Simplification workflow for hydraulically fractured reservoirs

Accepted Manuscript Simplification workflow for hydraulically fractured reservoirs Raul Velasco, Palash Panja, Milind Deo

PII:

S2405-6561(17)30072-X

DOI:

10.1016/j.petlm.2018.01.001

Reference:

PETLM 185

To appear in:

Petroleum

Received Date: 18 April 2017 Revised Date:

5 December 2017

Accepted Date: 4 January 2018

Please cite this article as: R. Velasco, P. Panja, M. Deo, Simplification workflow for hydraulically fractured reservoirs, Petroleum (2018), doi: 10.1016/j.petlm.2018.01.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Title: Simplification Workflow for Hydraulically Fractured Reservoirs

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Journal: Petroleum Manuscript ID: PETLM_2017_58 Authors:

Raul Velasco Energy & Geoscience Institute, The University of Utah, Salt Lake City, USA [email protected]

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Palash Panja† Energy & Geoscience Institute, The University of Utah, Salt Lake City, USA [email protected]

Milind Deo Department of Chemical Engineering, The University of Utah, Salt Lake City, USA [email protected]

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No conflicts.

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Grant information:

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Corresponding author: Palash Panja† Energy & Geoscience Institute, The University of Utah, 423 Wakara Way, Suite #300, Salt Lake City, UT 84108, USA [email protected] (801)-585-9829

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Key words: Hydraulic Fracture, Shale, Simulation Workflow, Multiwell, Field case Abbreviations: (in alphabetical order) Tables: ____8____

Figures: ____21___ (color – Yes, 20)

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Simplification Workflow Fractured Reservoirs

for

Hydraulically

†

[email protected]

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Corresponding author Department of Chemical Engineering, University of Utah

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Energy & Geoscience Institute, University of Utah

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Raul Velasco1, Palash Panja†2, Milind Deo1

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Abstract

Production from unconventional formations, such as shales, has significantly increased in recent years by stimulating large portions of a reservoir through the application of horizontal drilling and hydraulic fracturing. Although oil shales are heavily dependent on oil prices, production

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forecasts remain positive in the North-American region [1]. Due to the complexity of hydraulically fractured tight formations, reservoir numerical simulation has become the standard tool to assess and predict production performance from these unconventional resources. Many of

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these unconventional fields are immense, consisting of multistage and multiwell projects, which results in impractical simulation run times. Hence, simplification of large-scale simulation

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models is now common both in the industry and academia. Typical simplified models such as the “single fracture” approach do not often capture the physics of large-scale projects which results in inaccurate results. In this paper we present a simple, yet rigorous workflow that generates simplified representative models in order to achieve low simulation run times while capturing physical phenomena which is fundamental for accurate calculations. The proposed workflow is based on consideration of representative portions of a 1

ACCEPTED MANUSCRIPT large-scale model followed by post-process scaling to obtain desired full model results. The simplified models that result from the application of the proposed workflow for a single well and a multiwell case are compared to full-scale models and the “single fracture” model.

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Comparison of fluid rates and cumulative production show that accurate results are possible for simplified models if all important components for a particular case are taken into account. Finally, application of the workflow is shown for a heterogeneous field case where prediction

Introduction

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studies can be carried out.

Horizontal well technology combined with hydraulic fracturing has unleashed vast amounts of oil and gas production in the U.S. by stimulating formation rocks traditionally considered economically unfit for investment due to their low permeabilities. The glut created through fracturing of tight formation and shales has changed the domestic and global oil and gas industry.

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Even though oil prices have been declining in recent years, the amount of oil and gas produced from hydraulically fractured reservoirs has been steadily increasing when compared to

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production from traditional reservoirs.

Multistage and multiwell fracking projects in low-permeability formations (also known as tight

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formations) often result in complex systems that are difficult to predict using traditional analytical and semi-analytical solutions. As a result, reservoir numerical simulation has become the best way to comprehensively and quantitatively describe multiple phase fluid flow behavior in tight formations[2]. Over time, reservoir simulation has become a critical tool for determining hydrocarbon potential due to its flexibility and success in describing and predicting tight hydrocarbon production. 2

ACCEPTED MANUSCRIPT The well density necessary to attain acceptable recoveries in low-permeability oil producing reservoirs is high as seen in the Bakken. Spacing of 40 acres or lower has been discussed for Eagle Ford of Permian Basin wells. The length of horizontal wells in these plays is of the order

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of miles with 30-40 hydraulic fracturing stages. In reservoir simulation, full-scale models with horizontal wells of these scales are typically modeled as shown in Figure 1.

In addition, fine grids are often required in the near fracture areas to avoid numerical errors,

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convergence problems and to obtain accurate results [3]. Hence, the number of grid blocks to model even one well tend to be substantial, leading to large computational times (in the order of

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several hours to days for a single run) for full-field simulations. For this reason, it is desirable to simplify large-scale models to considerably lower simulation run times. Typical simplification approaches to full-scale models are shown in Figure 1, where the “single fracture” model is widely used in the industry and academia [4-13].

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The single fracture model is a result of a concept known as Stimulated Reservoir Volume (SRV). This concept contends that hydrocarbon production is primarily produced from the reservoir volume that is stimulated by hydraulic fracturing. This volume is therefore determined by

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fracture half lengths, fracture height and number of propped fractures [14]. The single fracture approach assumes that all fractures behave identically, hence only one fracture is modeled along

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with its corresponding stimulated volume. Typical dimensions of the single fracture model correspond to the length of the fracture and inter-fracture spacing as shown in Figure 1. The advantage of the single fracture approach is the substantially low computational requirements achieved by assuming all fractures in the field behave similarly. In this model, total fluid production of the entire horizontal well is simply calculated by multiplying production from a single fracture by the number of fractures in the well as described by Equation 1. 3

ACCEPTED MANUSCRIPT X  =  

1

Where, X is the well fluid rate or cumulative production;  is the number of fractures;  is the single fracture fluid rate or cumulative production, and  indicates the fluid phase.

X =   

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Where  corresponds to the number of wells.

2

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shown in Equation 2.

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Similarly, total production for a multiwell project is calculated from the single fracture model as

However, simplified models used for numerical expediency are often not accurate enough to represent the entire model. This is the case for the single fracture model which incurs errors by ignoring important contributions to production. For instance, while the single fracture may represent early transient flow properly for all identical fractures, as soon as interference and/or

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boundary dominated flow begins, the model fails to account for “external” fractures trends. Internal fractures are defined as the hydraulic fractures that perceive interference from other surrounding fractures, while external fractures perceive fracture interference on one side and

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boundary dominated flow on the other side. This means that the stimulated volume allocated for

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internal and external fractures may be different as visualized in Figure 2. Taking “Well 1” in Figure 2 as reference, single fracture models assume that all fractures behave like internal fractures. The error incurred by not including external fractures varies from case to case (depending on the total number of fractures, fracture spacing, etc.). However, as fracture configurations become increasingly more complex as shown by “Well 2” in Figure 2, single fracture models are no longer a valid representation of the field-scale model. 4

ACCEPTED MANUSCRIPT Due to different stimulated volumes and interference effects, internal and external fractures exhibit a different trend in production rates as shown numerically and analytically by several authors [15-17]. The difference in cumulative gas oil ratios from internal and external fractures is

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shown in Figure 3, where ratios are identical during transient flow, but deviate as soon as fracture interference begins. The difference between internal and external fracture highlights the importance of using the correct simplified model. The single fracture extrapolation leads to

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significant errors in all results, especially Gas Oil Ratios (GOR).

In this work, we propose a different approach to traditional simplified models by introducing a

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standard simplification workflow. The main concept of a simplification workflow is to come up with a practical representative portion of a full-scale model that can be scaled without incurring significant errors. The simplified models that result from this workflow should achieve low simulation run times while accounting for all flow phenomena taking place in a full-scale model. Although we use numerical simulation to solve these models, analytical and semi-analytical tools

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are also viable paths to solve these systems.

Simplification Workflow

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The proposed simplification workflow is meant to guide and provide the simulation engineer

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with educated and systematic criteria to simplify full-scale models. Its successful application results in a simplified representative portion of the reservoir. The workflow procedure is divided into three major steps applied to specific reservoir projects: 1. Identification of model elements, 2. Simplification, and 3. Post-processing. Descriptions of these steps are introduced next.

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Identification of Model Elements

Identification of pertinent characteristics of a full-field simulation model is the most engineering intensive step as it requires important analysis. In this step, we identify all physical elements that

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make up a full-scale simulation model such as boundary conditions, hydraulic fractures, wells, etc. Basically, model elements are physical features that contribute to hydrocarbon flow and that cannot be neglected. For example, the single fracture model represents a full-scale model with three key aspects: A number of internal fractures, interfracture interference, and a no-flow





= 0 at the boundary), and that internal

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mimicked by a no-flow boundary condition (i.e.

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boundary condition at the SRV boundaries. Upon realizing fracture interference can be

fractures behave almost identically, two elements are identified: Internal fracture, no-flow boundary condition. Thus, the single fracture model as shown in Figure 1 is a representative portion of a full-scale model with two elements.

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However, as discussed earlier, the single fracture model does not consider other important elements present in typical projects such as the external fractures. More complex models need careful examination in order to account for all important model elements.

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It is often helpful to organize all pertinent elements of a simulation in solution diagram. An example of solution diagram for a typical hydraulically fractured reservoir is shown in Figure 4.

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In this small example, important components such as the existence of internal and external fractures are acknowledged. 2.2

Simplification

Once full-scale model elements have been identified, a simplified model can be obtained by building a representative reservoir sub-model composed of essential elements. Representative reservoir sub-models consist of elements that take into account all components which define the 6

ACCEPTED MANUSCRIPT full-scale simulation model. These elements may be identified as fractures or wells depending on the particular case. For instance, based on the solution diagram in Figure 4, a representative reservoir sub-model needs to include internal, external and inter-stage fracture elements. These

scale results during the post-processing steps.

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elements are essential to keeping track meaningful representation of a full-scale model and to

A good reservoir sub-model should be able to be integrated back into a full-scale model using

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symmetry. The single fracture sub-model is a classic example of the use of symmetry where one element fracture represents all fractures in the horizontal well as shown in Figure 1. Hence, we

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can always go back from the simplified reservoir sub-model to the full-scale by simple multiplication.

The successful combination of model elements and symmetry lead to efficient simplified models. When all vital elements are identified, a symmetric analysis defines their configuration. This step

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is a more creative process than the previous one and may spam a number of models that should arrive to the same answer if correctly applied. In upcoming sections, we show examples for reservoir sub-models for more complex multistage and multiwell configurations. Post-Processing

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2.3

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After a representative portion of the reservoir (sub-model) is built by assembling elements using a symmetric scheme, the model is run using numerical simulation. The results of this simplified must be then scaled to represent a full-scale model in a process similar to the single fracture equations 1 and 2.

In this step, the importance of elements in a simplified model becomes evident. Each element carries essential information about the full-scale model and represents it in the form of results 7

ACCEPTED MANUSCRIPT such as fluid rates, pressures or performance. To scale simplified model to full-scale model, results from each element must be multiplied by the total number of elements in the full-scale model and by an integer depending on the use of symmetry.

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Equations that make these conversions are intuitive as in the case of the single fracture model. However, as models grow in complexity, these equations may become less obvious. Equation 3 is designed to help keep track of important elements in simple and complex simulation models.

X =   , 

3

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Where  is the number of elements;  is the symmetric factor for the ith element;  is the multiplier factor for the ith element, and , is fluid cumulative production or production rate from the ith element.

The single fracture equations 1 and 2 can be obtained from Equation 3, if the symmetric factor,

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 , is one (fracture element is not divided) and the multiplier factor,  , is equal to the number of fractures. It is important to note that bottom-hole pressures and field-averaged values such as

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reservoir pressure, average saturations, etc. do not need to be scaled. The simplification workflow steps can be visualized as shown in Figure 5. These steps are

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summarized as follows:

1. Build a solution process diagram by accounting for all representative elements in the fullscale model.

2. Prepare a reservoir sub-model by applying symmetry to representative model elements obtained in step 1. 3. Scale simplified results to full-scale results by applying Equation 3. 8

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Case Examples Single Well Case

In this instance, the single fracture model and a model prepared after application of the

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simplification workflow are compared to a full-scale, black-oil, single horizontal well model. To give some context of the problem at hand, the full-scale model is shown in Figure 6 with basic model properties as described in Table 1. The single fracture version of the full-scale model

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shown in Figure 6 consists of one fracture as shown in Figure 7 (bounded by red dashed lines). By application of equations 1 and 2, the results of this model are scaled in an attempt to represent

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the entire horizontal well model.

On the other hand, application of the proposed simplification workflow by following the steps described in the previous section results in a different model. Most importantly, the contribution of external fractures must be included in the reservoir sub-model. Under this requirement, the

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single fracture sub-model is insufficient and a new model is built as shown in Figure 7 (bounded by green dashed lines), where two fractures are present. The left fracture is representative of interior fracture behavior while the right fracture represents exterior fracture behavior as required

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by the simplified solution process diagram. The right side of the exterior fracture is deliberately left ‘open’ since the distance between the last fracture and the reservoir boundary can change

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from case to case. Note that in Figure 7, element 2 represents only half of an internal fracture to further reduce computational time. Full internal fracture results are then realized in the postprocessing portion of the workflow using equation 3 and the symmetric factor. The entire model is solved numerically with a commercial simulator. The final step of the simplification process consists of scaling results. As mentioned in previous sections, the single fracture results are scaled as shown in Equations 1 and 2. However, results 9

ACCEPTED MANUSCRIPT scaling from the proposed model are not as obvious. Since the number of fracture elements in the proposed model shown in Figure 7 is two, Equation 3 becomes: X =   , +   ,

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Taking into consideration the fracture element labels 1 and 2 in Figure 7, the symmetric factor for element 1 is 1 because the entire fracture element is modeled in the sub-model and its multiplier factor is 2 because there are only 2 external fractures. Similarly, the symmetric factor

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for fracture element 2 is 2 because only half of the fracture volume is considered in the

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simplified model and the multiplier factor is the number of internal fractures (the total number of fractures minus two). Table 2 lists value assignments for variables in Equation 4. From this example, it is clear that the symmetric factor accounts for element divisions and the multiplier factor represents the number of elements present in the full-scale model.

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Upon value substitution of Equation 4 based on Table 2, Equations 5 and 6 are used to determine scaled oil rates and oil cumulative production respectively.

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 = 2, + 2, ( − 2)



=2

,

+2

, (

− 2)

5

6

Where  is the total number of fractures. Equation 7 shows typical cumulative gas oil ratio calculation for this specific case.

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=

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− 2) , (n − 2)

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Three cases for each model (full-scale, single fracture, and proposed model) were run with reservoir horizontal permeabilities of 50, 100, 500 and 1000 nD. Based on the above equations and the results obtained by numerical simulation, oil rates, cumulative production and

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cumulative GOR were calculated for the proposed model.

As observed in Figure 8, cumulative oil production for all reservoir permeability cases is almost

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identical for an initial period of time (transient flow). However, as soon as boundary dominated flow and/or interference effects take place, there is a substantial difference between the full-scale model and the single fracture model. Since the proposed model does account for exterior fracture and fracture interference effects, it matches the full-scale model almost perfectly. A similar

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situation is observed in oil rates and cumulative GOR as shown in Figure 9 and Figure 10. Table 3 shows total simulation run times for all three models clearly proving that the proposed model retains valuable short computing times while keeping results accurate as shown by the

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coefficient of determination for cumulative GOR. As mentioned previously, the single fracture

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model is similar to the proposed model when only internal fractures are the target of analysis for a single well. However, this is no longer the case as the models become increasingly complex such as the case of multi-well cases. 3.2

Multi-well and Multi-Fracture Case

In this case, a more complex model is considered in the form of three horizontal wells. Fluid as well as rock properties are found in Table 1 and the model schematics are shown in Figure 11. 11

ACCEPTED MANUSCRIPT A simplified model is built by taking into consideration internal and external fractures as well as internal and external wells. The resulting model with four fracture elements is shown in Figure 12 bounded by green dashed lines.

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As shown in the simplification schematics, the single fracture model does not change substantially while the proposed model adapts to capture well interference. Fractures elements are labeled as 1, 2, 3 and 4 which represent external fractures in external wells, internal fractures

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in external wells, external fractures in internal wells, and internal fractures in internal wells respectively. In order to clarify the objective of these fracture elements, one must consider the

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fluid flow contributions made by internal and external fractures in the previous case example. Similarly, internal and external wells contribute to total field production. Hence, by application of symmetry, the proposed model shown in Figure 12 satisfies all key elements. Note that the proposed model only takes into account half a fracture for the internal well, and a

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full fracture for the external well. This was done with the purpose of capturing boundary flow perceived by the external wells.

The single fracture model’s results from the previous example can be used to calculate results for

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three wells. This is not the case for the proposed model which accounts for more fracture

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elements that represent different components of the full-scale model. Symmetric and multiplier factors are determined using fracture elements as labeled in Figure 12 and are shown in Table 4. Applying Equation 3 to this case, the oil rate expression becomes:  = 4, + 4(n − 2), + 4), + 4(n − 2)*,

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After solving this model numerically, cumulative oil production, oil rates and cumulative GOR are shown in Figure 13, Figure 14 and Figure 15 respectively. Eventhough the proposed model 12

ACCEPTED MANUSCRIPT results show some slight deviations from the full-scale model, it is still substantilly more accurate than the single fracture model. Computational run times and coefficient of determinations for the cumulative GOR results were also determined for each case as shown in

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Table 5. Again, it is shown that models resulting from the application of the proposed simplifcation workflow are realiably accurate and save valuable project time. Similarly, more complex systems

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can also be simplified by application of the present simplification workflow. A few sub-model examples are shown in Table 6 for for single well configurations and in Table 7 for multiwell

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configurations. Application of Equation 3 to these sub-models are also shown in the tables for each case. This workflow was proven to be helpful for well spacing optimization studies where modelling time and computational run times are impractical for full-scale models [18]. 3.3

Bakken Field Case

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Well and fracture spacing studies are very important when it comes to oil production optimization. An accurate and simple simulation model is crucial to conduct timely studies. Even though full-scale reservoir models are desirable for the most accurate description of fluid flow

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behavior, multiwell fields are usually handled separately, without considering well interference. This is crucial, especially because well interference has been deemed as an important aspect of

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field performance [19].

In this section, a Bakken field case model is considered for simplification by using the proposed workflow. Figure 16 shows the full case model which consists of three multi-stage horizontal wells. Reservoir properties for the model were populated according to statistical analyses provided in Table 8. Samples of model distribution maps for arbitrary layers are shown in Figure 17 for porosity and Figure 18 for permeability. “Well 1” is opened at a maximum 13

ACCEPTED MANUSCRIPT producing oil rate of 3500 STB/day at time 0, under the same operating conditions “Well 2” and “Well 3” are then opened 3 months and 1 year after “Well 1” correspondingly. All wells consist of 27 to 30 stages and 3 clusters per stage.

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Because all three wells are not opened at the same time, special consideration to potential well interference must be taken. Hence, the proposed simplified model that results from the application of the workflow is shown in Figure 19, where reservoir properties such as porosity

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and permeability are averaged. Maximum oil production rates were normalized for each well in the simplified model to accurately represent the full-scale model.

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Results comparing the field model total cumulative oil production and well oil rates are shown in Figure 20 and Figure 21 correspondingly. Both models generate almost virtually the same results, however, the computational times differ from hours (full-scale model) to a couple of minutes (simplified model). Successful engineering analysis applied during production of a

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simplified model can indeed be used to substantially increase results accuracy and reliability. Nevertheless, the design of full-scale models is encouraged to validate any simplified model. Simplified models can then be used to conduct history matching or sensitivity studies. It should

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also be noted that depending on the scenario, natural fracture and/or geomechanical effects may need to be considered and added to the solution process diagram in Figure 4 to achieve accurate

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results. Additionally, systems that show severe heterogeneity may pose a challenging simplification model. Results from this section were successful due to low porosity and permeability variance in the Bakken.

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Conclusions

Large scale simulation models have been notorious for their significant simulation times. 14

ACCEPTED MANUSCRIPT Traditionally, one option has been to use results from one fracture (single fracture model) and extrapolate to the field based on the number of fractures and wells in order to minimize simulation run times. However, errors incurred by the single fracture model only accumulate

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during extrapolation of results. A new standardized simplification workflow was presented and proven to greatly reduce simulation run times while achieving accurate results for production from low-permeability formations with hydraulically fractured wells. The simplification steps were explained by accounting for essential elements that contribute to fluid flow, building a

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representative reservoir sub-model, applying symmetry, and post-processing simulation results.

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Applicability of the workflow to represent full-field results was demonstrated for a multifracture single well case, a multifracture, multiwell case, and a heterogeneous Bakken dataset. It should be noted, however, that severe heterogeneity might present challenges to these models and that the effects of geomechanics are not account for, but may be included in the workflow.

Cumulative Production or Production Rate from ith Element

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Description Number of Clusters per Stage Number of Fractures in a Horizontal Well Number of Stages in a Horizontal Well Number of Horizontal Wells in one Section Symmetric Factor for ith Element

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Symbol +  &  

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Nomenclature

GOR

,, ,-, ,.  N1,p N2,p Np NRMSE q1,gas q1,oil

Gas Oil Ratio Formation Permeability in X, Y and Z direction respectively Multiplier for Full Model Representation Cumulative Oil Production from Element 1 Cumulative Oil Production from Element 2 Cumulative Oil Production Normalized Root Mean Square Error Gas Rate from Element 1 Oil Rate from Element 1

Units STB, SCF, STB/DAY, SCF/DAY SCF/STB nD STB STB STB SCF/day STB/day 15

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X

Gas Rate from Element 2 Oil Rate from Element 2 Oil Rate from Element 3 Oil Rate from Element 4 Total Gas Rate Total Oil Rate Coefficient of Determination Total Fluid (Oil, Gas or Water) Flowrate or Cumulative Production

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SCF/day STB/day STB/day STB/day SCF/day STB/day STB/day, SCF/day, STB, SCF

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q2,gas q2,oil q3,oil q4,oil qgas qoil R2

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Panja, P., R. Velasco, M. Pathak and M. Deo, Application of artificial intelligence to forecast hydrocarbon production from shales. Petroleum, DOI 10.1016/j.petlm.2017.11.003 (2017). Mayerhofer, M.J., E. Lolon, N.R. Warpinski, C.L. Cipolla, D. Walser and C.M. Rightmire, What is stimulated reservoir volume? SPE Production and Operations, 25(1) (2010), p. 89. ElRafie, E.A. and R.A. Wattenbarger. Comprehensive Evaluation of Horizontal Wells with Transverse Hydraulic Fractures in the Upper Bahariyia Reservoirs. in SPE Middle East Oil Show and Conference, Bahrain, (1997). Raghavan, R.S., C.C. Chen and B. Agarwal, An Analysis of Horizontal Wells Intercepted by Multiple Fractures. SPEJ, 2(03) (1997), p. 235 - 245. Meyer, B.R., L.W. Bazan, R.H. Jacot and M.G. Lattibeaudiere. Optimization of multiple transverse hydraulic fractures in horizontal wellbores. in SPE Unconventional Gas Conference. Pittsburgh, Pennsylvania, USA: Society of Petroleum Engineers, (2010). Velasco Guachalla, R., Advanced techniques for reservoir engineering and simulation, The University of Utah, (2016), p. 168. Elliot, G.R. Well Interference Supports Wide Spacing. in Drilling and Production Practice. American Petroleum Institute, (1951).

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Figures

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Figure 1 Aerial view of a typical multiwell and multistage reservoir simulation model and traditional simplification approaches

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Figure 2 Types of internal and external fractures and their corresponding volume assignments (Top view)

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Figure 3 Internal and external fracture cumulative GOR for matrix permeabilities of (a) 50 nD, and (b) 1000 nD

Figure 4 Example of a solution process diagram with model elements

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Figure 5 Simplification workflow visualization

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Figure 6 Horizontal well with 50 equally-spaced fractures

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Figure 7 Aerial view of a full-scale horizontal well and its simplification approaches: Single fracture model (red dashed lines), and the proposed model (green dashed lines) with fracture elements 1 and 2

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Figure 8 Cumulative oil production for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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Figure 9 Oil rate comparison for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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Figure 10 Cumulative GOR comparison for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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AC C

EP

TE D

Figure 11 Full-scale multiwell simulation model schematic

Figure 12 Aerial view of a full-scale multiwell model and its simplification approaches: Single fracture model (red dashed lines), and the proposed model (green dashed lines) with 4 fracture elements

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(b)

(c)

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(a)

(d)

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Figure 13 Cumulative oil production comparison for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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(b)

(c)

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(a)

(d)

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Figure 14 Oil rate comparison for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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(b)

(c)

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(a)

(d)

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Figure 15 Cumulative GOR comparison for matrix permeabilities of (a) 50 nD, (b)100 nD, (c) 500 nD, and (d) 1000 nD

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Figure 16 Bakken field case model well paths

Figure 17 Porosity distribution maps for two layers

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Figure 18 Permeability distribution maps for two layers

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Figure 19 Top-view for the Bakken case sub-model

Figure 20 Well cumulative production for the full-scale Bakken case and the proposed simplified model

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Figure 21 Well oil rates for the full-scale Bakken case and the proposed simplified model

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Table 1 Reservoir properties and operational parameters 9600

Matrix Permeability, kx, ky (nD)

50, 100, 500, 1000

Matrix Permeability, kz (nD)

0.1 * kx

Number of Fractures

SC

YZ plane

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Fracture Orientation

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Reservoir Top (ft)

TE D

Initial Reservoir Pressure (psi)

50

4500

4x10-6

Initial Oil Saturation (%)

84

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Rock Compressibility (psi-1)

Reservoir Porosity (%)

8

Bottom-Hole Pressure (psi)

500

Bubble Point Pressure (psi)

1965

33

Oil Gravity (API)

52

Reservoir Temperature (OF)

245

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Table 2 Symmetric and multiplier factors for elements 1 and 2 in the proposed model

1

k

2

2

nf - 2

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s

2

SC

1

Table 3 Model run times and error Single Fracture Model

Run Time

(nD)

(Minutes)

50

Full-Scale Model

R2

Run Time

R2

Run Time

(Cum. GOR)

(Minutes)

(Cum. GOR)

(Minutes)

EP

Permeability

Proposed Model

TE D

Reservoir

1.6

-3.741

1.6

0.923

17.9

1.4

-0.540

1.5

1.000

17.8

500

1.4

0.726

1.5

0.998

17.9

1000

1.4

0.823

1.4

1.000

19.7

AC C

100

Table 4 Symmetric and multiplier factors for a multiple horizontal well case 34

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2

3

4

s

1

2

2

4

k

4

2(nf - 2)

2(nw - 2)

(nw - 2) (nf - 2)

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1

R2

Run Time

R2

Run Time

(Minutes)

(Cum. GOR)

(Minutes)

(Cum. GOR)

(Minutes)

50

1.6

-4.005

2.7

0.981

81.3

100

1.4

-0.683

2.7

0.992

77.7

500

1.4

0.351

2.4

0.995

74.7

1000

1.4

-0.058

2.7

0.992

73.1

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(nD)

TE D

Permeability Run Time

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Table 5 Run time and goodness of model (compared to full-scale model) comparison for different models for the multiwell case Single Fracture Model Proposed Model Full-Scale Model Reservoir

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#

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Table 6 Examples of typical single well sub-models and post-processing equations Full-Scale Model

TE D

1

AC C

2

X =  ,

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Single well where /0 1 /

Simplified Model and Equation

Single well where /0 2 /

X = 2( − 2), + 2,

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X = (2& − 2), +

(+ − 2)& , + 2,)

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Single well with & stages and + clusters per stage. Note that  = & +

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3

Simplified Model and Equation

EP

Full-Scale Model

AC C

#

TE D

Table 7 Examples of typical multiwell sub-models and post-processing equations

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1

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2

TE D

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Two wells where /0 1 /

SC

X = 4( − 2), + 8,

X = 24 − 25, + 4, + 24 − 25( − 2),) + 4( − 2),*

Multiple wells ( ) where /0 2 /

Table 8 Field case property distributions (normal for porosity and log-normal for permeability) 38

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Permeability

Porosity

Mean (µD)

Std (µD)

Mean (%)

Std (%)

Upper Bakken Shale

0.010

0.003

1.500

0.375

Middle Bakken 1

8.125

2.031

7.375

1.844

Middle Bakken 2

120.000

30.000

10.500

2.625

Middle Bakken 3

32.500

8.125

Middle Bakken 4

107.500

26.875

10.000

2.500

Middle Bakken 5

18.750

4.688

9.000

2.250

Middle Bakken 6

0.625

0.156

7.875

1.969

0.031

3.375

0.844

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8.250

0.125

2.063

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Lower Bakken Shale

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Layer

39