^{1}

^{*}

^{1}

In this paper, the methods developed by [1] are used to analyze flowback data, which involves modeling flow both before and after the breakthrough of formation fluids. Despite the versatility of these techniques, achieving an optimal combination of parameters is often difficult with a single deterministic analysis. Because of the uncertainty in key model parameters, this problem is an ideal candidate for uncertainty quantification and advanced assisted history-matching techniques, including Monte Carlo (MC) simulation and genetic algorithms (GAs) amongst others. MC simulation, for example, can be used for both the purpose of assisted history-matching and uncertainty quantification of key fracture parameters. In this work, several techniques are tested including both single-objective (SO) and multi-objective (MO) algorithms for history-matching and uncertainty quantification, using a light tight oil (LTO) field case. The results of this analysis suggest that many different algorithms can be used to achieve similar optimization results, making these viable methods for developing an optimal set of key uncertain fracture parameters. An indication of uncertainty can also be achieved, which assists in understanding the range of parameters which can be used to successfully match the flowback data.

In recent years, as a result of low gas prices and relatively high oil prices, many producers have turned their attention to LTO reservoirs as a means of producing commercial wells. Much like shale gas reservoirs, LTO reservoirs are typically very low in permeability and require extensive hydraulic fracturing to allow for commercial production. As a result, operators are seeking new methods to estimate hydraulic fracture properties, particularly early in the well life. Over the past 5 years there have been numerous authors have identified quantitative flowback analysis as a suitable method which in most cases aligns well with more conventional long-term production data analysis (i.e.[

Although the majority of the literature has focused on shale gas reservoirs, there has been a substantial amount of research conducted in analyzing flowback from LTO wells. These methods have been applied to LTO plays across North America. A comprehensive literature review was given by [

[

Reference [

The base tool used in this work is a modified version of what was developed by [

The analysis procedure used in this work is shown in

Algorithms Used. In this work, six different algorithms were tested for the purpose of uncertainty quantification and assisted history-matching. The methods applied in this paper include: 1) MC MO simulation (Palisade^{®} @RISK^{TM}); 2) Microsoft^{®} Excel’s SO Gradient-based (GRG2) algorithm (GRG Nonlinear Solver); 3) Microsoft^{® }Excel’s SO Evolutionary Solver; 4) Palisade^{®} Evolver’s SO GA; 5) GAPS MO GA (based on the NSGA-II-non-dominated sorting genetic algorithm) algorithm; and 6) Palisade^{®} Evolver’s SO OptQuest^{TM} algorithm. Each

algorithm will be briefly discussed here, with more details available in the literature.

There are two main characteristics of all of these algorithms: 1) the OF; and 2) constraints. The OF is the key parameter that one is attempting to minimize or maximize (minimization in this case). Constraints are relationships which must be satisfied for a solution to be considered acceptable. The OFs used in this work are sum of squares OFs comparing measured water and oil rate with modeled rate. Cumulative production OFs can also be introduced to further constrain the problem. The OFs used in this work will be discussed in the coming sections. Since there are no hard constraints which are applicable to this problem, the only constraints used will be the input ranges of uncertain parameters, which will also be discussed in the coming sections.

Monte Carlo Simulation. Traditional deterministic analysis techniques combine single-point estimates of key input variables to provide a single-point estimate of the result. This type of analysis assumes that the true values of all inputs are known in order to derive an accurate solution. Often these single-point estimates may differ greatly from the actual result and can lead to negative outcomes such as financial loss. In the majority of real-life problems, certainty in all parameters is rarely the case; while some variables may be known precisely or can be estimated with a reasonable degree of accuracy (ex. from lab testing or other methods), others may contain a high degree of uncertainty [^{TM} add-in for Microsoft^{®} Excel^{TM}. As mentioned previously. MC simulation is conducted in such a way that multiple objectives are considered.

Microsoft^{®} Excel’s GRG Non-Linear Solver (GRG2 Gradient-Based Algorithm). This technique is based on the Generalized Reduced Gradient 2 (GRG2) algorithm which is an extension of a version of the GRG code developed by [

Microsoft^{®} Excel’s Evolutionary Solver. The version of the Evolutionary Solver available in the standard version of Excel^{TM} is a SO algorithm developed by [

As with other GAs there are four main steps applied within the algorithm: 1) Selection; 2) Crossover/Mating; 3) Mutation; and 4) Replacement. Reference [

Palisade^{®} Evolver’s Genetic Algorithm. Palisade^{®} Evolver’s GA is a SO GA which contains 5 potential solving methods: 1) Recipe; 2) Order; 3) Grouping; 4) Budget; and 5) Project. The Recipe method is the default method and is designed to be used when parameter values can be varied independently and can be applied to the majority of optimization problems, especially when the relationship between the adjustable variables are not well understood, or cannot be handled better by one of the other techniques. In this work, the Recipe solving technique is used. The GA used in Evolver^{TM} is unique, much like that used in Microsoft^{®} Excel’s Evolutionary Solver, in that it uses a steady-state approach, meaning that only one organism is replaced at a time rather that the entire generation. According to [

GAPS Multi-Objective Genetic Algorithm (Based on NSGA-II Algorithm). GAPS is a MO GA based on the NSGA-II algorithm developed by [

・ Utilizes a faster method for nondominated sorting.

・ Preserves elitism, meaning that the best solutions are maintained without modification.

・ Incorporates a parameter-less diversity preservation mechanism to replace the need for a sharing parameter, which is the traditional mechanism for maintaining diversity.

・ Utilizes parallelization to improve solution speed by allowing calculations to be spread out over multiple processors.

There are two key concepts to the algorithm, being: 1) nondominated sorting; and 2) diversity preservation and follows the same general concept of the other GAs discussed above, although is generational in nature. This algorithm has been shown to work well for three OFs by [

Palisade^{®} Evolver’s OptQuest Algorithm. OptQuest^{TM} is a “black box” optimizer first developed by [

A full analysis demonstrating each step of the analysis procedure (shown in

・ Cased-hole completion.

・ Hydraulically fractured with hybrid water fracs in 18 stages using plug and perf technology (single perforation cluster per stage).

・ Fracture stages spaced at ~330 ft.

・ 1350 STB of fracture fluid and 45 T of proppant pumped per stage.

Assessing the microseismic collected on this well, the assumption of circular bi-wing planar fractures appears to be reasonable and will be used in this analysis. Preceding the flowback data used for this analysis, plugs were drilled out with coil tubing following stimulation, after which the well was placed on flowback monitoring through a test separator. Rate and pressure data was gathered every 15 minutes for approximately 300 hours during flowback following a 12 day shut-in period.

Input common to the different flowback analysis techniques are shown below in

Raw Data and Diagnostic Plots. Water, oil and gas rates as well as bottom-hole flowing pressure and gas-oil ratio (GOR) are shown below in

From

Fracture Properties | Parameter Value |
---|---|

Initial Fracture Pressure (psia) | 5000 |

Initial Water Saturation (%) | 100 |

Fracture Porosity (proppant pack) (%) | 31 |

Fracture Compressibility (psi^{−}^{1}) | 1 × 10^{−4} |

Number of Hydraulic Fractures | 18 |

Individual Hydraulic Fracture Width (ft) | 0.0417 |

Total Hydraulic Fracture Width (ft) | 0.75 |

Reservoir Properties | Parameter Value |

Formation Pressure (psia) | 3700 |

Net Pay (ft) | 197 |

Matrix Porosity (%) | 4 |

Initial Mobile Oil Saturation (%) | 99 |

Initial Mobile Water Saturation (%) | 1 |

Formation Compressibility (psi^{−1}) | 4 × 10^{−6 } |

Matrix Permeability (md) | 0.0003 |

Reservoir Temperature (˚F) | 140 |

Fluid Properties | Parameter Value |

Fracture Water Salinity (%) | 50,000 |

Formation Water Salinity (%) | 200,000 |

Oil Gravity (˚API) | 52 |

Gas-Oil-Ratio (scf/stb) | 1250 |

Bubble-Point Pressure (psia) | 2858 |

Gas Gravity (air = 1) | 0.747 |

gas into the fractures. Therefore, only the first 8 days of production were considered for this analysis as this is the period where production is under two-phase (water + oil) flow in the formation and fractures (the tool cannot currently model three-phase flow). Over the first 8 days of production, water rate and bottom-hole flowing pressure generally decline, while the hydrocarbon rate generally increases following breakthrough as would be expected from a formation with minimal mobile water and constantly decreasing bottom-hole flowing (and fracture) pressure. From

Casing pressures were converted to sandface pressures using a wellbore model, and initial formation pressure was estimated from p* obtained from a Diagnostic Fracture Injection (DFIT) test which also yielded the estimate of matrix permeability. The GOR and bubble-point pressure are defined based on PVT analysis of the reservoir fluid from a group of off-setting wells. Initial fracture pressure was determined by a trial and error process conducted by [

Rate-Transient Analysis of BBT Single-Phase Data. To assist with the history-matching process, rate-transient analysis (RTA) is applied to the flow-regimes identified in

higher than expected even for a well with minimal natural fracturing (as inferred from microseismic and experience in the formation of interest), and may result from the impact of the other two wells being stimulated on the same pad prior to flowback of the well. These values will be used in the deterministic history-matching process. Finally, the fracture parameters estimated from radial flow analysis and the FMB can be confirmed by using the Fetkovich type-curve (_{Dd}, fracture depletion data falls down the harmonic stem, with a positive deviation indicating the breakthrough of formation fluid.

Parameters estimated from quantitative RTA of this flowback data are provided in

Deterministic History-Match. Deterministic history-matching was first conducted to validate the application of the conceptual model to this dataset, confirm selection of a fracture shape and geometry model, and confirm RTA-derived parameters for BBT fracture properties. For this analysis, a circular shape with a single bi-wing fracture being generated from each stage was selected for simplicity, as was done by [

Radial Flow Plot | Parameter Value |
---|---|

Fracture Conductivity, F_{cT} (md-ft) | 2625 |

Fracture Permeability (md) | 3500 |

Flowing Material Balance | Parameter Value |

Fracture Fluid-In-Place (STB) | 24,000 |

BBT Half-Length, x_{f_BBT} (ft) | 411 |

Fracture Conductivity, F_{cT} (md-ft) | 2550 |

Fracture Permeability (md) | 3400 |

Fetkovich Type-Curve | Parameter Value |

x_{f}/r_{wa} | ~1000 |

Fracture Conductivity, F_{cT} (md-ft) | 2625 |

Fracture Permeability (md) | 3500 |

The history-matches, guided by the BBT RTA-derived parameters are provided in

From

The key history-match parameters are given in

From

Stochastic Simulation and Assisted History-Matching. In this section, the results from multiple stochastic and assisted history-matching techniques will be

History-Match Parameter | Parameter Value |
---|---|

Fracture Permeability (md) | 3500 |

BBT Drainage Area (Ac) | 14 |

BBT Half-Length, x_{f_BBT} (ft) | 441 |

ABT Drainage Area (Ac) | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 425 |

Breakthrough Pressure (psi) | 3825 |

Corey Oil Exponent―Fracture, n’ | 1.2 |

Corey Water Exponent―Fracture, m’ | 5.0 |

discussed. As mentioned previously, the algorithms used include: 1) MC simulation (Palisade^{®} @RISK^{TM}); 2) Microsoft^{®} Excel’s Gradient-based (GRG2) algorithm (GRG Nonlinear Solver); 3) Microsoft^{® }Excel’s Evolutionary Solver; 4) Palisade^{®} Evolver’s GA; 5) GAPS MO GA (based on the NSGA-II) algorithm; and 6) Palisade^{®} Evolver’s OptQuest^{TM} algorithm. The results of each individual technique will be discussed followed by a comparison of the results of each of the techniques. As discussed previously, only the first 8 days of flow data were analyzed because, during this flow period, the flowing pressure remains above the bubble point, and therefore only two-phase flow exists in the matrix and fractures. During this period, the GOR is also relatively constant, as would be expected for flow above the bubble point.

Monte Carlo Simulation. As discussed previously, stochastic history-matching can be a multi-step process, with multiple refinement stages. For example, two sets of MC simulations were conducted in the presented example. Following the first stage, inputs including fracture compressibility and matrix properties were held constant for the final set of simulations, which will be discussed here. Further refinement stages could be conducted using information from past runs to adjust input distributions and increase the number of success cases. The parameter distributions for the first refinement stage are shown below in

Uncertain Parameter | Distribution Type | Low Value | High Value |
---|---|---|---|

Fracture Permeability (md) | Uniform | 3000 | 4000 |

BBT Drainage Area (Ac) | Uniform | 10 | 14* |

BBT Half-Length, x_{f_BBT} (ft) | Uniform | 372 | 441* |

Breakthrough Pressure (psi) | Uniform | 3700 | 4100 |

Corey Oil Exponent―Fracture, n’ | Uniform | 1 | 3 |

Corey Water Exponent―Fracture, m’ | Uniform | 1 | 10 |

*Upper bound on fluid in place given assumptions of fracture shape, width and porosity.

Because enough data was not available to construct proper input distributions, uniform distributions were used for each parameter between a reasonable low and high value. In some cases, the high and low value were constrained by physical limits (i.e. the upper limit of half-length, the lower limit of n’ and m’ and breakthrough pressure) whereas other limits were set at a reasonable range and then adjusted following the screening stage of iterations. The input parameter ranges were also further constrained by the initial screening phase of simulations (500,000 iterations with significantly wider parameter ranges), as well as reasonable limits on the uncertain parameters. The initial screening phase was used to rule out outlier matches which occurred with minimal frequency. The same limits were used for the application of the assisted-history matching techniques, which will be discussed in the following section. This analysis is comparable to that conducted by [

The following objective functions (OFs) were used in either the MC simulations, assisted history-matching algorithms or both. The OFs take the form of sum of squared residuals for the rate and cumulative production of the water and oil phases. Because the well is flowed above the bubble point throughout the analysis period of the flowback, and the GOR is relatively constant at approximately the solution gas level, the gas phase is not considered and is effectively lumped in with the oil phase.

Water Rate OF:

O F q w = ∑ i = 1 n ( q w , d a t a − q w , s i m ) 2 (1)

where, n is the number of data points collected during the portion off the flowback data being analyzed for each phase.

Oil Rate OF:

O F q o = ∑ i = 1 n ( q o , d a t a − q o , s i m ) 2 (2)

Cumulative Water OF:

O F Q w = ∑ i = 1 n ( Q w , d a t a − Q w , s i m ) 2 (3)

Cumulative Oil OF:

O F Q o = ∑ i = 1 n ( Q o , d a t a − Q o , s i m ) 2 (4)

Summed Rate OF:

O F q t = ∑ i = 1 n [ w w ( q w , d a t a − q w , s i m ) 2 + w o ( q o , d a t a − q o , s i m ) 2 ] (5)

where,

w w + w o = 1 (6)

The summed rate OF given by Equation 5 is used for the SO algorithms. There are however two issues associated with using summed OFs: 1) objective conflict leading to erroneous results; and; 2) weighting can have a significant impact on the results of the algorithm. In this work, a 1:1 weighting was used for direct comparison to the MO algorithms (equivalent to using w_{w} = w_{o} = 0.5).

Alternate criteria, such as those applied by [^{2} value of each phase (in terms of rate) greater than 0.9 (as suggested by [^{2} value existed but the match was poor, due to the inherent nature of R^{2} as a match fit indicator, especially when dealing with highly nonlinear problems with a large number of data points. For the current study, 100,000 iterations are conducted, and fairly strict criteria were enforced to obtain a successful match, including the following (solution has to be better than the deterministic solution for both phases):

・ O F q w < O F q w , d e t e r m i n i s t i c

・ O F q o < O F q o , d e t e r m i n i s t i c

・ O F Q w < O F q w , d e t e r m i n i s t i c

・ O F Q o < O F q o , d e t e r m i n i s t i c

Using all four of these criteria, only 79 matches were found (~0.1%), while if only the two rate criteria were used, as is usually done with the assisted history-matching techniques, ~10× the number of matches were found (~1%). Based on these results, it is clear that the random behaviour of MC simulation is not particularly efficient in finding optimal solutions, making the use of modern assisted history-matching techniques desirable, particularly when a deterministic solution is not available. For the remainder of this paper, only the two rate OFs will be used in the application of all algorithms, because adding more OFs can often cause these algorithms to converge slowly and creates further objective conflict, potentially leading to finding a less desirable solution. The parameters for the deterministic match, as well as the best 5 matches (in order of increasing OF value) when considering the summation of the two OFs, are shown below in

The history-match associated with the top 10 MC simulation history-matches as well as the deterministic history-match are shown in

For later comparison to the assisted history-matching results, the average of the top five iterations were assumed to represent the best solution found using MC simulation, as each of these solutions have a total OF within 1% of each other. The values for each of the uncertain parameters are provided in

Uncertain Parameter | Deterministic | Match 1 | Match 2 | Match 3 | Match 4 | Match 5 | Mean | Std. Dev. | P10/P90 |
---|---|---|---|---|---|---|---|---|---|

Fracture Permeability (md) | 3500 | 3225 | 3184 | 3167 | 3159 | 3216 | 3425 | 254.41 | 1.23 |

BBT Drainage Area (Ac) | 14 | 13.82 | 13.73 | 13.97 | 13.78 | 13.67 | 13.52 | 0.37 | 1.07 |

BBT Half-Length, x_{f_BBT} (ft) | 440 | 438 | 436 | 440 | 437 | 435 | 432 | 6.01 | 1.04 |

ABT Drainage Area (Ac) | 13 | 12.83 | 12.75 | 12.97 | 12.79 | 12.69 | 12.54 | 1.41 | 1.04 |

ABT Half-Length, x_{f_ABT} (ft) | 425 | 422 | 420 | 424 | 421 | 420 | 417 | 5.79 | 1.04 |

Breakthrough Pressure (psi) | 3825 | 4093 | 4096 | 4058 | 4099 | 4089 | 4024 | 56.75 | 1.04 |

Corey Oil Exponent―Fracture, n’ | 1.20 | 1.44 | 1.44 | 1.35 | 1.42 | 1.51 | 1.40 | 0.12 | 1.26 |

Corey Water Exponent―Fracture, m’ | 5.00 | 4.81 | 5.04 | 5.31 | 6.10 | 5.25 | 5.31 | 1.41 | 2.03 |

Total Objective Function (millions) | 41.5 | 35.4 | 35.4 | 35.5 | 35.6 | 35.7 | 39.2 | 0.14 | 1.10 |

Uncertain Parameter | Top 5 Average |
---|---|

Fracture Permeability (md) | 3190 |

BBT Drainage Area (Ac) | 13.78 |

BBT Half-Length, x_{f_BBT} (ft) | 437 |

ABT Drainage Area (Ac) | 12.80 |

ABT Half-Length, x_{f_ABT} (ft) | 421 |

Breakthrough Pressure (psi) | 4087 |

Corey Oil Exponent―Fracture, n’ | 1.43 |

Corey Water Exponent―Fracture, m’ | 5.30 |

Total Objective Function (millions) | 35.4 |

The parameter distributions generated from the stochastic history-matching exercise are provided in ^{2} values are shown to indicate the lognormal nature of the output distributions, and the deterministic and mean values are provided for reference. Note that the parameter distribution for BBT drainage area is not provided because the focus is on the half-length calculated from the drainage area (using the assumed shape and geometry constraints), since half-length is one of the key parameters controlling long-term production of the well.

From

・ Fracture permeability covers the entire input distribution, suggesting that fracture permeability may fall outside the search space. However, very few matches beyond the selected range were found during the screening phase.

・ Breakthrough pressures, including the deterministic match, are greater than

reservoir pressure estimated from DFIT analysis. Values (other than the deterministic match) also fell near the upper limit of 4100 psia, suggesting a better match to the available data could be achieved using a breakthrough pressure of 4100 psia. However, experimentation with a variety of fracture parameters suggested that breakthrough pressures greater than 4100 psia led to model breakthrough significantly BBT in the actual data, which led to setting the upper limit at the selected value. 4100 psia still yields a breakthrough earlier than the data―however, the fact that flow initiates at ~36 STB/D, which is higher than other similarly completed wells on the same pad, suggests that some early-time hydrocarbon data may not have been recorded. An improved late-time match was observed with earlier breakthrough. Overall the results suggest a near fracture supercharge of up to 10% following an 11 day shut-in between stimulation and the onset of flowback in which the bridge plugs were milled out. The supercharge has been shown to be much higher in some formations depending on factors such as the stimulation pumped, shut-in time between stimulation and flowback and other reservoir and fluid properties.

・ BBT half-length values fell in the range of 413 to 441 ft suggesting that a BBT half-length less than 400 ft is unlikely and that a high degree of fracture efficiency was achieved.

・ Fracture relative permeability exponents to oil fall in a tight band between 1.1 and 1.6, suggesting minimal potential variability in this parameter.

・ Fracture relative permeability exponents to water are far less constrained than those to oil falling between 2.1 and 9, although are significantly higher than those to oil. This has been observed in nearly all wells analyzed using these methods. In this case it can be seen that the values between the P10 (3.5) and P90 (7.2) follow a lognormal distribution and yield a P10/P90 ratio of ~2.20% of the solutions fell outside this range and may be considered as outliers.

Assisted History Matching. In addition to the MC simulation approach demonstrated above, five assisted history-matching algorithms were applied in an attempt to find the best possible history-match to the same flowback data set discussed above. Two types of algorithms were used in this analysis (gradient-based and evolutionary) with a total of five techniques being tested: 1) Microsoft^{®} Excel’s SO Gradient-based (GRG2) algorithm (GRG Nonlinear Solver); 2) Microsoft^{® }Excel’s SO Evolutionary Solver; 3) Palisade^{®} Evolver’s SO GA; 4) GAPS MO GA (based on the NSGA-II) algorithm; and 5) Palisade^{®} Evolver’s SO OptQuest^{TM} algorithm. For this analysis, the lower and upper bounds given above in

Uncertain Parameter | Value |
---|---|

Fracture Permeability (md) | 3100 |

BBT Drainage Area (Ac) | 13.96 |

BBT Half-Length, x_{f_BBT} (ft) | 440 |

ABT Drainage Area (Ac) | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 424 |

Breakthrough Pressure (psi) | 3900 |

Corey Oil Exponent―Fracture, n’ | 1 |

Corey Water Exponent―Fracture, m’ | 1 |

first population. As was discussed previously, and as will be demonstrated below, the initial guess is critical to achieving good results from the GRG algorithm because these algorithms will tend to find the closest local minima in the OF (downhill nature of the algorithm). The initial guesses were selected based on the following criteria:

・ Fracture permeability―RTA of early-time flowback data suggested a maximum fracture permeability of ~3500 psia (as was used in the deterministic history match) and therefore a slightly lower value was selected for this application.

・ Breakthrough pressure―DFIT analysis suggested an initial reservoir pressure of ~3700 psia and therefore a 200 psia supercharge effect was assumed (~5%).

・ Drainage area―set based on results of the FMB in the deterministic analysis.

・ Relative permeability exponents―straight-line relative permeability curves were assumed as may be expected for homogeneous perfectly planar fractures under ideal flowing conditions.

The results of each algorithm will be discussed, followed by a comparison of the results of each algorithm, as well as the average results of the top five MC simulations.

Microsoft^{®} Excel’s GRG Nonlinear Solver. As discussed previously, the initial guess is critical to the quality of the result using this type of algorithm due to the “downhill” nature of the algorithm and tendency to get trapped in local minima. This impact will be demonstrated in this section. Due to the deficiencies of this algorithm in solving complex problems with multiple minima, a poor result was expected using the initial guesses shown in ^{®} Excel^{TM}, allowing for fast and simple application following the deterministic history-matching exercise. To test the capacity of the algorithm for solving this problem, two runs were completed. In the first run the initial guesses shown in

Uncertain Parameter | Initial Guess | Final Solution |
---|---|---|

Fracture Permeability (md) | 3100 | 3000 |

BBT Drainage Area (Ac) | 14 | 14 |

BBT Half-Length, x_{f_BBT} (ft) | 441 | 441 |

ABT Drainage Area (Ac) | 13 | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 425 | 425 |

Breakthrough Pressure (psi) | 3900 | 3911 |

Corey Oil Exponent―Fracture, n’ | 1 | 1 |

Corey Water Exponent―Fracture, m’ | 1 | 1 |

Total Objective Function (millions) | 42.8 |

(see ^{TM} unless the algorithm is stopped at each iteration (which was not done in this case). The initial guess and final solution for the two sets of input parameters are provided in

From

Microsoft^{®} Excel’s Evolutionary Solver. In this section the results of the Excel’s Evolutionary Solver will be demonstrated. This Solver algorithm is the first of two SO GAs which will be tested in this work. This solver also uses several classical optimization methods to attempt to improve upon the solutions found by the GA, thus making it a hybrid GA. As discussed previously, details of how

Uncertain Parameter | Initial Guess | Final Solution |
---|---|---|

Fracture Permeability (md) | 3500 | 3102 |

BBT Drainage Area (Ac) | 14 | 14 |

BBT Half-Length, x_{f_BBT} (ft) | 441 | 441 |

ABT Drainage Area (Ac) | 13 | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 425 | 425 |

Breakthrough Pressure (psi) | 3825 | 4099 |

Corey Oil Exponent―Fracture, n’ | 1.2 | 1.45 |

Corey Water Exponent―Fracture, m’ | 5 | 5.61 |

Total Objective Function (millions) | 34.7 |

Excel’s Evolutionary Solver works are not readily available and very little assistance was provided by the developer to help understand exactly which techniques are employed. Given that this is a time-based, rather than a generation-based algorithm, the exact number of iterations conducted is unknown, although the algorithm converged significantly faster than the other algorithms tested, suggesting that significantly fewer 10,000 iterations were conducted in finding the best solution. The input parameters used for this algorithm are shown below in

As with other SO algorithms, a single best solution is found by the algorithm. The parameters resulting from the optimization are found in

Palisade® Evolver’s Genetic Algorithm. In this section the results of Palisade^{®} Evolver’s SO GA will be demonstrated. Evolver^{TM} is the second SO GA used in this work. Much like Excel’s Evolutionary Solver, Evolver^{TM} uses a steady-state approach, which the company has found to work as well or better than the generational approach. Further, given that this is a proprietary commercial tool, details on the exact workings of the algorithm are not readily available. Based on the information provided by the developer, the algorithm operates in a manner

Mutation Rate (%) | 15 |
---|---|

Population Size | 100 |

Max Time Without Improvement (s) | 10,000 |

Convergence Criteria for Max Change (%) | 0.01 |

Uncertain Parameter | Value |
---|---|

Fracture Permeability (md) | 3156 |

BBT Drainage Area (Ac) | 14 |

BBT Half-Length, x_{f_BBT} (ft) | 441 |

ABT Drainage Area (Ac) | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 225 |

Breakthrough Pressure (psi) | 4098 |

Corey Oil Exponent―Fracture, n’ | 1.46 |

Corey Water Exponent―Fracture, m’ | 5.66 |

Total Objective Function (millions) | 34.7 |

comparable to a basic GA, although several specialty operators are included to improve the results of the algorithm. The algorithm is trial-based rather than generational-based, and therefore to mimic the generational approach used by the GAPS MO GA, 10,000 trials were conducted (equivalent to 100 generations with a population of 100). A convergence criteria for maximum change in the OF is also used as an input for termination of the algorithm, although this was not achieved. The input parameters used for this algorithm are shown below in

As with other SO algorithms, a single best solution is found by the algorithm. The parameters resulting from the optimization are given in

The best solution was found in the 7455^{th} trial, although only 245 trials were required to get within less than 1% of the best solution, suggesting that significantly fewer trials could have been run for this particular scenario. Fewer trials, however, would limit the search extent of the algorithm, which may lead to poor results in some cases, as the majority of the early trials produce significantly higher OF numbers.

Crossover Rate (%) | 50% |
---|---|

Mutation Rate (%) | 15 |

Population Size | 100 |

Number of Iterations | 10,000 |

Convergence Criteria for Max Change (%) | 0.01 |

Uncertain Parameter | Value |
---|---|

Fracture Permeability (md) | 3102 |

BBT Drainage Area (Ac) | 14 |

BBT Half-Length, x_{f_BBT} (ft) | 425 |

ABT Drainage Area (Ac) | 13 |

ABT Half-Length, x_{f_ABT} (ft) | 4099 |

Breakthrough Pressure (psi) | 14 |

Corey Oil Exponent―Fracture, n’ | 1.45 |

Corey Water Exponent―Fracture, m’ | 5.61 |

Total Objective Function (millions) | 34.7 |

it can again be seen that values approaching the minimum are found quite quickly and continue to be found throughout the remainder of the optimization.

GAPS Multi-Objective Genetic Algorithm. In this section, the results of the only MO GA tested will be demonstrated. This is the GAPS algorithm developed by Mohammed Kanfar for the Tight Oil Consortium at the University of Calgary, and is based on the NSGA-II algorithm as discussed previously. The benefits of using MO algorithms were discussed previously, so in this section, the focus will solely be on the results of the algorithm. Although it is common practice in the application of GAs to run half the number of generations as the population size, in this application an equal number of generations and populations were conducted to allow the algorithm to “dig deeper” towards an absolute minimum. Note that larger population sizes allow the algorithm to explore further in the search space. The impact of running more generations will be discussed below. The algorithm was run with 100 generations with populations of 100 following the recommendations of [

As is the case with all MO Gas, the final generation does not converge to a single solution, but instead converges to a Pareto Front of nondominant mathematically-equivalent solutions. In this case, the Pareto Front is convex in nature, which suggests that two phase rate objectives are conflicting (a straight-line would suggest non-conflicting OFs). To converge on a single best solution, the solutions were filtered, removing solutions that have an OF higher than a certain threshold (with the threshold being continuously reduced until only several solutions remained around the corner point of the Pareto Front), and then visual inspection was used to pick the final solution. There are currently no methods available in the literature for selecting the single best solution, and therefore an approach similar to that used by [

Mutation Rate (%) | 15 |
---|---|

Population Size | 100 |

Number of Generations | 100 |

Number of Iterations | 10,000 |

be used in

Next, generation 100 will be investigated in greater detail, focusing primarily on the extent of variability in the key parameter estimates during this final generation. The parameters corresponding to the best match and the average, standard deviation and P10/P90 ratio for Generation 100 are given in

Palisade Evolver’s OptQuest^{TM} Algorithm. In this section the results of Palisade Evolver’s OptQuest^{TM} will be demonstrated. Much like Evolver’s GA, this is a SO algorithm. This algorithm has its basis in Scatter Search which draws many similarities to GAs, although also includes integer programming, Tabu Search and an Artificial Neural Network to improve its results and efficiency, as discussed previously. The algorithm is trial-based much like Evolver’s GA. In this case, since there is no basis for comparison of the algorithm, the maximum number of trials was set to a very large value (100,000) allowing the convergence criteria for maximum change in the OF to control the termination of the optimization. The input parameters used for this algorithm are shown below in

As with other SO algorithms a single best solution is found by the algorithm. The parameters resulting from the optimization are found in

In this particular case, 33,756 trials were required to reach the set criteria, although a value with a combined OF within 1% of the optimal value was found in 13,421 trials which equates to a ~60% reduction in optimization time, although many significantly higher OF values were found in the final 20,000 trials. To

Uncertain Parameter | Optimal Match | Average | Std. Dev. | P10/P90 |
---|---|---|---|---|

Fracture Permeability (md) | 3136 | 3,584 | 290 | 1.24 |

BBT Drainage Area (Ac) | 13.98 | 13.99 | 0.21 | 1.00 |

BBT Half-Length, x_{f_BBT} (ft) | 440 | 440 | 0.20 | 1.00 |

ABT Drainage Area (Ac) | 12.99 | 12.99 | 0.01 | 1.00 |

ABT Half-Length, x_{f_ABT} (ft) | 424 | 424 | 0.20 | 1.00 |

Breakthrough Pressure (psi) | 4099 | 4093 | 0.01 | 1.00 |

Corey Oil Exponent―Fracture, n’ | 1.45 | 1.50 | 0.03 | 1.05 |

Corey Water Exponent―Fracture, m’ | 5.87 | 7.89 | 1.57 | 1.57 |

Maximum Number of Iterations | 100,000 |
---|---|

Convergence Criteria for Max Change (%) | 0.01 |

Uncertain Parameter | Value |
---|---|

Fracture Permeability (md) | 3148 |

BBT Drainage Area (Ac) | 14 |

x_{f_BBT} (ft) | 441 |

ABT Drainage Area (Ac) | 13 |

x_{f_ABT} (ft) | 425 |

Breakthrough Pressure (psi) | 4099 |

Corey Oil Exponent―Fracture, n’ | 1.46 |

Corey Water Exponent―Fracture, m’ | 5.65 |

Fracture Permeability (md) | 34.7 |

allow comparison with the GAs, the results were filtered into “equivalent generations” of 100 trials. ^{th} “equivalent generation only contains 56 trials”). From the average curve, the differences between OptQuest’s performance and a generational GA become apparent. Unlike a generational GA, where one would expect to see the generational average go down over time, in this case the average decreases for approximately 10 “equivalent generations” prior to stabilization with four groups of “equivalent generations” with significantly higher values which occur when the algorithm tries radically different areas of the search space. This is characteristic of a Scatter Search Algorithm which utilizes Tabu Search and an Artificial Neural Network to stop the algorithm from going back to areas of the search space which either have, or are expected to yield inferior solutions. From the minimum curve, it can be seen that a value within 1% of the minimum is found in the 133^{rd} “equivalent generation” and remains relatively constant for the remainder of the “equivalent generations”. This result can be seen by plotting the algorithms improvement progress which is shown in

Summary of Results. In the previous sections the results of several techniques including: 1) Deterministic Analysis; 2) MC MO simulation (Palisade^{®} @RISK^{TM}); 3) Microsoft^{®} Excel’s SO Gradient-based (GRG2) algorithm (GRG Nonlinear Solver); 4) Microsoft^{® }Excel’s SO Evolutionary Solver; 5) Palisade^{®} Evolver’s SO GA; 6) GAPS MO GA (based on the NSGA-II) algorithm; and 6) Palisade^{®} Evolver’s SO OptQuest^{TM} algorithm were discussed individually. In this section the results of the different techniques will be compared. In

manipulation of the initial guess to achieve an acceptable history-match (although its key match parameters will be discussed below).

From

・ Fracture permeability ranges from 3102 md to 3190 md with the lowest value coming from Evolver’s GA and the highest coming from the average of the top five MC simulations. Each of the algorithms finds a fracture permeability ~350 md lower than the deterministic match (~10% difference). The percent variability from the five algorithms is approximately is ~2.5% when compared to the deterministic match.

・ Breakthrough pressure approaches the upper limit for each of the five algorithms and is significantly higher than the deterministic match (~7%). An earlier breakthrough yields a better late-time oil match, which is where oil rates are highest and therefore have greatest potential to add to the OF value. This is also the piece of data were the deterministic solution deviates most from the measured data. As mentioned previously, a breakthrough pressure of greater than 4100 psia leads to premature breakthrough, although also yields a better late-time history-match. A breakthrough pressure of 4100 psia suggests a 10% supercharge of the formation directly surrounding the fractures which results from pumping the fracture at significantly higher pressures than formation pressure (mini water flood effect).

・ BBT half-length is nearly constant using each of the six techniques, ranging from 437 - 441 ft which is to be expected given the rather definitive results of the FMB shown above. The deterministic history-match used the same BBT half-length as the four main assisted history-matching techniques.

・ Oil relative permeability exponent shows almost no variability from the five algorithms ranging from 1.43 - 1.46. This is ~20% higher than the value used in the deterministic history-match.

・ Water relative permeability exponent shows slightly more variability from each of the five algorithms, ranging from 5.30 - 5.87. Each of the algorithms predicted a water exponent exceeding that of the deterministic history-match by an average of ~4%. The percent variability from the five algorithms is approximately is ~11.4% when compared to the deterministic match.

・ The total OF for the four assisted history-matching algorithms was nearly identical, ranging from 34.7 - 35.4 million. The average of the top five MC simulations was ~2% higher than the other assisted history-matching techniques. The five different algorithms improved the total OF from 14.7% - 16.4%, although this suggests that the deterministic match still falls within the ±20% range often accepted in industry in this particular case.

The above results demonstrate that each of the algorithms find a very similar optimal value for each of the key parameters suggesting that this likely represents the global optimum. After reviewing the total OF, it is clear that there is significant benefit to applying these algorithms once bounds on key parameters can be estimated. Another interesting observation is that the deterministic history-match yielded values within 10% of the optimal values for three out of the five uncertain parameters. The only exceptions are the relative permeability exponent to oil water, which varied by ~20% and ~11% respectively. This higher differential can be attributed to the low values of these exponents, making them particularly sensitive when calculating percent difference (although the absolute value was within 0.25 and 0.66 of the average optimal values respectively).

Based on the results shown above, it would be expected that application of Excel’s GRG Non-Linear Solver with the use of multi-restart mode would likely yield the same results. This was not tested to its full extent in this analysis, although using the deterministic history-match as an initial guess led to similar parameters as those solved by the other algorithms. This result suggests that the multi-restart method would likely be successful in this problem and also demonstrates that there are no local minima between the deterministic match and global optimum.

The basis of this work is the tool developed by [^{TM} which combines several optimization techniques into a single algorithm. It was demonstrated that each technique could essentially locate the same optimal set of parameters, suggesting that this corresponds to the absolute minimum rather than a local minimum, which in turn led to a significant improvement in history-matching over the deterministic analysis. Despite the versatility of the methods described, there are several areas which warrant further discussion.

Two of the biggest challenges when applying MC simulation and other assisted history-matching techniques are: 1) selecting which variables to consider unknowns; and 2) developing an input distribution for the unknowns. These methods are typically most successful and converge faster when the number of inputs is limited to the minimum possible number with the smallest range to minimize the search space for the algorithm. In the case of flowback analysis, there are many uncertain inputs making this a difficult problem to solve using these methods, and therefore it is important to select the most important parameters as uncertain (i.e. fracture half-length and conductivity), while assuming that some less critical inputs that are constant (i.e. initial fracture pressure and fracture porosity). The next challenge is developing an input distribution for the uncertain parameters (particularly for MC simulation). In an ideal scenario, the input distributions can be developed from existing data allowing for greater precision and ultimately better output results, although this requires a significant amount of analogous data. For some scenarios, such as history-matching long-term production from wells with a significant number of analogs which have all been analyzed, this is feasible. Further, in many cases parameters such as matrix permeability have been demonstrated extensively in the literature to show a lognormal distribution. Unfortunately this is not the case with flowback analysis, where the data set is generally limited, or in many cases non-existent, due to the very new nature of industry interest in analyzing this data and lack of widespread (although rapidly growing) application. For example, the basic techniques used in this work have been applied by several companies including in an SPE paper written by [

Another challenge is determining an acceptable number of iterations (i.e. the number of generations and population size in the GAPS algorithm) to allow achieving reasonable results while minimizing run time to make the application of the techniques to a large number of wells more feasible. In this work, the purpose was to demonstrate the applicability of the different techniques used, and therefore run time was not a consideration, although this will become more important as these techniques continue to gain traction in industry. In this case, other than the Excel^{TM} Solver methods, each technique required multiple days of run time making the techniques not practically applicable to a large number of wells. Further, the tool is still in the research phase, and could be made significantly more efficient (~1 iteration per second comparable to other similar commercial tools), which would also help to significantly reduce run time. Trying to determine an acceptable number of iterations is an area of future work which will require application to more than the several wells which have been analyzed using these techniques.

In this paper, it was demonstrated that Excel’s GRG Non-Linear Solver is highly ineffective when a relatively generic initial guess is used, as this algorithm will find the closest local minima to the initial guess. When the deterministic solution was used as the initial guess, the algorithm converged to parameters similar to the other techniques applied, suggesting there is no local minima between the deterministic solution and the optimal solution. This may not always be the case, and in some applications a complete deterministic analysis may not be conducted prior to applying an assisted history-matching algorithm. The convergence speed of this algorithm makes it ideal, although its application clearly has limitations. One solution is to apply the multi-restart techniques discussed previously, where the algorithm will run for a series of different initial guesses in attempt to find the global minima. It is likely that a substantial number of restarts would be required to find the optimal solution for the flowback problem, and therefore extensive testing would be required before confidently applying this technique and determining how its run time compares to the other algorithms tested. The standard version of Solver available in Excel^{TM} does not offer a multi-restart option, although the developer of this solver (Frontline Solver’s) offers more advanced versions which include this option as well as further improvements and additional algorithms.

In this work, six techniques were applied for assisted history-matching purposes. These methods were selected as they were either developed within the research group (GAPS algorithm) or commercially available from reputable vendors that are used extensively in industry (Microsoft^{®} and Palisade^{®}).

Future work on this topic will focus on the application of addition algorithms available both commercially and in the literature. Testing of further MO GA’s would be of particular interest as they overcome the biggest challenge of SO algorithms which were the primary focus of this work. Testing further techniques is warranted, seeking algorithms which converge faster and/or are potentially more effective in consistently finding the optimal solution. A detailed investigation, using multiple examples (both simulated and field examples) will also be conducted to determine the number of iterations required to achieve the desired result. This will allow for a better comparison of both convergence speed and accuracy which was not directly addressed in this paper.

In this work, several algorithms were tested for the purpose of uncertainty analysis and assisted history-matching of flowback data. In previous work, [

・ MC simulation can effectively be applied for both uncertainty quantification and assisted-history matching, assuming enough trials are conducted to effectively cover the search space. For practical application, this limits the number of uncertain parameters and the distribution range for these parameters.

・ As anticipated, application of a gradient-based algorithm was not successful unless a very good initial guess was provided. This is due to the nature of the algorithm limiting its application in the absence of using the multi-restart feature.

・ Each of the techniques tested (excluding Excel’s GRG Non-Linear Solver), including both SO and MO techniques, was able to converge to a very similar optimal solution, suggesting that they were likely finding the global optima. There are often problems associated with applying SO algorithms to MO problems due to competing objectives, although this issue did not appear to arise in the analyzed well. It was demonstrated that each of these techniques provided a significant improvement in history-match quality over a single deterministic analysis, although deterministic history-matching is useful in determining which parameters should be considered uncertain and constraining the range of these uncertain parameters.

・ Further testing is warranted to determine the wide-spread applicability of these techniques, and to reduce run time making the application more desirable for industry applications.

・ Additional algorithms should be investigated for a larger number of wells to determine which techniques are most applicable to the flowback problem. Specifically, testing additional MO algorithms would be desirable as these algorithms tend to better represent the problem. It is possible that MO algorithms other than the GAPS algorithm could provide better results for flowback analysis.

Jesse Williams-Kovacs would like to thank the University of Calgary for supporting this research. Chris Clarkson would like to acknowledge Encana/Shell and Alberta Innovates Technologies Futures (AITF) for support of his Chair position in Unconventional Gas and Light Oil Research at the University of Calgary, Department of Geoscience. The authors would also thank Mohammed Kanfar for providing the GAPS algorithm as well as technical support. Finally, the sponsors of Tight Oil Consortium (TOC), hosted at the University of Calgary, are acknowledged for their support.

The authors declare no conflicts of interest regarding the publication of this paper.

Williams-Kovacs, J.D. and Clarkson, C.R. (2019) Stochastic Modeling and Assisted History-Matching Using Multiple Techniques of Multi-Phase Flowback from Multi-Fractured Horizontal Tight Oil Wells. Advances in Pure Mathematics, 9, 242-280. https://doi.org/10.4236/apm.2019.93012

ABT = After-breakthrough

BBT = Before-breakthrough

CDF = Cumulative distribution function

DFIT = Diagnostic fracture injection test

FMB = Flowing material balance

FR = Flow-regime

GA = Genetic algorithm

GOR = Gas-oil ratio

GRG = Generalized reduced gradient

LTO = Light tight oil

MBT = Material balance time

MC = Monte Carlo

MINC = Multiple interacting continua approach

MO = Multi-objective

NSGA = Nondominated sorting genetic algorithm

PDF = Probability density function

PVT = Pressure-Volume-Temperature

RNP = Rate-normalized pressure

RNP’ = Rate-normalized pressure derivative

RTA = Rate-transient analysis

SO = Single-objective

Field VariablesF_{c} = Fracture conductivity, md-ft

m’ = Corey water relative permeability constant for the fractures, dimensionless

n’ = Corey oil relative permeability constant for the fractures, dimensionless

p = Pressure, psia

p_{wf} = Sadface flowing pressure, psia

p^{*} = Extrapolated initial reservoir pressure, psia

q_{o} = Oil production (surface) flowrate, STB/D

q_{w} = Water production (surface) flowrate, STB/D

Q_{o} = Cumulative water production (surface), STB

Q_{w} = Cumulative production (surface), STB

r_{wa} = Apparent wellbore radius (r_{wa} = r_{w} e^{-s}), ft

w = Objective function weighting factor, dimensionless

x_{f} = Fracture half-length, ft

t_{Dd} = Dimensionless decline time

ABT = After-breakthrough

BBT = Before-breakthrough

BT = Breakthrough

D = Dimensionless variable

f = Fracture

o = Oil

T = Total

w = Water

wf = Sandface