Carbon dioxide source selection and supply planning for fracking operations in shale gas and oil wells

Accepted Manuscript Carbon dioxide source selection and supply planning for fracking operations in shale gas and oil wells Xin Li, Jose A. Ventura, Luis F. Ayala PII:

S1875-5100(18)30170-7

DOI:

10.1016/j.jngse.2018.04.014

Reference:

JNGSE 2536

To appear in:

Journal of Natural Gas Science and Engineering

Received Date: 24 October 2017 Revised Date:

10 April 2018

Accepted Date: 11 April 2018

Please cite this article as: Li, X., Ventura, J.A., Ayala, L.F., Carbon dioxide source selection and supply planning for fracking operations in shale gas and oil wells, Journal of Natural Gas Science & Engineering (2018), doi: 10.1016/j.jngse.2018.04.014. 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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Carbon Dioxide Source Selection and Supply Planning for Fracking Operations in Shale Gas and Oil Wells

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Xin Li1, Jose A. Ventura1*, and Luis F. Ayala2 1

Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802, USA 2

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John and Willie Leone Family Department of Energy and Mineral Engineering, The Pennsylvania State University, University Park, PA 16802, USA

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Abstract

As a promising alternative approach for shale gas and oil withdrawal, fracking using CO2 instead of water has been proved to be not only technically feasible but also environmentally friendly. This paper aims at evaluating CO2 fracking from an economic perspective by considering the collection, supply,

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transportation, and storage of CO2. More specifically, a CO2 source selection and supply planning problem is defined and discussed from three aspects: CO2 collection and storage at sources, CO2 transportation, and CO2 storage and usage at well pads. Cost models for the three aspects are built, based

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on which, a mixed-integer nonlinear programming model is developed to determine the optimal set of sources, the corresponding supply and transportation plan, and the investment in associated facilities and

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equipment. A two-stage algorithm is proposed, which decomposes the original model into a large set of simple sub-problems and a mixed-integer linear problem. A case study for three emerging well pads in North Dakota is presented to illustrate the implementation of the mathematical model and algorithm, and a sensitivity analysis is conducted to analyze the influences of four key parameters over the cost of using CO2. Keywords: CO2 fracking; source selection; supply planning; mathematical model; two-stage algorithm; sensitivity analysis.

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*Corresponding author. Tel.:+1 814 865 3841; fax: +1 814 863 4745. E-mail: [email protected].

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1. Introduction The U.S. has witnessed a rapid and remarkable growth of shale gas and oil production in the past decade. In 2004, shale gas production contributed only 5% of total U.S. dry gas production; but in 2013, the

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proportion reached nearly 40% (EIA, 2016a). Similarly, shale oil production has increased by more than ten times since 2006, accounting for about half of the total oil production today (EIA, 2016b). However, the withdrawal enabling method, hydraulic fracking, remains controversial. As currently practiced,

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hydraulic fracking uses large amounts of water injected into wells under high pressure to help free natural gas or oil from shale deposits. A typical fracking operation pumps approximately five million gallons of

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water and chemicals underground (Jenkins, 2013). Water pumped into a well carries fluid additives, high levels of total dissolved solids, metals and naturally occurring radioactive materials. 20 to 50% of the fracking fluid may flow back to the surface; and the rest stays in the rock formation, which can block the path of the natural gas or oil, hindering production and potentially decreasing the total amount a well may produce over its lifetime. Moreover, water that enters underground formations has been described as a

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potential contamination threat to aquifers and soils (Gordalla et al., 2013; Vengosh et al., 2014). As a result, water-related issues in hydraulic fracking are fueling the growing policy and regulatory problems, and environmental compliance hurdles that could potentially challenge shale gas production expansion

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and increase operational costs (Cupas, 2008; Rahm, 2011; Ridlington and Rumpler, 2013). However, fracking, when practiced responsibly, may not be the menace it has been described.

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Water-free fracking, particularly, fracking using pressurized CO2 (usually in liquid or super-critical form), instead of water, to crack open the formation, has been shown to be a promising alternative. The promise of waterless fracking using CO2 has indeed attracted a great deal of attention.

Middleton et al. (2014), Middleton et al. (2015), and Moridis (2017) have described potential advantages and disadvantages of using CO2 as the working fluid for fracturing operations in shales. Among those, elimination of flow hindrances due to water-induced productivity damage, enhanced fracture propagation and productivity, and elimination of water disposal and reutilization needs are often cited among clear advantages. CO2 unique properties such as the low viscosity, low lubricity, high compressibility, and high 3

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miscibility with formation fluids have been also added to the list of technical advantages (Gupta and Bobier, 1998; Fang et al., 2014; Asadi et al., 2015). Zhang et al. (2017), Li et al. (2015) and Li et al. (2016) have presented experimental evidence which demonstrates that the use of CO2 over water-based

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hydraulic fracturing can have absolute advantages in terms of required initiation pressures and resulting fracture morphology. Also, CO2 used in fracking can be recovered and used repeatedly. CO2 has been used in the past – in early applications, CO2 was pumped with water or hydrocarbon fluid; while in recent

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years, CO2-sand treatments have been developed that show significant improvements over other tested treatment types (Asadi et al., 2015). From the environmental perspective, fracking with CO2 is even more

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attractive. CO2 fracking eliminates the need for large amounts of water and reduces the possibility of contamination of drinking water and soil. Moreover, CO2 fracking presents a new way to utilize CO2. In the past few decades, researchers have been studying technologies to capture CO2 from anthropogenic sources such as power plants and other industrial processes, and have generated remarkable outcomes (IEA, 2013). At present, there are 8 large-scale CO2 capture and sequestration projects in the U.S.,

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capturing a total of 23.3 million metric tons of CO2 per year from natural gas processing, fertilizer production, hydrogen production, etc. (Global CCS Institute, 2015). The captured CO2 is either used for enhanced oil recovery or sequestrated in basalt formations, which can also be used for fracking in shale

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gas and oil wells if the infrastructure for getting CO2 to fracking sites is made available. Despite the clear technical advantages, the overall cost and operational safety issues are often

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cited as potentially significant drawbacks for using CO2 as a fracking fluid. Moridis (2017) and Zhang et al. (2017) highlighted that the need for additional surface equipment and transportation in pressurized containers can have a significant effect on costs, and logistical and operational challenges. Industry understands that these cost- and logistic-driven challenges would need to be overcome before CO2 can realistically be considered as a fracturing fluid of choice. Kohshour et al. (2016) summarizes this by indicating that, despite the technical promise, significant improvements would need to be made on logistics and reliability of suppliers for waterless technologies to become widely accepted. While costs and logistical problems are often cited as the main hindrance to wide acceptance of 4

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waterless fracking technologies, no previous studies have directly explored any of the salient economic and logistical aspects driving these decisions. This paper aims at shedding light at some of these salient aspects, by providing valuable guidance for estimating the cost of supplying CO2 for fracking operations.

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Although McCollum and Ogden (2006) propose technical-economical models for CO2 compression, transport, and storage, they focus on the application of CO2 sequestration, which is different from the scope of our paper. Pipelines are considered as the primary transportation mode in CO2 sequestration due

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to the continuous transportation of CO2 from the sources to the injection site; whereas in our application, since the active periods of fracking operations are relatively short and the demand occurs only when the

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fracking operations are taking place, instead of pipelines, trucks or rail tanks are used as the major CO2 transportation modes for CO2 fracking. More importantly, this study ultimately considers the optimization of the CO2 supply chain using a realistic case scenario while exploring some of the logistical issues involved with source and transportation mode selection. More specifically, cost models are presented to reliably estimate equipment, supply (incurred by collecting high purity CO2 from industrial sources),

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transportation (CO2 transportation from suppliers to well pads), and storage (for CO2 storage at sources and well pads) costs. Based on these cost models, an optimization model is proposed to select the optimal set of sources, determine the supply plan and transportation mode to fulfill the CO2 demand for a

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predetermined set of well pads within a given planning horizon, as well as the investment in related facilities and equipment, so that the present value of the overall cost is minimized. As will be shown later,

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the optimization model is a mixed-integer, nonlinear program, which can hardly be solved by existing software packages. To address this issue, a two-stage algorithm is proposed, which solves a discrete version of the model efficiently. The rest of the paper is organized as follows. Section 2 defines the CO2 source selection and

supply planning problem, based on which, the cost models for CO2 collection, storage, and transportation are built, and a mathematical model for the original problem is developed. In Section 3, a two-stage algorithm is proposed to solve a discrete version of the problem. A case study for three well pads in North Dakota is presented in Section 4 to illustrate the implementation of the model and algorithm; a sensitivity 5

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analysis is conducted over four key parameters of the problem. Some conclusions are drawn at the end of the paper.

2. Problem Statement and Formulation Problem Statement

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2.1.

In this problem we consider the scenario where there are m emerging well pads in a certain region in the U.S. Let W = {1,L, m} be the set of well pads. In well pad j , j ∈ W , there are l j wells to be fracked and

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each well requires D j metric tons (Mt) of CO2 for fracking. We need to generate a supply plan of CO2 for

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all well pads in W , assuming that each well will be fracked within a period of T days. Without loss of generality, it is assumed that in periods 1 to l1 , the wells in well pad 1 are fracked; then the remaining wells are fracked sequentially from well pad 2 to m . That is, the wells in well pad j are fracked in periods from ∑ kj −=11 lk + 1 to ∑ kj =1 lk . Let S = {1,L, n} be the set of candidate CO2 sources from industry, such as ammonia manufacturing, hydrogen production, and natural gas processing. The objective is to

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select the supply sources and determine the supply and transportation plan, and the investment in related facilities and equipment, such that the present value of the total cost throughout the planning horizon is minimized.

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Let J WT be the maximum number of storage tanks of capacity cWT (in Mt/tank) that need to be set up at any well pad to hold the inventory of CO2 and act as direct supply of CO2 for the fracking

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operation. Note that tanks of capacity cWT are the largest tanks that can be transported empty by truck from one well pad to the next. The fracking operation at any well in well pad j , j ∈W , lasts for t jf days. The CO2 transported to the well pad is first transferred to the storage tanks from truck tanks with a loss rate of CO2 (LRC) ∆ TW (in %/transfer). The storage tanks at the well pad have an LRC of ∆W (in %/filling period). Then, when the fracking operation starts, the CO2 used for fracking is pumped from the storage tanks. As shown in Figure 1, the filling operation begins t jfb days before the fracking

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operation starts to build up safety stock and is completed t jfe days before the fracking operation ends. Thus, the duration of the filling operation Pj (in days) is calculated as follows:

Pj = t jfb + t jf − t jfe , j ∈ W .

Pj

t jfe t jf

tiSI, j

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t jfb

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(1)

Figure 1. Illustration of a filling and fracking period.

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It is assumed that storage tanks at well pad j , j ∈W , are filled at a constant rate of D j Pj Mt/day. Then, there should be enough storage tanks to hold the CO2 that is filled in the first t jfb days, which is   storage tanks. Besides, we assume that 

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 t jfb D j equal to t jfb D j Pj Mt. To hold this much CO2, we need  WT  Pj c

the consumption rate is constant, which is equal to D j t jf Mt/day. Since the filling operation ends t jfe

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days before the fracking operation ends, the storage tanks should also hold adequate inventory for the  t jfe D j  fracking operation in the last t jfe days, which is equal to t jfe D j t jf Mt. At this time point,  f WT   t j c 

storage tanks are required. In other words, there exist two time points when the CO2 storage in the tanks at the fracking site can reach the maximum level. The first time point is when the fracking operation starts, as there is only feeding in and no consumption by this time point; and the second is when the filling operation ends, as after this time points, there is only consumption of CO2. If the filling rate is lower than

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the consumption rate, then the CO2 storage reaches its maximum level when the fracking operation starts; otherwise, the CO2 level starts to decrease only after the filling operation ends. When all fracking operations are completed at one well pad, the empty storage tanks are moved to the next well pad for the

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same usage. To calculate the maximum number of storage tanks to be set up, denoted by J WT , we need to consider the requirements at all sites. Therefore,

fb fe   t j D j   t j D j    = max max  WT  ,  f WT   , j ∈W  .   Pj c   t j c   

(2)

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J

WT

For the purpose of transportation, CO2 needs to be liquefied and kept under certain conditions,

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usually -20℃, 2 MPa. Therefore, when source i is selected, i ∈ S , the corresponding CO2 liquefaction facility and buffer storage tanks need to be built at the source. The facility and equipment incur capital, and operating and maintenance (O&M) costs, which are determined by the daily flow rate (or production rate) of CO2 at source i , denoted by fi (in Mt/day). Since each source i has limited daily capacity ciS (in

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Mt/day) and a minimum filling rate flb (in Mt/day), the daily CO2 flow rate should satisfy:

flb yi ≤ fi ≤ ciS yi , i ∈ S ,

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where yi is a binary variable to determine if source i is selected ( yi = 1 ) or not ( yi = 0 ). Source i

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collects tiSI, j days of CO2 at the daily flow rate fi , as the inventory (kept in storage tanks) in well pad j ,

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j ∈ W , before the filling operation begins, which should satisfy:

Pj + tiSI, j ≤ T , i ∈ S , j ∈ W .

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Let xi , j (in Mt/period) denote the amount of CO2 collected at source i for well pad j per period, i ∈ S ,

j ∈W . Then, xi, j should satisfy:

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xi , j ≤ fi Pj + tiSI, j , i ∈ S , j ∈W .

(5)

Two transportation modes are considered in the delivery of CO2 from sources to well pads: (i) trucks and (ii) truck-railcar-truck intermodal (Verma and Verter, 2010). The first mode uses trucks to 8

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carry CO2 from a source to a well pad directly. Let ∆ T be the LRC by truck (in %/hour) and cTT be the capacity of a truck tank (in Mt/truck). The LRC from a storage tank at a source to a truck tank is ∆ ST (in %/transfer). Due to the truck’s towing capacity limitations and the high demand of CO2 at the well pad

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during the fracking operation, when the travel distance is long, this mode would require a large fleet of trucks. To address this issue, in the second mode, the liquefied CO2 is first moved to an intermodal (origin) terminal by truck and then transferred to larger tanks carried by freight trains; let cR be the rail tank

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capacity (in Mt/tank) and ∆ R be the LRC of a rail tank (in %/hour). After the train arrives at an intermodal (destination) terminal, the CO2 is transferred to another fleet of trucks that deliver the CO2 to

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the well pad. The LRC is ∆TR (in %/transfer) when CO2 is transferred from a truck tank to a rail tank and ∆RT (in %/transfer) from a rail tank to a truck tank. Figure 2 illustrates the flow of CO2 throughout the

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entire process.

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Figure 2. Illustration of the two transportation modes. Additional assumptions considered in the proposed model are: (i) O&M costs occur at the end of

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each period, (ii) capital costs occur at the beginning of the first period, (iii) the filling operation of CO2 at a well pad can be done continuously throughout the day, (iv) sources can fill the tank with CO2 continuously throughout the day, (v) trucks can move the tanks from a source to a well pad continuously throughout the day, (vi) all sources use truck tanks of the same size to transport CO2, and (vii) each

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source should be able to keep a minimum level of inventory as safety stock, which is denoted by Iilb , i∈S .

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The proposed model focuses on the supply and transportation of CO2, where the total cost consists of three components: (i)

cost of collecting CO2, including the capital and O&M costs of liquefaction facilities and storage tanks at the CO2 sources,

cost of transportation, including the cost incurred by renting trucks and railcars, and

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(ii)

(iii) cost of additional equipment, including the capital and O&M costs of storage tanks at the well pads and truck tanks.

The objective is to minimize the present value of the total cost over a given planning horizon by

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determining the following (variables): set of selected sources,

(ii)

amount of CO2 collected from each selected source to each well pad,

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(i)

(iii) amount of inventory that each source keeps for each well pad,

(iv)

transportation mode from each selected source to each well pad,

(v)

capacity of the liquefaction facility at each selected source,

(vi)

combination of the storage tanks at the sources,

(vii) number of storage tanks to be set at each well pad during fracking operations, and (viii) number of truck tanks required.

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2.2.

Problem Formulation

2.2.1. CO2 Capture and Storage at Sources In order to transport CO2 efficiently, CO2 should be liquefied and held in tanks. To achieve this,

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liquefaction facilities need to be set up at the CO2 sources. The capital cost for the liquefaction facility at source i , denoted by CAPEX iLF (in $M), can be determined by a fitted quadratic function of the daily flow rate fi (in Mt/day) provided by one of our sponsors that manufacturers gas processing and

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compression equipment (Holtz and Neumayer, 2016):

CAPEX iLF = −10−5 fi 2 + 0.0208 fi + 1.9746 , i ∈ S .

(6)

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The O&M cost OMEX iLF (in $M/period) can be assumed to be proportional to the capital cost (Holtz and Neumayer, 2016), i.e.,

OMEX iLF = λ LF CAPEX iLF , i ∈ S ,

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where λ LF is the portion of capital cost corresponding to the O&M cost for liquefaction facilities. The

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CO2 liquefaction cost is included the O&M cost.

For storage tanks at the sources, we need to find the optimal combination of storage tanks such

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that the corresponding capital cost is minimized. Let J iST , w be the number of type w storage tanks built at source i , pwST and cwST be the price (in $M/tank) and capacity (in Mt/tank) of a type w storage tank,

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ST ST i ∈ S , w ∈ C , where C is the set of all storage tank types at the sources. Since the storage tanks

should be capable of holding the maximum inventory level, we have SI   xi , j ti , j   ST ST c J ≥ max max : j ∈W  , I ilb  , i ∈ S . ∑ST w i ,w   SI w∈C  Pj + ti , j   

(8)

The capital cost for the storage tanks at source i , i ∈ S , denoted by CAPEX iST (in $M), can be obtained by solving the following model: Model (ST):

CAPEX iST = min ∑ pwST J iST ,w , w∈C ST

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∑

s.t.

w∈C ST

cwST J iST ,w

SI   xi , j ti , j  lb  ≥ max max  : j ∈ W  , Ii  , SI  Pj + ti , j   

(10)

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ST J iST . , w ≥ 0, integer, w ∈ C

The O&M cost of the storage tanks at the sources (in $M/period) is also proportional to the capital cost, i.e.,

OMEX iST = λ ST CAPEX iST , i ∈ S ,

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PA

r 1 − (1 + r )

be the interest factor that converts a given present worth P into its

−N

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Let µr , N =

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where λ ST is the portion of capital cost corresponding to O&M cost for storage tanks at the sources.

uniform end-of-period payment A for multiple periods, given that r is the interest rate per period and N is the total number of periods to pay off the present worth (Park, 2012). In addition, let

1 − (1 + r ) r

−N

be the interest factor to convert all end-of-period payments to their present worth.

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µrAP ,N =

Then, provided that N% LF ( N% ST ) is the number of periods to pay off the capital cost of the liquefaction facility (storage tanks) at a source, the present value of the cost of CO2 collection can be calculated as

(

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follows:

PA LF PV Collection = µrAP + OMEX iLF + µrPA CAPEX iST + OMEX iST . , N ∑ µr , N% LF CAPEX i , N% ST

(12)

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i∈S

2.2.2. Transportation by Truck Given that the amount of CO2 collected at source i for well pad j is xi , j (in Mt/period), i ∈ S , j ∈W , the total number of truck trips required to transport CO2 from source i to well pad j per period, denoted by KiT, j , is calculated as follows: xi , j   K iT, j =  1 − ∆ ST TT  , i ∈ S , j ∈W . c  

(

)

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The number of trips between source i and well pad j a truck can do during Pj days is  24 Pj 

(t

ST

)

TW  , where t ST and t TW are the times to transfer CO2 from a storage tank to a + 2tiSW ,j +t 

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truck tank and from a truck tank to a storage tank (in hours), respectively, and tiSW , j is the time to travel from source i to well pad j (in hours). Let J iSW , j denote the number of truck tanks needed to transport

otherwise,

ST

)

SW T TW  + 2tiSW < 1 , J i, j = Ki, j ; ,j +t 

    .    

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J iSW ,j

  KiT, j  = 24 Pj    t ST + 2t SW + t TW i, j  

(t

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CO2 from source i to well pad j per period. Then, if  24 Pj 

(14)

Equivalently, J iSW , j can be calculated as follows:

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J iSW ,j

     T  K iT, j = min  K i , j ,  24 Pj      t ST + 2t SW + t TW i, j   

      , i ∈ S , j ∈W .       

(15)

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The cost of transporting xi , j Mt CO2 from source i to well pad j using trucks, denoted by CiT, j (in $M/ period), consists of two components, the cost when trucks are waiting, and the cost when trucks are

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traveling. Thus, CiT, j is calculated as follows:

(

)

TM  CiT, j =  t ST + t TW v TW + 2tiSW K T × 10 −6 , i ∈ S , j ∈W , ,j v   i, j

(16)

where vTM ( vTW ) is the cost rate (in $/hour/truck) when a truck is travelling (waiting).

2.2.3. Intermodal Transportation Let t TR be the time to transfer CO2 from a truck tank to a rail tank (in hours) and tiSR , j be the time to travel from source i to the nearby intermodal terminal for well pad j (in hours). Then, given that the amount of

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CO2 collected at source i for well pad j is xi, j (in Mt/period), i ∈ S , j ∈W , the number of railcar trips required to transport the CO2 from source i to well pad j per period, denoted by K iR, j , is calculated as

 K iR, j =  1 − ∆ ST 1 − ∆T 

(

)(

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follows: xi , j  , i ∈ S , j ∈W . R  

) (1 − ∆ ) c tiSR ,j

TR

(17)

Suppose that the frequency of trains from the intermodal terminal near source i to that near well pad j is

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θi, j trips/day. Then, the total number of train trips during the Pj days of filling operation is θ i , j Pj  ; and

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the maximum number of railcars needed per train is  K iR, j θi , j Pj   . To eliminate the idle time between two consecutive transfer operations, the number of trucks assigned to transport CO2 from source i to one

(

)

railcar in the intermodal terminal per period is equal to the minimum between  1 − ∆TR c R cTT  , the  

(

number of truck tanks of CO2 needed to fill a rail tank, and  t ST + t TR + 2tiSR ,j 

)

t TR  , the minimum 

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number of truck tanks required for each rail truck that ensures no idle time of truck tanks regardless of tank capacities. Then, the total number of truck tanks assigned to transport the CO2 from source i to the nearby intermodal terminal for well pad j per period, denoted by J iSR , j , is calculated as follows: R  t ST + t TR + 2tiSR    K iR, j    ,j  TR c   , i ∈ S , j ∈W . = min   1 − ∆   , cTT   t TR    θ i , j Pj    

(

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J iSR ,j

(18)

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Similarly, the number of trucks assigned to transport CO2 from each rail car in the intermodal

(

)

terminal to well pad j per period is equal to the minimum between  1 − ∆ RT c R cTT  and  

(

 t RT + t TW + 2t RW j 

)

RT RW t RT  , where t is the time to transfer CO2 from a rail tank to a truck tank and t j 

is the time to travel from the intermodal terminal near well pad j to well pad j . Denote the total number of truck tanks assigned to transport the CO2 from the intermodal terminal near well pad j to well pad j per period by J RW . For each well pad, the number of truck tanks required depends on the source i ∈ S j

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with the largest supply. Then, J RW is calculated as follows: j R  K R    t RT + t TW + 2t RW      i, j j RT c    J RW = max min 1 − ∆ ,    , j ∈W .     j i∈S cTT   t RT  θi , j Pj      

)

(19)

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(

The total number of truck trips from source i to its nearby intermodal terminal is

(

)

(

)

 1 − ∆ TR c R c TT  K iR, j per period. The number of trips from the intermodal terminal to well pad j is  

SC

 1 − ∆ RT c R c TT  K iR, j per period. Therefore, the total cost of transporting xi , j Mt CO2 from source i to  

well pad j using intermodal approach per period, denoted by CiI, j , is calculated by:

((

)(

R TM  TR c  CiI, j = t ST + t TR vTW + 2tiSR v K R × 10−6 1 − ∆  ,j TT  i , j c  

)

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)

    cR  cR  RM +    1 − ∆TR TT  t TR +  1 − ∆ RT TT  t RT  v RW + tiRR ,j v    c c      R TM  RT c  + t RT + t TW vTW + 2t RW K R × 10−6 ,  1− ∆ j v TT  i , j c  

(

(

)

)(

)

 R  K i , j × 10−6  

)

i ∈ S , j ∈W ,

(20)

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((

)

where vRM ( vRW ) is the cost rate (in $/hour/railcar) when a railcar is travelling (waiting) and tiRR , j is the time to travel from the intermodal terminal near source i to the intermodal terminal near well pad j by

EP

train (in hours). The transportation cost to transport the CO2 from source i to well pad j per period,

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denoted by CiTsp , j (in $M/period), is determined by either one the two transportation modes, i.e.,

(

)

T I , C iTsp , j = Ci , j z i , j + Ci , j 1 − zi , j

(21)

where zi, j is a binary variable to determine whether truck transportation ( zi, j = 1 ) or intermodal transportation ( zi , j = 0 ) is selected for the ( i, j ) pair. Let µrFP = ,N

1

(1 + r )

N

be the interest factor that

converts a future value F at the end of period N into its present value P given that the interest rate per period is r . Then, the present value of the total transportation cost, denoted by PV Tsp , can be calculated as follows: 15

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lj

PV Tsp = ∑ ∑ µ FP j∈W q =1

j −1

∑ CiTsp ,j .

(22)

r , q + ∑ lk i∈S k =1

2.2.4. Additional Equipment

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Additional equipment includes the truck tanks and the storage tanks at the well pad. The number of truck tanks to be purchased, denoted by J TT , should satisfy the demand at all well pads:

(

))

  SR J TT = max  J RW + ∑ J iSW j , j zi , j + J i , j 1 − zi , j  . j∈W  i∈S 

SC

(

(23)

Then, the present value of additional equipment is calculated as follows:

(

)

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PA WT WT PV AE = µrAP J + OMEX WT J WT + µrPA CAPEX TT J TT + OMEX TT J TT , , N µ r , N% WT CAPEX , N% TT

(24)

where CAPEX WT ( CAPEX TT ), OMEX WT ( OMEX TT ), N% WT ( N% TT ), are the capital cost (in $M), O&M cost (in $M/period), and the number of periods to pay off the capital cost of the storage tanks at the well pads (truck tanks), respectively. The O&M costs are proportional to the corresponding capital costs, i.e.,

OMEX X = λ X CAPEX X , X = TT , W T .

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(25)

where λWT ( λ TT ) is the corresponding portion for storage tanks at the well pads (truck tanks).

2.2.5. Demand and Supply

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Let FiT, j and Fi ,I j , i ∈ S , j ∈W , be the factors to convert the amount of CO2 collected at source i to the amount of CO2 pumped into the storage tanks at well pad j when using trucks and intermodal

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transportation, respectively:

(

)(

FiT, j = 1 − ∆ ST 1 − ∆T

(

)(

Fi ,I j = 1 − ∆ ST 1 − ∆T

) (1 − ∆ ) , i ∈ S , tiSW ,j

TW

j ∈W ,

) (1 − ∆ )(1 − ∆ ) (1 − ∆ )(1 − ∆ ) (1 − ∆ ) , i ∈ S , tiSR ,j

TR

R

tiRR ,j

RT

T

t RW j

TW

j ∈W ,

Since the demand of CO2 needs to be met, then

  F W  ∑  FiT, j zi , j + Fi ,I j 1 − zi , j  xi , j  ≥ D j , j ∈W . i∈S 

(

)

16

(26)

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where F W < 1 is an adjustment factor that considers the loss of CO2 in the storage tanks at the well pads.

2.2.6. Complete Model The complete mathematical model for source selection and supply planning, named Model (SSSP), is

planning horizon, which consists of three major components:

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provided in the Appendix. The objective is to minimize the present value of the total cost throughout the

(i) the cost of collecting CO2, i.e., the capital and O&M costs of the liquefaction facilities and

SC

storage tanks at sources,

(ii) the transportation cost, and

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(iii) the cost incurred by additional equipment, i.e., the capital and O&M costs of storage tanks at well pads and truck tanks.

The main constraints include the requirement that demand at the well pads should be fulfilled, such as inequality set (26), and the restriction that supply at each selected source should not exceed its capacity, such as inequality sets (3) to (5). Besides, the equalities and inequalities introduced in

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Subsections 2.2.1 to 2.2.5 are also placed as constraints in the model to describe the transportation and storage requirements of CO2. It should be noted that, to simplify the model, some of these equalities and inequalities are replaced by their equivalent linearized form, such as equation set (13); but still, some of

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the constraints cannot be linearized as such and thus retain their nonlinear form, such as equation set (6). A comprehensive mathematical formulation of Model (SSSP) is provided in the Appendix.

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Model (SSSP) is defined by integer variables, such as the number of truck tanks, binary variables to determine the selection of sources, and continuous variable to find CO2 flow rates from selected sources to well pads. On top of that, non-linear constraints, including the cost function for liquefaction facilities, transportation, etc., further complicate the model. The resulting mixed-integer, nonlinear programming model can hardly be solved using existing optimization software packages. In the next section, a two-stage algorithm is proposed to solve the problem.

3. Two-Stage Algorithm for Model (SSSP) 17

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In this section, we propose a new approach to efficiently solve a discrete version of the problem. We first discretize some of the continuous variables by restricting them to take a finite number of values within a given range, which will enable us to linearize the model. Then, the problem can be solved in two

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stages. In the first stage, we solve a large number of sub-problems considering all possible combinations of (CO2 source, well pad) pairs, transportation modes, a pre-determined number of CO2 supply levels at the sources, and a pre-determined number of CO2 inventory levels at the sources. As will be shown later,

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these sub-problems have simple forms and can be solved speedily. The results of these sub-problems are fed as input parameters into a linearized version of the original model, which is called the master problem.

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Then, in the second stage, optimization software can be used to solve the master problem. The solution of the master problem is essentially a combination of solutions from a subset of sub-problems that satisfy the demand requirements at the well pads and the CO2 source capacity restrictions at minimum cost.

3.1.

Linearized Model

Although a source may be capable of supplying CO2 to different well pads at any level of its capacity, in

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practice, the change of supply level from one well pad to another is not necessarily continuous. For example, suppose the maximum amount of CO2 a certain source can produce is 5,000 Mt/period; then, instead of supplying well pads with any amount of CO2 between 0 and 5,000 Mt/period, the supply level

EP

that the source can provide is restricted to be a multiple of 1,000 Mt. Let k represent the supply level. In this case, there exist five supply levels: 1,000 Mt ( k = 1 ), 2,000 Mt ( k = 2 ), 3,000 Mt ( k = 3 ), 4,000 Mt

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( k = 4 ), and 5,000 Mt ( k = 5 ). For the inventory level, which is also not necessarily continuous, given a well pad and the supply level from a source, we need to determine how much time is used to build up the inventory at the source before the filling operation begins at the well pad in a given period,. For example, given that a source needs to supply CO2 to a well pad within a 30-day period and the filling operation lasts for 10 days, we can choose a sub-period from 1 to 20 days to build up inventory at the source. For the convenience of presentation, we use ( i, j , k ,τ ) to represent the scenario where source i supplies well pad j at level k when βτ days of buffer storage is generated before the filling operation

18

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at well pad j begins, where β is a constant (in days) that determines the difference between two inventory-building time values and τ represents the level of inventory building time. For example, suppose the maximum time to build inventory is 8 days, and the interval is chosen as β = 2 days. Then,

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the inventory building time has 5 levels, which are: 0 day ( τ = 0 ), 2 days ( τ = 1 ), 4 days ( τ = 2 ), 6 days ( τ = 3 ), and 8 days ( τ = 4 ). For each ( i, j ) pair, i ∈ S , j ∈W , each k level corresponds to a certain amount of CO2 collected at source i for well pad j per period, denoted by xi , j ,k (in Mt/period). Let Li , j

SC

be the set of all possible k levels for ( i, j ) , and Ti ,SIj ,k be the set of all possible τ values for ( i, j , k ) .

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Now, we can define the binary variables that determine whether ( i, j , k ,τ ) occurs under a certain transportation mode:

1, when truck transportation is used for ( i, j , k ,τ ) ,

α iT, j ,k ,τ = 

0, otherwise,

i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k ,

1, when intermodal transportation is used for ( i, j , k ,τ ) , i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k . 0, otherwise,

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α iI, j ,k ,τ = 

When the transportation mode is truck, the range of xi , j ,k is (27)

EP

 D j  flb Pj ≤ xi , j ,k ≤ min  W T , ciS T  , i ∈ S , j ∈W , k ∈ Li , j ,  F Fi , j 

where the first inequality on the left states that the amount of CO2 collected at a source is bounded from

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below by the minimum flow rate at the source times the duration of the filling operation. The two terms in the parenthesis are the amount of CO2 to be collected at a source if the demand at well pad j is fulfilled by this source alone considering the loss during transportation, and the maximum amount that can be collected within a fracking period considering the source capacity, respectively. Similarly, when the transportation mode is intermodal, the range of xi , j ,k is

 D j  flb Pj ≤ xi , j ,k ≤ min  W I , ciS T  , i ∈ S , j ∈W , k ∈ Li , j .  F Fi , j 

19

(28)

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Then, we have:

   D j   D j  f lb Pj ≤ xi , j ,k ≤ max min  W T , ciS T  , min  W I , ciS T  , i ∈ S , j ∈W , k ∈ Li , j .  F Fi , j   F Fi , j  

( Pj + βτ )

(in Mt/day).

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Given xi , j ,k and τ values, the corresponding daily flow rate is f i , j , k ,τ = xi , j , k

(29)

Since the minimum flow rate and the capacity constraints for sources must be satisfied, we have:

xi , j ,k Pj + βτ

≤

ciS

⇒

xi, j ,k

− Pj ≤τ ≤

β

filb

β

− Pj

.

(30)

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flb ≤

xi , j ,k

ciS

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Also, considering constraint set (4), i.e., the maximum time to build inventory should not exceed the difference between fracking period and the filling time, the range of τ for a given xi , j ,k value is xi , j , k  xi , j , k    − Pj   S − Pj  T − P lb f  c    j max  i ,0  ≤ τ ≤ min  , i  , i ∈ S , j ∈W , k ∈ Li , j . β β    β     

(31)

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In scenario ( i, j , k ,τ ) , i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k , the capital cost of the liquefaction LF

facility, denoted by CAPEX i , j , k ,τ , given that fi , j ,k is the maximum daily flow rate, is calculated as

EP

follows: LF

CAPEX i , j ,k ,τ = −10−5 f i ,2j ,k + 0.0208 fi , j ,k + 1.9746 , i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k .

(32)

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Since the inventory to be kept at source i for well pad j is βτ fi , j ,k ,τ (in Mt), the corresponding

capital cost for storage tanks at source i , denoted by CAPEX i , j , k ,τ , given that βτ f i , j ,k ,τ is the highest ST

inventory level that source i keeps, can be determined by: ST  ST ST lb ST  CAPEX i , j ,k ,τ = min  ∑ pwST J iST , J iST , , w : ∑ cw J i , w ≥ max βτ f i , j ,k ,τ , I i , w ≥ 0, integer, w ∈ C w∈C ST  w∈C ST 

{

}

i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k . (33)

Equation (33) is a transformation of Model (ST). 20

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Given ( i, j , k ) , i ∈ S , j ∈ W , k ∈ Li , j , and the transportation mode being truck, the number of truck trips required to transport CO2 from source i to well pad j , denoted as K iT, j ,k , is calculated as

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follows: xi , j ,k   K iT, j ,k =  1 − ∆ ST TT  , i ∈ S , j ∈W , k ∈ Li , j . c  

(

)

(34)

pad j , denoted as J iSW , j , k , is calculated as follows:

      , i ∈ S , j ∈W , k ∈ Li , j .       

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J iSW , j ,k

     T  K iT, j , k = min  K i , j , k ,  24 Pj    ST   t + 2t SW + t TW i, j   

SC

From equation (15), the number of truck tanks per period required to transport CO2 from source i to well

(35)

And the transportation cost per period when trucks are used for ( i, j , k ) , denoted as CiT, j ,k , is calculated as follows:

(

)

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TM  CiT, j , k =  t ST + t TW vTW + 2tiSW K T × 10 −6 , i ∈ S , j ∈ W , k ∈ Li , j . ,j v   i, j ,k

(36)

Similarly, when intermodal is implemented, the total number of railcar trips transport CO2 from

EP

source i to well pad j , denoted as K iR, j ,k , can be determined by:  K iR, j , k =  1 − ∆ ST 1 − ∆T 

)(

) (1 − ∆ ) tiSR ,j

TR

xi , j , k   , i ∈ S , j ∈ W , k ∈ Li , j . cR 

(37)

AC C

(

The maximum number of rail tanks per train given ( i, j, k ) to transport CO2 from source i to

well pad j , given that the supply level is k , denoted as Ri , j ,k , is equal to  KiR, j ,k θi , j Pj   . From equation (18), the number of truck tanks required per period to transport CO2 from source i to the intermodal terminal when supplying well pad j , denoted as J iSR , j , k , is calculated as follows:

21

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R  t ST + t TR + 2tiSR    K iR, j , k    ,j  TR c   , i ∈ S , j ∈W , k ∈ Li , j . J iSR ,     , j ,k =   1 − ∆ cTT   t TR    θ i , j Pj    

(

)

(38)

j , denoted as J iRW , j , k , is calculated as follows: R  K iR, j , k     t RT + t TW + 2t RW j RT c    J iRW = min 1 − ∆ ,  , j ,k TT   RT   c t  θ i , j Pj      

(

)

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The number of truck tanks required per period to transport CO2 from the intermodal terminal to well pad

    , i ∈ S , j ∈W , k ∈ Li , j .  

(39)

SC

And the transportation cost per period, when intermodal is implemented for ( i, j, k ) , denoted as CiI, j ,k , can be determined by:

((

)(

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R TM  TR c  CiI, j ,k = t ST + t TR vTW + 2tiSR v 1 − ∆ K R × 10−6  ,j TT  i , j , k c   R R     c  c  RM +    1 − ∆TR TT  t TR +  1 − ∆ RT TT  t RT  v RW + tiRR ,j v    c  c    

)

(

)

((

(

)

)

)(

 R  Ki , j ,k × 10−6  

R TM  RT c  R + t RT + t TW vTW + 2t RW v 1 − ∆ K × 10−6 ,  j TT  i , j , k c  

)

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)

i ∈ S , j ∈ W , k ∈ Li , j , (40)

which is adapted from equation set (20).

Model (LSSSP):

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Then, Model (SSSP) can be reformulated as follows:

(

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PA LF min PV total = µ rAP + OMEX iLF + µ rPA CAPEX iST + OMEX iST , N ∑ µ r , N% ST CAPEX i , N% LF i∈S

lj

+ ∑ ∑ µ FP j∈W q =1

(

j −1



)



∑  ∑ CiT, j ,k ∑SI α iT, j ,k ,τ + ∑ CiI, j ,k ∑SI α iI, j ,k ,τ 

r , q + ∑ lk i∈S k =1

 k∈Li , j

τ ∈Ti , j , k

k∈Li , j

τ ∈Ti , j , k



)

PA WT WT + µ rAP J + OMEX WT J WT + µ rPA CAPEX TT J TT + OMEX TT J TT , (41) , N µ r , N% WT CAPEX , N% TT

s.t.

F W ∑ ∑ xi , j , k ∑ i∈S k ∈Li , j

τ ∈TiSI , j,k

(F

T i, j

)

α iT, j , k ,τ + Fi ,I jα iI, j , k ,τ ≥ D j , j ∈W ,

22

(42)

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∑

∑ (α iT, j ,k ,τ + α iI, j ,k ,τ ) ≤ 1 ,

k ∈Li , j τ ∈TiSI , j ,k

i ∈ S , j ∈W ,

CAPEX iLF ≥ ∑

∑SI CAPEX i, j ,k ,τ (αiT, j ,k ,τ + αiI, j ,k ,τ ), i ∈ S , j ∈W ,

CAPEX iST ≥ ∑

∑SI CAPEX i, j ,k ,τ (αiT, j ,k ,τ + αiI, j ,k ,τ ) , i ∈ S , j ∈W ,

LF

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k∈Li , j τ ∈Ti , j , k

ST

k∈Li , j τ ∈Ti , j , k

(43)

(44)

(45)

OMEX iX = λ X CAPEX iX , X = L F , ST , i ∈ S ,

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SC

    I SW T SR I    J TT ≥ ∑  JiRW α + J α + J α ∑ ∑ ∑ ∑ ∑ , j ,k i , j , k ,τ i ′, j , k i′, j , k ,τ , integer, i ∈ S , j ∈W , (46)  i′∈S k∈Li′, j  i′, j ,k τ ∈T SI i′, j ,k ,τ SI SI  k∈Li , j  τ τ ∈ ∈ T T i j k , , i ′, j , k i ′, j , k    

α iT, j ,k ,τ ,α iI, j ,k ,τ ∈ {0,1} , i ∈ S , j ∈W , k ∈ Li , j , τ ∈ Ti ,SIj ,k ,

(47)

The objective function (41) is equivalent to (49). Constraint set (42) is the demand constraint. Constraint set (43) ensures that, if source i is selected, for one well pad

j

, only one particular ( i , j , k ,τ ) scenario is

selected. Constraint set (44) determines the capital costs for the liquefaction facilities at the sources.

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Constraint set (45) calculates the capital cost for the storage tanks at the sources. Constraint set (46) calculates the total number of truck tanks to be purchased. Constraint set (47) imposes the binary restrictions.

Two-stage Algorithm

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3.2.

The proposed algorithm solves a discrete version of the problem. It can be divided into two stages. In the

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first stage, for each pair of source i and well pad j , i ∈ S , j ∈W , we first determine the ranges of xi, j under the two different transportation modes using set of inequalities (29). Then, we define the values of

xi , j ,k using the following equations: xi , j , k = k ×

η cTT Pj 1 − ∆ ST

, i ∈ S , j ∈W , k ∈ Li , j ,

(48)

where η is a constant term that determines the step size between two supply levels. The difference between two consecutive levels of supply is equal to the amount of CO2 collected at source i during the 23

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Pj days of the filling operation if η truck tanks of CO2 are transported from source i to well pad j per day.

LF

ST

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After the values of xi , j ,k are determined, for each xi , j ,k value, the range of τ values is determined from T SR RW I inequality (31), and used to calculate CAPEX i , j , k ,τ , CAPEX i , j , k ,τ , J iSW , j , k , Ci , j , k , J i , j , k , J i , j , k , and Ci , j , k ,

SI i ∈ S , j ∈ W , k ∈ Li , j , τ ∈ Ti , j ,k , with equation sets (32) to (40). The results of the first stage are used as

SC

parameters for Model (LSSSP) in the second stage. Model (LSSSP) is solved using an optimization software package. A diagram showing the step-by-step progression of the algorithm is provided in Figure

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EP

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3.

24

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EP

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SC

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Figure 3. Flow chart of the two-stage algorithm

4. Case Study: North Dakota 4.1.

Overview of the Case Study

In this section, a case study with three emerging well pads in North Dakota is presented to illustrate the implementation of the proposed model and two-stage algorithm. This is a realistic case study prepared in collaboration with one of our sponsors (Holtz and Neumayer, 2016) using precise parameters and cost

25

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functions. The data for candidate sources are obtained from the online tool FLIGHT developed by EPA (EPA, 2016). Figure 4 shows the locations of the well pads and the seven candidate sources in North Dakota. Each period lasts for one month (30 days). Detailed information on well pads and candidate

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sources is shown in Tables 1 and 2, respectively.

Well Pad 3

Well Pad 2

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7. Tioga Gas Processing Plant (72.01 Mt/day)

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3. BPE GPRP Stateline (101.97 Mt/day)

6. Robinson Lake Gas Plant (74.47 Mt/day)

5. BPE GPRP Garden Creek (74.52 Mt/day)

1. Great Plains Gasification Plant (682.86 Mt/day)

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4. BPE GPRP Grasslands Gas Plant (95.50 Mt/day)

Well Pad 1

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2. Badlands Gas Plant (159.84 Mt/day)

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Figure 4. The locations of well pads and sources (EPA, 2016)

Well pad j Location (coordinates) 1 2 3

(46.50, -102.81) (48.30, -103.59) (48.87, -101.18)

Table 1. Data on well pads

D j (Mt/period)

l j (wells)

t jf (days)

t jfb (days)

t jfe (days)

4,000 5,000 6,000

12 12 6

7 7 8

5 5 6

2 2 2

Table 2. Data on candidate sources (EPA, 2016) 26

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tiSW , j (hours/trip)

1 2 3 4 5 6 7

(47.36, -101.83) (46.09, -103.66) (48.23, -103.94) (47.59, -104.00) (48.03, -102.35) (47.85, -103.18) (48.39, -102.92)

682.86 159.84 101.97 95.5 74.52 74.47 72.01

j =1

j=2

j =3

2.00 1.25 2.85 2.43 2.60 2.08 3.28

2.90 3.43 0.55 1.38 1.28 1.35 0.72

2.35 5.20 2.77 3.62 1.72 2.82 1.87

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ci (Mt/day)

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Source i

Location (coordinates)

The seven candidate sources considered in this case study are sorted in decreasing order of the

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capacity in Table 2. The largest source has a capacity of 682 Mt/day while the smallest has a capacity of only 72 Mt/day. The travel time by truck from a source to a well pad ranges from 0.55 to 5.20 hours/trip. A number of assumptions are applied to generate the data for rail road transportation. It is assumed that the time to travel from the intermodal terminal near source i to the intermodal terminal near the well pad by freight train is twice of that to travel from the source to the well pad by truck, i.e.,

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SW tiRR , j = 2ti , j , i ∈ S , j ∈ W ; the time to travel from source i to its nearby intermodal terminal a is 0.5 hour,

i ∈ S , and from the intermodal terminal near well pad j to well pad j is also 0.5 hour, j ∈W . The

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number of train trips per day between each pair of source i and well pad j , θi, j , i ∈ S , j ∈W , is assumed to be uniformly distributed over 0.5, 1, 1.5, and 2. Table 3 shows the train trip frequency data.

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Table 3. Number of train trips between sources and well pads per day (trips/day) Source i

Well pad j 1 2 3

1

2

3

4

5

6

7

2 2 1.5

0.5 1 1

2 2 2

2 0.5 1

0.5 0.5 0.5

1.5 1.5 1.5

1.5 1 0.5

The capacity of storage tanks for the well pads ( cWT ) is 88 Mt/tank, and the unit price 27

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( CAPEX WT ) is $0.26M/tank. These are the largest storage tanks that can be moved empty by truck from one well pad to the next. The capacity of truck tanks ( cTT ) is 20 Mt/tank, and the unit price ( CAPEX TT )

Table 4. Storage tanks at sources

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is $0.1M/tank. There are 5 types of storage tanks for sources; the detailed information is shown in Table 4.

Storage tank type Capacity ( cwST ) Unit price ( pwST ) ($M/tank) 0.50 0.35 0.30 0.20 0.15

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1 2 3 4 5

(Mt/tank) 150 100 80 50 30

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w

Besides, it is assumed that the sources should keep at least 25 Mt of CO2 as the minimum inventory level. The LRC is 5%/transfer for all transfers ( ∆ ST , ∆ TW , ∆ TR , and ∆RT ), 2%/hour in the truck tank ( ∆ T ), 1%/hour in the rail tank ( ∆ R ), and 8%/filling period in the storage tank at the well pad ( ∆W ).

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It is assumed that the total time to pay off the capital cost of the liquefaction facility is 15 years and for other equipment is 5 years. Besides, the annual interest rate is 10%; then, the interest rate for a period of 12

(1 + 10% ) − 1 = 0.8% . The annual O&M cost is assumed to be 4% of the

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one month ( r ) is equal to

capital cost for all facilities and equipment. The ratio between capital cost and periodical O&M cost is

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calculated as follows:

λ X = 4% /12 = 0.33% ,

X = LF , ST , W T , T T .

4.2. Computational Results In this case study, the model was coded in MATLAB. CPLEX’s “cplexmilp” solver was used to solve Model (LSSSP). The code was run on an Intel Core i7-3610QM 2.30GHz PC with 8 GB of RAM and 64bit Windows 7 operating system. Table 5 shows the computational results under two different settings: (i) =2, =2, and (ii) =1, =1. Under the first setting, 1,181 scenarios were generated in Stage I; while 28

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under the second setting, over four times more scenarios were generated. As a result, it took more time to solve the problem under the second setting. Since the second setting led to a larger solution space, it produced a better solution than the first. In this case study, the total cost under the second setting is 1.4%

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lower than that under the first. Table 5. Computational results when (i) =2, =2, and (ii) =1, =1

=1, =1

1,181

4,795

1,285

5,161

13.52

13.24

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Number of scenarios generated in Stage I

=2, =2

Total number of variables in Stage II

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Total cost ($M) Total number of storage tanks for well pads to be purchased

31

31

Total number of truck tanks to be purchased

9

9

1, 3

1, 5

4.97

13.79

0.20

0.68

5.17

14.47

Sources selected CPU time of Stage I (sec)

Total CPU time (sec)

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CPU time of Stage II (sec)

In the following analysis, we focus on the results under the second setting, where sources 1 and 5

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are selected to supply CO2. Nine (9) truck tanks for CO2 transportation and 31 storage tanks for well pads need to be purchased. Detailed results are shown in Tables 6 to 8.

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Table 6 shows the information for each scenario selected. In the optimal solution, Source 1 acts as

the major CO2 supplier, which provides 96.31% of the total CO2. The selected sources are capable of producing adequate CO2 for the well pads during the filling operation, and only the minimum inventory level (25 Mt) is kept at the selected sources. Truck is the only transportation mode that is selected.

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Table 6. Information for each well pad and selected source pair

βτ

Well pad j

(Mt/period)

(days)

1 5 1 5 1 5

1 1 2 2 3 3

4,631.58 210.53 5,894.74 210.53 6,947.37 231.58

0 1 0 1 0 1

uiP, j **

0.68 0.28 0.86 0.28 0.92 0.28

0.23 0.09 0.29 0.09 0.34 0.10

5 1 8 1 8 1

Ri, j ***

J iSR ,j

J iRW ,j

CiI, j

($K/ period) 61.60 3.52 108.64 1.94 106.26 2.71

(railcars/ train)

(trucks/ period)

(trucks/ period)

3 1 3 1 4 1

12 4 12 4 16 4

12 4 12 4 16 4

($K/ period) 112.14 4.37 141.51 4.37 157.82 6.95

uiD, j : utilization rate of source i while CO2 is being collected for well pad j,

**

uiP, j : utilization rate of source i for the entire period when CO2 is used in well pad j,

***

Ri, j : maximum number of railcars per train to transport CO2 from the intermodal terminal near source

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*

uiD, j *

CiT, j

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Source i

J iSW ,j (trucks/ period)

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xi, j

i to the intermodal terminal near well pad j.

Table 7. Information for the selected sources

1 5

CAPEX iLF

Ii *

CAPEX iST

(Mt/day)

($M)

(Mt)

($M)

631.58 19.30

11.12 2.37

25 25

0.15 0.15

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Ii : maximum inventory level at source i

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*

fi

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Source i

30

w=1 0 0

JiST ,w (storage tanks)

w=2 0 0

w=3 0 0

w=4 0 0

w=5 1 1

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Table 8. Information for well pads *

LSC j **

(Mt/period) 1 2 3

C Tsp j

($K/period) 406.45 406.45 406.45

+

($K/period) 65.12 110.58 108.97

C Total j

++

($K/period) 471.57 517.03 515.42

UCC j +++ ($/Mt) 117.89 103.41 85.90

X j : total amount of CO2 collected from selected sources for well pad j per fracking period,

**

LSC j : lost rate of CO2 for well pad j,

***

C FE j : cost for facility and equipment (including capital and O&M costs) at well pad j per fracking

period, C Tsp : transportation cost from selected sources to well pad j per fracking period, j

++

SC

+

0.17 0.18 0.16

***

C Total : total cost at well pad j per fracking period, j

+++

UCC j : unit CO2 cost at well pad j.

+++

UCC j : unit CO2 cost at well pad j.

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*

4,842.11 6,105.26 7,178.95

C FE j

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Well pad j

Xj

Although the periodic demand varies among well pads, the same facilities and equipment are shared by all well pads. Therefore, as shown Table 8, the periodic costs for facility and equipment are the

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same for all well pads. Notice that the transportation cost of a well pad depends on its demand and the travel times from the sources to the well pad. For example, although well pad 1 has the highest demand, the travel time from its major source (source 1) is less than that from source 1 to well pad 2; thus, its

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transportation cost is lower than that of well pad 2. The unit cost of using CO2 (CUC) at a well pad is equal to the total cost per period divided by the amount of CO2 used at the well pad per period. For

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example, the demand of in well pad 1 is 4,000 Mt/period and the total cost is $471.57K/period; thus, the corresponding CUC is $471.57K / 4,000Mt = $117.89/Mt. In this problem, the CUCs at the three well pads are $117.89, $103.41, and $85.90 per Mt; and the overall (average) CUC is equal to the overall cost divided by the overall demand, i.e.,

m l j ∑ mj =1 D j l j = $103.86 / Mt . Figure 5 shows the ∑ j =1CTotal j

breakdown of the overall CUC and CUC at each well pad. Note that the major components of the overall CUC are the costs for the liquefaction facility, storage tanks, and transportation, contributing 36.89%, 38.86%, and 18.47% of the total cost, respectively.

31

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Figure 5. Breakdown of CUC for each well pad in each period

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4.3. Sensitivity Analysis

It can be inferred from the results of the case study that four parameters play a major role in determining the CUCs: length of the fracking period, starting time of the filling operations, overall LRC, and payoff

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time. In the following two subsections, sensitivity analysis is performed for the first two factors and the last two factors, respectively.

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4.3.1. Fracking period and starting time As shown in Table 6, only 9% to 34% of the source capacities are used, and the liquefaction facilities and tanks are left idle for nearly two thirds of the entire planning horizon. Therefore, if the period length decreases, the utilization of the selected CO2 sources, liquefaction and storage equipment can be improved, and the CUCs will decrease accordingly. The starting time of the filling operation also has a significant impact on the CUCs. It is assumed that the filling operations start 5 days before the fracking operations begin. As a result, a total of 31 storage tanks are required at the well pads, leading to a large

32

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capital cost. By reducing the starting time of the filling operations, the number of storage tanks at the well pads will decrease and thus the capital cost will be reduced. In this subsection, the problem is solved repeatedly with four period length values: 30, 25, 20, and 15 days, and four starting time values: 5, 4, 3,

and the results are shown in Table 9 and Figure 6.

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and 2 days (assuming all well pads share the same starting time). Hence, a total of 16 cases are examined

Table 9. CUC values in different test cases (in $/Mt)

60.94 76.32 91.07 103.89

60.00 74.04 88.17 101.89

4 days

5 days

60.52 74.44 88.38 102.35

61.39 75.57 89.71 103.86

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3 days

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2 days

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Starting time Period length 15 days 20 days 25 days 30 days

Figure 6. Breakdown of overall CUC under different period lengths and starting times 33

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The lowest CUC is $60.00/Mt, which is achieved when the period length is 15 days and the starting time is 3 days; and the highest is $103.89/Mt, which occurs when the period length is 30 days and the starting time is 2 days (see Table 9). Although the period length has little impact on the transportation

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cost, with a shorter period length, less capital costs for facilities and equipment are counted into the planning horizon, leading to lower CUC. It should be noted that the period length should be long enough for the sources to produce adequate CO2 and build inventory for the well pads. For the same period length,

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as the starting time decreases, the number of storage tanks required at the well pads also decreases, so does the associated capital cost. However, with less time for the filling operation, the selected sources

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have to either increase the flow rate or the inventory level so that the demand at the well pad can be fulfilled. As shown in Figure 6, when the starting time declines, the cost for storage tanks at the well pads also declines; but the sum of the costs for the liquefaction facilities and buffer storage tanks at sources rises.

4.3.2. LRC and payoff time

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In this case study, the overall loss of CO2 is 17.45%. The influence of LRC on UUC is multi-fold. Firstly, with higher LRC values, more CO2 has to be collected at the sources to satisfy the demand at the well pads. Thus, liquefaction facilities with larger capacities or extra storage tanks are required at the selected

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sources, which lead to extra costs. Additionally, more truck (or train) trips will be needed to transport CO2 to the well pads; hence, higher transportation cost will occur and more truck tanks may be required. In the

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proposed model, we only count the capital costs that are paid during the planning horizon. Therefore, the longer the payoff time, less capital costs are counted into the overall cost. In this subsection, we examine the results when the LRCs change by -20%, -10%, 10%, and 20%, and when the times to pay off the capital costs change by -20%, -10%, 10%, and 20%. The results are shown in Figures 7 and 8. Obviously, as the LRC values increase, the overall LRC also increases. It should be noted that there exists a jump of the overall LRC from 0.1382 to 0.1744 when the change of LRCs goes from -10% to 0. This is caused by the discretization of the supply amounts. As LRCs increase, more CO2 needs to be collected at the sources so that the demand can be met. Thus, the selected sources need to reach the next 34

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level of supply to satisfy the demand once the current supply level violates the demand constraint. As a result, some CO2 will be wasted since in most cases not all additional CO2 collected is required. With

η = 1 , the interval between two supply levels is between 200 to 240 Mt CO2 per period (the exact amount

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could be wasted per period (4% to 6% of the demand per period).

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depends on the change of LRC values), which implies that, in some extreme cases, nearly 240 Mt CO2

Figure 7. Overall LRC under different LRC settings.

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Figure 8 shows the change of CUC under different settings of LRCs and payoff times when period length is 30 days and starting time is 5 days. The lowest overall CUC is $95.26/Mt of CO2, which is achieved

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when the LRCs are 20% less than their original values and the payoff times are 20% more than their original values; while the highest overall CUC is $115.98/Mt CO2, which is obtained when the LRCs are 20% more than their original values and the payoff times are 20% less than their original values. As shown in Figure 8, there exists a clear trend that the overall CUC decreases when the LRCs decrease and the payoff times increase. With lower LRCs, less amount of CO2 is required; and with longer payoff time, the overall capital cost per period is lower.

35

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Figure 8. Breakdown of overall CUC under different LRC and payoff time settings.

5. Conclusions and Recommendations for Future Research

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This paper provides a solution for the CO2 source selection and supply planning problem for fracking operations in shale gas and oil wells. Cost models for CO2 collection, storage and transportation have been built. A mathematical model for the original problem has been formulated as a mixed-integer and

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highly nonlinear program. Since a global solution for the problem is difficult to find, a two-stage algorithm has been proposed to solve a discrete version of the problem. The algorithm first requires the

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solution of a large set of simple sub-problems (scenarios) that consider all pairs of sources and well pads at different CO2 supply levels and inventory levels at the sources. Then, the subset of scenarios that defines the optimal solution is determined by solving a large mixed-integer linear programming model. A case study with three emerging well pads and seven candidate sources in North Dakota is

analyzed and solved. The results show that the two-stage algorithm solves the problem efficiently. Sensitivity analysis has also been conducted on the period length, starting time of the filling operation, overall LRC, and payoff times. It is shown that decreasing the period length reduces the CUC, but it

36

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should be long enough for the sources to produce the necessary CO2 and build initial inventory for the fracking operations. Postponing the filling operation reduces the number of storage tanks required at the well pad; however, this requires to either raise the flow rates or inventory levels at the sources to fulfill

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the demand, which incurs extra cost in liquefaction facilities and storage tanks at the sources. In the case study, starting the filling operation three days before the fracking operation begins achieves the minimum cost. Increasing payoff times or decreasing LRCs will reduce the CUC.

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There exist several potential extensions of the current work. Firstly, the computational efficiency of the proposed algorithm heavily depends on the number of scenarios generated in the first stage. To

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improve the efficiency of the algorithm, further study should be conducted to identify and remove nonoptimal scenarios involved in the second stage. Secondly, it should be noted that CO2 emissions occur during every operation from CO2 collection, to transfer, to transportation, and to storage. The control of CO2 emissions should be regarded as an additional criterion when selecting sources and evaluating transportation modes. A multi-criteria model should be formulated and an efficient algorithm should be

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developed to estimate the Pareto front. Thirdly, the current model does not consider the case that a portion of the CO2 injected into the wells can be recovered. In this situation, lower supply levels at the sources may be required and the transportation cost could be significantly reduced.

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Acknowledgements

This work was supported in part by the GE Global Research Oil & Gas Technology Center in Oklahoma

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City. The authors thank the five referees for providing useful comments that helped to improve the paper significantly.

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fracturing in shale with H2O, CO2 and N2. In 49th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association.

Li, X., Feng, Z., Han, G., Elsworth, D., Marone, C., Saffer, D., and Cheon, D. S. (2016). Breakdown

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Appendix: Complete Mathematical Model The proposed model includes the following variables:

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CAPEX WT capital cost for storage tanks at the well pad (in $M),

OMEX WT O&M cost for storage tanks at the well pad (in $M), daily flow rate of source i (in Mt/day), i ∈ S ,

fi

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CAPEX iLF capital cost of the liquefaction facility at source i (in $M), i ∈ S ,

OMEX iLF O&M cost of the liquefaction facility at source i (in $M/period), i ∈ S , tiSI, j

t jfb D j Pj Mt, i ∈ S , j ∈W ,

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number of days source i collects as initial CO2 inventory for well pad j , equivalent to

number of type w storage tanks to be built at source i , i ∈ S , w ∈ C ST ,

J iST ,w

CAPEX iST capital cost for the storage tanks at source i (in $M), i ∈ S ,

O&M cost of the storage tanks for source i (in $M/period) , i ∈ S ,

xi , j

amount of CO2 collected at source i for well pad j (in Mt/period), i ∈ S , j ∈W ,

K iT, j

number of truck trips required to transport CO2 from source i to well pad j per period, i ∈ S ,

j ∈W ,

number of truck tanks needed to transport CO2 from source i to well pad j per period, i ∈ S ,

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J iSW ,j

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OMEX iST

j ∈W ,

CiT, j

cost of transporting CO2 from source i to well pad j using trucks (in $M/period), i ∈ S ,

j ∈W , K iR, j

number of railcar trips required to transport the CO2 from source i to well pad j per period, i ∈ S , j ∈W ,

41

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J iSR ,j

number of truck tanks assigned to transport CO2 from source i to the intermodal terminal nearby for well pad j per period, i ∈ S , j ∈W ,

J RW j

pad j to the well pad j per period, i ∈ S , j ∈W ,

CiI, j

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number of truck tanks assigned to transport CO2 from the intermodal terminal near the well

cost of transporting CO2 from source i to well pad j using intermodal approach per period

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(in $M/period), i ∈ S , j ∈W ,

binary variable to determine whether truck transportation ( zi, j = 1 ) or intermodal

zi, j

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transportation ( zi , j = 0 ) is selected to transport CO2 from source i to well pad j , i ∈ S ,

j ∈W , CiTsp ,j

transportation cost to transport CO2 from source i to well pad j per period (in $M/period), i ∈ S , j ∈W ,

J TT

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number of truck tanks to be purchased,

CAPEX TT capital cost for truck tanks (in $M), O&M cost for truck tanks (in $M).

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OMEX TT

By considering the objective of minimizing the present value of the total cost throughout the planning

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horizon, and the constraints for the demand at the well pads, the capacities of the sources, and the transportation and storage of CO2, the complete model is formulated as follows: Model (SSSP):

min PV total = PV Collection + PV Tsp + PV AE

(

PA LF = µrAP + OMEX iLF + µrPA CAPEX iST + OMEX iST , N ∑ µr , N% LF CAPEX i , N% ST i∈S lj

+ ∑ ∑ µ FP

j −1

) (49)

∑ CiTsp ,j

j∈W q =1 r , q + ∑ lk i∈S

+ µrAP ,N

(

k =1

µrPA CAPEX WT J WT , N% WT

)

+ OMEX WT J WT + µrPA CAPEX TT J TT + OMEX TT J TT , , N% TT 42

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  F W  ∑  FiT, j zi , j + Fi ,I j 1 − zi , j  xi , j  ≥ D j , j ∈W , i∈S 

(

s.t.

)

(

)

0 ≤ xi , j ≤ fi Pj + tiSI, j , i ∈ S , j ∈W , 0 ≤ tiSI, j ≤ T − Pj , i ∈ S , j ∈W ,

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flb yi ≤ fi ≤ ciS yi , i ∈ S ,

(51)

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CAPEX iLF = −10−5 fi 2 + 0.0208 fi + 1.9746 , i ∈ S ,

(50)

CAPEX iST = ∑ pwST J iST ,w , i ∈ S , w∈C ST

xi , j tiSI, j

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∑ w∈C ST

ST J iST , w cw

≥

Pj + tiSI, j

, i ∈ S , j ∈W ,

∑ J iST, w cwST ≥ Iilb ,

w∈C ST

i∈S ,

ST J iST , , w ≥ 0 , integer, i ∈ S , w ∈ C

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T I CiTsp , j = Ci , j + Ci , j , i ∈ S , j ∈ W ,

(

K iT, j ≥ 1 − ∆ ST

xi , j

, integer, i ∈ S , j ∈W ,

EP (

      − M (1 − zi , j ) , i ∈ S , j ∈W ,       

)

(

(

)(1 − ∆ ) (1 − ∆ ) c SR T ti , j

TR

xi , j R

(53)

)

(54)

, integer, i ∈ S , j ∈W ,

(55)

TM  CiT, j ≥  t ST + t TW vTW + 2tiSW K T × 10 −6 − M 1 − zi , j , i ∈ S , j ∈ W , ,j v   i, j

K iR, j ≥ 1 − ∆ ST

(52)

T

    K iT, j  T  ≥ min  K i , j ,  24 Pj      t ST + 2t SW + t TW i, j   

AC C

J iSW ,j

)c

R  t ST + t TR + 2tiSR    K iR, j    ,j  TR c   − M (1 − zi , j ) , integer, i ∈ S , j ∈W , (56) J iSR ≥ min 1 − ∆ ,      ,j cTT   t TR    θi , j Pj    

(

)

43

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RT TW RW   c R   t + t + 2t j J RW ≥ min   1 − ∆ RT TT  ,  j c   t RT  

((

)

)(

   K iR, j   − Mzi , j , integer, i ∈ S , j ∈W ,      θ i , j Pj  

R TM  TR c  1 − ∆ CiI, j ≥ t ST + t TR vTW + 2tiSR v K R × 10−6  ,j TT  i , j c       cR  cR  RM +    1 − ∆TR TT  t TR +  1 − ∆ RT TT  t RT  v RW + tiRR ,j v    c c     

)

(

((

)

(

)

)(

 R  K i , j × 10−6  

R TM  RT c  + t RT + t TW vTW + 2t RW − ∆ v 1 K R × 10−6 − Mzi , j ,  j TT  i , j c  

)

(

)

i ∈ S , j ∈W ,

SC

)

(57)

)

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(

(58)

SR J TT ≥ J RW + ∑ J iSW j , j + J i , j , integer, j ∈ W ,

(59)

 D j t jfb D j t jfe  J WT ≥ max  W , f W  , integer, j ∈W ,  Pj c t j c 

(60)

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i∈S

OMEX iX = λ X CAPEX iX , i ∈ S , X = L F , ST ,

yi ∈{0,1} , i ∈ S ,

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zi , j ∈ {0,1} , i ∈ S , j ∈W .

(61) (62)

Constraint set (50) represents inequality set (5) considering the non-negativity amount of CO2 collected at each source. Constraint sets (51) and (52) represent equation sets (6) and (13), respectively. Constraint set

inactive when zi, j = 0 . Constraint set (54) calculates the cost of transporting CO2 from each source

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( i, j )

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(53) stands for equation set (15), where M is a arbitrarily large number intended to make constraint

to each well pad using trucks. Constraint sets (55), (56), and (57) characterize equation sets (17), (18), and (19), respectively. Constraint set (58) calculates the cost of transporting CO2 from each source to each well pad by intermodal. Constraint sets (59) and (60) stand for equation set (23) and (2), respectively. Model (SSSP) is a mixed-integer nonlinear (non-convex) model, which can hardly be solved by existing optimization software packages. Regarding the nonlinearity, it should be noted that some of the constraints with ceiling or flooring functions, such as (56) and (57), can be linearized by introducing additional variables; however, other constraints, like (50) and (51), are difficult to linearize. 44

Table 1. Data on well pads

(46.50, -102.81) (48.30, -103.59) (48.87, -101.18)

l j (wells)

t jf (days)

t jfb (days)

t jfe (days)

4,000 5,000 6,000

12 12 6

7 7 8

5 5 6

2 2 2

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1 2 3

D j (Mt/period)

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Well pad j Location (coordinates)

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1

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Table 2. Data on candidate sources (EPA, 2016)

tiSW , j (hours/trip)

1 2 3 4 5 6 7

(47.36, -101.83) (46.09, -103.66) (48.23, -103.94) (47.59, -104.00) (48.03, -102.35) (47.85, -103.18) (48.39, -102.92)

682.86 159.84 101.97 95.5 74.52 74.47 72.01

j =1 2.00 1.25 2.85 2.43 2.60 2.08 3.28

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j=2

j =3

2.90 3.43 0.55 1.38 1.28 1.35 0.72

2.35 5.20 2.77 3.62 1.72 2.82 1.87

SC

ci (Mt/day)

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Source i

Location (coordinates)

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Table 3. Number of train trips between sources and well pads per day (trips/day) 1

2

3

4

2 2 1.5

0.5 1 1

2 2 2

2 0.5 1

5

6

7

0.5 0.5 0.5

1.5 1.5 1.5

1.5 1 0.5

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Well pad j 1 2 3

SC

Source i

3

Table 4. Storage tanks at sources

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Storage tank type Capacity ( cwST ) Unit price ( pwST ) ($M/tank) 0.50 0.35 0.30 0.20 0.15

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1 2 3 4 5

(Mt/tank) 150 100 80 50 30

SC

w

4

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Table 5. Computational results when (i) =2, =2, and (ii) =1, =1

=1, =1

Number of scenarios generated in Stage I

1,181

4,795

Total number of variables in Stage II

1,285

5,161

13.52

13.24

31

31

9

9

1, 3

1, 5

4.97

13.79

0.20

0.68

5.17

14.47

Total cost ($M)

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Total number of storage tanks for well pads to be purchased Total number of truck tanks to be purchased Sources selected CPU time of Stage I (sec) CPU time of Stage II (sec)

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Total CPU time (sec)

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=2, =2

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Source i

Well pad j

(Mt/period)

(days)

1 5 1 5 1 5

1 1 2 2 3 3

4,631.58 210.53 5,894.74 210.53 6,947.37 231.58

0 1 0 1 0 1

CiT, j ($K/ period) 61.60 3.52 108.64 1.94 106.26 2.71

uiD, j *

uiP, j **

(trucks/ period)

0.68 0.28 0.86 0.28 0.92 0.28

0.23 0.09 0.29 0.09 0.34 0.10

5 1 8 1 8 1

Ri, j ***

J iSR ,j

J iRW ,j

CiI, j

(railcars/ train)

(trucks/ period)

(trucks/ period)

($K/ period)

3 1 3 1 4 1

12 4 12 4 16 4

12 4 12 4 16 4

112.14 4.37 141.51 4.37 157.82 6.95

uiD, j : utilization rate of source i while CO2 is being collected for well pad j,

uiP, j : utilization rate of source i for the entire period when CO2 is used in well pad j,

Ri, j : maximum number of railcars per train to transport CO2 from the intermodal terminal near source i to the intermodal terminal near well pad j.

TE D

***

EP

**

AC C

*

J iSW ,j

SC

βτ

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xi , j

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Table 6. Information for each well pad and selected source pair

6

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1 5

CAPEX iLF

Ii *

CAPEX iST

(Mt/day)

($M)

(Mt)

($M)

631.58 19.30

11.12 2.37

25 25

0.15 0.15

JiST ,w (storage tanks)

w=1 0 0

w=2 0 0

EP

TE D

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Ii : maximum inventory level at source i

AC C

*

fi

w=3 0 0

SC

Source i

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Table 7. Information for the selected sources

7

w=4 0 0

w=5 1 1

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1 2 3

LSC j **

(Mt/period) 4,842.11 6,105.26 7,178.95

0.17 0.18 0.16

C FE j

***

C Tsp j

($K/period) 406.45 406.45 406.45

+

($K/period) 65.12 110.58 108.97

C Total j

++

($K/period) 471.57 517.03 515.42

UCC j +++ ($/Mt) 117.89 103.41 85.90

X j : total amount of CO2 collected from selected sources for well pad j per fracking period,

**

LSC j : lost rate of CO2 for well pad j,

***

+

*

C FE j : cost for facility and equipment (including capital and O&M costs) at well pad j per fracking

C Tsp j

++

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*

Xj

SC

Well pad j

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Table 8. Information for well pads

period, : transportation cost from selected sources to well pad j per fracking period,

: total cost at well pad j per fracking period, C Total j

UCC j : unit CO2 cost at well pad j.

+++

UCC j : unit CO2 cost at well pad j.

AC C

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+++

8

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Table 9. CUC values in different test cases (in $/Mt) 3 days

4 days

5 days

60.94 76.32 91.07 103.89

60.00 74.04 88.17 101.89

60.52 74.44 88.38 102.35

61.39 75.57 89.71 103.86

SC

2 days

AC C

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Starting time Period length 15 days 20 days 25 days 30 days

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tiSI, j

t jfb

t jfe

SC

t jf

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Pj

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Figure 1. Illustration of a filling and fracking period.

1

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Figure 2. Illustration of the two transportation modes.

2

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Figure 3. Flow chart of the two-stage algorithm

3

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Well Pad 3

Well Pad 2

7. Tioga Gas Processing Plant (72.01 Mt/day)

SC

3. BPE GPRP Stateline (101.97 Mt/day)

6. Robinson Lake Gas Plant (74.47 Mt/day)

2. Badlands Gas Plant (159.84 Mt/day)

1. Great Plains Gasification Plant (682.86 Mt/day)

TE D

4. BPE GPRP Grasslands Gas Plant (95.50 Mt/day)

M AN U

5. BPE GPRP Garden Creek (74.52 Mt/day)

Well Pad 1

AC C

EP

Figure 4. The locations of well pads and sources (EPA, 2016)

4

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AC C

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Figure 5. Breakdown of CUC for each well pad in each period

5

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Figure 6. Breakdown of overall CUC under different period lengths and starting times

6

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Figure 7. Overall LRC under different LRC settings.

7

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Figure 8. Breakdown of overall CUC under different LRC and payoff time settings.

8

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Research Highlights A mathematical model for the CO2 source selection and supply planning is provided. The model optimizes the CO2 supply chain for fracking operations.

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The objective is to minimize the present value of equipment, transportation, and O&M costs. A two-stage algorithm is proposed to solve a discrete version of the problem efficiently. A case study in North Dakota is presented to illustrate the model and algorithm.

AC C

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Sensitivity analysis is conducted to analyze the influences of four key parameters.