# Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production in the Bakken Shale

^{*}

## Abstract

**:**

## 1. Introduction

**8**billion barrels in total.

**regions**in which reservoir quality is similar, see Figure 1. In each region, we identify well

**classes**by different completion technologies. Finally, a well class in a region constitutes a well

**sample**. We ensure that oil production from all wells in each sample is statistically uniform, that is, has a unimodal distribution. For each well sample, we then identify well

**cohorts**with

**at least**$1,2,\cdots $ years on production. In general, well cohorts contain different sets of wells that satisfy the minimum time on production required for each cohort. It turns out that each cohort of wells is superbly characterized by its unique Generalized Extreme Value (GEV) distribution (see Appendix A) of annualized well rates or cumulative well production. Different cohorts in the same sample have different GEV distributions, each with its unique expected value, median and mode. Here we choose the somewhat better GEV fits of the production rate distributions. Each GEV distribution is statistically superior to the corresponding log-normal distribution at the 95% confidence level. When we plot the expected values of the GEV distributions of all wells cohorts in a sample versus elapsed time of production, we obtain this sample’s average ${P}_{50}$ statistical well prototype that is purely field data-driven.

## 2. Results

- Divide all 14,678 horizontal oil wells in the Bakken shale into 12 samples in which oil production is statistically uniform;
- Fit a generalized extreme value distribution to all wells in every sample and obtain 12 stable mean ${P}_{50}$ well prototypes;
- Fit the physics-based scaling curves to every statistical well prototype and extend these prototypes smoothly to 30 years on production;
- Replace oil production from all existing wells with the 12 extended well prototypes and obtain a ‘base’ forecast;
- Calculate the infill potential for each of the 12 regions in the Bakken; and
- Create the plausible infill drilling schedules and forecast total field oil production rate up to the year 2050.

#### 2.1. Design of Well Regions

#### 2.2. Statistical Well Prototypes

#### 2.3. Physical Scaling Fits

#### 2.4. Base or ‘Do Nothing’ Forecasts

#### 2.5. Infill Potentials

#### 2.6. Future Drilling Scenarios and Infill Forecasts

## 3. Discussion

## 4. Materials and Methods

- Define play regions in which oil production is statistically uniform. In each region, follow the steps below:
- (a)
- Divide all wells in a play into $i=1,2,\cdots ,n$ well groups, where n is the number of reservoirs. In the Bakken play, for example, there are two main reservoirs. Thus, at this step, we identify two groups of wells,

and $n=2$ in the Bakken shale.**Well Sample****Reservoir****Sample Size**1 Middle Bakken 9860 2 Three Forks 4818 - (b)
- Further subdivide each of these well groups into ${j}_{i}$, $i=1,2,\cdots ,n$, areas with different reservoir qualities. For example, in the Bakken play, the center of the basin has the thickest oil-prolific layer and hence the highest oil production [3,22,28,29,30]. Thus, we have delineated an area at the center of the basin (with maximum oil rate >750 bopd) as core area and the rest as $noncore$ area, see Figure 2 for more details. At this step, we have four well groups that fall into four distinct, static play regions,

In Bakken, ${j}_{1}=2$ and ${j}_{2}=2$, that is, there are two reservoir qualities (core and noncore) for each of the reservoirs (Middle Bakken and Three Forks).**Reservoir****Region****Sample Size**1 Middle Bakken core 5732 2 Middle Bakken noncore 4128 3 Three Forks core 2672 4 Three Forks noncore 2146 - (c)
- Subdivide wells in each of the four regions (two reservoirs × two reservoir qualities each) by time interval classes that encompass significant changes in well completion technology. In the Bakken play, for instance, the newly completed wells have longer lateral lengths, bigger hydraulic fractures and more fracture stages [22,31,32,33,34]. Thus, we classify the wells in each of the four regions by three completion date intervals: [2000–2012], [2013–2016] and [2017–2019]. In the end, we have divided all 14,678 horizontal wells in the Bakken into 12 well groups (4 regions × 3 completion date classes) listed in Table 1. In Bakken, each of the 4 play regions has 3 completion classes, finally yielding 12 static well samples.

- For each well sample, obtain a ${P}_{50}$ well prototype by fitting a Generalized Extreme Value (GEV) distribution to all qualifying sample wells as follows:
- (a)
- From a given static well sample-k (a region further subdivided by completion dates), consider a dynamic cohort ${l}_{k}$ that contains all wells that have at least ${l}_{k}$ years on production (${l}_{k}=1,2,\cdots ,{t}_{{max}_{k}}$ and $k=1,2,\cdots ,{N}_{\mathrm{sample}}$). For example: (i) There are 2550 wells in Middle Bakken noncore [2000–2012] group. However, there are only 2540 wells with at least one year on production (the other 10 wells have production records with less than 12 months). Thus, we retain these 2540 wells as cohort-1 of this particular group, see Figure 4. (ii) There are 428 wells in Three Forks core [2000–2012] group. But, only 197 wells have production records of at least seven years. As such, these 197 wells are qualified as cohort-7 of this particular well group, see Figure 5. For detailed GEV fits for all dynamic well cohorts and static well groups in the Bakken play, see Supporting Online Materials-1. For Bakken, ${N}_{\mathrm{sample}}=12$.
- (b)
- Define a set ${X}_{{l}_{k}}$ that takes values of oil production rate (kbbl/year) from all wells with at least ${l}_{k}$ years on production in sample k.
- (c)
- For every ${X}_{{l}_{k}}$, fit a Generalized Extreme Value (GEV) distribution using Equation (A1) and obtain the location parameter, $\mu $, scale parameter, $\sigma $ and shape parameter, $\xi $. (All of these parameters are further subscripted by ${l}_{k}$ but we will skip this complication in the notation.)
- (d)
- Calculate ${\overline{x}}_{{l}_{k}}$ as the GEV mean of ${X}_{{l}_{k}}$ for each year-${l}_{k}$, ${l}_{k}=1,2,\cdots ,{t}_{{max}_{k}}$, using Equation (A3).
- (e)
- For each well sample k, obtain the ${P}_{{50}_{k}}$ well prototype. We can use the same procedure to obtain other statistic parameters for each well cohort. The ${P}_{{10}_{k}}$ and ${P}_{{90}_{k}}$ values can be calculated as the ninetieth and tenth percentiles of ${X}_{{l}_{k}}$, respectively. The median and mode can be calculated using Equations (A5) and (A6). by connecting GEV mean values of all years ${l}_{k}=1,2,\cdots ,{t}_{{max}_{k}}$$${P}_{{50}_{k}}=({\overline{x}}_{1},{\overline{x}}_{2},\cdots ,{\overline{x}}_{{t}_{{max}_{k}}}),k=1,2,\cdots ,{N}_{\mathrm{sample}},\phantom{\rule{1.em}{0ex}}\mathrm{kbbl}/\mathrm{year}.$$

- Extend ${P}_{{50}_{k}}$ (we will subsequently skip the implied subscript k) well prototypes by fitting to them physical scaling curves as follows:
- (a)
- For a each well sample k, calculate the observed cumulative mass produced, $\mathbf{m}$ (ktons) from Equation (A13). Fix ${q}_{o,\mathrm{ST}}={P}_{{50}_{k}}$ in kbbl/year and $\Delta {t}_{i}=1$ year.
- (b)
- Adjust $\tau $ in Equation (A22) or Equation (A7) (with or without exterior flow) and $\mathcal{M}$ in Equation (A11), so that the physical scaling curve matches the observed $\mathbf{m}$. Use ${c}_{t}/{S}_{oi}({p}_{i}-{p}_{f})$ and a values from the average Bakken reservoir properties listed in Table A1 and Table A2. The matching values of $\tau $ and $\mathcal{M}$ for two both scaling curves and for all well samples are detailed in Table 2.
- (c)
- Get the extended cumulative mass produced, $\widehat{\mathbf{m}}$ (ktons) by multiplying the fitted $\mathcal{M}$ and $\mathrm{RF}(t/\tau )$ from the master curve where t is now calculated each month The benefit of matching ${P}_{50}$ with a physics-based scaling curve is that we can interpolate and extrapolate production data precisely. Thus, we can change time intervals from years to months and forecast production decades into the future. We recommend to use monthly intervals for precise forecasts that is, $t=\frac{1}{12},\frac{2}{12},\cdots ,50$ years.
- (d)
- Obtain the extended ${P}_{50}$ well prototypes for well sample k by differencing $\widehat{\mathbf{m}}$ converted to volume$${\widehat{P}}_{50}=\frac{\Delta \widehat{\mathbf{m}}}{\Delta t({\rho}_{o,\mathrm{ST}}+{R}_{s}{\rho}_{g,\mathrm{ST}})},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kbbl}/\mathrm{month}$$

- Obtain base forecast of total field oil production for existing wells as follows:
- (a)
- Create a calendar date series with monthly intervals for example, (1/1/2000, 2/1/2000, ⋯, 12/1/2050) and assign ${N}_{\mathrm{dates}}$ as the length of this series.
- (b)
- Create an empty vector $\mathbf{s}$ with the length of ${N}_{\mathrm{dates}}$: ($\mathbf{s}=[{s}_{1}\phantom{\rule{3.33333pt}{0ex}}{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}{s}_{{N}_{\mathrm{dates}}}]=[0\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}0]$).
- (c)
- For each well, find index $ii$ in the calendar date series that brackets well completion date.
- (d)
- For each well, add to vector $\mathbf{s}$ its corresponding ${\widehat{P}}_{50}$ right-shifted by $ii$.$$\mathbf{s}=\mathbf{s}+[\underset{ii\phantom{\rule{3.33333pt}{0ex}}\mathrm{zeros}}{\underbrace{0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\widehat{P}}_{50}\left(1\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\widehat{P}}_{50}({N}_{\mathrm{dates}}-ii+1)]$$
- (e)
- Calculate total field oil rate, $\mathbf{q}$ and total field cumulative oil, $\mathbf{Q}$ for existing wells as follows:$$\mathbf{q}=\mathbf{s}/(365.25/12)\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{million}\phantom{\rule{3.33333pt}{0ex}}\mathrm{bbl}\phantom{\rule{3.33333pt}{0ex}}\mathrm{of}\phantom{\rule{3.33333pt}{0ex}}\mathrm{oil}\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\phantom{\rule{3.33333pt}{0ex}}\mathrm{day}$$$$\mathbf{Q}=\left(\right)open="["\; close="]">{s}_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{1}+{s}_{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\sum _{i=1}^{{N}_{\mathrm{dates}}}{s}_{i}$$

- Calculate infill potential as follows:
- (a)
- Create a one-square-mile fishnet inside the boundary of each area defined in step I(b).
- (b)
- Search all existing wells located inside the boundary.
- (c)
- Calculate wellhead density by counting the number of wells on each of the one-square-mile squares.
- (d)
- Calculate approximate well density following algorithm below.
- For each well, calculate the number of squares, n, intercepted by the lateral For example, a 5000 ft lateral occupies one square, a 9000 ft lateral occupies two squares and a 14,000 ft lateral occupies three squares because one mile is 5280 ft.
- Search for the least-occupied n squares in all possible directions.
- Increase the value of well density by 1 well/mi${}^{2}$ for every least dense n square found.
- Repeat points i–iii until all wells in the area of interest are exhausted.

- (e)
- Calculate infill potential by subtracting the calculated well density from the maximum allowable number of wells that still avoid frac hits. For example, in the Bakken play, the tip-to-tip hydraulic fracture length is roughly 1200 ft. 5280 ft/1200 ft ≈ 4. Therefore, the maximum allowable number of wells to avoid frac hits is 4 wells per square mile. Suppose that at some location, the well density already is 3 wells per square mile. Then, the infill potential is $4-3=1$ well per square mile.

- Obtain an infill forecast of total field oil production after adding future wells.
- (a)
- Create monthly drilling schedules for every infill potential and assume a constant drilling rate based on current rig availability, see Figure 9b and SOM-2.
- (b)
- Use the same calendar date series as in step IV(a) with ${N}_{\mathrm{dates}}$ as the length of date series.
- (c)
- Create an empty vector ${\mathbf{s}}_{\mathrm{f}}$ with the length of ${N}_{\mathrm{dates}}$ (${\mathbf{s}}_{\mathrm{f}}=[{s}_{\mathrm{f},1}\phantom{\rule{3.33333pt}{0ex}}{s}_{\mathrm{f},2}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}{s}_{\mathrm{f}},\mathrm{N}]=[0\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}0]$).
- (d)
- For every drilling schedule, find index $ii$ in the calendar date series that brackets infill schedule date. Let ${N}_{\mathrm{f}}$ be the number of wells to be drilled on that date.
- (e)
- For every drilling schedule, add to vector ${\mathbf{s}}_{\mathrm{f}}$ its corresponding ${\widehat{P}}_{50}$ after right-shifting it by $ii$ and multiplying by ${N}_{\mathrm{f}}$.$${\mathbf{s}}_{\mathrm{f}}={\mathbf{s}}_{\mathrm{f}}+{N}_{\mathrm{f}}\times [\underset{ii\phantom{\rule{3.33333pt}{0ex}}\mathrm{zeros}}{\underbrace{0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\widehat{P}}_{50}\left(1\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}{\widehat{P}}_{50}({N}_{\mathrm{dates}}-ii+1)]$$
- (f)
- Calculate total field oil rate, ${\mathbf{q}}_{\mathrm{f}}$ and total field cumulative oil, ${\mathbf{Q}}_{\mathrm{f}}$ after infill drilling as follows$${\mathbf{q}}_{\mathrm{f}}=\mathbf{q}+{\mathbf{s}}_{\mathrm{f}}/(365.25/12)\times {10}^{-3}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{million}\phantom{\rule{3.33333pt}{0ex}}\mathrm{bbl}\phantom{\rule{3.33333pt}{0ex}}\mathrm{of}\phantom{\rule{3.33333pt}{0ex}}\mathrm{oil}\phantom{\rule{3.33333pt}{0ex}}\mathrm{per}\phantom{\rule{3.33333pt}{0ex}}\mathrm{day}$$$${\mathbf{Q}}_{\mathrm{f}}=\mathbf{Q}+\left(\right)open="["\; close="]">{s}_{\mathrm{f},1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{s}_{\mathrm{f},1}+{s}_{\mathrm{f},2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\sum _{i=1}^{{N}_{\mathrm{dates}}}{s}_{\mathrm{f},\mathrm{i}}$$

## 5. Conclusions

- We have provided a transparent hybrid method of forecasting oil production at shale basin scale.
- Our statistical approach generates the non-parametric well prototype templates that are used to calibrate our physics-based flow scaling with late-time radial inflow.
- In particular, our average ${P}_{50}$ well prototypes follow the physics of linear transient flow and are used to calibrate the physics-based scaling extensions to 30 years on production.
- A combination of GEV statistics with physical scaling matches historical production data almost perfectly and gives a smooth, physics-based estimate of future production.
- Regulators may want to consider our approach as a prerequisite to booking reserves in oil shales.
- Newly completed wells have almost the same ultimate recovery as the older ones, despite their much higher initial oil rates.
- Ultimately, we predict that the 14,678 existing wells in the Bakken will produce 5 billion bbl of oil by 2050 (∼340 kbbl/well).
- After drilling additional 4400 new wells at the rate of 120 wells/month, the core area of the Bakken will be drilled out by 2021 and ultimate recovery will be 7 billion barrels of oil.
- With 26,500 more wells to be drilled in the noncore area until 2041, ultimate recovery in the Bakken might be 13 billion barrels of oil but drilling of such scale is unlikely to happen.
- Policy-makers should not of assume that oil boom in the Bakken shale will last decades longer.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CDF | Cumulative Distribution Function |

EUR | Estimated Ultimate Recovery |

GEV | Generalized Extreme Value |

Probability Density Function | |

SRV | Stimulated Reservoir Volume |

## Appendix A. The Generalized Extreme Value (GEV) Distributions

**Figure A1.**Examples of three GEV distributions with the location parameter, $\mu =0$, the scale parameter, $\sigma =1$. and three values of the shape parameter i.e., $\xi <0$ (Weibull), $\xi =0$ (Gumbel) and $\xi >0$ (Fréchet).

## Appendix B. Physical Scaling Approach to Forecasting Oil Production in Hydrofractured Shales

#### Appendix B.1. Physical Scaling without Exterior Flow

**Figure A2.**(

**a**) Schematic of bi-linear flow towards hydraulic fractures in a shale well inside the SRV with volume H × 2d × 2L (interior problem). The early radial flow is neglected because the hydrofracture permeability is much higher than that of the matrix. Reproduced from Patzek et al. [25] (

**b**) Illustration of physical scaling approach of the interior oil production problem. Reproduced from Saputra et al. [27].

#### Appendix B.2. Physical Scaling with Exterior Flow

**Figure A3.**(

**a**) Illustration of interior flow towards the hydraulic fractures of a shale well inside SRV and exterior flow from the reservoir beyond SRV. Adapted from Patzek et al. [25] and Eftekhari et al. [26] (

**b**) Illustration of physical scaling approach of the interior and exterior oil production problem. Adapted from Saputra et al. [27].

## Appendix C. Reservoir Properties of the Bakken Shale

**Table A1.**Reservoir properties used in scaling oil production in the Middle Bakken and Three Forks (part-1).

Parameter | Middle Bakken | Three Forks | Data Source | ||
---|---|---|---|---|---|

SI Units | Field Units | SI Units | Field Units | ||

Initial pressure, ${p}_{i}$ | 36.8 Mpa | 5340 psia | 37.1 Mpa | 5380 psia | [22] |

Fracture pressure, ${p}_{f}$ | 3.4 Mpa | 500 psia | 3.4 Mpa | 500 psia | [22] |

Connate water saturation, ${S}_{wc}$ | 0.57 | 0.57 | 0.65 | 0.65 | [22] |

Initial oil saturation, ${S}_{oi}$ | 0.43 | 0.43 | 0.35 | 0.35 | [22] |

Rock porosity, $\varphi $ | 0.046 | 0.046 | 0.058 | 0.058 | [22] |

Rock permeability, k | 4.4 × ${10}^{-17}$ m${}^{2}$ | 0.045 md | 4.6 × ${10}^{-17}$ m${}^{2}$ | 0.047 md | [22] |

Rock compressibility, ${c}_{\varphi}$ | 4.3 × ${10}^{-10}$ Pa${}^{-1}$ | 3.0 × ${10}^{-6}$ psi${}^{-1}$ | 4.3 × ${10}^{-10}$ Pa${}^{-1}$ | 3.0 × ${10}^{-6}$ psi${}^{-1}$ | [45] |

Water compressibility, ${c}_{w}$ | 4.3 × ${10}^{-10}$ Pa${}^{-1}$ | 3.0 × ${10}^{-6}$ psi${}^{-1}$ | 4.3 × ${10}^{-10}$ Pa${}^{-1}$ | 3.0 × ${10}^{-6}$ psi${}^{-1}$ | [45] |

Oil compressibility, ${c}_{o}$ | 1.4 × ${10}^{-9}$ Pa${}^{-1}$ | 1.0 × ${10}^{-5}$ psi${}^{-1}$ | 1.4 × ${10}^{-9}$ Pa${}^{-1}$ | 1.0 × ${10}^{-5}$ psi${}^{-1}$ | [45] |

Total compressibility, ${c}_{t}$ | 6.3 × ${10}^{-10}$ Pa${}^{-1}$ | 9.0 × ${10}^{-6}$ psi${}^{-1}$ | 5.9 × ${10}^{-10}$ Pa${}^{-1}$ | 8.5 × ${10}^{-6}$ psi${}^{-1}$ | |

Oil viscosity, ${\mu}_{o,i}$ | 3.9 × ${10}^{-5}$ Pa s | 0.392 cp | 2.8 × ${10}^{-5}$ Pa s | 0.276 cp | [46] |

API gravity | 42.2${}^{\circ}$API | 42.2${}^{\circ}$API | 38.7${}^{\circ}$API | 38.7${}^{\circ}$API | [46] |

$({c}_{t}/{S}_{oi})({p}_{i}-{p}_{f})$ | 0.1014 | 0.1014 | 0.1178 | 0.1178 |

**Table A2.**Reservoir properties used in scaling oil production in the Middle Bakken and Three Forks (part-2).

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**Figure 2.**Maps of all 14,678 active horizontal wells completed in the Middle Bakken formation (

**a**) and Three Forks formation (

**b**) colored by maximum daily oil rate. The red lines define the core areas for each reservoir and delineate best producing wells with more than 750 barrels of oil per day. The blue lines define the effective areas for drilling neglecting a few poor producing wells outside. We define the noncore areas as the difference between the effective and core areas. In the Middle Bakken, there are currently 5732 and 4128 wells located in the core and noncore areas, respectively. The other 2672 and 2146 wells are located in the core and noncore areas of Three Forks, respectively.

**Figure 3.**The procedure of arriving at the Generalized Extreme Value (GEV) well prototypes in a given shale play. (

**a**) Define static regions in which oil production is statistically uniform. (

**b**) For each region, gather the dynamic cohorts of wells with ≥i years on production, i = 1, 2, 3, .... (

**c**) Fit GEV probability density function (PDF) to each well cohort. (

**d**) From the corresponding cumulative distribution function (CDF) pick the P

_{10}, P

_{50}, and P

_{90}values for each cohort. (

**e**) Construct time-lapse P

_{50}well prototype for each region, by connecting all P

_{50}values of all cohorts. (

**f**) Time-shift and superpose the GEV well prototypes to match past production. Reproduced from our previous work [21].

**Figure 4.**Distribution of oil production rates for 2540 horizontal wells with one year on production, completed between the year 2000 and 2012 in the Middle Bakken noncore area. (

**a**) GEV PDF: ξ = −0.0274, μ = 43.9421, σ = 24.0635. (

**b**) Maximum Likelihood Estimate, 95% confidence interval (CI) for μ and σ. (

**c**) GEV CDF with the 95% CI on the residual for the P

_{10}well.

**Figure 5.**Distribution of oil production rates for 197 horizontal wells with seven years on production, completed between the year 2000 and 2012 in the Three Forks core area. (

**a**) GEV PDF: ξ = 0.3368, μ = 14.2825, σ = 8.1499. (

**b**) Maximum Likelihood Estimate, 95% confidence interval (CI) for μ and σ. (

**c**) GEV CDF with the 95% CI on the residual for the P

_{10}well.

**Figure 6.**Average wells in 12 regions in the Middle Bakken and Three Forks. These wells are located in both the core and noncore areas and have three different completion periods, (2000–2012), (2013–2016) and (2017–2019). Each year in the past, every average well traces the expected values of the Generalized Extreme Value (GEV) distributions of all active horizontal wells in each well cohort, which have at least 1, 2, ..., 15 years on production. The dashed lines labeled ${P}_{10}$ and ${P}_{90}$ denote wells whose cumulative production is exceeded by 10% and 90% of wells in each region. The red and green lines are the physics-based scaling curves that match each average well, respectively, with and without exterior flow during late time production. In general, ultimate oil recovery from the core areas is higher than that from the noncore ones. The Middle Bakken wells are slightly more productive than the Three Forks wells. The newly completed wells have much higher initial oil production but they decline faster, resulting in more or less the same ultimate recovery as older wells.

**Figure 7.**The actual and forecasted total field rate (

**a**) and cumulative oil (

**b**) in the Bakken shale. Total field cumulative production curves were obtained by stacking calendar-shifted average wells. Total production rates were obtained by differencing cumulative production. The red and green curves are the physical scaling forecasts with and without exterior flow, while the black line shows the historical production from the existing 14,678 wells. The red physical scaling gives the lower bound of EUR estimate with about 4.5 billion bbl by 2050. Assuming reasonable exterior flow, EUR prediction becomes slightly more than 5 billion bbl.

**Figure 8.**A procedure to calculate infill well potential for each part of the Bakken play. This figure shows only the Three Forks core area. For other areas and reservoirs, please see the Supporting Online Materials. The procedure is as follows: (

**a**) Create a one-square-mile fishnet inside the boundary of each area. In this case, there are 1914 grid squares that translate into the total area of 1914 mi

^{2}. (

**b**) Search all existing wells located inside the boundary. The black dots show the surface locations of 2672 existing wells in Three Forks core area. (

**c**) Calculate wellhead density by counting the number of wells on each of the one-square-mile squares. This map is not the real well density map, because it only shows the wellhead density. (

**d**) Calculate an approximate well density map from wellhead density map. The algorithm is as follows: (1) For each well, record its lateral length and calculate the number of squares, n, intercepted by the lateral. For example, a 5000 ft lateral will occupy one square and a 10,000 ft lateral will occupy two squares because one mile is 5280 ft. (2) Search for the least occupied n squares in all possible directions (i.e., north, northeast, east, southeast, south, southwest, west and northwest). (3) Increase the value of well density by 1 well/mi

^{2}for every least dense square found. (4) Repeat the process until all wells in the area of interest are exhausted. (

**e**) Calculate an infill potential map by subtracting the calculated well density from the maximum number of wells, N

_{max}(e.g., N

_{max}= 4 wells/mi

^{2}to avoid frac hits). The summation of all values in the map is the infill potential for the one-square-mile grid. In this case, we obtain 3650 wells. As most wells have 10,000 ft laterals and occupy two one-square-mile squares, we divide 3650 by 2 to obtain 1825 infill potential wells in the Three Forks core area.

**Figure 9.**(

**a**) The number of drilled wells, completed wells and rig count per month for the Bakken play from the U.S. Energy Information Administration (EIA). The drilled and completed wells are almost the same for each month and are strongly correlated with the number of rigs available. These data reveal an increase in drilling efficiency. In 2015, one rig could drill about 1.2 wells per month, while in 2019 so far, one rig could drill two wells per month. The current drilling rate is constant at about 120 wells/month. (

**b**) A future drilling scenario for the Bakken region up to 2041. The plan is to continue at the current drilling rate of 120 wells/month for both core and noncore areas. The numbers of wells to be infilled in each region were previously calculated using the procedure detailed in Figure 8. The results show that the core areas will be fully drilled by mid 2022, leaving the less productive noncore areas for infilling until 2041.

**Figure 10.**The forecasted total field rate (

**a**) and cumulative oil (

**b**) for Bakken based on the drilling scenario in Figure 9b. Since we plan to infill the core areas first, the field oil rate will reach the all-time production peak of about 1.6 million bbl/d by mid 2022, leaving no ‘sweet spots’ in the core areas. The continuous infill of the less productive wells in the noncore areas will decline the production to a plateau of 1 million bbl/d. After 2041, no more drilling locations will be left in the Bakken region and the oil rate will steeply decline to half a million bbl/d by 2042 and 0.2 million bbl/d by 2050. Ultimately, the 14,898 existing wells will give an EUR of 5 billion bbl. By adding 4402 wells in the core areas, EUR will increase to 7 billion bbl. By adding another 26,512 wells in the noncore areas, EUR will increase to 13 billion bbl.

Well | Reservoir | Area | Completion | Sample |
---|---|---|---|---|

Sample | Date | Size | ||

1 | Middle Bakken | Noncore | 2000–2012 | 2550 |

2 | Middle Bakken | Noncore | 2013–2016 | 1355 |

3 | Middle Bakken | Noncore | 2017–2019 | 223 |

4 | Three Forks | Noncore | 2000–2012 | 735 |

5 | Three Forks | Noncore | 2013–2016 | 1204 |

6 | Three Forks | Noncore | 2017–2019 | 207 |

7 | Middle Bakken | Core | 2000–2012 | 2086 |

8 | Middle Bakken | Core | 2013–2016 | 2534 |

9 | Middle Bakken | Core | 2017–2019 | 1112 |

10 | Three Forks | Core | 2000–2012 | 428 |

11 | Three Forks | Core | 2013–2016 | 1502 |

12 | Three Forks | Core | 2017–2019 | 742 |

TOTAL | 14,678 |

Well Sample | Physical Scaling | Physical Scaling with Exterior Flow | ||||||
---|---|---|---|---|---|---|---|---|

Shift Years | $\mathit{\tau}$ Years | $\mathcal{M}$ kbbl/Well | EUR kbbl/Well | $\mathit{\tau}$ Years | $\mathcal{M}$ | a | EUR kbbl/Well | |

Middle Bakken noncore [2000–2012] | 0.25 | 24.5 | 454.3 | 271.1 | 19.9 | 286.7 | 0.050 | 282.2 |

Middle Bakken noncore [2013–2016] | 0.00 | 11.8 | 341.3 | 211.7 | 10.9 | 267.2 | 0.026 | 236.4 |

Middle Bakken noncore [2017–2019] | 0.00 | 5.0 | 346.9 | 215.5 | 5.0 | 306.6 | 0.015 | 259.5 |

Three Forks noncore [2000–2012] | 0.15 | 13.3 | 288.4 | 207.5 | 10.7 | 181.2 | 0.058 | 238.3 |

Three Forks noncore [2013–2016] | 0.00 | 10.8 | 271.9 | 196.1 | 9.9 | 212.4 | 0.030 | 221.1 |

Three Forks noncore [2017–2019] | 0.00 | 5.0 | 270.8 | 195.4 | 5.0 | 238.4 | 0.018 | 236.5 |

Middle Bakken core [2000–2012] | 0.20 | 25.0 | 810.8 | 482.5 | 22.6 | 536.3 | 0.050 | 512.2 |

Middle Bakken core [2013–2016] | 0.00 | 9.1 | 543.8 | 337.7 | 8.2 | 420.7 | 0.026 | 389.7 |

Middle Bakken core [2017–2019] | 0.00 | 5.0 | 581.9 | 361.5 | 5.0 | 513.6 | 0.015 | 434.6 |

Three Forks core [2000–2012] | 0.27 | 25.0 | 681.1 | 470.9 | 25.0 | 473.4 | 0.058 | 511.6 |

Three Forks core [2013–2016] | 0.00 | 10.3 | 450.1 | 324.6 | 9.6 | 353.1 | 0.030 | 369.7 |

Three Forks core [2017–2019] | 0.00 | 5.0 | 462.8 | 334.0 | 5.0 | 406.9 | 0.018 | 403.5 |

Reservoir | Region | Total Area | Existing | Infill |
---|---|---|---|---|

Type | sq Miles | Wells | Potential | |

Middle Bakken | Noncore | 8534.5 | 4128 | 14,171 * |

Three Forks | Noncore | 6905.9 | 2146 | 12,341 * |

Middle Bakken | Core | 3382.5 | 5732 | 2577 |

Three Forks | Core | 1914.1 | 2672 | 1825 |

TOTAL | 20,737 | 14678 | 30,914 |

© 2019 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**

Saputra, W.; Kirati, W.; Patzek, T.
Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production in the Bakken Shale. *Energies* **2019**, *12*, 3641.
https://doi.org/10.3390/en12193641

**AMA Style**

Saputra W, Kirati W, Patzek T.
Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production in the Bakken Shale. *Energies*. 2019; 12(19):3641.
https://doi.org/10.3390/en12193641

**Chicago/Turabian Style**

Saputra, Wardana, Wissem Kirati, and Tadeusz Patzek.
2019. "Generalized Extreme Value Statistics, Physical Scaling and Forecasts of Oil Production in the Bakken Shale" *Energies* 12, no. 19: 3641.
https://doi.org/10.3390/en12193641