# Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production from All Vertical Wells in the Permian Basin

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Exploratory Data Analysis

#### 2.2. Design of Well Cohorts

#### 2.3. GEV Statistics and Historical Well Prototypes

#### 2.4. Physical Scaling and Extended Well Prototypes

#### 2.5. Probability of Well Survival

#### 2.6. Total Field Forecast

## 3. Discussion

## 4. Materials and Methods

- Design of well cohorts: We divide nearly half a million vertical wells in the Permian into 192 spatiotemporal well cohorts. The number 192 is the multiplication of four reservoir ages, six sub-plays, and eight completion date intervals detailed in Table 2.
- Perform GEV statistics and historical well prototypes: For each cohort, we sample $i=1,2,\cdots ,n$ years on production and fit a generalized extreme value distribution using Equation (1) to find the location parameter, $\mu $, scale parameter, $\sigma $, and shape parameter, $\xi $. Using Equations (2) and (3), we calculate the expected value or mean, ${P}_{50}$, the upper bound, ${P}_{10}$, and the lower bound, ${P}_{90}$, of the GEV distributions. We then connect each value of annual ${P}_{50}$’s to construct a historical well prototype, see Figure 7.$$\begin{array}{cc}\hfill \mathrm{PDF}:\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}& f\left(x\right)=\frac{1}{\sigma}{\left(\right)}^{1}-\frac{1}{\xi}-1\hfill & {\mathrm{e}}^{-{\left(\right)}^{1}}\end{array}$$$$\begin{array}{cc}\hfill \mathrm{CDF}:\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}& F\left(x\right)={\mathrm{e}}^{-{\left(\right)}^{1}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \mathrm{Mean}:\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}& E\left(x\right)=\mu -\frac{\sigma}{\xi}+\frac{\sigma}{\xi}\mathsf{\Gamma}(1-\xi ),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\xi \ne 0.\hfill \end{array}$$
- Perform physical scaling and extended well prototypes: We use the physical scaling approach to extend the historical well prototypes for several more decades. In Appendix A, we derive the new physical scaling for conventional vertical wells. First, we convert the annual oil rate into the cumulative mass produced using Equation (A31). Next, we scale the cumulative mass with $\tau $ along the x-axis and by $\mathcal{M}$ along the y-axis, so that the scaling result matches the master curve in Equation (A34).
- Estimate probability of well survival: We calculate the survival probability of each sub-region of the Permian using Equation (4), where ${N}_{\mathrm{active},i}$ and ${N}_{\mathrm{inactive},i}$ are the numbers of active and inactive wells in year-i:$${P}_{\mathrm{survival},i}=\frac{{N}_{\mathrm{active},i}}{{N}_{\mathrm{active}}+{N}_{\mathrm{inactive},i}}$$To estimate the maximum time of well survival, we fit the probability of survival with Equation (5) and find the intercept of the curve fit to the probability equal to zero:$$y\left(t\right)=(1+\beta )({e}^{-{t}^{\beta}}-\beta )$$
- Complete total field forecast: We replace the actual reported field production rate from all existing vertical wells in the Permian with the corresponding extended well prototypes. The summation of all prototypes becomes the total basin-wide forecast of the conventional Permian wells.

## 5. Conclusions

- We have provided a transparent hybrid method of forecasting conventional oil production at a basin scale.
- A combination of GEV statistics of very large data sets with physical scaling matches historical production data almost perfectly and gives a smooth, optimal prediction of the future in the least-square sense.
- Our spatiotemporal well cohorts are a combination of different reservoir ages, sub-plays, and completion date intervals.
- The estimated ultimate recovery (EUR) from all 484,759 existing vertical wells in the Permian is about 34 billion barrels of oil.
- We observed that the vertical wells in the Permian can last between 10 and 100 years, depending on which sub-play and reservoir these wells penetrate.
- In practice, no large reservoir has been found in the Permian since the 1970s.
- Today, operators need to drill wells that are twice as deep as the 1930s’ wells but that produce 4–12 times less.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EUR | Estimated Ultimate Recovery |

GEV | Generalized Extreme Value |

GOR | Gas to Oil Ratio |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

RF | Recovery Factor |

## Appendix A. Physical Scaling for Conventional Wells

#### Appendix A.1. One-Dimensional Pressure Diffusion Equation in Radial Coordinates: Constant Pressure—Bounded Reservoir

#### Appendix A.2. Discretization Techniques and Numerical Solutions

**Figure A3.**Numerical solution of the radial pressure diffusion for constant pressure—bounded reservoir (${r}_{e}=500$ ft).

#### Appendix A.3. Calculating Recovery Factor

**Figure A4.**(

**a**) Differential dimensionless pressure at the wellbore; (

**b**) Time function, $f\left(\tilde{t}\right)$, for simulation result in ft Figure A3 (${r}_{e}=500$).

**Figure A5.**Oil recovery factor for various values of reservoir radius, ${r}_{e}$, in (

**a**) the linear scale, (

**b**) the square root of time scale.

#### Appendix A.4. Semi-Analytical Solution of Oil Recovery Factor in Radial Flow: Bounded-Reservoir Constant-Pressure

**Figure A6.**Comparison between the numerical solutions and the corresponding fits from Equation (A28), for different reservoir radii, ${r}_{e}=$ (

**a**) 500 ft, (

**b**) 10,000 ft, (

**c**) 250,000 ft, and (

**d**) 10,000,000 ft.

${\mathit{r}}_{\mathit{e}}$ (ft) | ${\mathit{r}}_{\mathit{w}}/{\mathit{r}}_{\mathit{e}}$ | $ln({\mathit{r}}_{\mathit{w}}/{\mathit{r}}_{\mathit{e}})$ | a | b |
---|---|---|---|---|

500 | 0.00050 | −7.6 | −0.00296 | 0.284 |

1000 | 0.00025 | −8.3 | −0.00235 | 0.257 |

2500 | 0.00010 | −9.2 | −0.00179 | 0.228 |

5000 | 0.000050 | −9.9 | −0.00149 | 0.210 |

10,000 | 0.000025 | −10.6 | −0.00126 | 0.195 |

25,000 | 0.000010 | −11.5 | −0.00103 | 0.178 |

50,000 | 0.0000050 | −12.2 | −0.00090 | 0.167 |

100,000 | 0.0000025 | −12.9 | −0.00079 | 0.157 |

250,000 | 0.0000010 | −13.8 | −0.00068 | 0.145 |

500,000 | 0.00000050 | −14.5 | −0.00061 | 0.138 |

1,000,000 | 0.00000025 | −15.2 | −0.00055 | 0.131 |

2,500,000 | 0.00000010 | −16.1 | −0.00048 | 0.123 |

5,000,000 | 0.000000050 | −16.8 | −0.00043 | 0.117 |

10,000,000 | 0.000000025 | −17.5 | −0.00040 | 0.112 |

**Figure A8.**Linear relationship between (

**a**) $ln(-a)$ and (

**b**) $ln\left(b\right)$ versus $ln(ln({r}_{w}/{r}_{e}))$.

#### Appendix A.5. Physical Scaling for Conventional Wells and Validations

^{2}floods in some of the Permian reservoirs. Otherwise, if reservoir-specific production data are known, these effects could be taken into account by adjusting the constant C in Equation (A27) and by the method of superposition.

**Figure A9.**(

**a**) A cylindrical reservoir model using the commercial reservoir simulator, CMG; (

**b**) Comparison between the physical scaling curve (black) and simulation result from CMG (red).

**Figure A10.**Comparison between the master curves in Equation (A34) and field data from Permian Basin: (

**a**) ${\mathcal{K}}_{E}=0.022$ and (

**b**) ${\mathcal{K}}_{E}=0.035$.

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**Figure 1.**(

**a**) Annual oil rate in kbbl/yr and (

**b**) Cumulative oil in kbbl from all 484,759 vertical wells in the Permian Basin.

**Figure 3.**The map of all 484,759 vertical wells in the Permian Basin, colored by four different reservoir ages.

**Figure 4.**The map of all 484,759 vertical wells in the Permian Basin, colored by three classes of maximum daily oil rate.

**Figure 7.**Procedure of arriving at a historical well prototype using GEV statistics. (Example: Midland Basin—Leonardian [2000–2009]). Each row shows different sampling of annual production after the first, second, …, n-th year on production. The first column is the probability density function (PDF) of annual production fitted with lognormal distribution (blue) and GEV distribution (red). The corresponding $\xi $, $\mu $, and $\sigma $ are the fitting parameters for the GEV distribution. The second column shows the maximum likelihood estimate, 95% confidence interval (CI) for $\mu $ and $\sigma $. The third column displays the cumulative distribution function (CDF), including the lower bound, ${P}_{90}$, median, ${P}_{50}$, and upper bound, ${P}_{10}$, with its confidence interval. The last column shows the resulting historical well prototype, which is constructed by connecting the expected GEV values (means) of each row.

**Figure 8.**Historical well prototypes (solid lines) and the corresponding physical scaling forecasts (dashed lines) for the 192 well cohorts. The columns represent the four reservoir ages, while the rows denote the eight different completion date intervals. The six different sub-plays are encoded with different colors. The horizontal bar charts show the percentage of each subset sub-play from the total n sample of each reservoir-age × completion-date combination. In addition, the small square at the right boundary of each graph shows the average ultimate recovery for the six sub-play scenarios.

**Figure 9.**Distribution of fitting parameters for physical scaling: (

**a**) $\tau $, GEV mean = 2.37 years and (

**b**) $\mathcal{M}$, GEV mean = 106.37 ktons.

**Figure 10.**Survival probability plot for Permian region: (

**a**) Central Basin—Leonardonian and (

**b**) Midland Basin—Wolfcampian. The fitting function is $y\left(t\right)=(1+\beta )({e}^{-{t}^{\beta}}-\beta )$.

**Figure 11.**(

**a**) Total field rate, (

**b**) Total field cumulative forecasts from existing 484,759 wells in the Permian Basin.

**Figure 12.**Evolution of completion depth in the Permian Basin. The upper bound, ${P}_{10}$, the expected value, ${P}_{50}$, and the lower bound, ${P}_{90}$ are all calculated with GEV statistics.

**Table 1.**Summary of four reservoir ages and list of reservoir names included. The cross-ticked boxes indicate the existence of each reservoir layer in six different areas (sub-plays).

No | Reservoir Age | Time (m.y.) | Reservoir Name | Areal Extent | ||||
---|---|---|---|---|---|---|---|---|

Delaware Basin | Northwest Shelf | Central Basin | Eastern Shelf | Midland Basin | ||||

1 | Guadalupian | 251 | Bell Canyon | ⊠ | ☐ | ☐ | ☐ | ☐ |

Delaware | ⊠ | ☐ | ☐ | ☐ | ☐ | |||

Capitan | ☐ | ☐ | ⊠ | ☐ | ☐ | |||

Tansill | ☐ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Yates | ☐ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Seven Rivers | ☐ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Queen | ☐ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Grayburg | ☐ | ⊠ | ⊠ | ⊠ | ⊠ | |||

San Andres | ☐ | ⊠ | ⊠ | ⊠ | ☐ | |||

Holt | ☐ | ☐ | ⊠ | ☐ | ☐ | |||

Brushy Canyon | ⊠ | ☐ | ☐ | ☐ | ⊠ | |||

2 | Leonardian | 275 | Bone Spring | ⊠ | ☐ | ☐ | ☐ | ☐ |

Spraberry | ☐ | ☐ | ☐ | ☐ | ⊠ | |||

Dean | ☐ | ☐ | ☐ | ☐ | ⊠ | |||

Glorieta | ☐ | ⊠ | ⊠ | ⊠ | ☐ | |||

Paddock | ☐ | ⊠ | ☐ | ☐ | ☐ | |||

Blinebry | ☐ | ⊠ | ☐ | ☐ | ☐ | |||

Clear Fork | ☐ | ⊠ | ⊠ | ⊠ | ☐ | |||

Tubb | ☐ | ⊠ | ⊠ | ⊠ | ☐ | |||

Drinkard | ☐ | ⊠ | ☐ | ☐ | ☐ | |||

Yeso | ☐ | ⊠ | ☐ | ☐ | ☐ | |||

Abo | ☐ | ⊠ | ⊠ | ⊠ | ☐ | |||

3 | Wolfcampian | 290 | Wolfcamp | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ |

4 | Pre-Permian | 302–495 | Pennsylvanian | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ |

Cisco | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Canyon | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Strawn | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Atoka | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Morrow | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Barnett | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Mississippian | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Devonian | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Silurian | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Fusselman | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Ordovician | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Montoya | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | |||

Simpson | ⊠ | ☐ | ⊠ | ☐ | ⊠ | |||

Ellenburger | ⊠ | ⊠ | ⊠ | ⊠ | ☐ |

4 Reservoir Ages | 6 Sub-Plays | 8 Completion Dates |
---|---|---|

Guadalupian | Central Basin | 1930–1949 |

Leonardian | Midland Basin | 1950–1959 |

Wolfcampian | Delaware Basin | 1960–1969 |

Pre-Permian | Northwest Shelf | 1970–1979 |

Eastern Shelf | 1980–1989 | |

Others | 1990–1999 | |

2000–2009 | ||

2010–2021 |

**Table 3.**Summary of maximum survival times, ${t}_{\mathrm{surv}}$, in years for the Permian regions.

Guadalupian | Leonardian | Wolfcampian | Pre-Permian | Mean | |
---|---|---|---|---|---|

Delaware Basin | 70 | 47 | 35 | 20 | 43 |

Northwest Shelf | 95 | 78 | 47 | 31 | 63 |

Central Basin | 78 | 87 | 44 | 57 | 67 |

Eastern Shelf | 50 | 91 | 48 | 39 | 57 |

Midland Basin | 94 | 77 | 40 | 50 | 65 |

Others | 45 | 27 | 10 | 19 | 25 |

Mean | 72 | 68 | 37 | 36 | 53 |

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**MDPI and ACS Style**

Saputra, W.; Kirati, W.; Patzek, T.
Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production from All Vertical Wells in the Permian Basin. *Energies* **2022**, *15*, 904.
https://doi.org/10.3390/en15030904

**AMA Style**

Saputra W, Kirati W, Patzek T.
Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production from All Vertical Wells in the Permian Basin. *Energies*. 2022; 15(3):904.
https://doi.org/10.3390/en15030904

**Chicago/Turabian Style**

Saputra, Wardana, Wissem Kirati, and Tadeusz Patzek.
2022. "Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production from All Vertical Wells in the Permian Basin" *Energies* 15, no. 3: 904.
https://doi.org/10.3390/en15030904