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Optimal Water Management under Uncertainty for Shale Gas Production Luis Fernando Lira-Barragán, José María Ponce-Ortega, Gonzalo Guillén-Gosálbez, and Mahmoud M El-Halwagi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b02748 • Publication Date (Web): 14 Jan 2016 Downloaded from http://pubs.acs.org on January 26, 2016

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Industrial & Engineering Chemistry Research

Optimal Water Management under Uncertainty for Shale Gas Production Luis Fernando Lira-Barragán,a,b José María PonceOrtega,b* Gonzalo Guillén-Gosálbezc,d and Mahmoud M. El-Halwagia,e a

Chemical Engineering Department, Texas A&M University, College Station TX, 77843, USA

b

Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mich., 58060, México c

Departament d’Enginyeria Quimica (DEQ), Escola Técnica Superior

d’Enginyeria Quimica (ETSEQ), Universitat Rovira i Virgili (URV), 43007 Tarragona, Spain d

Centre for Process Integration, School of Chemical Engineering and

Analytical Science, The University of Manchester, Manchester M13 9PL, UK e

Adjunct Faculty at the Chemical and Materials Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia

* Corresponding author, [email protected]; Tel. +52 443 3223500 ext. 1277; Fax. +52 443 3273584

ACS Paragon Plus Environment


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Abstract

This paper presents a mathematical programing formulation for synthesizing water networks associated with shale gas hydraulic fracturing operations while accounting for the system uncertainty. The proposed formulation yields a strategic planning that minimizes the cost considering water requirements as well as equipment capacities for treatment technologies, storage units, and disposals. The key uncertainties pertain to the water usage for fracturing and the time-based return of flowback water. The objective function is aimed at the minimization of the total annual cost, which accounts for the operating and capital costs associated with the water network. The developed model addresses the scheduling problem associated with shale gas production, which provides as output the completion phases for all the projected wells. This information is used in estimating the periods with water requirements and where flowback water can be collected. The proposed methodology includes an analysis of the optimal equipment size. An illustrative example is presented to show the capabilities of the proposed methodology.

Keywords: Water networks; Uncertainty; Shale gas; Hydraulic fracturing (fracking); Flowback water reuse.


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1. Introduction Recent advances in directional drilling have led to a significant growth in the extraction and production of shale gas in North America. The Energy Information Administration (EIA) has estimated that 46% of the total natural gas production in 2035 will be provided by shale gas in U.S.1 An essential step to properly extract the shale gas is hydraulic fracturing, which is also called fracturing. This process demands huge amounts of water to formulate the fracturing fluid that is injected into the wells to release the gas. Different challenges are open and still merit further attention. These include water issues associated with shale gas production, such as satisfying the water needed by the completion phase, the collection of the flowback water, the optimal equipment design and operation for treating and storing used water, the optimal disposition of wastewater streams and finally the optimization of the transportation tasks. If flowback water is treated and stored, it can be reused to reduce the fresh water consumption and the wastewater streams, which can in turn help decrease the corresponding costs. These concerns are very similar to those addressed in the synthesis of water networks in the downstream processing industries. Several works for the synthesis of recycle and reuse water networks in industrial facilities have been proposed. These techniques have been classified as graphical, algebraic, or mathematical approaches. Among the graphical methodologies, we find the pinch point techniques (see for example Wang and Smith,2 Kazantzi and El-Halwagi,3 Foo,4 Deng et al.5 and Bandyopadhyay et al.6). Examples of algebraic approaches include the works of Almutlaq et al.,7 and Qin et al.8 Mathematical approaches make use of formal mathematical programming techniques (Gabriel and El-Halwagi,9 Yang and Grossmann,10 Yu et al.,11 Ponce-Ortega et al.,12,13 Nápoles-Rivera et al.14 and Lira-Barragán et al.15-17). All previous formulations have been developed for traditional industrial facilities and cannot be readily applied to shale gas production. Uncertainty is an inherent condition in the design and planning of energy production systems and water management networks. In this sense, the uncertainty has been considered in the supply chain planning to determine the involved risk.18,19 Furthermore, this important aspect has been widely studied for the synthesis of water networks. In this context, Nápoles-Rivera et al.20 proposed a methodology for the sustainable use of water under parametric uncertainty in macroscopic water networks, Broad et al.21 incorporated



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uncertainty in a systematic approach for the water distribution systems, whereas Khor et al.22 integrated the synthesis of a water network under uncertainty with risk management. Furthermore, Ahn and Kang23 introduced a long-term plan for sustainable water supply considering future uncertainties in water demands and source availability. Islam et al.24 examined the uncertainties in traditional water supply systems. Whereas, Yang et al.25 developed an approach for water networks accounting for realistic and unit-specific shortcut models as well as the uncertainty in mass load of contaminants. Even, Gao and You26 designed and planned shale gas supply chains under uncertainty including the drilling, production, processing and transportation. Additionally, owing to the shale gas field represents a novel topic, some attempts have dealt with the optimization for the most important tasks in shale gas such as the following. He and You27 integrated the shale gas processing with ethylene production to increase the profitability and He and You28 presented a novel process design for a greener chemicals production integrating shale gas with bioethanol dehydration. Nevertheless, none of the above-mentioned approaches has dealt with the water network synthesis under uncertainty for the optimal water management in fracturing processes related to shale gas. Shale gas extraction is a modern technology, so there is little published on water reuse strategies for shale gas processing. In this context, Jiang et al.29 assessed the life cycle water consumption and wastewater generation impacts of a Marcellus shale gas well from its construction to end of life. Theodori et al.30 reported empirical information about the hydraulic fracturing and the management, disposal and reuse of fracturing wastewater. Clark et al.31 determined the water consumed over the life cycle of conventional and shale gas production, accounting for different stages of production and for flowback water reusing. Best and Lowry32 estimated the potential effects of high-volume water extractions applying groundwater flow modelling for the Marcellus shale play. However, previous methodologies are empirical studies and these do not include formal optimization approaches. In this context, Gao and You33 presented a methodology for the optimal design of a shale gas supply chain network including the life cycle analysis through an MINLP model and a global optimization algorithm. Nevertheless, this work did not account for the water network associated to hydraulic fracturing operations in order to determine the optimal treatment, storage and disposal to exploit shale


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gas resources. Additionally, Yang et al.34 developed an optimization approach for scheduling fracturing operations while minimizing the costs associated with transportation, treatment, storage and disposal, and accounting for reusing flowback water and Gao and You35 proposed a mathematical programming formulation for the optimal design of water supply chain networks for shale gas production including multiple transportation modes and water management options. But the most important aspect that was not considered in previous approaches is that these methodologies rely on deterministic models that neglect the different uncertainty sources affecting the calculations in water management in shale gas in addition to other significant differences such as the capital costs for treatment and storage systems and for disposal as well as the non-linearity in the capital cost function, the wastewater streams were disposed without treatment when the wastewater streams are disposed in injection wells (this has generated the contamination of groundwater), and the number of treatment technologies, storage units and disposals required in the solution have not been determined. On the other hand, the most important uncertainties in water management for shale gas are the water requirements to complete each well and the percentage of this water collected as flowback water once the completion has finished. Notice in Table 1 that there are significant differences for the values of water requirements and flowback water found by several authors. In fact, this information is expressed as intervals (in other words, these values cannot be perfectly known beforehand, so a most likely interval within which they may fall is provided instead of a single nominal value). This poses a significant difficulty when one needs to predict the amount of water that will be required and the quantity of water that can be collected as flowback. This information is needed to determine the total costs or the total water requirements to complete a determined number of wells. It has also an impact on the equipment capacities in terms of treatment technologies, storage units and disposals required by the optimal water network in order to properly manage the water (see Figure 1). Decision-makers must determine the optimal equipment size to be acquired and installed prior to the fracturing operations, so the main aim of this work is to develop a methodology for facilitating the associated decision-making problem. To overcome the limitations of the foregoing works in this area and motivated by the inherent uncertainties in hydraulic fracturing, we propose here an optimization model



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for the synthesis of water networks that calculates the water requirements and the flowback water considering the most critical uncertainties present in fracturing operations related to shale gas. The uncertainties are modeled through stochastic distributions from which representative scenarios showing different profiles for costs and water requirements as well as equipment capacities for treating, storing and disposing wastewater are generated via sampling methods. This methodology represents a useful tool for decision-makers to select the final capacities. Then, the objective function consists of minimizing the total annual cost, which is composed by capital costs (due to the equipment acquisition) and operating costs (due to fresh water costs, operating costs for water treatment and transportation costs). Also, it is worth noting that this work considers that all the flowback water is treated prior to be stored (and reused) or disposed, generating an environmentally friendly way to mitigate the contamination of groundwater bodies. Finally, the methodology considers a fixed scheduling for the completion operations of each well in order to implement the approach, which can incorporate simultaneous operations by several fracturing crews. This paper is organized as follows: Section 2 formally states the problem of interest, and provides relevant information on shale gas processing. Section 3 presents the proposed mathematical model, while the results and discussion are described in Section 4, and the generated conclusions are drawn in the last section.

2. Problem Statement. The problem studied here can be formally stated as follows. Given are: •

A set of shale gas wells to be completed using hydraulic fracturing operations, where the water requirements and the flowback water obtained operate under uncertainty. This process can be carried out simultaneously by one or several fracturing crews. The operation time needed to frack each well is known.

•

A fresh water source available to be the primary resource supplied to wells. The unit costs for fresh water consumption and transportation (considering the distance among the fresh water source location and the geographical location for the wells) are also given.

•

A set of treatment technologies available to treat all the flowback water collected in order to accomplish acceptable conditions for its reuse or comply with the


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environmental regulations in terms of pollutant concentrations for its final disposal. Each treatment unit has associated a unit operating cost, as well as fixed and variable charges for the capital cost function. Finally, the maximum capacity or size for each unit considered is provided. •

Given is also a set of storage/pits to save the treated streams coming from the interception network. The storage equipment is required to accurately reuse the flowback water in hydraulic fracturing. Each storage unit includes fixed and variable costs for the capital cost function, the maximum capacity existing for each unit and a volumetric efficiency (flowrate leaving the unit with respect to flowrate at the inlet).

•

A set of disposals for the treated flowback water. Similarly, each disposal considers fixed and variable costs and a maximum volume.

•

Also, the unit transportation costs for the rest of the trajectories obtained in each step is provided (wells-treatments, treatments-disposals, treatments-storage and storage-wells) as well as a fixed scheduling for the fracturing operation of all wells.

•

Finally, the uncertain parameters (i.e., the water requirements to complete each well and the percentage of water collected as flowback water) are described through probability functions that are discretized via sampling methods. Hence, a set of scenarios that are equally likely are generated to represent the uncertainties inherent to water management in shale gas operations. Then, the goal is to find a strategic planning that considers the uncertainties and

associated risk in the water management implemented in shale gas operations. The objective function consists of minimizing the total annual cost (TAC), which is composed by the total operating cost (TOC) and the total capital cost (TCC). The TOC includes the fresh water costs, the operating costs for treatment units and the transportation costs; whereas the TCC considers the capital costs associated with the acquisition of treatment, storage and disposal units. Note that shale gas represents an emerging resource and together with natural gas have lower carbon footprints compared with other fossil fuels such as oil and carbon. Highlighting the importance and novelty of this resource (whose exploitation just started



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very recently) only a few works have addressed optimization strategies for shale gas, particularly in the optimal water management.

3. Model Formulation. To properly develop and implement the proposed methodology, we consider the superstructure shown in Figure 2. A certain number of wells to be exploited Nn must be established in addition to a fixed scheduling for the operations to complete each well (which can be carried out by one or several fracturing crews). The water required to _ in _ out ) and the flowback water ( Fnwell ) are given parameters, complete each well ( Fnwell ,t , s ,t , s

where their associated uncertainties are described through scenarios. It should be noted that if these values change according to the scenario, then the total water required, costs and all the internal flowrates involved in the proposed configuration depend on each scenario. Hence, the model is a two-stage stochastic programming one, where first stage decisions taken before the uncertainty is unveiled represent. The goal is to find the optimal values of the first-stage and second-stage variables that optimize the expected economic performance of the system. As shown in Figure 2, the water demanded by the fracturing operations is provided mainly by the fresh resources as well as by the reused streams leaving the storage/pits. Also, notice that all the flowback water collected is treated in the interception network in order to avoid future contamination of groundwater; however, the treated streams have two options: the fluid can be stored (and later it is reused in the wells) or disposed. In both cases the water has been previously treated. On the other hand, the sets used in the model formulation are defined for a better understanding. The subscript n represents a well, Nn is the total number of wells, and N is the set used to denote the wells; i is the subscript for the treatment units, Ni corresponds to the total number of treatment units and I is the set for treatment technologies; the subscript j denotes the storage units, whereas the total number of storage units corresponds to Nj and its set is represented through J; in addition, d is employed to represent the disposals, the total number of disposals is Nd and D is the set for the disposals; while the subscript that symbolizes the time period is t, Nt and T represent the total time periods and the associated set for the time periods, respectively. Finally, s indicates the subscript for a scenario that


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considers the uncertainty associated with the water requirements for the completion of each well and the portion of flowback water obtained, Ns is the total number of scenarios and the set representing the scenarios is S. The mathematical formulation is composed by mass balances, capacity limitations and design constraints, total water requirements and the objective function (which accounts for the economic aspects). These relationships are described as follows.

3.1. Mass balances based on the superstructure. This section presents the relationships used to model the mixers (yellow marked) and divisors (pink marked) considered in the proposed superstructure as well as the balances for the storage units, treatment technologies and disposals. Additionally, owing to the water flowrate required by the wells and the flowback water change for each scenario, all the flowrates and equations belonging to the superstructure model vary in each scenario. a) Segregation of fresh water. All the fresh water resources come from a water reservoir available to extract the liquid required to prepare the fracturing fluid and release the gas. Then, the water stream ) is separated and sent extracted from the water body in time period t and scenario s ( Ft ,fresh s to each well n that requires water during the same time period ( ff n fresh ,t , s ): Ft ,fresh = ∑ ff n fresh s ,t , s ,

∀t , ∀s

(1)

n

b) Water supply to wells. In addition to the fresh resources ( ff nfresh ,t , s ), the water required by well n in time _ in period t and scenario s ( Fnwell ) can be provided by the reused streams coming from the ,t , s

storage j ( ff jstorage , n ,t , s ): _ in storage Fnwell = ff nfresh ,t , s ,t ,s + ∑ ff j ,n ,t ,s , ∀n, ∀t , ∀s j

c) Flowback water.


(2)


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According to the literature on shale gas, once the completion phase has finished in each well, a certain amount of water returns to the surface mainly in the first two weeks, which is called “flowback water”; however, these reports have stated important variations for the percentage of flowback water obtained with respect to the total water injected even for adjacent wells. _ out Then, the flowback water in well n during time period t and for scenario s ( Fnwell ) ,t , s

can be separated to be sent to any treatment i ( ff nwell ,i ,t , s ): _ out Fnwell = ∑ ff nwell ,t ,s ,i ,t , s ,

∀n, ∀t , ∀s

(3)

i

It should be noticed that this configuration proposes that all flowback water is treated and can be stored or disposed. Also, recall that the water requirement and the flowback water in the wells are parameters under uncertainty, so their values change from one scenario to another. d) Water inlet to treatment units. Firstly, it is important to mention that this work only can consider treatment technologies for the interception network with proven efficiency to treat the flowback water and generate outlet water streams under good quality and adequate conditions to be disposed or reused without generating environmental issues or technical drawbacks in the process, where each treatment unit can involve different costs and volumetric efficiency factors. In this regard, in existing shale gas plays there are some treatment technologies that meet the restriction required by the proposed methodology.37 Also, note that all the flowback water is treated, which represents an environmental improvement that avoids disposal of untreated flowback. Thus, the following relationship establishes that the stream _ in entering the treatment unit i in time period t and scenario s ( Fi treat ) is supplied by the ,t , s

outlet streams from the wells n ( ff nwell ,i ,t , s ): _ in Fi ,treat = ∑ ff nwell t ,s ,i ,t , s ,

∀i , ∀t , ∀s

n

e) Water outlet from treatment units.


(4)

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_ out The stream treated with technology i in time period t and scenario s ( Fi ,treat ) is t ,s _ sto _ dis split to be sent to any storage unit j ( ff i treat ) and to any disposal d ( ff i treat ): , j ,t , s ,d ,t , s

_ out _ sto _ dis Fi ,treat = ∑ ffi ,treat + ∑ ff i ,treat , ∀i, ∀t, ∀s t ,s j ,t , s d ,t ,s j

(5)

d

f) Water balance for waste disposal and discharge. All the segregated flowrates in the trajectory treatment i-disposal d in the time _ dis ) are collected to generate the waste streams ( Fdwaste period t for the scenario s ( f i treat ,d ,t , s ,t , s ):

treat _ dis Fdwaste , ,t ,s = ∑ ff i ,d ,t , s

∀d , ∀t , ∀s

(6)

i

g) Water inlet to storage units. The streams coming from treatment i and sent to storage/pits j in time period t and _ sto _ in scenario s ( f i treat ) are gathered to obtain the inlet flowrate to each storage unit ( F jstorage , j ,t , s ,t , s

): _ in _ sto F jstorage = ∑ ff i ,treat , ,t , s j ,t , s

∀j, ∀t , ∀s

(7)

i

h) Water outlet from storage units. The water is stored in pits (owing to their large capacities), which show a large residence time; however, water needs to be available for reuse according to the fracturing fluid requirement for the wells. In this regard, the flowrate at the exit of storage j over time _ out period t and scenario s ( F jstorage ) is divided to be transported to any well n demanding ,t , s

water ( ff jstorage , n ,t , s ): _ out F jstorage = ∑ ff jstorage ,t , s , n ,t , s ,

∀j, ∀t , ∀s

(8)

n

i) Balances in treatment units. Additionally, during the treatment there are typically significant water losses; this aspect is modeled through an efficiency factor in terms of the volume associated with the technology i ( αitreat ):



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_ out _ in Fi ,treat = α itreat Fi ,treat , t ,s t ,s

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∀i , ∀t , ∀s

(9)

It should be noted that this factor can be determined prior to the optimization based on experimental reports. j) Balances in storage units. Mass balances are defined for the storage units, where it is considered an initial volume for each unit ( V jstorage _ initial ) and a fixed density (which allows defining the balances on a volumetric basis). Then, the volume for storage/pit j over time period t in scenario s ( ) is calculated from the previous volume in time period t-1 ( V jstorage V jstorage ,t , s ,t −1, s ) plus the product _ in ) and of the time conversion factor (Htime) and the difference among the inlet ( F jstorage ,t , s _ out outlet ( F jstorage ) flowrates, which is modeled as follows: ,t , s

_ in _ out V jstorage = V jstorage _ initial + H time ( F jstorage − F jstorage ) , ∀j, ∀t = 1, ∀s (10) ,t , s ,t , s ,t , s time _ in _ out V jstorage = V jstorage − F jstorage ( F jstorage ), ,t , s ,t −1, s + H ,t , s ,t , s

V jstorage _ initial = V jstorage , ∀j, ∀t = t final , ∀s ,t , s

∀j, ∀t > 1, ∀s

(11) (12)

It should be noted that equation (12) is included to ensure the continuity of the cycles.

3.2. Capacity and design constraints. The mathematical formulation must decide whether a treatment technology, storage unit and disposal unit exists or not. Also, the model optimizes the capacity for each unit required, which must be lower than the maximum capacity available in each case (i.e., the highest capacity of each unit). Thus, the optimization determines the total number of treatment technologies, storage/pits and disposals needed. Then, the capacities of the treatment, storage and disposal units are calculated as follows. a) Treatment units. For the treatment units, the next expressions determine their existence and capacity in the optimal solution.


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_ in Fi treat _ cap ≥ Fi ,treat , t ,s

∀i , ∀t , ∀s

Fi treat _ cap ≤ Fi treat _ max yitreat ,

∀i

TotCap treatment = ∑ Fi treat _ cap

(13) (14) (15)

i

where Fi treat _ cap is the optimal capacity for technology i, Fi treat _ max is the upper limit for the capacity of unit i, whereas yitreat is a binary variable that models the existence of treatment unit i. Finally, TotCaptreatment represents the total capacity of treatment technologies that must be installed for the operation of the project. It is worth to mention that the maximum capacity of the treatment technologies can be obtained by the equipment manufacturers. b) Storage units. For the storage/pits the next relationships are needed: V jstorage _ cap ≥ V jstorage , ,t , s

∀j , ∀t , ∀s

(16)

V jstorage _ cap ≤ V jstorage _ max y storage , ∀j j

(17)

TotCap storage = ∑V jstorage _ cap

(18)

j

where V jstorage _ cap represents the capacity for the storage/pit unit j, V jstorage _ max is the upper capacity of the unit j, while y storage is a binary variable used to model the existence of j storage unit j and TotCapstorage represents the total capacity of storage/pit units required to store the fluid. c) Disposals. Finally, the next relationships are used to determine the existence of disposals, total number of disposals, the capacity for each one and their total capacity. Equation (19) _ scen calculates the optimal capacity for disposal d in scenario s ( Vdwaste ): ,s

_ scen Vdwaste = H time ∑ Fdwaste ,s ,t , s ,

∀d , ∀s

(19)

t _ scen Vdwaste _ cap ≥ Vdwaste , ∀ d , ∀s ,s


(20)


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Vdwaste _ cap ≤ Vdwaste _ max ydwaste ,

∀d

TotCap disposal = ∑Vdwaste _ cap

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(21) (22)

d

where Vdwaste _ cap is the optimal capacity for disposal d, Vdwaste _ max is the maximum capacity of disposal d, ydwaste is a binary variable used to model the existence of disposal d and TotCapdisposal represents the total capacity for disposals. It should be noticed that in the mathematical programming formulation must be suggested a determined potential number of treatment technologies, a number of storage units and other number of disposals; however the optimization process determines the existence or not for each type of units as well as their capacity required in the optimal solution. Thus, it is possible that the optimal capacity for some units is zero; in other words the number of units for treatment, storage and disposals in the optimal solution can be lower to the originally proposed potential number of units. Consequently, for some units there is not inlet flowrate. Also, note that the existence, number, unit and total capacity for treatment and storage units as well as for disposals do not depend on the scenario, as these decisions are first-stage ones that must be made here and now before the uncertainty is unveiled. The model must then determine the optimal values of these first-stage variables considering the uncertainty in the input parameters, whose values vary from one scenario to another.

3.3. Total water requirements. In this project is important to quantify the total fresh water requirements (TWRs) to exploit properly the fracturing wells, which represents an environmental concern. Therefore, the fresh water requirement is determined by the next equation: TWRs = H time ∑ Ft ,fresh , ∀s s

(23)

t

TWRs is directly influenced by the water requirement of the well and by the amount of flowback water that can be obtained to be reused; then, considering that these two parameters are uncertain, the profile of water consumption is expected to vary significantly


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among the scenarios. Finally, the expected total fresh water requirement (ETWR) is determined through the following relationship: ETWR = ∑ probsTWRs

(24)

s

3.4. Objective function. The proposed model is a mixed-integer linear programing (MILP) problem, whose goal is to minimize the expected total annual cost (economic criteria). In addition, the model can minimize the total fresh water required (environmental goal). However, for the proposed scheme, both objectives tend to be equivalent (i.e., fresh water can be reduced by minimizing the cost). It should be noted that this might not hold for all possible cases. Hence, when these two objectives contradict each other, multi-objective optimization tools must be employed to generate Pareto curves and visualize the tradeoffs among them. Expected Total Annual Cost (ETAC). The main target for the proposed methodology is to minimize the expected total annual cost (ETAC). Since a different economic performance is obtained for each scenario s, the resulting cost distribution is determined as follows: Min ETAC = ∑ probsTAC s

(25)

s

where probs is the probability of scenario s and TACs is the total annual cost associated with the same scenario s, which is composed by the total operating cost (TOCs) and the total capital cost (TCC): TAC s = TOC s + TCC , ∀s

(26)

It should be noted that the operating costs change according to the scenario, while the capital costs do not depend on the scenario. Additionally, the operating costs include the fresh water cost, operating costs for treatment units and the transportation costs for all the trajectories considered (fresh water-wells, flowback water-treatment, treatment-storage, storage-wells and treatment-disposals). Then, the total operating costs are calculated through the following relationship:



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  fresh fresh treat treat _ in + ∑∑UTCnfresh ff n fresh UC ∑ Ft ,s + ∑∑UOCi Fi ,t ,s ,t , s +  t i t n t   time  used well treat _ sto treat _ sto , TOCs = H UTCn ,i ff n ,i ,t ,s + ∑∑∑UTCi , j ff i , j ,t ,s + ∑∑∑   n i t i j t   sto _ well treat _ dis treat _ dis ff jstorage + UTC ff  ∑∑∑UTC j ,n  ∑∑∑ ,n ,t , s i ,d i ,d ,t , s j n t i d t  

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∀s

(27) In the previous expression, UCfresh represents the unit cost for the fresh water,

UOCitreat is the unit operating cost for the treatment unit i; whereas UTCnfresh , UTCnused ,i , _ sto _ well _ dis , UTC sto and UTCitreat are the unit transportation costs associated with UTCitreat ,j ,d j ,n

sending material between fresh water source-wells, flowback water-treatment, treatmentstorage, storage-wells and treatment-disposals, respectively. It is worth noting that the transportation capital cost can be annualized and included in the transportation operating costs; whereas the operational cost of storage is negligible compared with the other terms in the total operating cost. Furthermore, the capital cost corresponds to the equipment acquisition associated with the treatment technologies, storage units, as well as to the creation of final disposals. Thus, the total capital cost is computed as follows:

  treat treat treat treat _ cap ) +   ∑  FC yi + VC ( Fi i     storage storage storage storage _ cap TCC = k F  ∑  FC yj + VC (V j ) +   j   waste waste waste waste _ cap  FC y d + VC (Vd )   ∑   d where FCtreat, FCstorage and FCwaste are fixed charges included in the capital cost functions for treatment units, storage units and disposals, respectively; while the variable charges are represented by VCtreat, VCstorage and VCwaste.

4. Results and Discussion. The capabilities of the proposed approach are illustrated in the next example problem. It is important to mention that this example employs technical information


(28)

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collected by some of the most important existing shale plays in United States (mainly Marcellus and Barnett).36,37,40 Additionally, according to Table 1, the mean values of 15,000 m3 and 25% for the amount of water required to complete each well and the portion of the water injected that returns as flowback water during the following three weeks, respectively, are used in this work. As can be seen in Slutz et al.37 (most of the information provided in this case study is based on this work), they reported a range of 12,700-19,000 m3 for the amount of water required to complete each well. Then, if the mean value (15,000 m3) has a standard deviation of 10%, it means that approximately the 96% of the data can be found between 12,000-18,000 m3; while for a deviation of 20% the equivalent range is 9,000-21,000 m3. Then, if these values are compared with the intervals contained in Table 1, it is possible to visualize that the selected values for the standard deviations (i.e., 10% and 20%) are adequate to model the common uncertainties under study (note that for some reports these values seem conservative). However, the model is general enough to deal with any type of scenarios regardless of how they are generated. 100 equally likely (i.e., all with the same probability of occurrence) scenarios were generated via Monte Carlo sampling and used to build the multi-scenario optimization model. It should be noted that when the number of scenarios increases, the accuracy of the method increases; however, when the number of scenarios increases, also the computation time increases making it impossible to get feasible solutions due to the computation capabilities. Then, the number of scenarios was fixed by gradually increasing it and stopping whenever no significant changes were detected in the simulation results. Furthermore, for the sampling approach, based on reported data, the mean and standard deviation for the uncertain data were determined. Then, normal distributions using the Monte Carlo method for the number of scenarios considered were generated. Moreover, twenty wells to be completed by three fracturing crews according to the scheduling shown in Figure 3 were considered. The time horizon is a year and each time period consists of a week. Notice that the time required by the fracturing crew for the ends and starts of operations well-well is at least one week owing to the transition time to move from a well to the next one. This figure shows a period where water scarcity leads to low fracturing operations (approximately during weeks 23-35).



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The unit operating cost in the treatment units is 12.50 $/m3 and the fresh water cost is 2 $/m3. Subsets n1, n2 and n3 are defined to facilitate the modeling of the transportation costs: n1 involves wells 1-8, while n2 and n3 correspond to wells 9-15 and 16-20, respectively. Thus, the unit transportation costs are 4, 7 and 12 $/m3 from the fresh source to wells n1, n2 and n3, respectively. The transportation costs from the wells to the interception network are 0.39, 0.72 and 0.94 $/m3, respectively. Besides, the unit transportation costs for the paths treatment-storage/pits and treatment-disposals are 1.41 and 8.3 $/m3, respectively. Finally, the transportation costs from the storage to wells n1, n2 and n3 are 0.98, 1.23 and 1.65 $/m3, respectively. On the other hand, the interception network of this case study is composed by a set of identical treatment units, where the characteristics of the considered technologies have been previously reported.37 The solution of this example indicates the number treatment units required. Remember that each treatment unit must generate clean water streams available to be reused or disposed under environmental conditions (this type of treatment technology is typically employed in shale gas plays and it accomplishes the mentioned requirements). The disposals considered are typical Class II wells; however, it is important to remark that the proposed configuration only considers treated wastewater streams. The fixed and variable costs used in the capital cost functions as well as the maximum capacities are shown in Table 2. Finally, the conversion factor Htime and the annualization factor kF are 7 days/week and 0.1 year-1, respectively; whereas the volumetric efficiency for the treatment units αitreat and the initial volume for the storage/pits units Vsstorage _ initial are 0.9 and 0, respectively. The MILP problem associated with the case study has 17 binary variables, 1,629,342 continuous variables and 465,923 constraints and each solution takes approximately 1,019 s using a computer with an i7 processor at 2.1 GHz with 8 MB of RAM. The methodology was implemented in General Algebraic Modeling System (GAMS41) and solved using the solver CPLEX. Figures 4 and 5 show the cumulative probability curves associated with the total cost and freshwater consumption, respectively, for standard deviations of 10% and 20% in the uncertain parameters. Hence, the vertical axis of the figures provides the probability of


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a performance less or equal to the target value given in the horizontal axis. As an example, the 20% curve has a probability of 43% of being below $3.2 MM/y, while the 10% curve has a probability of 77%. Finally, if the goal is more ambitious (i.e., $3 MM/y), the probabilities are 1% and 8% for the 10% and 20% curves, respectively. Similarly, for a standard deviation of 20%, the probability to satisfy the total water requirements with an amount of 0.23 hm3 is 28%; however, if the standard deviation is 10%, the probability drops to 5%. These figures depict as well the expected performance of each solution and the maximum and minimum cost values attained by them over all the scenarios. It should be noted that the expected total annual costs are $3.16 MM/y and $3.22 MM/y for standard deviations of 10% and 20%, respectively. Then, if a comparison of these values with respect to the extreme cases, can be established that the worst scenarios augment the costs 6.65% ($3.37 MM/y) and 13.04% ($3.64 MM/y), respectively; whereas the best scenarios can obtain reductions of 5.06% ($3 MM/y) and 9.63% ($2.91 MM/y), respectively. Additionally, although 20% of standard deviation curve is wider than the other one, the expected values are very close (0.238 hm3 for the 10% curve and .0239 hm3 for the 20% curve). Finally, as shown, as the level of uncertainty increases, we get solutions with more variability and worse expected performance. Finally, Table 3 contains the equipment capacities for this solution. Notice that an increase in the uncertainty (from 10% to 20%), generates larger capacities for treatment and disposals by 24% and 25%, respectively as well as a negligible reduction in the total capacity for storage. The above results were obtained considering that the equipment capacity (i.e., the capacity for treatment technologies, storage units and disposals) is fixed. In this case, the economic and water profiles fluctuate according to the changes in water requirements for each of the wells and to the amount of flowback water (i.e., the uncertainties considered). This approach yields equipment sizes adequate for the worst scenarios; however, at the same time this configuration could generate oversized equipment capacities for most of the scenarios and consequently undesired solutions in the economic performance. Accounting for this aspect and with the purpose to carry out a detailed analysis for the optimal



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equipment capacities, the value of full information for this approach is included. In this scheme, the total capacities for treatment technologies, storage units and for disposal are determined in each scenario (i.e., the equipment capacities are optimized according to the scenario). In other words, the design found in the optimal solution is different for each scenario. Therefore, the previous model formulation that consists of constraints (1)-(28) is modified to obtain the second methodology, which requires to remove relationships (13)(22), (24) and (26) to incorporate the following equations: Treatment units. Constraints assigned to model the capacity and existence for the _ cap treatment units are modified to allow that the capacity ( Fi treat ) and existence ( yitreat ,s , s ) for

the treatment unit i as well as the total capacity of treatment units ( TotCapstreatment ) can be determined in each scenario s: _ cap _ in Fi ,treat ≥ Fi ,treat , s t ,s

∀i , ∀t , ∀s

_ cap Fi treat ≤ Fi treat _ max yitreat ,s ,s ,

∀i, ∀s

_ cap TotCapstreatment = ∑ Fi ,treat , ∀s s

(29) (30) (31)

i

ETotCap treatment = ∑ probsTotCapstreatment

(32)

s

It should be noted that it is included the expected total capacity (ETotCapstorage) in analogy to the expected costs to determine a mean value for the total capacity of treatment technologies that must be installed. Storage units. Similarly for the storage/pits: _ cap V jstorage ≥ V jstorage , ,s ,t , s

∀j , ∀t , ∀s

(33)

_ cap V jstorage ≤ V jstorage _ max y storage , ∀j, ∀s ,s j ,s

(34)

_ cap TotCapsstorage = ∑V jstorage , ∀s ,s

(35)

j

ETotCap storage = ∑ probsTotCapsstorage s


(36)

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Disposals. Finally, for the disposals: _ cap Vdwaste = H time ∑ Fdwaste ,s ,t , s ,

∀d , ∀s

(37)

∀d , ∀s

(38)

∀s

(39)

t

_ cap Vdwaste ≤ Vdwaste _ max ydwaste ,s ,s , _ cap TotCapsdisposal = ∑Vdwaste , ,s d

ETotCap disposal = ∑ probsTotCapsdisposal

(40)

s

Objective function. Additionally, the capital cost values (TCCs) included in the objective function change for each scenario s:   treat treat treat treat _ cap ) +   ∑  FC yi , s + VC ( Fi , s i     storage storage _ cap  TCC s = k F  ∑  FC storage y storage VC V + + ( j ,s )  j ,s  j   waste waste waste waste _ cap  FC y d , s + VC (Vd ,s )    ∑ d 

(41)

TAC s = TOC s + TCC s , ∀s

(42)

Notice that constraints (24) and (26) do not consider the fluctuation scenarioscenario for the capital costs. Therefore, the second mathematical programming model is composed by equations (1)-(12), (23), (25) and (27)-(42). The generated approach is also a MILP model that consists of 1,700 binary variables, 1,633,104 continuous variables, 468,002 constraints and each solution requires approximately 281.54 s of CPU time. The results for the economic performance are illustrated in Figure 6. In this case, the expected costs show a small improvement with respect to the previous case (for σ=10%, $3.12 MM/y vs $3.16 MM/y and for σ=20%, $3.15 MM/y vs $3.22 MM/y) and even for the extreme scenarios, which is owing to the equipment capacities are not fixed. It should be noted that the extreme and intermediate solutions (with a probability of 50%) have been highlighted through the Points A, B and C for σ=10%, and A’, B’ and C’ for σ=20% to make a wider discussion among these cases.



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In this regard, Table 4 shows the differences in TAC, TWR as well as the optimal total capacities for treatment units, storage units and disposals. Notice that an increment in the uncertainty (from σ=10% to σ=20%) can affect minimally the cost by 1.28% (the TAC augments $0.04 MM/y) for the solutions with 50% of probabilities (B vs B’) and at the same time the total capacity required for storage units is significantly diminished by 24.30% (13.95 vs 10.56 dam3). Also, it should be noted that these extreme and intermediate cases do not correspond to their equivalent extreme and intermediate solutions for the TWR curves (see the values shown in Figure 5 and Table 4). Another important aspect in the results found by the optimization process is the total water volume transported for each trajectory considered in the proposed design, which can evidence the differences among the scenarios A, A’, B, B’ C, and C’. Thus, Figures 7, 8 and 9 illustrate the amount of water moving in each trajectory included by the superstructure for points A, B and C, respectively (notice that these points are the extreme and intermediate solutions for the curve with σ=10%). Whereas the corresponding solutions A’, B’ and C’ (points belonging to solutions with a profile of uncertainty σ=20%) are shown through Figures 10, 11 and 12, respectively. As can be seen in these figures, all the configurations are different among them and the water required in each step yields different values. Afterwards, a similar analysis of cumulative probabilities is carried out for the optimal equipment capacities with the purpose that this study can be useful for the decisionmakers associated with the selection of total size for treatment technologies, storage units and disposals that must be acquired and installed around the fracturing wells for proper operations. In this sense, Figure 13 shows the cumulative probability curves for the total capacity of treatment units with the same two cases for the standard deviations. In this case, for σ=10% the minimum capacity required is 696 m3/d, the maximum case requires 894 m3/d and the expected value is 792 m3/d; whereas for σ=20% the minimum, expected and maximum capacities are 608, 867 and 1,109 m3/d, respectively. Additionally, if the decision-maker selects a total capacity for treatment units of 800 m3/d, it means that there is 57% of possibilities that this equipment is enough to treat the flowback water if the uncertainties related to the water requirements and flowback water follow a σ=10%; however, if this uncertainty follows a σ=20%, then the probability that 800 m3/d of


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treatment technologies are adequate to treat the flowback water decreases to 27%. Also, notice that if a value of 900 m3/d is chosen, then it is completely guaranteed (probability of 100%) the flowback water treatment if σ=10%; however if σ=20%, then the possibility diminishes to 60%. Figure 14 shows the cumulative probability curves for the total capacity of storage units. In this figure can be seen two details: for the area of the 20% curve where it is not overlapped with the other one, it conducts to lower total capacities for storage units and even for almost all overlapped zone the 20% curve has a higher probability to require lower total capacities for storage units than the 10% curve. For this reason, it can be stated that the 20% curve leads to total capacities for storage units lower than the 10% curve. In order to explain these results, the following reasons are suggested: •

Several scenarios generated under a standard deviation of 20% have a lower amount (with respect to the scenarios with a deviation standard of 10%) of water entering to the treatment units and as consequence the flowrates at the inlet of storage units are smaller than for the case of standard deviation of 10% (remember that the scenarios were generated via the Monte Carlo).

•

Notice that the objective function consists in minimizing the costs and it is possible that the solutions associated to the scenarios with a standard deviation of 10% require higher volume for storage equipment (compared with the solutions for scenarios under a deviation of 20%) in order to reduce other costs (i.e., disposal cost or freshwater cost).

•

Also, it is possible that the water management associated to the solutions in the 20% curve is better than the one of solutions in the 10% curve, specifically the reuse of the water entering to storage equipment is faster to reduce the accumulation in the storage units and consequently the required capacity for the storage unis is lower. Additionally, if a total capacity for storage units of 12 dam3 is selected, then the

10% curve has 9% of possibilities to store properly the flowback water; however, the 20% curve increases this value to reach 54% of probability. A similar analysis can be carried out for the total capacity for disposals according to the cumulative probability curves shown in



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Figure 15. It should be noted that for all the cumulative probabilities presented, the curve for σ=20% is always wider than the σ=10% curve. Also, notice that the extreme and intermediate solutions highlighted in TAC curves (points A, A’, B, B’, C and C’) do not correspond to their analogous solutions for any of the total capacity curves shown (comparing Table 4 and Figures 13, 14 and 15). Finally, this methodology represents a useful tool for the decision-makers to determine and select adequate equipment capacities for the treatment technologies, storage units and disposals for the water management in wells (notice that this decision must be taken prior to the start of operations in the wells) according to the historic or a projected uncertainty in the water requirements to carry out the fracturing process as well as the flowback water returned.

5. Conclusions. This paper has proposed mathematical programming models for the synthesis of water networks in shale gas production while accounting for uncertainties. The considered uncertainties correspond to the water requirements to fracture each well and the portion of this water that returns as flowback water. These uncertainties were manipulated through the generation of scenarios. The proposed superstructure for the water integration allows the efficient management of flowback water to reduce the fresh water consumption and the wastewater streams are discharged in a way that can avoid the contamination of groundwater. The methodology generates probability curves of costs and water requirements. The solution also determines equipment capacities for the treatment technologies, storage units and disposal systems. The probability curves are used to visualize the risks associated with the uncertainties and to determine adequate equipment sizes. The introduced approach can be used to guide the decision makers in managing water resources and in acquiring properly-size water-management systems. Furthermore, the risk curves offer valuable insights based on solutions that span a broad domain including the worst and best cases and the expected values of each considered criterion. A case study was solved based on technical data from the Marcellus and Barnett shale plays. Each aspect discussed in the results was evaluated considering that uncertainties follow normal distributions with standard deviations of 10% and 20% in order


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to examine the differences and sensitivities in the economics, water requirements and equipment capacity performances when the uncertainty is increased. Finally, the proposed approach is general and its solution procedure does not represent significant numerical and convergence problems.

Acknowledgments. The authors acknowledge the financial support from the Mexican Council for Science and Technology (CONACyT) and the Council for Scientific Research of the Universidad Michoacana de San Nicolás de Hidalgo.

Nomenclature. Parameters FCtreat

fixed charge for the treatment unit in the capital cost function, $

FCstorage

fixed charge for the storage unit in the capital cost function, $

FCwaste

fixed charge for the disposal in the capital cost function, $

_ in Fnwell ,t , s

flowrate entering in the wells n in the time t for the scenario s, m3/d

_ out Fnwell ,t , s

flowback water leaving the well n in the time t for the scenario s, m3/d

Htime

time conversion factor, d/ week

kF

factor used to annualize the inversion, y-1

probs

probability of the scenario s

UCfresh

unit cost for the fresh water, $/m3

UOCitreat

unit operating cost for the treatment i, $/m3

UTCnfresh

unit transport cost for the fresh water to the well n, $/m3

_ well UTC sto j ,n

unit transport cost for the reused water from the storage j to well n, $/m3

_ dis UTCitreat ,d

unit transport cost for the treated water from the treatment i to disposal d,

$/m3



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_ sto UTCitreat ,j

unit transport cost for the treated water from the treatment i to storage j, $/m3

UTCnused ,i

unit transport cost for the flowback water from well n to treatment i, $/m3

VCtreat

variable charge for the treatment unit in the capital cost function, $/m3/d

VCstorage

variable charge for the storage unit in the capital cost function, $/m3

VCwaste

variable charge for the disposal in the capital cost function, $/m3

Greek Symbols

αitreat

volumetric efficiency factor for the treatment i

Variables ETAC

expected total annual cost, $

ETotCaptreatment

expected total capacity of treatment technologies that must be

installed, m3/d ETotCapstorage

expected total capacity of storage units that must be installed, m3

ETotCapdisposal

expected total capacity of disposals that must be installed, m3

ETWR

expected total water requirements, m3

Ft ,fresh s

fresh water flowrate consumed in the time t for the scenario s, m3/d

_ in F jstorage ,t , s

flowrate entering to the storage j in the time t for the scenario s, m3/d

_ out F jstorage ,t , s

flowrate leaving the storage j in the time t for the scenario s, m3/d

Fi treat _ cap

capacity flowrate for the treatment i, m3/d

_ cap Fi treat ,s

capacity flowrate for the treatment i for the scenario s, m3/d

_ in Fi treat ,t , s

inlet flowrate to the treatment i in the time t for the scenario s, m3/d

Fi treat _ max

maximum flowrate for the treatment units, m3/d

_ out Fi treat ,t , s

flowrate leaving the treatment i in the time t for the scenario s, m3/d


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Fdwaste ,t , s

flowrate sent to disposal d in the time t for the scenario s, m3/d

ff n fresh ,t , s

segregated flowrate of fresh water to well n in the time t for the scenario s,

m3/d ff jstorage , n ,t , s

segregated flowrate from storage j to well n in the time t for the scenario s,

m3/d _ dis ff i ,treat d ,t , s

segregated flowrate from treatment i to disposal d in the time t for scenario s,

m3/d _ sto ff i ,treat j ,t , s

segregated flowrate from treatment i to storage j in the time t for scenario s,

m3/d ff nwell ,i ,t , s

segregated flowrate from well n to treatment i in the time t for scenario s,

m3/d TACs

total annual cost for scenario s, $

TCC

total capital cost, $

TCCs

total capital cost for scenario s, $

TOCs

total operating cost for scenario s, $

TotCaptreatment total capacity of treatment technologies that must be installed, m3/d

TotCapstreatment total capacity of treatment technologies that must be installed for scenario s, m3/d TotCapstorage total capacity of storage units that must be installed, m3

TotCapsstorage total capacity of storage units that must be installed for scenario s, m3 TotCapdisposal total capacity of disposals that must be installed, m3

TotCapsdisposal total capacity of disposals that must be installed for scenario s, m3 TWRs

total water requirements for the scenario s, m3

V jstorage ,t , s

storage/pit j volume in the time t for the scenario s, m3



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V jstorage _ cap

capacity for storage/pit j, m3

_ cap V jstorage ,s

capacity for storage/pit j for the scenario s, m3

V jstorage _ initial

initial storage/pit j volume, m3

V jstorage _ max

maximum volume for storage/pit j, m3

Vdwaste _ cap

capacity for disposal d, m3

_ cap Vdwaste ,s

capacity for disposal d for the scenario s, m3

Vdwaste _ max

maximum volume for disposals, m3

_ scen Vdwaste ,s

capacity for disposal d in for scenario s, m3

y storage j

binary variable used to model the existence of the storage j

y storage j ,s

binary variable used to model the existence of the storage j in for scenario s

yitreat

binary variable used to model the existence of the treatment i

yitreat ,s

binary variable used to model the existence of the treatment i in for scenario

s

ydwaste

binary variable used to model the existence of the disposal d

ydwaste ,s

binary variable used to model the existence of the disposals d in for scenario

s Sets I

{i | i is a treatment unit}

J

{j | j is a storage/pit}

D

{d | d is a disposal}

N

{n | n is a well}

S

{s | s is a scenario}


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{t | t is a time period}

T

Subscripts and Superscripts i

treatment unit

j

storage/pit

d

disposal

n

well

s

scenario

t

time period

References (1) U.S. Energy Information Administration (EIA). U.S. 2012 Annual Energy Outlook with Projects to 2035. Washington, DC: US Department of Energy, 2012. (2) Wang, Y. P.; Smith, R. Wastewater minimisation. Chemical Engineering Science 1994, 49, 981-1006. (3) Kazantzi, V.; El-Halwagi, M. M. Targeting material reuse via property integration. Chemical Engineering Progress 2005, 101 (8), 28-37. (4) Foo, D. C. Y. Process integration for resource conservation, CRC Press, Boca Raton, Florida, 2012. (5) Deng, C.; Feng, X.; Ng, D. K. S.; Foo, D. C. Y. Process-based graphical approach for simultaneous targeting and design of water network. AIChE Journal 2011, 57 (11), 3085-3104. (6) Bandyopadhyay, S.; Sahu, G. C.; Foo, D. C. Y.; Tan, R. R. Segregated targeting for multiple resource networks using decomposition algorithm. AIChE Journal 2010, 56 (5), 1235-1248. (7) Almutlaq, A.; Kazantzi, V.; El-Halwagi, M. M. An algebraic approach to targeting waste discharge and impure-fresh usage via material recycle/reuse networks. Clean Technologies and Environmental Policy 2005, 7 (4), 294-305.



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(8) Qin, X.; Gabriel, F.; Harell, D.; El-Halwagi, M. M. Algebraic techniques for property integration via componentless design. Industrial and Engineering Chemistry Research 2004, 43, 3792-3798. (9) Gabriel, F.; El-Halwagi, M. M. Simultaneous synthesis of waste interception and material reuse networks: Problem reformulation for global optimization. Environmental Progress 2005, 24 (2), 171-180. (10) Yang, L.; Grossmann, I. E. Water targeting models for simultaneous flowsheet optimization. Industrial and Engineering Chemistry Research 2012, 52 (9), 32093224. (11) Yu, J. Q.; Chen, Y.; Shao, S.; Zhang, Y.; Liu, S. L.; Zhang, S. S. A study on establishing an optimal water network in a dyeing and finishing industrial park. Clean Technologies and Environmental Policy 2014, 16 (1), 45-57. (12) Ponce-Ortega, J. M.; Hortua, A. C.; El-Halwagi, M.M.; Jiménez-Gutiérrez, A. A property-based optimization of direct recycle networks and wastewater treatment processes. AIChE Journal 2009, 55 (9), 2329-2344. (13) Ponce-Ortega, J. M.; El-Halwagi, M. M.; Jiménez-Gutiérrez, A. Global optimization of property-based recycle and reuse networks including environmental constraints. Computers and Chemical Engineering 2010, 34 (3), 318-330. (14) Nápoles-Rivera, F.; Ponce-Ortega, J. M.; El-Halwagi, M. M.; Jiménez-Gutiérrez, A. Global optimization of mass and property integration networks with in-plant property interceptors. Chemical Engineering Science 2010, 65 (15), 4363-4377. (15) Lira-Barragán, L. F.; Ponce-Ortega, J. M.; Serna-González, M.; El-Halwagi, M. M. An MINLP model for the optimal location of the new industrial plant with simultaneous consideration of economic and environmental criteria. Industrial Engineering and Chemical Research 2011, 50 (2), 953-964. (16) Lira-Barragán, L. F.; Ponce-Ortega, J. M.; Serna-González, M.; El-Halwagi, M. M. Synthesis of water networks considering the sustainability of the surrounding watershed. Computers and Chemical Engineering 2011, 35, 2837-2852.


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(17) Lira-Barragán, L. F.; Ponce-Ortega, J. M.; Nápoles-Rivera, F.; Serna-González, M.; El-Halwagi, M. M. Incorporating property-based water networks and surrounding watersheds in site selection of industrial facilities. Industrial and Engineering Chemistry Research 2013, 52 (1), 91-107. (18) Gebreslassie, B. H.; Yao, Y.; You, F. Design under uncertainty of hydrocarbon biorefinery supply chains: multiobjective stochastic programming models, decomposition algorithm, and a comparison between CVaR and downside risk. AIChE Journal, 2012, 58 (7), 2155-2179. (19) You, F.; Wassick, J. M.; Grossmann, I. E. Risk management for a global supply chain planning under uncertainty: models and algorithms. AIChE Journal, 2009, 55 (4), 931-946. (20) Nápoles-Rivera, F.; Rojas-Torres, M. G.; Ponce-Ortega, J. M.; Serna-González, M.; El-Halwagi, M. M. Optimal design of macroscopic water networks under parametric uncertainty. Journal of Cleaner Production 2015, 88, 172-184. (21) Broad, D. R.; Dandy, G. C.; Maier, H. R. A systematic approach to determining metamodel scope for risk-based optimization and its application to water distribution system design. Environmental Modelling and Software 2015, 69, 382395. (22) Khor, C. S.; Chachuat, B.; Shah, N. Fixed-flowrate total water network synthesis under uncertainty with risk management. Journal of Cleaner Production 2014, 77, 79-93. (23) Ahn, J.; Kang, D. Optimal planning of water supply system for long-term sustainability. Journal of Hydro-Environment Research 2014, 8 (4), 410-420. (24) Islam, S.; M., Sadiq, R.; Rodriguez, M. J.; Najjaran, H.; Hoorfar, M. Reliability assessment for water supply systems under uncertainties. Journal of Water Resources Planning and Management 2014, 140 (4), 468-479. (25) Yang, L.; Salcedo-Diaz, R.; Grossmann, I. E. Water network optimization with wastewater regeneration models. Industrial and Engineering Chemistry Research 2014, 53 (45), 17680-17695.



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(26) Gao, J.; You, F. Deciphering and handling uncertainty in shale gas supply chain design and optimization: Novel modeling framework and computationally efficient solution algorithm. AIChE Journal 2015, 61 (11), 3739-3755. (27) He, C.; You, F. Shale gas processing integrated with ethylene production: novel process designs, exergy analysis, and techno-economic analysis. Industrial and Engineering Chemistry Research 2014, 53 (28), 11442-11459. (28) He, C; You, F. Toward more cost effective and greener chemicals production from shale gas by integrating with bioethanol dehydration: Novel process design and simulation based optimization. AIChE Journal 2015, 61 (4), 1209-1232. (29) Jiang, M.; Hendrickson, C. T.; VanBriesen, J. M. Life cycle water consumption and wastewater generation impacts of a Marcellus shale gas well. Environmental Science and Technology 2014, 48 (3), 1911-1920. (30) Theodori, G. L.; Luloff, A. E.; Willits, F. K.; Burnett, D. B. Hydraulic fracturing and the management, disposal, and reuse of frac flowback waters: views from the public in the Marcellus Shale. Energy Research and Social Science 2014, 2, 66-74. (31) Clark, C. E.; Horner, R. M.; Harto, C. B. Life cycle water consumption for shale gas and conventional natural gas. Environmental Science and Technology 2013, 47 (20), 11829-11836. (32) Best, L. C.; Lowry, C. S. Quantifying the potential effects of high-volume water extractions on water resources during natural gas development: Marcellus Shale, NY. Journal of Hydrology: Regional Studies 2014, 1, 1-16. (33) Gao J, You F. Shale gas supply chain design and operations toward better economic and life cycle environmental performance: MINLP model and global optimization algorithm. ACS Sustainable Chemistry and Engineering. 2015;3(7):1282-1291. (34) Yang, L.; Grossmann, I. E.; Manno, J. Optimization models for shale gas water management. AIChE Journal 2014, 60 (10), 3490-3501. (35) Gao J, You F. Optimal design and operations of supply chain networks for water management in shale gas production: MILFP model and algorithms for the water energy nexus. AIChE Journal. 2015;61(4):1184-1208.


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(36) Hayes, T. Sampling and analysis of water streams associated with the development of Marcellus shale gas. Marcellus Shale Initiative Publications Database, 10. 2009. http://energyindepth.org/wp-content/uploads/marcellus/2012/11/MSCommissionReport.pdf (37) Slutz, J; Anderson, J; Broderick, R; Horner, P. Key shale gas water management strategies: an economic assessment tool. SPE/APPEA International Conference on Health, Safety, and Environment in Oil and Gas Exploration and Production, Perth, Australia. 2012. (38) Vidic, R. D.; Brantley, S. L.; Vandenbossche, J. M.; Yoxtheimer, D.; Abad, J. D. Impact of shale gas development on regional water quality. Science 2013, 340 (6134) 1235009. (39) Zammerilli, A.; Murray, R. C.; Davis, T.; Littlefield, J. Environmental impacts of unconventional natural gas development and production. DOE/NETL-2014/1651, 2014, (800) 553-7681. (40) Silva, J. M. RPSEA Produced water pretreatment for water recovery and salt production.

Final

report

2012,

08122-36.

http://www.rpsea.org/media/files/project/18621900/08122-36-FRPretreatment_Water_Mgt_Frac_Water_Reuse_Salt-01-26-12.pdf (41) Brooke, A.; Kendrick, D.; Meeruas, A.; Raman, R. GAMS: A Users Guide. GAMS Development Corporation, Washington, DC, 2015.



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Tables. Table 1. Previous reports for the main uncertainties in shale gas processing. Amount of water required to complete each well

Report/Concept Hayes36

Portion of the water injected returning as flowback water

3-4 MM gallons (11,356-15,142 m3)

25%

Slutz et al.37

80,000-120,000 bbl (12,700-19,000 m3)

10-40 %

Vidic et al.38

2-7 MM gallons (7,570-26,500 m3)

9-53 %

Zammerilli et al. Silva

39

3

40

2-6 MM gallons (7,570-22,712 m )

30-70 %

3

15-25 %

2-6.5 MM gallons (7,570-24,605 m )

Table 2. Data used in the example problem. Concept/Case case

Fixed cost (FC

Variable cost (V

Treatment Storage Disposal

), $

case

95,000

10,000

52,000

4.1

6.3

13.5

500

12,000

20,000

3

), $/m

Maximum capacity (V

case_max

3

3

), m /d or m

Table 3. Details for solutions when the equipment capacity is fixed. Concept/Case

σ=10%

σ=20%

894.37

1,108.77

14.38

14.20

5.90

7.37

3

Total capacity for treatments, m /d 3

Total capacity for storage/pits, dam 3

Total capacity for disposals, dam

Table 4. Details for the extreme and intermediate solutions. Concept/Case

A 3

Total capacity for treatments, m /d

B

C

A'

B'

C'

787.00 754.90 758.11 736.10 759.84 926.64 3

Total capacity for storage/pits, dam

13.31

13.95

13.89

11.85

10.56

15.76

4.17

4.55

5.30

3.84

4.65

5.44

TWR, hm

0.230

0.240

0.251

0.218

0.237

0.266

TAC, $ MM/y

2.97

3.12

3.33

2.85

3.16

3.58

3

Total capacity for disposals, dam 3


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Caption for Figures. Figure 1. Schematic representation for the uncertainties associated with hydraulic fracturing and their impact on the determination of equipment size. Figure 2. Proposed superstructure for the water management in fracturing operations. Figure 3. Scheduling for the completion phase in each well. Figure 4. Cumulative probability curves for the economic performance when the equipment capacity is fixed. Figure 5. Cumulative probability curves for the total water requirements. Figure 6. Cumulative probability curves for the economic performance when the equipment capacity can change in each scenario. Figure 7. Total water volumes in the solution for Case A. Figure 8. Total water volumes in the solution for Case B. Figure 9. Total water volumes in the solution for Case C. Figure 10. Total water volumes in the solution for Case A’. Figure 11. Total water volumes in the solution for Case B’. Figure 12. Total water volumes in the solution for Case C’. Figure 13. Cumulative probability curves for the total capacity required of treatment units. Figure 14. Cumulative probability curves for the total capacity required of storage units. Figure 15. Cumulative probability curves for the total capacity required of disposals.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Optimal Size for Uncertainties

Fresh Water

Hydraulic Fracturing

Treatment Units Flowback Water

Total Costs? Total Water Used?

Figure 1. Schematic representation for the uncertainties associated with hydraulic fracturing and their impact on the determination of equipment size.


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Ft fresh ff nfresh ,t _ in Fnwell ,t

_ out Fnwell ,t

ff nwell ,i ,t

_ in Fi treat ,t

Fdwaste ,t

_ out Fi ,treat t

_ dis ff i ,treat d ,t

_ sto ff i ,treat s ,t

ff sstorage , n ,t

_ out Fsstorage ,t

_ in Fsstorage ,t

Figure 2. Proposed superstructure for the water management in fracturing operations.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3. Scheduling for the completion phase in each well.


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Page 39 of 51

TACup = $3.37 MM

100 90 Cumulative Probability (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80

TACup = $3.64 MM

70

ETAC = $3.16 MM

60 50

ETAC = $3.22 MM TAClo = $2.91 MM

40

σ=20%

30

σ=10%

20 10 TAClo = $3 MM

0 2.9

3

3.1

3.2 3.3 3.4 TAC, $/y (x106)

3.5

3.6

3.7

Figure 4. Cumulative probability curves for the economic performance when the equipment capacity is fixed.



TWRup = 0.252 hm3


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Page 40 of 51

80

TWRup = 0.268 hm3

70 60

ETWR = 0.238 hm3

50 40

ETWR = 0.239 hm3 TWRlo = 0.216 hm3

σ=20%

30 σ=10% 20 10 TWRlo = 0.228 hm3

0 0.21

0.22

0.23

0.24 0.25 TWR, m3 (x106)

0.26

0.27

Figure 5. Cumulative probability curves for the total water requirements.


Page 41 of 51

C

TACup = $3.33 MM

100

C'

90 Cumulative Probability (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


80

TACup = $3.58 MM

70

ETAC = $3.12 MM ETAC = $3.15 MM

60 50

B

B'

40 30

TAClo = $2.85 MM

σ=20% σ=10%

20 10 0

TAClo = $2.97 MM A' 2.8

2.9

A

3

3.1 3.2 3.3 TAC, $/y (x106)

3.4

3.5

3.6

Figure 6. Cumulative probability curves for the economic performance when the equipment capacity can change in each scenario.



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Page 42 of 51

Figure 7. Total water volumes in the solution for Case A.

42 ACS Paragon Plus Environment

Page 43 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Figure 8. Total water volumes in the solution for Case B.



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Figure 9. Total water volumes in the solution for Case C.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Figure 10. Total water volumes in the solution for Case A’.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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Figure 11. Total water volumes in the solution for Case B’.


Page 47 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Figure 12. Total water volumes in the solution for Case C’.



TotCapup = 894 m3/d


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Page 48 of 51

80 70

TotCapup = 1,109 m3/d ETotCap = 792 m3/d

60 ETotCap = 867 m3/d

50 40

σ=20% 30 TotCaplo = 608 m3/d

20 10

σ=10%

TotCaplo = 696 m3/d

0 600

700

800 900 1,000 1,100 Total Capacity for Treatment Units, m3/d

1,200

Figure 13. Cumulative probability curves for the total capacity required of treatment units.


Page 49 of 51

100 σ=20%

TotCapup = 15.37 dam3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


σ=10%

80 70


ETotCap = 11.69 dam3

60 50 40

TotCaplo = 7.23 dam3

ETotCap = 13.32 dam3

30 20 10


0 7

8

9 10 11 12 13 14 15 Total Capacity for Storage Units, m3 (x103)

16

17

Figure 14. Cumulative probability curves for the total capacity required of storage units.



100



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 50 of 51

80


70 ETotCap = 4.65 dam3

60

ETotCap = 4.71 dam3

50 40


σ=20%

30

σ=10%

20 10


0 2

3

4 5 6 Total Capacity for Disposals, m3 (x103)

7

8

Figure 15. Cumulative probability curves for the total capacity required of disposals.


Page 51 of 51

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For Table of Contents Use Only

Optimal Water Management under Uncertainty for Shale Gas Production

Luis Fernando Lira-Barragán, José María Ponce-Ortega, Gonzalo Guillén-Gosálbez and Mahmoud M. El-Halwagi.

Synopsis: This article presents a mathematical programming model for the water management in hydraulic fracturing operations to exploit shale gas considering that the water required to complete each well and the amount of flowback water are uncertain data. The objective consists in minimizing the costs; also, it generates a relevant analysis to select adequately the capacities for equipment.


Optimal Water Management under Uncertainty for ... - ACS Publications

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