominates the production in tight gas reservoirs, the longer the wellbore length, the higher the production rate. Since comparing to the drawdown of the reservoir, the pressure drop along the wellbore is usually not significant, we neglect the pressure drop along the wellbore in this example. Figs. 12 and 13 show the production performance for the horizontal well under transient flow condition for relatively low permeability (0.01-md, in Fig. 12), and moderate permeability (0.5-md, Fig. 13). The strategy of well structure should be different considering the reservoir permeability. At the same

Table 5 Input data for case study. Reservoir type

Tight gas

Formation type

Sandstone

Parameter

Value

Unit

Reservoir length (assumed), b Reservoir width (assumed), a Reservoir thickness, h Horizontal wellbore length, l Porosity, f Reservoir initial pressure, pi Reservoir temperature, T Bottomhole pressure, pwf Gas specific gravity, gg Water specific gravity, gw 8 < H2 S Gas component CO2 : N2

4000 2000 200 2000, 3000, 4000 9.0% 2335 146 1886 0.836 1.005 0% 1% 1.4%

ft ft ft ft

Reveal Data Horizontal permeability, kH Vertical permeability, kv Gas viscosity, mg

0.5, 0.1, 0.01, 0.001 0.25, 0.05, 0.005, 0.0005 0.0156

md md cp

drawdown, with low permeability, longer wellbore keeps significant advantage through the production period of 500 days (Fig. 12), but for higher permeability, the benefit of longer wellbore diminishes and rate declines much faster for longer wellbore compared with short wellbore (Fig. 13). It is necessary to point out that when permeability is higher than 0.1-md, the initial rate is significant and pressure drop in the wellbore should be considered in this case. 4.2. Horizontal well with single transverse fracture The most effective way to produce low permeability gas formation is to create transverse fractures along a horizontal wellbore. To isolate the effect of fracture on well performance, we assumed in this section that all the production comes from the fracture, and the horizontal wellbore does not contribute to total flow in the cases of fractured horizontal wells (assume the wellbore is cased and perforated for fracturing only). We put the fracture in the middle of the reservoir perpendicular to the horizontal wellbore. The dimension of the fracture is 500-ft in length, 150-ft in height and 0.01-ft in width. In this study, we assume there is no pressure drop inside the fracture. Fig. 14 shows the production performance for different reservoir permeability. Comparing Fig. 12 with Fig. 14, the production rate for 0.01-md case is 700 Mscf/day initially if the wellbore is 3000-ft long without fracture (Fig. 11), but if we place a

psi  F psi

Fig. 13. Effect of wellbore length on production rate for kh ¼ 0.5 md, kv ¼ 0.25 md.

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

187

Fig. 16. Cases comparison for kh ¼ 0.01 md and kv ¼ 0.005 md.

Fig. 14. Performance of single fracture.

fracture to the well, the initial rate is 1000 Mscf/day. In low permeability formation, hydraulic fracturing adds significant improvement to horizontal well production meanly because two facts; the contact area is increased by the fracture and the flow regime is changed from radial/linear flow in horizontal well only to dominated linear flow in fracture/horizontal well. Depending on the completion of the well, the wellbore usually also contribute to the total production in the fractured well case. The assumption that wellbore does not produce underestimates the flow rate in the case of fractured wells. A single fracture delivers lower production than an openhole horizontal well, because the reservoir-wellbore contact is better distributed in the case of openhole horizontal wells. 4.3. Multiple transverse fractures along a horizontal well Superposition method has is applied to study the cases of multiple transverse fractures. We compare the production rate for horizontal well without fracture, and with one, two, and five fractures. All the fractures have the same dimensions, and each fracture is treated as one source Figs. 15e17 show the results of well performances for three different reservoir permeabilities. For a highperm gas reservoir, fracture number is a sensitive parameter to make hydraulic fracturing attractive. When the number of fractures is too high, benefit of adding more fractures diminishes. For low-

Fig. 17. Cases comparison for kh ¼ 0.001 md and kv ¼ 0.0005 md.

perm gas formations, fractured wells are more likely to perform better than non-fractured wells. The more fractures that can be placed along the wellbore, the higher production rate will be. As studied before (Magalhaes et al., 2007; Bagherian et al., 2010), there may be a maximum number of fractures, and above that number, more fractures will not bring more benefit to the well production. The optimal fracture spacing also depends on the stress field change while fracturing each stage (Roussel and Sharma, 2010; Suri and Sharma, 2009), and completion limitations (how many

Table 6 Permeability summary for layers.

Fig. 15. Cases comparison for kh ¼ 0.1 md and kv ¼ 0.5 md.

Layer no.

Layer thickness (ft)

kh (md)

kv (md)

Average perm, kh (md)

1 2 3 4 5 6 7 8 9

12 6 5 26 2 116 10 5 18

0.05e0.5 10.64 0.003 0.2e0.6 2.32 0.1e0.8 1.92 7.55 0.5e2.3

0.14 5.53 0.003 0.16 2.32 0.12 1.92 4.07 0.89

0.26 10.64 0.003 0.36 2.32 0.25 1.92 7.55 1.67

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J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

Layer 1

kh=0.26 md

k v= 0.14 md

h=12 ft

Layer 2 Layer 3

kh =10.64 md

kv=5.53 md

h=6 ft

kh =0.003 md

kv =0.003 md

h=5 ft

Layer 4

kh=0.36 md

kv=0.16 md

h=26 ft

Layer 5

kh=2.32 md

kv=2.32 md

h=2 ft

Layer 6

kh=0.25 md

k v=0.12 md

h=116 ft

Layer 7

kh=1.92 md

kv=1.92 md

h=10 ft

Layer 8

kh=7.55 md

kv=4.07 md

h=5 ft

Layer 9

kh=1.67 md

kv=0.89 md

h=18 ft

Fig. 18. Schematic of the formation.

Table 7 Input data for case study. Reservoir type

Tight gas

Formation type

Sandstone

Parameter

Value

Unit

Reservoir length (assumed), b Reservoir width (assumed), a Reservoir thickness, h Horizontal wellbore length, l Porosity, f Reservoir initial pressure, pi Reservoir temperature, T Bottomhole pressure, pwf Gas specific gravity, gg Water specific gravity, gw 8 < H2 S Gas component CO2 : N2

4000 2000 200 3000 9.0% 2335 146 1886 0.836 1.005 0% 1% 1.4%

ft ft ft ft psi  F psi

Table 8 Parameter list for slanted well. Zone no.

Layer no.

Kh, md

Kv, md

1 2 3

1e4 5e6 7e9

0.72 0.26 1.67

0.34 0.2 0.92

reason is that for more fractures, the sources are distributed in the reservoir, and traveling distance for the reservoir fluid to research the sources (one of the fractures) becomes smaller, implying that flow efficiency is improved with more fractures placed along the wellbore. 5. Case study A gas reservoir is used in the example. The reservoir has multiple layers with significant contraction in permeability in each layer. The permeability profile is shown in Table 6 and Fig. 18. Layer 3 has the lowest permeability, serving as a vertical isolation in the formation. In such a reservoir, horizontal well may lose its attraction compared with vertical well because of low vertical communication. The example compares the well structure plans including vertical well, vertical well with fracture, slanted well, horizontal well without fracturing, and horizontal well with multiple transverse fractures. The reservoir properties are shown in Table 7 and the permeability profile for slanted well is shown in Table 8. 5.1. Well structure design Even vertical wells are considered as conventional and less aggressive, since this reservoir has multiple layers, vertical well should not be simply eliminated from consideration. The initial

fractures can actually be placed along a wellbore). All these should be considered when design a multi-stage fracture treatment in additional to maximum production rate. Noticed that the results show that a horizontal well with five fractures produces more than five times the production rate of a horizontal well with one fracture at early time of production. The Fig. 20. Fractured Vertical well schematic.

Fig. 19. Vertical well schematic.

Fig. 21. Horizontal well schematic.

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

189

Fig. 22. Slanted well with 82 and wellbore treatment.

Fig. 23. Cases comparisons for production rate.

comparison considers a vertical well fully perforated in the pay zone as shown in Fig. 19, a hydraulic fractured vertical well with 500 ft fracture half-length (Fig. 20), and a 3000-ft long horizontal well (Fig. 21). One disadvantage of horizontal wells is that they could only produce from one pay zone. In such a heterogeneous

reservoir, a slanted well with a deviated angle of 82 (Fig. 22) is considered here. To simplify the problem, the slanted well is approximated with three horizontal sections. The sum of the production from these sections provides us an estimated flow rate of a slanted well.

Fig. 24. Cases comparisons for cumulative production.

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J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

that the method is conceptually correct and can be applied for optimization study. With the case comparison, we have drawn the conclusions that optimal number of fracture stages is strongly depends on reservoir permeability. For moderate permeability formation, higher number of fracture stages does not necessarily yield justified benefit in production. For low permeability formation, the higher the fracture number, the better the production will be. In such a formation, the optimal number of fractures should be determined by other constrains than production rate.

Acknowledgments

Fig. 25. Cumulative production for kh ¼ 0.001 md.

The last well plan is multiple transverse fractures along the horizontal well. Transverse fractures are equally placed along the horizontal well. Fracture numbers varied from 5 to 20. All the fractures have the same dimension of 500-ft in half length, 150 ft in height, and 0.033 ft in width. 5.2. Result comparisons The results of production rate for different well structures are shown in Fig. 23. From Fig. 23 it can be seen that the horizontal well with 20 fractures has the highest production rate since 20 fractures increase the contact with the reservoir most effectively and create the communication of the upper and the lower pay zones. However, the production rate declines significantly after 50 days. It also shows that the slanted well has the challenge production compared with the horizontal well and the fractured vertical well. Since the reservoir permeability is moderate for this gas formation, slanted well has the potential of improved production with relatively low cost compared with horizontal well and fracturing. Fig. 24 shows the cumulative production. There are significant increases in production from horizontal well/ slanted well to horizontal well with 5 fractures. But increasing the fracture number from 10 to 20 does not add sufficient additional production, especially when considering the complexity and cost associated with high number of fracture stages. Economic evaluation is necessary to make final decision. This observation is directly related to the moderate permeability of the reservoir. For low-permeability gas formation, the observation can be different. Fig. 25 shows the production for different fracture numbers if the average permeability is 0.001 md. The benefit of adding more fractures is obvious, In such a case, the number of fractures should be selected based on other considerations such as stress field alternation and fracturing operation constrains.

The authors thank the sponsors of the Crisman Research Institute at Texas A&M University for providing the financial support for this study.

Nomenclature As cross-section area of source term, ft2 a reservoir width, ft B formation volume factor b reservoir length, ft ct compressibility, psi1 h reservoir height, ft kx permeability in x-direction, md ky permeability in y-direction, md kz permeability in z-direction, md l wellbore length, ft L dimensionless reservoir position pD dimensionless pressure pint initial reservoir pressure, psi pR average reservoir pressure, psi pwf bottomhole pressure, psi Dp pressure drawdown, psi q flow rate, Mscf/day T temperature,  R tD dimensionless time xD dimensionless reservoir length xf fracture width ftyD dimensionless reservoir width zD dimensionless reservoir height gg specific gas gravity, dimensionless r density, lb/ft3 f porosity, fraction m viscosity, cp s time

Appendix A. Dimensionless parameters The following dimensionless parameters are used throughout the paper. The dimensionless pressure (pD) for a constant production rate (q) is defined as

6. Conclusions The slab source method has been developed in this study. It is a viable alternate to generate transient and pseudo-steady state solutions for horizontal well with or without fractures. The method yields a semi-analytical solution, which is easier to apply for complicated geometrics. It also provides a way to include flow effect inside fracture. Simple examples illustrate

pD ¼

kLðpi  pÞ qBm

where k ¼

(A1)

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 kx ky kz and L ¼ 3 xe ye ze .

The dimensionless times, reservoir dimension is defined as

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

191

"

kt ¼ fmct L2

(A2)

pffiffiffi x k xD ¼ pffiffiffiffiffi$ kx L

(A3)

tD

sz ¼

pffiffiffi y k yD ¼ pffiffiffiffiffi$ ky L

zD

sy ¼

N yf mpyf 4b X 1 mpyo mpy sin 1þ cos cos pyf m ¼ 1 m b b 2b b !# m2 p2 ky s  exp  b2

(B2)

"  # N zf lpzf 4h X 1 lpzo lp z l2 p2 kz s sin exp  1þ cos cos pzf l h b 2h h h2 l¼1

(A4)

(B3) For the continuous slab source to calculate the fracture performance, the equation is

pffiffiffi z k ¼ pffiffiffiffiffi$ L kz

(A5)  pint  pðx; y; z; tÞ ¼

B0 m0 q0 abha

Appendix B

 Zz2 Zy2 Zx2 Z t z1

y1

x1

  sx sy sz ds dx dy dz

0

(B4) Consider a segment of the real axes between x ¼ 0 and x ¼ a, the diffusivity equation describes the behavior of the reservoir. For the reservoir is completely bounded or no-flow across the reservoir boundary, and Sx, Sy, and Sz in this case are

"

sx ¼

 pint  pðx; y; z; tÞ ¼

N X

xf npxf 4a 1 npxo npx sin 1þ cos cos pxf n ¼ 1 n a a 2a a #   n2 p2 kx s  exp  a2

To integrate Eq. B4, The solution is

 Bo mo qo Fðx; y; z; tÞ aabhðx2  x1 Þðy2  y1 Þðz2  z1 Þ (B5)

(B1) Where F(x, y, z, t) is equal to

N npxf  4aa4 ðy2  y1 Þðz2  z1 Þ X 1 npx npx2 npx1 sin * sin  sin cos 4 4 a 2a a a p kx xf n n¼1    N mpyf  n2 p2 kx t 4ab4 ðx2  x1 Þðz2  z1 Þ X 1 mpy mpy2 mpy1 sin * sin  sin þ cos  1  exp  4 b 2b b b aa2 p4 ky yf m m¼1      N npzf m2 p2 kx t 4ah4 ðx2  x1 Þðy2  y1 Þ X 1 lpz lpz2 lpz1 sin * sin  sin þ cos  1  exp  4 h 2h h h ab2 p4 kz zf l l¼1    N N l2 p2 kz t 16aa2 b2 ðz2  z1 Þ X X  npx2 npx1  mpy2 mpy1 sin  sin sin  sin þ  1  exp  a a b b ah2 p6 xf yf m¼1 n¼1 ( " !#) npxf p m y x sin 2 sin 2a cos np sin mbpx p2 t n2 kx m2 ky 16aa2 h2 ðy2  y1 Þ 2b a 2 1  exp  þ 2 þ * 2k 2 m n k y a p6 xf zf a b n2 m2 2 x þ 2

Fðx; y; z; tÞ ¼ ðx2  x1 Þðy2  y1 Þðz2  z1 Þt þ

a



N X N  X

sin

n¼1 l¼1

b

     n pxf x sin lpz2 sin lpz  p2 t n2 kx l2 kz npx2 npx1 lpz2 lpz1 sin 2a cos np h 2h  a2  sin sin  sin * 1  exp  þ 2 a a a h h a2 h2 n2 l2 n kx þ l kz a2

h2

  mpyf mpy lpz N X N   x1 Þ X mpy2 mpy1 lpz2 lpz1 sin 2b cos b sin 2h2 sin lphz  2 sin  sin sin  sin * þ 2 m k b b h h m2 l2 b2 y þ lhk2z m¼1 l¼1 !#) ( " N X N X N  p2 t m2 ky l2 kz 64aa2 b2 h2 X npx2 npx1  mpy2 mpy1 sin  sin sin  sin þ þ 8  1  exp  2 2 a a a b b p xf yf zf n ¼ 1 m ¼ 1 l ¼ 1 b h " !#) (  p p n x m y p m y p l z f f p n z l 2 sin 2a cos a sin 2b cos b sin 2h sin phz p2 t n2 kx m2 ky l2 kz lpz2 lpz1  2  sin 1  exp  þ þ * sin * 2 m2 k a h h a2 b2 h2 n2 m2 l2 na2kx þ b2 y þ lhk2z 16ab2 h2 ðx2 p6 yf zf

(B6)

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J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

At late time or stabilized flow, the exponential terms in Eq. (B6) becomes zero and it reduced to

 N Bo mo qo 4aa4 ðy2  y1 Þðz2  z1 Þ X 1 npx fðx2  x1 Þðy2  y1 Þðz2  z1 Þt þ cos 4 4 aabhðx2  x1 Þðy2  y1 Þðz2  z1 Þ a p kx xf n n¼1 N   4 X p p n xf m yf npx2 npx1 4ab ðx2  x1 Þðz2  z1 Þ 1 mpy mpy2 mpy1 sin sin  sin þ sin  sin cos  sin 4 4 b 2a a a 2b b b p ky yf m m¼1   N N X N  4 2 2 X X p n z 4ah ðx2  x1 Þðy2  y1 Þ 1 lpz lpz2 lpz1 16aa b ðz2  z1 Þ npx2 f sin sin sin  sin þ cos þ 4 6x y h 2h h h a p4 kz zf p l f f m¼1 n¼1 l¼1

 pint  pðx; y; z; tÞ ¼

 sin

npx1  mpy2 mpy1 sin sin  sin * a b b

npxf 2a

x sin mpy2 sin mpx N X N  cos np 16aa2 h2 ðy2  y1 Þ X npx2 npx1 b 2b a 2 sin  sin þ 2k 6 m a a p xf z f n2 m2 na2kx þ b2 y n¼1 l¼1

  npxf x sin lpz2 sin lpz N X N  lpz2 lpz1 sin 2a cos np 16ab2 h2 ðx2  x1 Þ X mpy2 mpy1 2h h  a2 sin  sin *  sin þ  sin 2 6 h h b b p yf zf n2 l2 n kx þ l kz m¼1 

  sin

a2

l¼1

h2

mpy sin 2b f

lpz2 lpz1  sin * h h

N X N X N  cos mbpy sin lp2hz2 sin lphz 64aa2 b2 h2 X npx2  2 sin þ 8 2 m ky l kz a p x y z 2 2 f f f n¼1 m¼1 l¼1 m l þ h2 b2

 sin

npx1 a

9   npxf mpyf mpy lpz2 z lpz= cos sin sin mpy2 mpy1 lpz2 lpz1 sin 2a cos np a sin h b  22b 2h  sin * sin  sin  sin 2 m2 ky ; b b h h n2 m2 l2 n kx þ þ l kz 

a2

b2

h2

(B7)

For stabilized flow under pseudo-steady-state condition, the average reservoir pressure can be written as

pint ¼ p þ

mo Bo qo t abha

(B8)

Substituting Eq. B8 into Eq. B7, we have

( N N X X npxf  Bo mo qo 4aa4 1 np x npx2 npx1 4ab4 1 mpy sin  sin p  pðx; y; z; tÞ ¼ * cos cos sin þ aabh a b 2a a a ðx2  x1 Þp4 kx xf n ¼ 1 n4 ðy2  y1 Þp4 ky yf m ¼ 1 m4   N  X1 mpyf npzf mpy2 mpy1 4ah4 lpz lpz2 lpz1  sin sin  sin sin þ sin  sin cos h b b 2b 2h h h ðz2  z1 Þp4 kz zf l4 

þ

þ

þ þ

16aa2 b2

N X N  X

p6 xf yf ðx2  x1 Þyðz2  y1 Þ m ¼ 1 n ¼ 1 16aa2 h2

N X N  X

p6 xf zf ðx2  x1 Þðz2  z1 Þ n ¼ 1 l ¼ 1 16ab2 h2 64aa2 b2 h2

sin

npxf x sin mpy2 sin mpx npx2 npx1  mpy2 mpy1 sin 2a cos np b 2b a 2  sin * sin  sin m2 ky a a b b n2 m2 n kx þ a2

a2

npxf 2a

h2

  mpyf mpy lpz mpy2 mpy1 lpz2 lpz1 sin 2b cos b sin 2h2 sin lphz  2  sin * sin  sin sin 2 m k y b b h h m2 l2 þ l kz b2

N X N X N  X

p8 xf yf zf ðx2  x1 Þðy2  y1 Þðz2  z1 Þ n ¼ 1 m ¼ 1 l ¼ 1

sin *

b2

  npxf x sin lpz2 sin lpz npx2 npx1 lpz2 lpz1 sin 2a cos np h 2h  a2 sin  sin * sin  sin 2 a a h h n2 l2 n kx þ l kz

N X N  X

p6 yf zf ðy2  y1 Þðz2  z1 Þ m ¼ 1 l ¼ 1

l¼1

9 mpyf z cos np cos mbpy sin lp2hz2 sin lphz= a sin 2b  2 2 m2 k ; n2 m2 l2 na2kx þ b2 y þ lhk2z

h2

  npx2 npx1  mpy2 mpy1 lpz2 lpz1 sin  sin * sin  sin sin  sin a a b b h h

(B9)

J. Lin, D. Zhu / Journal of Natural Gas Science and Engineering 18 (2014) 180e193

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