**2. Production mechanisms, production modeling techniques and simulators**

Having outlined the pertinent characteristics of unconventional gas reservoirs, we now document the likely production mechanisms in the various shale plays based on our under‐ standing of their geology and the underlying geophysics.

The matrix permeability of shale gas reservoirs is extremely small, probably on the order of one tenth of a microdarcy or 100nd. It is virtually impossible to produce gas from these reservoirs in commercial quantities unless the wells are hydraulically fractured and even then, or so it is commonly believed, production is really only possible because a network of natural fractures is opened up. (It is interesting to note that gas has been produced from the ultra-tight Devonian shale plays of the North Eastern USA from more conventionally-fractured vertical wells, which implies that multi-stage hydraulic fracturing was unnecessary for these plays. This is the first hint that the role of the natural fractures may be quite different for the Devonian plays and the deeper shale plays.)

An essential element of a mathematical model of gas production from shales is therefore the ability to describe flow in a very tight rock matrix and flow in a network of fractures. In most gas shale reservoirs most of the reservoir fluid is stored in the matrix and the primary flow path is from the matrix into the fractures and thence into the wellbore. There are essentially two methods of characterizing a multiply-fractured reservoir:


#### **2.1. Discrete fracture network models**

Many commercial numerical reservoir simulators have the capability of simulating flow through a complex network consisting of pores and fractures. However, one of the greatest drawbacks and limitations of simulating a discrete fracture network model is the a priori assumption that all relevant properties of the fracture network are known. Nevertheless great insights can be obtained into the impact of the essential physical processes by examining simple fracture configurations. We note that in principle many different physical and petro-physical components can be included in numerical simulations. However, in practice it is quite common to see results presented only for the special cases:


The simplest fracture network that has been applied to shale gas production consists of a number of planar fractures placed transversely to a horizontal wellbore as illustrated sche‐ matically in Figure 3. It is apparent from many published numerical studies that under these circumstances flow from the reservoir can be described in terms of a number of identifiable flow regimes. The following account is taken from a recent paper by Luo et al [10]. These authors used a commercial reservoir simulator to calculate the flow into a horizontal well with six infinitely-conductive transverse fractures as shown in Figure 4.

acting regime in the sense that the neighboring fractures are effectively at infinity. The duration of this regime depends, as we shall see later, on many parameters including the matrix permeability, the fluid compressibility and the fracture spacing. The illustrations in

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**•** Early radial/elliptical flow: flow into the fracture tips is present, but weak; flow into the fracture surfaces is still predominantly linear, but fracture interference is just beginning to impact the flow. Note that the pressure drawdown in the matrix has almost reached the mid-line between the fractures. At this point the flow regime may be described as pseudo-

**•** Compound formation linear flow (CFL flow): here the fractures are fully interactive and the reservoir drainage area is dominated by the area defined by the length of the well and the

length of the fractures. Flow from beyond this area grows in importance.

**Figure 4.** Numerical solutions for flow into six infinitely-conductive transverse fractures (taken [10]).

Figure 4 are for infinite conductivity fractures.

steady-state or fracture-boundary-dominated.

**Figure 3.** Idealized discrete fracture network showing multiple transverse fractures originating from a horizontal well.

The flow behavior can be conveniently discussed in terms of five flow regimes as follows.

**•** Bilinear or linear flow: soon after the well is placed on production reservoir fluid flows normal to the fracture planes and along the fractures into the well. The streamlines are shown in yellow in Figure 4; reservoir pressure is in red and the constant bottomhole or well pressure is in blue. Note that flow into the fracture tips is negligible and each fracture behaves independently of the other fractures. This regime may also be termed the infinite-

**2.1. Discrete fracture network models**

334 Effective and Sustainable Hydraulic Fracturing

to see results presented only for the special cases:

**•** Reservoir fluid of small and constant compressibility.

six infinitely-conductive transverse fractures as shown in Figure 4.

**•** Production under constant drawdown conditions.

**•** Darcy flow in fractures and matrix.

conductivity is essentially infinite.

**•** No desorption.

Many commercial numerical reservoir simulators have the capability of simulating flow through a complex network consisting of pores and fractures. However, one of the greatest drawbacks and limitations of simulating a discrete fracture network model is the a priori assumption that all relevant properties of the fracture network are known. Nevertheless great insights can be obtained into the impact of the essential physical processes by examining simple fracture configurations. We note that in principle many different physical and petro-physical components can be included in numerical simulations. However, in practice it is quite common

**•** Matrix and fracture permeability independent of pressure; it is often assumed that fracture

The simplest fracture network that has been applied to shale gas production consists of a number of planar fractures placed transversely to a horizontal wellbore as illustrated sche‐ matically in Figure 3. It is apparent from many published numerical studies that under these circumstances flow from the reservoir can be described in terms of a number of identifiable flow regimes. The following account is taken from a recent paper by Luo et al [10]. These authors used a commercial reservoir simulator to calculate the flow into a horizontal well with

**Figure 3.** Idealized discrete fracture network showing multiple transverse fractures originating from a horizontal well.

**•** Bilinear or linear flow: soon after the well is placed on production reservoir fluid flows normal to the fracture planes and along the fractures into the well. The streamlines are shown in yellow in Figure 4; reservoir pressure is in red and the constant bottomhole or well pressure is in blue. Note that flow into the fracture tips is negligible and each fracture behaves independently of the other fractures. This regime may also be termed the infinite-

The flow behavior can be conveniently discussed in terms of five flow regimes as follows.

**Figure 4.** Numerical solutions for flow into six infinitely-conductive transverse fractures (taken [10]).

acting regime in the sense that the neighboring fractures are effectively at infinity. The duration of this regime depends, as we shall see later, on many parameters including the matrix permeability, the fluid compressibility and the fracture spacing. The illustrations in Figure 4 are for infinite conductivity fractures.


**•** Pseudo-radial/elliptical flow: flow from beyond the wellbore/fracture area grows in importance and appears to be radial or elliptical.

the order of several years for a fracture spacing of 100 ft or more. This is quite consistent with typical simulation results described above. If the fracture spacing was as small as 10ft, we should expect to see fracture interference or the onset of PSS flow after about 10 days. For shale gas reservoirs, more complex models (unsteady state or fully transient) are needed to resolve

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A detailed discussion of the Warren-Root model, its background and similar contemporary models can be found in the excellent monograph by van Golf-Racht [11]. We note in particular that Kazemi [12] was one of the first to extend the Warren-Root model to include transient flow in the matrix blocks. Some seventeen years after Warren and Root published their seminal work, Kucuk and Sawyer [13] adapted their model for flow in shale reservoirs by incorporating effects such as desorption from the organic matrix material and Knudsen flow in the pores

In the years following the formulation of the dual porosity model for naturally fractured reservoirs, solutions of the coupled partial differential equations for the pore and fracture fluid pressures were obtained using finite difference techniques. While these simulations can provide accurate solutions, often in a complicated geometry covering the entire reservoir, the large number of computations involved made them cumbersome for analysis of large data sets. In response, an alternative, faster, method of solution was developed in the 1980s. For a simplified geometry, Laplace transform solutions were developed, in which the transformed

Several authors have noted that analytic approximations can be developed for certain ranges of parameter values (referring to the Warren-Root dimensionless parameters defined below) appropriate for shale gas reservoirs. It will become apparent later in this paper that for typical shale gas reservoirs the interporosity flow coefficient (or transmissivity), *λ* , is very small and this allows asymptotic approximations to be derived for the Laplace-transformed solutions. Since these models still require numerical inversion of the transformed solution, it would be more accurate to label them semi-analytic models. They have advantages over full numerical solutions in terms of speed of calculation and in the added value they bring to understanding the flow characteristics and the impact of the reservoir and completion parameters on

Recently, we have taken the idea of developing asymptotic solutions one stage further. We have developed perturbation solutions for *λ* < <1 directly from the dual porosity partial differential equations, thereby removing the necessity for Laplace transforms altogether. The greater simplicity and enhanced understanding afforded by these solutions will become apparent as we proceed. (Full details are available in an internal EGI report [14].) The result is similar to the simple linear flow model that is currently gaining favor in the literature, but has

**•** The model does not make the *a priori* assumption of linear flow into a sequence of transverse

and, of course, incorporating full transient effects in the matrix blocks.

solutions were inverted numerically, using, typically, the Stehfast algorithm.

the flow in the matrix in more detail.

production.

some notable advantages:

fractures.

**2.3. Development of new semi-analytic solution**

**•** Reservoir-boundary-dominated flow: ultimately the outer boundary of the reservoir begins to impact the flow.

It is difficult to infer from these simulations the time scales and duration of these flow regimes for parameter values other than those used in the particular simulation. Indeed, this highlights one of the severe drawbacks to the full numerical approach to modeling production flow in these reservoirs: it is difficult to make general conclusions about the characteristics of the flow and their dependence on the input data without undertaking very many numerical simula‐ tions; this is a formidable task even for a restricted input data space. However, based on the semi-analytic models that are described below, we believe that for many shale gas wells it would be unusual to expect to encounter the compound formation linear flow regime for at least 10 years after the well was placed on production.

#### **2.2. Dual porosity/dual permeability models**

The conventional view of a naturally-fractured reservoir is that it is a complex system com‐ posed of irregular matrix blocks surrounded by a network of more highly permeable fractures. In reality in tight gas shales some or most of the fractures may not be open to flow or they are opened up only during the hydraulic fracturing process. Warren and Root [2] were among the first to recognize that the simple model of reservoir flow based on single values of the permeability and porosity does not apply to naturally-fractured reservoirs, though they had in mind reservoirs quite different from gas shale reservoirs. In order to handle the problems associated with lack of detailed information on the structure of the fracture network they proposed a dual-porosity model in which a primary porosity associated with inter-granular pore spaces is augmented by a secondary porosity related to that of the network of natural fractures. At each point in space there are two overlapping continua—one for the matrix and one for the fracture network. The detailed geometry of the fracture system need not be specified in this model, but can include as particular examples any of the discrete fracture models described above. In typical shale gas applications the matrix has high storage capacity but low permeability and the fractures have relatively low storage capacity and higher permeability. It is quite possible (or, indeed, likely) that in many shale gas reservoirs no gas is stored in the fractures, though they may become filled with frac fluid during the hydraulic fracturing process.

In the dual porosity formulation flow from the matrix to the fractures is described by a transfer function with Darcy flow characteristics. The original Warren-Root models incorporated the pseudo-steady-state assumption in the matrix blocks and assigned a single value to the pressure within the blocks; the mass transfer rate from the matrix to the fractures depends then on the pressure differential between the matrix and the fracture. Thus these models assumed, almost implicitly, that sufficient time had elapsed that the flow in the matrix blocks between the fractures was already fracture-boundary-dominated. Later in this paper we estimate the time scale on which inter-fracture pseudo-steady-state begins and find that it is typically of the order of several years for a fracture spacing of 100 ft or more. This is quite consistent with typical simulation results described above. If the fracture spacing was as small as 10ft, we should expect to see fracture interference or the onset of PSS flow after about 10 days. For shale gas reservoirs, more complex models (unsteady state or fully transient) are needed to resolve the flow in the matrix in more detail.

A detailed discussion of the Warren-Root model, its background and similar contemporary models can be found in the excellent monograph by van Golf-Racht [11]. We note in particular that Kazemi [12] was one of the first to extend the Warren-Root model to include transient flow in the matrix blocks. Some seventeen years after Warren and Root published their seminal work, Kucuk and Sawyer [13] adapted their model for flow in shale reservoirs by incorporating effects such as desorption from the organic matrix material and Knudsen flow in the pores and, of course, incorporating full transient effects in the matrix blocks.

In the years following the formulation of the dual porosity model for naturally fractured reservoirs, solutions of the coupled partial differential equations for the pore and fracture fluid pressures were obtained using finite difference techniques. While these simulations can provide accurate solutions, often in a complicated geometry covering the entire reservoir, the large number of computations involved made them cumbersome for analysis of large data sets. In response, an alternative, faster, method of solution was developed in the 1980s. For a simplified geometry, Laplace transform solutions were developed, in which the transformed solutions were inverted numerically, using, typically, the Stehfast algorithm.

Several authors have noted that analytic approximations can be developed for certain ranges of parameter values (referring to the Warren-Root dimensionless parameters defined below) appropriate for shale gas reservoirs. It will become apparent later in this paper that for typical shale gas reservoirs the interporosity flow coefficient (or transmissivity), *λ* , is very small and this allows asymptotic approximations to be derived for the Laplace-transformed solutions. Since these models still require numerical inversion of the transformed solution, it would be more accurate to label them semi-analytic models. They have advantages over full numerical solutions in terms of speed of calculation and in the added value they bring to understanding the flow characteristics and the impact of the reservoir and completion parameters on production.

#### **2.3. Development of new semi-analytic solution**

**•** Pseudo-radial/elliptical flow: flow from beyond the wellbore/fracture area grows in

**•** Reservoir-boundary-dominated flow: ultimately the outer boundary of the reservoir begins

It is difficult to infer from these simulations the time scales and duration of these flow regimes for parameter values other than those used in the particular simulation. Indeed, this highlights one of the severe drawbacks to the full numerical approach to modeling production flow in these reservoirs: it is difficult to make general conclusions about the characteristics of the flow and their dependence on the input data without undertaking very many numerical simula‐ tions; this is a formidable task even for a restricted input data space. However, based on the semi-analytic models that are described below, we believe that for many shale gas wells it would be unusual to expect to encounter the compound formation linear flow regime for at

The conventional view of a naturally-fractured reservoir is that it is a complex system com‐ posed of irregular matrix blocks surrounded by a network of more highly permeable fractures. In reality in tight gas shales some or most of the fractures may not be open to flow or they are opened up only during the hydraulic fracturing process. Warren and Root [2] were among the first to recognize that the simple model of reservoir flow based on single values of the permeability and porosity does not apply to naturally-fractured reservoirs, though they had in mind reservoirs quite different from gas shale reservoirs. In order to handle the problems associated with lack of detailed information on the structure of the fracture network they proposed a dual-porosity model in which a primary porosity associated with inter-granular pore spaces is augmented by a secondary porosity related to that of the network of natural fractures. At each point in space there are two overlapping continua—one for the matrix and one for the fracture network. The detailed geometry of the fracture system need not be specified in this model, but can include as particular examples any of the discrete fracture models described above. In typical shale gas applications the matrix has high storage capacity but low permeability and the fractures have relatively low storage capacity and higher permeability. It is quite possible (or, indeed, likely) that in many shale gas reservoirs no gas is stored in the fractures, though they may become filled with frac fluid during the hydraulic fracturing

In the dual porosity formulation flow from the matrix to the fractures is described by a transfer function with Darcy flow characteristics. The original Warren-Root models incorporated the pseudo-steady-state assumption in the matrix blocks and assigned a single value to the pressure within the blocks; the mass transfer rate from the matrix to the fractures depends then on the pressure differential between the matrix and the fracture. Thus these models assumed, almost implicitly, that sufficient time had elapsed that the flow in the matrix blocks between the fractures was already fracture-boundary-dominated. Later in this paper we estimate the time scale on which inter-fracture pseudo-steady-state begins and find that it is typically of

importance and appears to be radial or elliptical.

least 10 years after the well was placed on production.

**2.2. Dual porosity/dual permeability models**

to impact the flow.

336 Effective and Sustainable Hydraulic Fracturing

process.

Recently, we have taken the idea of developing asymptotic solutions one stage further. We have developed perturbation solutions for *λ* < <1 directly from the dual porosity partial differential equations, thereby removing the necessity for Laplace transforms altogether. The greater simplicity and enhanced understanding afforded by these solutions will become apparent as we proceed. (Full details are available in an internal EGI report [14].) The result is similar to the simple linear flow model that is currently gaining favor in the literature, but has some notable advantages:

**•** The model does not make the *a priori* assumption of linear flow into a sequence of transverse fractures.


For simplicity we restrict attention in this paper to single-phase flow in the matrix and assume that gas is produced at constant bottom hole pressure; we shall also neglect the impact of gas desorption. For the purposes of the present discussion the most important part of the solution is the leading order solution for the reservoir pseudo-pressure, which satisfies a standard diffusion equation.

$$\frac{\left\|\hat{\mathcal{C}}^2 m\_{Dm\_0} - \hat{\mathcal{C}}\right\|\_{Dm\_0}}{\left\|\hat{\mathcal{C}} z\_D\right\|^2} = \frac{\left\|\hat{\mathcal{C}}\right\|\_{Dm\_0}}{\left\|\hat{\mathcal{C}} t\_D\right\|}\tag{1}$$

0

*<sup>D</sup> <sup>z</sup>*

The dimensionless flowrate is defined in equation (4) and the characteristic mass flowrate is

*m ch*

*w ch*

Here A denotes the productive fracture surface area. The total mass flow rate measured at

Here subscript *s* denotes surface conditions. Note that *qs* is actually a volumetric flowrate and

The dimensionless flowrate defined in equation (4) may be readily calculated in terms of the dimensionless pseudo-pressure, either from the full numerical solution of the diffusion equation, (equation (1)), or from the early-time infinite-acting approximation to it. Both solutions provide very useful information and insights into the variation of the production

> 0 0 0 *Dt*

*qD*<sup>0</sup> <sup>=</sup> <sup>1</sup> *πtD*

The diffusion equation (1) is readily solved using standard numerical schemes as made available in mathematical software such as MATLAB. To complement this solution we have obtained an analytic approximation valid while the change in pressure has not been impacted by neighboring boundaries or fractures—often referred to as the infinite-acting approximation.

*<sup>z</sup>* <sup>=</sup>

*Dm*

*m*

At downhole conditions the mass flowrate from the matrix into the fracture network is

2

*km Zch Ts T <sup>w</sup> Zs*

*ch*

*qs* <sup>=</sup> *qm <sup>ρ</sup><sup>s</sup>* = *A*

rate with time. The dimensionless cumulative production is defined by

The early-time approximation to the dimensionless inflow rate is

0 *<sup>D</sup>*

¶ <sup>=</sup> ¶ (4)

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*qm* =*qch qD*<sup>0</sup> (5)

*Mk m q A RT Z* <sup>=</sup> (6)

*Q q dt D DD* <sup>=</sup> ò (8)

(9)

*ps mch* (7)

0

*D*

*q*

defined by

surface conditions is

is measured typically in units such as scf/s.

Here we have used dimensionless parameters (denoted by subscript D); full definitions are provided in the Nomenclature section later in this paper. Dimensionless distance normal to the fracture face, *zD* , is scaled on L/2 (i.e. half the inter-fracture spacing), and dimensionless time, *tD* , is scaled on a time scale,

$$t\_m = \frac{c\rho\_m \mu \left(\frac{L\_f}{2}\right)^2}{k\_m} \tag{2}$$

that characterizes pressure diffusion in the matrix

The appropriate initial and boundary conditions are

$$\begin{aligned} \label{eq:SDm} &m\_{Dm\_0} = 0 &\quad at & t\_D = 0\\ \begin{array}{ll} m\_{Dm\_0} = m\_{D\_0^f} = 1 &\quad at & z\_D = 0, \quad \frac{\partial m\_{Dm\_0}}{\partial z\_D} = 0 &\quad at & z\_D = 1 \end{array} \end{aligned} \tag{3}$$

The leading order influx from the matrix into the fracture network is given by

$$\eta\_{D0} = \left| \frac{\stackrel{\mathcal{D}}{\mathcal{O}} \, m\_{Dm\_0}}{\stackrel{\mathcal{D}}{\mathcal{O}} \, z\_D} \right|\_{z\_D = 0} \tag{4}$$

At downhole conditions the mass flowrate from the matrix into the fracture network is

**•** The model does not make the *a priori* assumption of infinite fracture conductivity, and allows

**•** Identification of the end of the linear infinite-acting flow period and development of the

**•** Identification of the reservoir and completion parameters that are the greatest (primary)

**•** Solution scheme that permits rational extension to include other physical processes, such

For simplicity we restrict attention in this paper to single-phase flow in the matrix and assume that gas is produced at constant bottom hole pressure; we shall also neglect the impact of gas desorption. For the purposes of the present discussion the most important part of the solution is the leading order solution for the reservoir pseudo-pressure, which satisfies a standard

0 0

Here we have used dimensionless parameters (denoted by subscript D); full definitions are provided in the Nomenclature section later in this paper. Dimensionless distance normal to the fracture face, *zD* , is scaled on L/2 (i.e. half the inter-fracture spacing), and dimensionless

> ( ) 2

> > 0

1 0, 0 1

*z*

*m*

*Dm*

*D*

2 *<sup>m</sup>*

*k*

*c L*

j m

*Dm Df D D*

¶ == = = = ¶

*m m at z at z*

The leading order influx from the matrix into the fracture network is given by

*m*

¶ ¶ <sup>=</sup> ¶ ¶ (1)

= (2)

(3)

*Dm Dm D D*

*m m z t*

2

*m*

*t*

that characterizes pressure diffusion in the matrix

0 0

*m at t*

0

The appropriate initial and boundary conditions are

*Dm D*

0 0

= =

2

**•** Provision of a solution form that facilitates fast and easy production data analysis.

an estimate to be made for the fracture pressure loss.

ensuing PSS solution is made explicit.

determinants of productivity.

338 Effective and Sustainable Hydraulic Fracturing

time, *tD* , is scaled on a time scale,

as desorption.

diffusion equation.

$$
\sigma\_m = q\_{ch} \, q\_{D0} \tag{5}
$$

The dimensionless flowrate is defined in equation (4) and the characteristic mass flowrate is defined by

$$q\_{ch} = A \frac{Mk\_m}{2RT\_w} \frac{m\_{ch}}{Z\_{ch}} \tag{6}$$

Here A denotes the productive fracture surface area. The total mass flow rate measured at surface conditions is

$$q\_s = \frac{q\_m}{\rho\_s} = A \frac{k\_m}{Z\_{ch}} \frac{T\_s}{T\_w} \frac{Z\_s}{p\_s} m\_{ch} \tag{7}$$

Here subscript *s* denotes surface conditions. Note that *qs* is actually a volumetric flowrate and is measured typically in units such as scf/s.

The dimensionless flowrate defined in equation (4) may be readily calculated in terms of the dimensionless pseudo-pressure, either from the full numerical solution of the diffusion equation, (equation (1)), or from the early-time infinite-acting approximation to it. Both solutions provide very useful information and insights into the variation of the production rate with time. The dimensionless cumulative production is defined by

$$Q\_{D0} = \int\_0^{t\_D} q\_{D0} \, dt\_D \tag{8}$$

The diffusion equation (1) is readily solved using standard numerical schemes as made available in mathematical software such as MATLAB. To complement this solution we have obtained an analytic approximation valid while the change in pressure has not been impacted by neighboring boundaries or fractures—often referred to as the infinite-acting approximation. The early-time approximation to the dimensionless inflow rate is

$$q\_{D0} = \frac{1}{\sqrt{\pi t\_D}}\tag{9}$$

And the early-time approximation to the dimensionless cumulative production is

$$Q\_{D0} = 2\sqrt{\frac{t\_D}{\pi}}\tag{10}$$

2

*ch ch m m*

Analogous to equation (9) we define cumulative production at surface conditions by

**Figure 5.** A comparison of the full and early- time solutions for the dimensionless flow rate and cumulative produc‐

tion.

The time scale *tm* is defined in equation (2) and *QD*0 is defined in equation (8).

*km Zch Ts T <sup>w</sup> Zs*

*Qs* <sup>=</sup> *Qm <sup>ρ</sup><sup>s</sup>* = *A* *m ch*

*Mk m Q qt A t RT z* = = (12)

*ps mch tmQD*<sup>0</sup> <sup>=</sup>*Qch* , *<sup>s</sup>QD*<sup>0</sup> (13)

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*w ch*

Figure 5 compares the full and early-time solutions for the dimensionless flow rate and cumulative production against dimensionless time. For convenience we have used a log-log plot here. There are several major features of this plot that are worthy of further comment:


Figure 6 shows the full and infinite-acting solutions for cumulative production plotted against the square-root of dimensionless time. As expected from equation (10) the early-time solution is well represented by a straight line with slope <sup>2</sup> *<sup>π</sup>* or 1.128. Later in this paper we develop this plot as the basis of our production data interpretation technique.

In anticipation of the application of these results to analysis of production data, it is useful to provide an expression for the cumulative production in dimensional terms. Analogous to equation (6) we define cumulative production at downhole conditions by

$$Q\_m = Q\_{ch} \ Q\_{D0} \tag{11}$$

The characteristic cumulative production scale is defined as

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$$Q\_{ch} = q\_{ch} t\_m = A \frac{M k\_m}{2 R T\_w} \frac{m\_{ch}}{z\_{ch}} t\_m \tag{12}$$

The time scale *tm* is defined in equation (2) and *QD*0 is defined in equation (8).

And the early-time approximation to the dimensionless cumulative production is

*QD*<sup>0</sup> =2

in dimensional terms.

340 Effective and Sustainable Hydraulic Fracturing

equations.

to produce 90% of the gas available.

is well represented by a straight line with slope <sup>2</sup>

*tD*

Figure 5 compares the full and early-time solutions for the dimensionless flow rate and cumulative production against dimensionless time. For convenience we have used a log-log plot here. There are several major features of this plot that are worthy of further comment:

**•** The early-time (infinite-acting) solution provides a good approximation to the full numerical solution for dimensionless time, *tD* , less than about 0.15, which translates to about 3 years

**•** During this time frame, the flowrate is represented by a straight line of slope – ½ and the

**•** Boundaries (or in this case neighboring fractures) begin to influence the flow after this point in time and the solution departs from the simple linear dependence on the square-root of time. This is also evident in the numerical solutions presented by Bello and Wattenbarger

**•** The dimensionless cumulative production approaches a final value of 1 as it should because of the way we have defined the characteristic scales in our non-dimensionalization of the

**•** The error incurred in estimating the cumulative production by extrapolating the infinite-

**•** Almost 90% of the total gas that can be produced has been produced by the time *tD* =1 . This then provides a simple interpretation of the matrix diffusion time as the time (in real terms)

Figure 6 shows the full and infinite-acting solutions for cumulative production plotted against the square-root of dimensionless time. As expected from equation (10) the early-time solution

In anticipation of the application of these results to analysis of production data, it is useful to provide an expression for the cumulative production in dimensional terms. Analogous to

[15] in their Figures 5, 7, 8 and 10 and in the field data shown in their Figure 1.

acting solution beyond its region of validity is apparent from Figure 5.

this plot as the basis of our production data interpretation technique.

equation (6) we define cumulative production at downhole conditions by

The characteristic cumulative production scale is defined as

cumulative production is represented by a straight line of slope + ½.

*<sup>π</sup>* (10)

*<sup>π</sup>* or 1.128. Later in this paper we develop

*Qm* =*Qch QD*<sup>0</sup> (11)

Analogous to equation (9) we define cumulative production at surface conditions by

$$Q\_s = \frac{Q\_m}{\rho\_s} = A \frac{k\_m}{Z\_{ch}} \frac{T\_s}{T\_w} \frac{Z\_s}{p\_s} m\_{ch} \, t\_m Q\_{D0} = Q\_{ch\_s,s} Q\_{D0} \tag{13}$$

**Figure 5.** A comparison of the full and early- time solutions for the dimensionless flow rate and cumulative produc‐ tion.

**Figure 6.** A comparison of the full and early- time solutions for the dimensionless cumulative production.

In view of the wide applicability of the early-time solution, it is useful to state the form taken by equation (13) during the infinite-acting period. Using the early-time approximation given in equation (9), we see that

$$\mathbf{Q}\_s = A \frac{k\_m}{Z\_{ch}} \frac{T\_s}{T\_w} \frac{Z\_s}{p\_s} m\_{ch} \mathbf{2} \sqrt{\frac{t\_m}{\pi}} \mathbf{\sqrt{t}} \tag{14}$$

*Qs* =*CP t* (15)

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2 *mss m*

The coefficient *CP* represents the slope of the dimensional equivalent of the straight line in Figure 6 and is of fundamental importance in the subsequent development of this paper. For now we will observe that *CP* characterizes the early-time solution in a form that is easy to estimate from production data. We shall refer to it as the "Production Coefficient". (We have adopted this terminology in recognition of the similarity of this result to a well-known expression for the leakoff rate of a compressible fluid from a fracture to a reservoir filled with

The analysis described above suggests that for a substantial part of a shale gas well's production history, the cumulative production varies linearly with the square-root of time. The coefficient *CP* represents the slope of the straight line in a plot of Q against *t* and characterizes the early-time solution in a form that is easy to compare with production data The time scale, *Tm* , defined in equation (2) defines the upper limit of the applicability of the linear flow regime and allows us to characterize the production rate once boundary-

We illustrate this production analysis technique by comparing production from a group of wells in the Barnett shale. Figure 7 shows a conventional plot of production rate against time for several wells that had been producing for at least 5 years in an area of the Barnett field. The data was obtained from a public data base and we have plotted the production rates at yearly intervals. For clarity of presentation we have connected the data points by smooth lines. This "conventional" plot reveals nothing about the relative decline rates of the wells or provides much insight into the flow regime(s). The same data sets have been plotted in the new format in Figure 8. It is immediately apparent that for most of these wells the data falls on straight lines as expected from our analysis. The slope of these lines is readily measured and provides an estimate for the production coefficient, *CP* . Estimation of *CP* is quick and easy and provides us with a new metric with which we can quantify the productivity of these wells. Again, we explore these results in more detail later, but for now we note that the linear flow regime extends beyond at least 5 years, since there is no indication at this point of departure from the straight line in this plot. This fact alone sets some bounds on the fracture spacing and the matrix

*t*

<sup>=</sup> (16)

p

*s ch wch s k TZ t QA m <sup>Z</sup>*

*T p*

Where

the same fluid (see, for example, [15]).

**3. Production data interpretation**

effects have become important.

permeability.

If we now use the definition of *Tm* (equation (2)), we can express the cumulative production at surface conditions as

$$Q\_s = C\_P \sqrt{t} \tag{15}$$

Where

**Figure 6.** A comparison of the full and early- time solutions for the dimensionless cumulative production.

*Qs* = *A*

*km Zch Ts T <sup>w</sup> Zs ps mch* 2

in equation (9), we see that

342 Effective and Sustainable Hydraulic Fracturing

at surface conditions as

In view of the wide applicability of the early-time solution, it is useful to state the form taken by equation (13) during the infinite-acting period. Using the early-time approximation given

If we now use the definition of *Tm* (equation (2)), we can express the cumulative production

*tm*

*<sup>π</sup> <sup>t</sup>* (14)

$$\mathbf{Q}\_s = A \frac{k\_m}{Z\_{ch}} \frac{T\_s}{T\_w} \frac{Z\_s}{p\_s} m\_{ch} 2 \sqrt{\frac{\mathbf{t}\_m}{\pi}} \sqrt{\mathbf{t}} \tag{16}$$

The coefficient *CP* represents the slope of the dimensional equivalent of the straight line in Figure 6 and is of fundamental importance in the subsequent development of this paper. For now we will observe that *CP* characterizes the early-time solution in a form that is easy to estimate from production data. We shall refer to it as the "Production Coefficient". (We have adopted this terminology in recognition of the similarity of this result to a well-known expression for the leakoff rate of a compressible fluid from a fracture to a reservoir filled with the same fluid (see, for example, [15]).

#### **3. Production data interpretation**

The analysis described above suggests that for a substantial part of a shale gas well's production history, the cumulative production varies linearly with the square-root of time. The coefficient *CP* represents the slope of the straight line in a plot of Q against *t* and characterizes the early-time solution in a form that is easy to compare with production data The time scale, *Tm* , defined in equation (2) defines the upper limit of the applicability of the linear flow regime and allows us to characterize the production rate once boundaryeffects have become important.

We illustrate this production analysis technique by comparing production from a group of wells in the Barnett shale. Figure 7 shows a conventional plot of production rate against time for several wells that had been producing for at least 5 years in an area of the Barnett field. The data was obtained from a public data base and we have plotted the production rates at yearly intervals. For clarity of presentation we have connected the data points by smooth lines. This "conventional" plot reveals nothing about the relative decline rates of the wells or provides much insight into the flow regime(s). The same data sets have been plotted in the new format in Figure 8. It is immediately apparent that for most of these wells the data falls on straight lines as expected from our analysis. The slope of these lines is readily measured and provides an estimate for the production coefficient, *CP* . Estimation of *CP* is quick and easy and provides us with a new metric with which we can quantify the productivity of these wells. Again, we explore these results in more detail later, but for now we note that the linear flow regime extends beyond at least 5 years, since there is no indication at this point of departure from the straight line in this plot. This fact alone sets some bounds on the fracture spacing and the matrix permeability.

the matrix permeability.

�� � � ��

Where

2

*P ch W S*

*ch z*

�� �� �� �� ���2���

*S S mm*

**3. Production data interpretation** 

*TZ c k CA m*

�� � ��√� (15)

 

*T p* (16)

In Figure 8 we have omitted the first year's cumulative production data. Generally, early-time production data is quite severely impacted by variable drawdown conditions and so we should not expect a good straight line fit at that time. Analysis of this regime is discussed at length elsewhere [14]. analysis. The slope of these lines is readily measured and provides an estimate for the production coefficient, ��. Estimation of �� is quick and easy and provides us with a new metric with which we can quantify the productivity of these wells. Again, we explore these results in more detail later, but for now we note that the linear flow regime extends beyond at least 5 years, since there is no indication at this point of departure from the straight line in this plot. This fact alone sets some bounds on the fracture spacing and

production rate once boundary-effects have become important.

from a fracture to a reservoir filled with the same fluid (see, for example, [15]).

In view of the wide applicability of the early-time solution, it is useful to state the form taken by equation (13) during the infinite-

The coefficient �� represents the slope of the dimensional equivalent of the straight line in Figure 6 and is of fundamental importance in the subsequent development of this paper. For now we will observe that �� characterizes the early-time solution in a form that is easy to estimate from production data. We shall refer to it as the "Production Coefficient". (We have adopted this terminology in recognition of the similarity of this result to a well-known expression for the leakoff rate of a compressible fluid

The analysis described above suggests that for a substantial part of a shale gas well's production history, the cumulative production varies linearly with the square-root of time. The coefficient �� represents the slope of the straight line in a plot of Q against √� and characterizes the early-time solution in a form that is easy to compare with production data The time scale, �� , defined in equation (2) defines the upper limit of the applicability of the linear flow regime and allows us to characterize the

We illustrate this production analysis technique by comparing production from a group of wells in the Barnett shale. Figure 7 shows a conventional plot of production rate against time for several wells that had been producing for at least 5 years in an area of the Barnett field. The data was obtained from a public data base and we have plotted the production rates at yearly intervals. For

new format in Figure 8. It is immediately apparent that for most of these wells the data falls on straight lines as expected from our

**4. Identification of production drivers: Nature versus nurture**

"nurture" the reservoir. Specifically the parameters are:

Nature:

Nurture:

**•** Matrix permeability, *km*

**•** Matrix porosity, *φ<sup>m</sup>*

**•** Gas compressibility, *c*

**•** Initial reservoir pressure, *pi*

**•** Bottom hole flowing pressure, *pw* **•** Productive fracture surface area, *A*

**•** Reservoir temperature, *T*

**•** Gas viscosity, *μ*

sufficiently large.

We have established the applicability of this new analysis technique to data from very many wells in many shale gas plays across the US and Canada, though there is neither space nor time available to discuss this in detail in the present paper. Following on from that analysis, we will now go on to discuss the results in more depth and begin to draw some tentative conclusions about the production drivers, by examining the parameters that together consti‐ tute the formula for *CP* in equation (15). We may divide these parameters broadly into two groups—those that characterize the nature of the reservoir and those that characterize how we

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345

In developing these results we are constrained by the requirement that *λ* <<1, where

*<sup>λ</sup>* <sup>=</sup> <sup>12</sup>*kmrw* 2 *L c <sup>f</sup>*

This requirement sets some bounds on the fracture network characteristics, but these are generally easily met for shale gas reservoirs. For given values of the matrix permeability and the wellbore radius, the combination of fracture spacing and fracture conductivity must be

It is apparent that given these conditions production for a large part of the production history of these wells depends upon the parameters listed above. We note in particular that history matching production data over this flow period furnishes only one parameter and that is the production coefficient, *CP* . That is all. The square-root of time behavior is inherent to the physics of the flow: i.e. linear flow into a network of (effectively infinitely-conductive) fractures. It is not at all surprising that conventional history-matching techniques using reservoir simulators give non-unique answers: many different values of the parameters in the list above can together constitute the same value of the production coefficient. Moreover this

(17)

In Figure 8 we have omitted the first year's cumulative production data. Generally, early-time production data is quite severely impacted by variable drawdown conditions and so we should not expect a good straight line fit at that time. Analysis of this

We have established the applicability of this new analysis technique to data from very many wells in many shale gas plays across the US and Canada, though there is neither space nor time available to discuss this in detail in the present paper. Following on from that analysis, we will now go on to discuss the results in more depth and begin to draw some tentative conclusions about the production drivers, by examining the parameters that together constitute the formula for �� in equation (15). We may divide these parameters broadly into two groups—those that characterize the nature of the reservoir and those that characterize how we

This requirement sets some bounds on the fracture network characteristics, but these are generally easily met for shale gas reservoirs. For given values of the matrix permeability and the wellbore radius, the combination of fracture spacing and fracture

If we now use the definition of �� (equation (2)), we can express the cumulative production at surface conditions as

acting period. Using the early-time approximation given in equation (9), we see that

� √� (14)

Figure 7. Production data from a group of Barnett wells plotted in the conventional format **Figure 7.** Production data from a group of Barnett wells plotted in the conventional format

**4. Identification of production drivers: Nature versus nurture** 

In developing these results we are constrained by the requirement that � <<1, where

Figure 8. Same production data from Figure 7 plotted in the new format. **Figure 8.** Same production data from Figure 7 plotted in the new format.

regime is discussed at length elsewhere [14].

"nurture" the reservoir. Specifically the parameters are:

*m*

Nature:

Nurture:

� � ������ � ���

Matrix permeability, ��

Matrix porosity,

Gas viscosity,

Gas compressibility, *c*

Initial reservoir pressure, *<sup>i</sup> p*

Bottom hole flowing pressure, *<sup>w</sup> p*

Productive fracture surface area, *A*

(17)

conductivity must be sufficiently large.

Reservoir temperature, *T*

#### analysis. The slope of these lines is readily measured and provides an estimate for the production coefficient, ��. Estimation of �� is quick and easy and provides us with a new metric with which we can quantify the productivity of these wells. Again, we explore **4. Identification of production drivers: Nature versus nurture**

these results in more detail later, but for now we note that the linear flow regime extends beyond at least 5 years, since there is no indication at this point of departure from the straight line in this plot. This fact alone sets some bounds on the fracture spacing and We have established the applicability of this new analysis technique to data from very many wells in many shale gas plays across the US and Canada, though there is neither space nor time available to discuss this in detail in the present paper. Following on from that analysis, we will now go on to discuss the results in more depth and begin to draw some tentative conclusions about the production drivers, by examining the parameters that together consti‐ tute the formula for *CP* in equation (15). We may divide these parameters broadly into two groups—those that characterize the nature of the reservoir and those that characterize how we "nurture" the reservoir. Specifically the parameters are:

Nature:

In Figure 8 we have omitted the first year's cumulative production data. Generally, early-time production data is quite severely impacted by variable drawdown conditions and so we should not expect a good straight line fit at that time. Analysis of this regime is discussed at length

Figure 7. Production data from a group of Barnett wells plotted in the conventional format

**Figure 7.** Production data from a group of Barnett wells plotted in the conventional format

Figure 8. Same production data from Figure 7 plotted in the new format.

**4. Identification of production drivers: Nature versus nurture** 

0 0.5 1 1.5 2 2.5

**sqrt(time) years^0.5**

In developing these results we are constrained by the requirement that � <<1, where

regime is discussed at length elsewhere [14].

**Figure 8.** Same production data from Figure 7 plotted in the new format.

"nurture" the reservoir. Specifically the parameters are:

*m*

Nature:

0

0.2

0.4

**cumulative**

**production**

**(bscf)**

0.6

0.8

1

1.2

Nurture:

� � ������ � ���

Matrix permeability, ��

Matrix porosity,

Gas viscosity,

Gas compressibility, *c*

Initial reservoir pressure, *<sup>i</sup> p*

Bottom hole flowing pressure, *<sup>w</sup> p*

Productive fracture surface area, *A*

(17)

conductivity must be sufficiently large.

Reservoir temperature, *T*

0123456

**time (years)**

production rate once boundary-effects have become important.

from a fracture to a reservoir filled with the same fluid (see, for example, [15]).

In view of the wide applicability of the early-time solution, it is useful to state the form taken by equation (13) during the infinite-

The coefficient �� represents the slope of the dimensional equivalent of the straight line in Figure 6 and is of fundamental importance in the subsequent development of this paper. For now we will observe that �� characterizes the early-time solution in a form that is easy to estimate from production data. We shall refer to it as the "Production Coefficient". (We have adopted this terminology in recognition of the similarity of this result to a well-known expression for the leakoff rate of a compressible fluid

The analysis described above suggests that for a substantial part of a shale gas well's production history, the cumulative production varies linearly with the square-root of time. The coefficient �� represents the slope of the straight line in a plot of Q against √� and characterizes the early-time solution in a form that is easy to compare with production data The time scale, �� , defined in equation (2) defines the upper limit of the applicability of the linear flow regime and allows us to characterize the

We illustrate this production analysis technique by comparing production from a group of wells in the Barnett shale. Figure 7 shows a conventional plot of production rate against time for several wells that had been producing for at least 5 years in an area of the Barnett field. The data was obtained from a public data base and we have plotted the production rates at yearly intervals. For clarity of presentation we have connected the data points by smooth lines. This "conventional" plot reveals nothing about the relative decline rates of the wells or provides much insight into the flow regime(s). The same data sets have been plotted in the new format in Figure 8. It is immediately apparent that for most of these wells the data falls on straight lines as expected from our

In Figure 8 we have omitted the first year's cumulative production data. Generally, early-time production data is quite severely impacted by variable drawdown conditions and so we should not expect a good straight line fit at that time. Analysis of this

We have established the applicability of this new analysis technique to data from very many wells in many shale gas plays across the US and Canada, though there is neither space nor time available to discuss this in detail in the present paper. Following on from that analysis, we will now go on to discuss the results in more depth and begin to draw some tentative conclusions about the production drivers, by examining the parameters that together constitute the formula for �� in equation (15). We may divide these parameters broadly into two groups—those that characterize the nature of the reservoir and those that characterize how we

This requirement sets some bounds on the fracture network characteristics, but these are generally easily met for shale gas reservoirs. For given values of the matrix permeability and the wellbore radius, the combination of fracture spacing and fracture

If we now use the definition of �� (equation (2)), we can express the cumulative production at surface conditions as

acting period. Using the early-time approximation given in equation (9), we see that

� √� (14)

elsewhere [14].

the matrix permeability.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

production **rate**

**(bsc/yearf)**

344 Effective and Sustainable Hydraulic Fracturing

�� � � ��

Where

2

*P ch W S*

*ch z*

�� �� �� �� ���2���

*S S mm*

**3. Production data interpretation** 

*TZ c k CA m*

�� � ��√� (15)

 

*T p* (16)


Nurture:


In developing these results we are constrained by the requirement that *λ* <<1, where

$$
\lambda = \frac{12k\_m r\_w^2}{L \cdot c\_f} \tag{17}
$$

This requirement sets some bounds on the fracture network characteristics, but these are generally easily met for shale gas reservoirs. For given values of the matrix permeability and the wellbore radius, the combination of fracture spacing and fracture conductivity must be sufficiently large.

It is apparent that given these conditions production for a large part of the production history of these wells depends upon the parameters listed above. We note in particular that history matching production data over this flow period furnishes only one parameter and that is the production coefficient, *CP* . That is all. The square-root of time behavior is inherent to the physics of the flow: i.e. linear flow into a network of (effectively infinitely-conductive) fractures. It is not at all surprising that conventional history-matching techniques using reservoir simulators give non-unique answers: many different values of the parameters in the list above can together constitute the same value of the production coefficient. Moreover this formulation tells us what parameters have little effect on the history-matching process, including the precise value of the fracture conductivity. Even if history-matching were attempted in terms of dimensionless parameters, it is apparent that the result is insensitive to *λ* provided that it is small enough.

**3.** Estimate the Production Coefficient form the slope of the best straight line fit to the data. The Barnett shale is a good starting point for a more in-depth data analysis, since production data is readily available from public databases and, moreover, that data extend over many wells for long periods of time. The Barnett shale occupies several counties in North Texas. It is broadly bounded by geologic and structural features and may be divided according to estimates of maturity into a gas window and oil window. Historically the major development has been in a core area located to the north of Fort Worth (Figure 9), but more recently expansion has occurred to the south and to the west. To date many thousands of wells have been drilled and completed in the Barnett, initially vertical, but now almost entirely horizontal.

The Role of Natural Fractures in Shale Gas Production

http://dx.doi.org/10.5772/56404

347

It has become common practice to sub-divide the Barnett play into three areas, described as the Core, Tier 1 and Tier 2. For convenience we may define these areas according to county as

**•** Core region: Denton, Tarrant and Wise counties, comprising 2974 horizontal wells and 3886

**•** Tier 1 region: Hood, Johnson and Parker counties, comprising 3865 horizontal wells and 251

It is apparent form this cursory division that the fraction of wells that were completed horizontally shifts from 43% in the Core area to 94% in the Tier 1 area, which reflects the development of technology with time and the spread of drilling with time to the outer areas.

The result of this detailed analysis (Figure 10) allows us to quantify the production variations in the Barnett Core, Tier 1 and Tier 2 areas and to distinguish the impact of horizontal and vertical well completions on the productivity. In a sense this represents a first, somewhat crude, pass at distinguishing the impact of nature (in the sense that reservoir properties depend on location, with the core area providing more fertile ground than Tier 1 or Tier 2) and nurture (in the anticipation that horizontal well technology provides more productive fracture surface

In Figure 10 we have shown the cumulative distribution of production coefficient for each of the six categories defined above. The plots should be interpreted as follows. For each category the probability that a well has a specified value of the production coefficient in excess of the value on the x-axis can be read off the y-axis. For example the probability of a horizontal well

It is apparent that, as is to be expected, wells in the Core have better production characteristics than wells in Tier 1 and wells in Tier 2 and that in general horizontal wells have better production characteristics then vertical wells. It is interesting in this context to examine the variation of production coefficient in the core area in more detail. Figure 11 shows the location of ten of the wells in the core area with high values of the production coefficient (in green), 10 of the wells with medium values (in blue) and 10 wells with low values (in red). It is apparent

in the Core having a production coefficient in excess of 0.75 (bcf/yr^0.5) is about 8%.

**•** Tier 2 region: all other counties, comprising 687 horizontal wells and 401 vertical wells

At the date of these figures (2009) Tier 2 was relatively unexploited.

area than does vertical well technology).

follows

vertical wells

vertical wells

We need to elaborate at this point on the parameter *A* defined above as the "productive fracture surface area". This is the area of the fractures in contact with the reservoir that serve as the channels that convey gas from the matrix to the wellbore. We can make no assertion at this point about whether these fractures are natural fractures or propped or unpropped hydraulic fractures and nor can we say anything (yet) about their spacing or their lengths or indeed their number and location. *All we can infer from the production data analysis is the total productive fracture surface area.*

Some insight about the spacing of these productive fractures can be obtained by examining the time scale of pressure diffusion in the matrix. We demonstrated earlier that we may expect the root-time solution to be valid until neighboring fractures begin to compete with one another for production. In other words, until pressure diffusion in the matrix can no longer be considered to be independent of the fracture spacing. According to the analysis presented above we should expect the cumulative data to deviate from a straight-line in the root-time plot for t> 0.15 *tm* .

If we could detect the time at which this departure occurs then, we have some information with which to estimate the productive fracture spacing. Even if the entire production history to date is in the linear flow regime, we can make an estimate of a lower bound on the fracture spacing. As we see later, the fracture spacing is surprisingly large for typical shale gas plays.

We have now analyzed many shale gas production data sets using our proposed technique and have found the square-root fit to be very good. Based on this and on the mathematical analysis that supports that technique we have concluded that the production rate declines inversely with the square root of time. As we have discussed above, this is a consequence of the dominant production process of linear flow into a network of fractures. The decline rate is therefore fully determined by the physics of the production process. We should not expect to see any significant variation from well to well, from vertical to horizontal wells or indeed form play to play. What does vary is the multiplier, the production coefficient , *CP* , which as we have demonstrated elsewhere depends on many factors, principally the reservoir quality, the reservoir and bottom hole pressure and the productive fracture surface area.

#### **4.1. Example: Barnett shale production data analysis**

As we have indicated above, it is relatively straightforward to use this new technique to analyze production data whether it is on a well-by-well basis or averaged over a play or area within a play. In essence, the process consists of three steps:


**3.** Estimate the Production Coefficient form the slope of the best straight line fit to the data.

formulation tells us what parameters have little effect on the history-matching process, including the precise value of the fracture conductivity. Even if history-matching were attempted in terms of dimensionless parameters, it is apparent that the result is insensitive to

We need to elaborate at this point on the parameter *A* defined above as the "productive fracture surface area". This is the area of the fractures in contact with the reservoir that serve as the channels that convey gas from the matrix to the wellbore. We can make no assertion at this point about whether these fractures are natural fractures or propped or unpropped hydraulic fractures and nor can we say anything (yet) about their spacing or their lengths or indeed their number and location. *All we can infer from the production data analysis is the total productive fracture*

Some insight about the spacing of these productive fractures can be obtained by examining the time scale of pressure diffusion in the matrix. We demonstrated earlier that we may expect the root-time solution to be valid until neighboring fractures begin to compete with one another for production. In other words, until pressure diffusion in the matrix can no longer be considered to be independent of the fracture spacing. According to the analysis presented above we should expect the cumulative data to deviate from a straight-line in the root-time

If we could detect the time at which this departure occurs then, we have some information with which to estimate the productive fracture spacing. Even if the entire production history to date is in the linear flow regime, we can make an estimate of a lower bound on the fracture spacing. As we see later, the fracture spacing is surprisingly large for typical shale gas plays. We have now analyzed many shale gas production data sets using our proposed technique and have found the square-root fit to be very good. Based on this and on the mathematical analysis that supports that technique we have concluded that the production rate declines inversely with the square root of time. As we have discussed above, this is a consequence of the dominant production process of linear flow into a network of fractures. The decline rate is therefore fully determined by the physics of the production process. We should not expect to see any significant variation from well to well, from vertical to horizontal wells or indeed form play to play. What does vary is the multiplier, the production coefficient , *CP* , which as we have demonstrated elsewhere depends on many factors, principally the reservoir quality, the

reservoir and bottom hole pressure and the productive fracture surface area.

As we have indicated above, it is relatively straightforward to use this new technique to analyze production data whether it is on a well-by-well basis or averaged over a play or area

**1.** From the daily (or monthly or yearly) production data calculate the cumulative produc‐

**4.1. Example: Barnett shale production data analysis**

tion for each well at different points in time.

within a play. In essence, the process consists of three steps:

**2.** Plot cumulative production against the square-root of time.

*λ* provided that it is small enough.

346 Effective and Sustainable Hydraulic Fracturing

*surface area.*

plot for t> 0.15 *tm* .

The Barnett shale is a good starting point for a more in-depth data analysis, since production data is readily available from public databases and, moreover, that data extend over many wells for long periods of time. The Barnett shale occupies several counties in North Texas. It is broadly bounded by geologic and structural features and may be divided according to estimates of maturity into a gas window and oil window. Historically the major development has been in a core area located to the north of Fort Worth (Figure 9), but more recently expansion has occurred to the south and to the west. To date many thousands of wells have been drilled and completed in the Barnett, initially vertical, but now almost entirely horizontal.

It has become common practice to sub-divide the Barnett play into three areas, described as the Core, Tier 1 and Tier 2. For convenience we may define these areas according to county as follows


It is apparent form this cursory division that the fraction of wells that were completed horizontally shifts from 43% in the Core area to 94% in the Tier 1 area, which reflects the development of technology with time and the spread of drilling with time to the outer areas. At the date of these figures (2009) Tier 2 was relatively unexploited.

The result of this detailed analysis (Figure 10) allows us to quantify the production variations in the Barnett Core, Tier 1 and Tier 2 areas and to distinguish the impact of horizontal and vertical well completions on the productivity. In a sense this represents a first, somewhat crude, pass at distinguishing the impact of nature (in the sense that reservoir properties depend on location, with the core area providing more fertile ground than Tier 1 or Tier 2) and nurture (in the anticipation that horizontal well technology provides more productive fracture surface area than does vertical well technology).

In Figure 10 we have shown the cumulative distribution of production coefficient for each of the six categories defined above. The plots should be interpreted as follows. For each category the probability that a well has a specified value of the production coefficient in excess of the value on the x-axis can be read off the y-axis. For example the probability of a horizontal well in the Core having a production coefficient in excess of 0.75 (bcf/yr^0.5) is about 8%.

It is apparent that, as is to be expected, wells in the Core have better production characteristics than wells in Tier 1 and wells in Tier 2 and that in general horizontal wells have better production characteristics then vertical wells. It is interesting in this context to examine the variation of production coefficient in the core area in more detail. Figure 11 shows the location of ten of the wells in the core area with high values of the production coefficient (in green), 10 of the wells with medium values (in blue) and 10 wells with low values (in red). It is apparent

outer areas. At the date of these figures (2009) Tier 2 was relatively unexploited.

provides more productive fracture surface area than does vertical well technology).

It is apparent form this cursory division that the fraction of wells that were completed horizontally shifts from 43% in the Core area to 94% in the Tier 1 area, which reflects the development of technology with time and the spread of drilling with time to the

The result of this detailed analysis (Figure 10) allows us to quantify the production variations in the Barnett Core, Tier 1 and Tier 2 areas and to distinguish the impact of horizontal and vertical well completions on the productivity. In a sense this represents a first, somewhat crude, pass at distinguishing the impact of nature (in the sense that reservoir properties depend on location, with the core area providing more fertile ground than Tier 1 or Tier 2) and nurture (in the anticipation that horizontal well technology

2 and that in general horizontal wells have better production characteristics then vertical wells. It is interesting in this context to

**Figure 11.** Preliminary identification of sweet spots in the core area of the Barnett. Each black dot represents a well; 10 wells with high Production Coefficients are identified in green, 10 medium performers are in blue and 10 poor per‐

The Role of Natural Fractures in Shale Gas Production

http://dx.doi.org/10.5772/56404

349

One of the advantages of the semi-analytic method outlined in this paper is that it enables us to make certain deductions about the magnitude of the parameters that drive the productivity of the well. In particular we can make some inferences about the magnitude of the fracture surface area through which the gas is produced and about the likely spacing of the productive

Our analytic solution allows us to relate the Productivity Coefficient *CP* to a group of parameters that may be roughly divided into those that characterize the nature of the reservoir and those that characterize the impact of the completion and stimulation strategy. In our formula for *CP* (equation (16)) perhaps the parameter that has the greatest uncertainty is the productive fracture surface area, A. This is the area of the fractures in contact with the reservoir that serve as the channels that convey gas from the matrix to the wellbore. We can make no assertion at this time point about whether these fractures are natural fractures or propped or unpropped hydraulic fractures and nor can we say anything (yet) about their spacing or their lengths or indeed their number and location. We can, however, make an estimate from the

production data analysis of the total surface area of these productive fractures.

formers are identified in red.

fractures.

**5. Implications and deductions**

**5.1. Productive fracture surface area**

It is apparent that, as is to be expected, wells in the Core have better production characteristics than wells in Tier 1 and wells in Tier **Figure 9.** Development of the Barnett shale in North Texas

that there appear to be sweet spots even within the core area, but there are substantial outliers and there are some relatively poor wells close to better wells. examine the variation of production coefficient in the core area in more detail. Figure 11 shows the location of ten of the wells in the core area with high values of the production coefficient (in green), 10 of the wells with medium values (in blue) and 10 wells with low values (in red). It is apparent that there appear to be sweet spots even within the core area, but there are substantial outliers

and there are some relatively poor wells close to better wells.

Figure 10.Distribution of values of the Production Coefficient for horizontal and vertical wells in the Core region, Tier 1 region and Tier 2 region of the Barnett shale. **Figure 10.** Distribution of values of the Production Coefficient for horizontal and vertical wells in the Core region, Tier 1 region and Tier 2 region of the Barnett shale.

It is apparent that, as is to be expected, wells in the Core have better production characteristics than wells in Tier 1 and wells in Tier 2 and that in general horizontal wells have better production characteristics then vertical wells. It is interesting in this context to examine the variation of production coefficient in the core area in more detail. Figure 11 shows the location of ten of the wells in the **Figure 11.** Preliminary identification of sweet spots in the core area of the Barnett. Each black dot represents a well; 10 wells with high Production Coefficients are identified in green, 10 medium performers are in blue and 10 poor per‐ formers are identified in red.
