# The Key Factors That Determine the Economically Viable, Horizontal Hydrofractured Gas Wells in Mudrocks

^{*}

## Abstract

**:**

## 1. Significance Statement

## 2. Introduction

- SRV is composed of a fracture network and organic matrix.
- SRV is nonuniform because of the power law distribution of properties of the fracture network.
- The single phase, compressible, and isothermal flow of gas is from right to left (see Figure 5b).
- The fracture permeability is 2–3 orders of magnitude higher than the matrix permeability.
- The matrix is disconnected from the hydrofracture and the wellbore, and only transfers gas into the fracture network.
- Gas flow from the fracture network is parallel to the hydrofracture plane (x-y plane).
- Depletion occurs only through the SRV, and the boundary flow is neglected.

## 3. Results

#### 3.1. Relation between the Optimized Parameters: Fracture Permeability ${k}_{f}$ and Microscale Source Terms

#### 3.2. Effects of Interference and Distance between Frac Stages

## 4. Materials and Methods

#### 4.1. Diffusivity Equation in the SRV Matrix

#### 4.2. Diffusivity Equation in SRV Fracture Network

#### 4.3. Match of Production Data

#### Constraints on SRV Parameters

**Constraint on the volume and surface area of discrete fracture network:**O’Malley et al. [61] presented a power law distribution of fracture length developed in a sheared shale sample. The volume and surface area of the generated fractures are given by:

**Constraint on D:**As shown in the above section, a portion of the injected water goes in macrofractures. The distribution of this water with distance from the hydrofracture face is used to constrain the value of D. The fractal dimension used in simulation is $D=1.8$. Figure 13a shows the strong effect of D on the injected water distribution. As we move from $D=1.7$ to $D=1.8$, the fracture porosity, ${\varphi}_{{f}_{0}}$ at the fracture face decreases from $1.8\%$ to $1.4\%$. For D = 1.7–1.8, 85–93% of the injected water is retained within $2d=85$ m, which is close to the average literature value of $2d$. However, $D=1.9$ decreases the fracture porosity, ${\varphi}_{{f}_{0}}$ to $0.1\%$ at the fracture face, thus making it an unreasonable choice for modeling. Such a low value of porosity at the fracture face is impossible. Expected porosity close to $1\%$ makes $D=1.8$ a good choice with $60\%$ of the water retained in the first few meters away from hydrofracture.

**Constraint on the value of microscale source term, s, from the optimization function:**The value of microscale source term, s, obtained after numerical fitting of production data is constrained with the power law distribution of total surface area created after hydrofracturing (see Table 2, total surface area). The continuous line in Figure 13 shows the power law distribution of the microscale source term, s (calculated from total surface area) for $D=1.8$. The values of s, obtained from simulation at the fracture face, cannot exceed the maximum value of the distribution at the fracture face. Values of s smaller than than the maximum value for the simulated wells reflect the fact that hydrofracturing is never $100\%$ efficient. For most wells, only a small part of the created surface area is well-connected and contributes to production.

## 5. Conclusions and Discussion

- The classical ($\tau ,\phantom{\rule{0.166667em}{0ex}}\mathcal{M}$) scaling of annual gas volumes produced from all non-refractured wells in the Barnett does an excellent job (see Figure 2), but the current (${k}_{f},\phantom{\rule{0.166667em}{0ex}}s,\phantom{\rule{0.166667em}{0ex}}D$) scaling gives deeper operational insights into well completion strategies (see Figure 6).
- The macroscale and microscale connectivities are represented by the fracture permeability, ${k}_{f}$, and the microscale source term, s. The higher the values of these two parameters are, the better flow connectivity at the respective scale is. In general, at higher fracture permeability, the microscale source term is lower, because of the balance of hydrofracturing energy (see Figure 7).
- The overall well quality depends on the ratio of macroscale and microscale connectivities, physically represented by the ratio ${k}_{f}/s$. A higher value of ${k}_{f}/s$ causes slow pressure decline in the matrix and results in high cumulative production. Poor microscale scale connectivity is compensated by better macroscale connectivity. A low value of ${k}_{f}/s$ causes high initial production rate resulting from fast gas discharge at the microscale. The matrix pressure depletes faster and limits overall cumulative production.
- There can be indirect ways of accessing the ratio of ${k}_{f}/s$ using the magnitude of microseismic events during hydrofracturing. The distribution of event magnitudes can be used to estimate the size distribution of microfractures/macrofractures created in the source rock. Eventually, hydrofracturing can be optimized by interpreting microseismicity.
- Stimulated mass is an important factor in controlling the overall production. The non-monotonic increase in the cumulative production histories for the 13 Barnett wells in Figure 9 highlights the importance of initial mass in place, which is the product of gas density, porosity, gas saturation, and volume stimulated after hydrofracturing.
- Efficient hydrofracturing is crucial to maximizing production. An important factor is the average distance between hydrofractures, $2d$. Small values of $2d$ can cause strong interference between two consecutive hydrofractures and can accelerate pressure loss in the matrix, thus limiting cumulative production (see Figure 10). Another key factor is the correct estimation of the Effective Propped Volume (EPV) width using microseismic data. The EPV width can give an estimate for the number of hydrofractures one should generate without causing a strong pressure interference and decreasing production.
- In addition to the correct estimation of EPV width from a cloud of microseismic events, it is important to understand configuration of the new/reactivated fracture network after stimulation. The same EPV width can give different EURs, depending on the ratio of ${k}_{f}/s$ that controls the efficiency of the backbone fracture network that drains gas at different scales (see Figure 9).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EPV | Effective Propped Volume |

EUR | Estimated Ultimate Recovery |

RF | Recovery Factor |

SRV | Stimulated Reservoir Volume |

## Nomenclature

Parameters | |

$2L$ | Tip-to-tip length [m] |

$\mathcal{M}$ | mass of gas in a reservoir rock volume [kg] |

$\mu $ | Viscosity [kg/m.s] |

$\varphi $ | Porosity |

$\mathrm{R}\phantom{\rule{-1.0pt}{0ex}}\mathrm{F}$ | recovery factor [-] |

$\rho $ | Density [kg/m${}^{3}$] |

$\theta $ | Tortuosity |

c | Compressibility [1/N/m${}^{2}$] |

D | Fractal dimension |

E | Euclidean dimension |

H | Hydrofracture height [m] |

K | Adsorption coefficient [m${}^{3}$(gas)/m${}^{3}$(rock)] |

k | Permeability [m${}^{2}$] |

M | Molar mass [kg] |

$p,P$ | Pressure [N/m${}^{2}$] |

S | Saturation |

s | Source term [m${}^{-2}$] |

T | Temperature [K] |

t | Time [s] |

u | Velocity [m/s] |

V | Volume [m${}^{3}$] |

w | Width [m] |

x | Distance [m] |

CI | confidence interval |

Subscripts | |

0 | Fracture face |

a | Adsorbed gas |

a | power law exponent |

f | Fracture |

g | Gas |

$HF$ | Hydrofracture |

L | Langmuir |

m | Matrix |

$SC$ | Standard conditions |

$tot$ | Total |

eff | Effective |

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**Figure 1.**In conventional reservoirs, ultimate recovery factor for a well (the ratio of gas produced from the stimulated reservoir volume (SRV), ${\mathcal{M}}_{\mathrm{stimulated}}$, to the original gas in place in the reservoir compartment assigned to a well, ${\mathcal{M}}_{\mathrm{geometric}}$) is proportional to the compartment size. This is not so in mudrock reservoirs, where produced gas is only partially connected to the reservoir compartment. In this case, produced gas volume is proportional to the compartment size raised to a fractional (fractal) power exponent $a<1$. The weaker is the mudrock connectivity, the smaller is the exponent. (

**a**) The power law scaling for three intervals of times on production (three age groups) of Barnett shale wells with the ages shown in the legend. The three solid lines are the result of a joint linear regression of production from these three groups of wells. The 95% CI that the regression explains the linear trend in the blue data points are drawn as the dotted lines above and below the blue fit. The 95% CI that a new well will belong to the blue dataset is shown as the outer dotted lines. The single power law exponent $a=0.72$ explains all data. (

**b**) The corresponding three linear trends for Haynesville almost overlay one another. Most of the poorest producers are on the Texas side of the play. Notice that the power exponent explaining gas production from the Haynesville wells $a=0.56$ is smaller than that in Barnett. Hence, on average, SRV connectivity in Haynesville is weaker than that in Barnett. Source: The unpublished calculations for [1,2]. The Haynesville play productivity will be addressed in a separate paper.

**Figure 2.**(

**a**) Up to fifteen years on production from 11,312 non-refractured wells in the Barnett were matched with hyperbolic DCA between January 1998 and December 2019. For each well, we normalize the predicted cumulative gas production with the last recorded value of cumulative production (max cum.) and plot it versus the normalized actual cumulative production. The increasingly hotter colors code the elapsed years on production from 1 to ≥11 years. Ideally, the annual cumulative production dots would track the black dashed diagonal ($x=y\to \mathrm{actual}=\mathrm{predicted}$). In contrast, the least squares fit of our hyperbolic DCA calculations is offset above the diagonal and is biased, overestimating EUR. The 95% confidence interval (CI) that a new well will belong to this dataset is plotted as the blue dotted lines. (

**b**) The same wells matched with an extension of the scaling in [3]. Consecutive hydrofractures in 4877 of these wells pressure-interfere with one another. There are 96,548 dots colored by each year on production, i.e., on average 8.2 years on production are plotted for each well. Time on production progresses along the diagonal away from the origin. The least squares fit of this cross-plot has less bias and offset compared with the hyperbolic DCA on the left. The 95% CI that a new well will belong to this dataset is plotted as the two red dotted lines. The red CI band on the right is narrower than the blue band on the left. In addition, the two color distributions are quite different. We conclude that the predicted annual production volumes from both the non-interfering and exponentially declining Barnett wells are accurate.

**Figure 3.**Probability distribution functions for the parameters of the hyperbolic DCA matches of 11,312 non-refractured wells in the Barnett plotted in Figure 2a. For each well, we apply the algorithm from the Reservoir Engineering Handbook (4$\mathrm{th}$ edition) [13] to solve for the Arp’s decline-curve exponent, b, and the initial decline rate, ${D}_{i}$. The mean values of b and ${D}_{i}$ are 1.31 and 0.35 month${}^{-1}$, respectively. In comparison, a recent published DCA analysis for 10,237 Barnett wells [14] gives the average values of b and ${D}_{i}$ as 1.477 and 0.1362 month${}^{-1}$, respectively. Even though the original formulation of hyperbolic DCA by Arps [15] requires $b\in [0,1]$, for many hydrofractured gas wells, $b>1$, and sometimes b is as high as 3.5 [16]. Therefore, we did not constrain the b exponents in Figure 2. Constraining them can result in the unrealistic shapes of decline curves that deviate from production rates observed over finite time intervals [17].

**Figure 4.**The fractures, represented by the dotted lines, initiate at the center of the cylinder and follow a branching pattern, spreading through the rock. Away from the center, density of the branched fractures decreases following a power law, as do the fracture porosity, $\varphi $, and fracture permeability, k. In the equation, D is the fractal dimension, E is the Euclidean dimension ($E=1$ for one fracture; $E=2$ for more fractures), and $\theta $ is the tortuosity index of the fracture network. Adapted from [38].

**Figure 5.**(

**a**) A horizontal well with eight hydrofracture stages. The hydrofractures are perpendicular to the wellbore and gas flows into each hydrofracture from both sides. The hydrofracture permeability is infinite relative to the stimulated matrix. Adapted from [56]. (

**b**) Planar view of half-SRV after hydrofracturing. The branching fracture network starts at the fracture face and its density decreases away from it. This fracture network interacts with the organic matrix and drains gas into the hydrofracture. The fracture porosity, ${\varphi}_{f}$, and permeability, ${k}_{f}$, also decrease as we move away from the fracture face. The Effective Propped Volume (EPV) is defined by the distance ${d}_{EPV}$ up to which the fractures are connected well enough to drain gas from the local organic matrix.

**Figure 6.**(

**a**) Predicted vs. actual annual cumulative gas production from 4877 pressure-interfering wells in the Barnett play scaled with an extension of the method in [1]. There are 37,942 dots, i.e., on average 7.6 years on production are plotted for each well (see the caption of Figure 2). The least squares fit of this cross-plot has some bias and offset ($\mathrm{predicted}=0.92\phantom{\rule{0.166667em}{0ex}}\mathrm{actual}+0.07$), which is not surprising for such a large heterogeneous set of wells. (

**b**) The multiscale (“${k}_{f},\phantom{\rule{0.166667em}{0ex}}s,\phantom{\rule{0.166667em}{0ex}}D$”) well performance model proposed in this paper. We plot the predicted cumulative gas production in one year increments vs. the one measured in 45 horizontal wells in the Barnett, each with 13.3 years on production on average, and each pressure-interfering. There are 597 dots of 11 colors. Time is progressing along the diagonal away from the origin. The least squares fit of this cross-plot has no bias and offset ($\mathrm{predicted}=0.99\phantom{\rule{0.166667em}{0ex}}\mathrm{actual}+0.01$). The 95% CI that this fit explains the linear trend in data is so tight we do not plot it. The 95% CI that a new well will belong to this dataset is plotted as the two red dotted lines. A mixture of simple scaling models published elsewhere and this paper’s multiscale model can provide crucial economic insights about well completions.

**Figure 7.**The microscale source term, s, represents microscale connectivity and the fracture permeability, ${k}_{f}$, represents macroscale connectivity for 45 Barnett wells. There is a tradeoff between the macroscale and microscale stimulation because of energy conservation. More energy spent on the macroscale stimulation leaves less energy available for the microscale stimulation. The result is higher values of ${k}_{f}$ and lower values of s, and vice versa. The extreme values in the red and blue ellipse represent two different types of fracture networks after hydrofracturing that result in two different types of production profiles.

**Figure 8.**(

**a**) Stimulated rock with low ${k}_{f}$ and high s. Density of the discrete macroscale fracture network is low and density of the microscale fractures (dotted lines) in the organic matrix is high. This situation results in a faster pressure decline in the organic matrix. Low permeability at the macroscale is compensated by higher permeability at the microscale. (

**b**) Stimulated rock with high ${k}_{f}$ and low s. Density of the discrete macroscale fracture network is high and density of microscale fractures (dotted lines) in the organic matrix is low, causing a slower pressure decline in the organic matrix. Low permeability at the microscale is compensated by higher permeability at the macroscale.

**Figure 9.**Gas production rate (

**a**) and cumulative production (

**b**) depend on the ratio ${k}_{f}/s$. The thickness of each line increases with the increasing ${k}_{f}/s$. This ratio represents the contributions from the macroscale and microscale discharge processes. For the low values of this ratio, initial production rate can be very high with a steep production decline (blue curves). Faster production decline caused by the low ${k}_{f}/s$ gives lower ultimate cumulative production. The two sets of curves do not increase monotonically with ${k}_{f}/s$, because of uncertainty in the initial mass in place, which depends on the porosity, gas saturation and the stimulated volume after hydrofracturing.

**Figure 10.**(

**A**–

**C**) represent three cases with different EPV length, tip-to-tip (2d) length, and number of fracking stages. The corresponding cumulative production for the three cases are also shown above. Curve (A) represents cumulative production of a Barnett well with the 660 m lateral, 6 frac stages, and the EPV width of 49 m. For 8 frac stages, the EPV width decreases to 41 m, and there is small interference, denoted by the shaded area, with a small effect on the flow properties. The cumulative production, Curve (B), increases by 2 Bscf. For 9 frac stages, the EPV width is 36 m and pressure interference is strong, thus affecting flow properties at the microscale. The matrix-fracture source term increases causing a steeper decline in pressure support. The incremental increase in cumulative production of Curve (C) with 9 frac stages compared with Curve (B) with 8 frac stages is negligible.

**Figure 11.**Cumulative production of two wells with the length of 640 m and the number of stages of 5. The ${k}_{f}/s$ ratio defines the fracture network efficiency of draining gas from the kerogen pore network. ${k}_{f}/s$ of 1.5 causes a faster pressure loss in the system, limiting cumulative production to 3 Bscf in 180 months. For the same well parameters (EPV width, number of stages, and hydrofracture height), ${k}_{f}/s$ of 5 or 6 gives 3.4 Bscf in 180 months.

**Figure 12.**(

**a**) Total amount of injected water in single frac stage is ∼750 m${}^{3}$. We assume that the water is retained in the SRV, which has the tip-to-tip width of 180 m, fracture height of 30 m, and fracture half-length of 50 m. The x-axis is the volume of injected water retained in the large discrete fracture network and the y-axis is the created surface area. The volume of proppant injected during each hydrofracture stage creates equal volume of discrete fractures, thus an average $40\%$ (275 m${}^{3}$) of the injected water goes into large fractures. The surface area created by this injected water volume is $2.7\times {10}^{5}$ m${}^{2}$, requiring fracture spacing of 2–4 m in the SRV. This spacing is the characteristic distance for the branching, complex fracture network. (

**b**) Hydrofracture frequencies inferred from the information contained in the paper on the Eagle Ford Connoco Phillips pilot by Raterman et al. [64]. The dotted horizontal lines represent the range of effective fracture spacing postulated by the authors of [1,2,56].

**Figure 13.**(

**a**) Effect of D on injected water distribution and fracture porosity, ${\varphi}_{{f}_{0}}$, at the fracture face. $D=1.9$ forces an unreasonably low porosity, ${\varphi}_{{f}_{0}}=0.1\%$, at the fracture face. The expected porosity close to $1\%$ at the fracture face gives $D=1.8$ as a good choice for the fracture network distribution in the SRV and $2d$ = 85–90 m, sufficient to retain $85\%$ of the injected water. (

**b**) The admissible values of the source term, s, for the simulated wells are limited by the maximum depicted as the continuous line at the fracture face. This upper bound helps in constraining the values of s obtained from optimization. A low value of s from the simulation means that only a small part of the stimulated surface is connected well enough for efficient gas production.

Parameters | SI Units | Field Units |
---|---|---|

Matrix porosity, ${\varphi}_{m}$ | $0.08$ | 8% |

Matrix Permeability, ${k}_{m}$ | $5\times {10}^{-20}$ m${}^{2}$ | 50 nanodarcy |

Gas saturation, ${S}_{g}$ | 0.6–0.8 | 0.6–0.8 |

Initial pressure, ${p}_{i}$ | 24.1 MPa | 3500 psi |

Reservoir temperature, T | 361 K | 190 °F |

Hydrofracture pressure | 3.44 MPa | 500 psi |

Fracture height, H | 30 m | 100 ft |

Tip-to-tip fracture length, $2L$ | 200 m | 590 ft |

Half SRV width, d | 40–50 m | 131–164 ft |

Parameters | Small Fractures | Large Fractures |
---|---|---|

${R}_{0}$ | 1 mm | 5 m |

${R}_{1}$ | 5 m | 50 m |

Volume | 400 m${}^{3}$ | 275 m${}^{3}$ |

Area | $3\times {10}^{7}$ m${}^{2}$ | $2.6\times {10}^{5}$ m${}^{2}$ |

Fractures | ${10}^{13}$ | 360 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Haider, S.; Saputra, W.; Patzek, T. The Key Factors That Determine the Economically Viable, Horizontal Hydrofractured Gas Wells in Mudrocks. *Energies* **2020**, *13*, 2348.
https://doi.org/10.3390/en13092348

**AMA Style**

Haider S, Saputra W, Patzek T. The Key Factors That Determine the Economically Viable, Horizontal Hydrofractured Gas Wells in Mudrocks. *Energies*. 2020; 13(9):2348.
https://doi.org/10.3390/en13092348

**Chicago/Turabian Style**

Haider, Syed, Wardana Saputra, and Tadeusz Patzek. 2020. "The Key Factors That Determine the Economically Viable, Horizontal Hydrofractured Gas Wells in Mudrocks" *Energies* 13, no. 9: 2348.
https://doi.org/10.3390/en13092348