3.5 Appendix A: Detailed model formulation for the hybrid LCO model of shale gas supply chains model of shale gas supply chains
3.5.1 Objective functions
3.5.2.2 Environmental constraints
oper power t
C indicates the total operating cost associated with electric power generation in time period t, calculated by,
, , ,
where veg denotes the unit cost for electricity generation from natural gas at power plant g.
3.5.2.2 Environmental constraints
The process emissions indicate the GHG emissions generated during all the activities in the process systems, calculated by the following equation.
m
pro pro
TE =e Qm (A16)
where empro is the environmental impact factors of basic process m. Qm is the total net input of process m from all activities, including water management, well drilling, shale gas production, processing, transportation, and electricity generation, in the process systems.
Based on the definition of Qm, it can be calculated by the following equation.
, ,. , , , ,
m water m drill m prod m proc m trans m power m
Q =Q +Q +Q +Q +Q +Q (A17)
Qwater,m is the total input of process m associated with water management activities,
including wastewater treatment at CWT facilities and onsite treatment units.
,
, i t _ i m, i o t, , _ o m,
i I t T i I o O t T
water m
Q WTC inv cwt WTO inv onsite
∈ ∈ ∈ ∈ ∈
=
∑∑
⋅ +∑∑∑
⋅ (A18)where inv_cwti,m is the amount of input from process m for treating unit amount of wastewater from shale site i at CWT facilities. inv_onsiteo,m indicates the amount of input from process m for treating unit amount of wastewater with onsite treatment technology o.
Qdrill,m is the total input of process m regarding drilling activities, which can be
where inv_drilli,m indicates the amount of input from process m for drilling a shale well at potential shale site i.
Qprod,m represents the total input of process m associated with shale gas production
activities, given by,
where inv_prodi,m indicates the amount of input from process m for producing unit amount of shale gas at shale site i.
Qproc,m represents the total input of process m associated with shale gas processing
activities, calculated by,
where inv_procp,m indicates the amount of input from process m for processing unit amount of raw shale gas at processing plant p.
Qtrans,m represents the total input of process m associated with gas transportation
activities, which is calculated by,
, , ,
Q STP lsp inv trans STPG lpg inv trans
∈ ∈ ∈ ∈ ∈ ∈
=
∑∑∑
⋅ ⋅ +∑ ∑∑
⋅ ⋅ (A22)where inv_transm indicates the amount of input from process m for transporting unit amount of shale gas for unit distance.
Qpower,m represents the total input of process m associated with electricity generation
where inv_powerg,m indicates the amount of input from process m for consuming unit amount of shale gas to generate electricity at CCGT power plant g.
The IO emissions indicate the GHG emissions generated in the EIO systems, calculated by the following equation.
IO IO
ns ns
TE =e P (A24)
where ensIO is the environmental impact factors of industrial sector ns. Pns is the total output of sector ns in the EIO systems.
The total output of each industrial sector Pns minus the direct requirement of all sectors in the EIO systems should be no less than the upstream inputs required by the process systems, given by,
where aions,ns’ is the technical coefficient connecting industrial sector ns and ns’ in the EIO table. UPns indicates the upstream input from industrial sector ns to the process systems.
The upstream input from industry sector ns to the process systems UPns can be calculated by the following equation,
,
where cns,m is the upstream technical coefficient linking industrial sector ns and process m. pricem indicates the unit price input from process m.
3.5.2.3 Mass balance constraints
The total water supply at each shale site comprises of freshwater from water sources and reused water from onsite treatment.
,
where FWi,t stands for the amount of freshwater acquired from water sources to shale site i in time period t. loo denotes the recovery factor for treating wastewater of onsite treatment technology o. WTOi,o,t denotes the amount of wastewater treated by onsite treatment technology o at shale site i in time period t. FDWi,t denotes the freshwater demand of shale site i in time period t.
The amount of freshwater required at each shale site in each time period equals the summation of water usage for drilling and hydraulic fracturing. The drilling water usage is proportional to the number of wells being drilled, and the hydraulic fracturing water usage is proportional to the amount of wastewater produced at shale site i.
,
where WPi,t denotes the wastewater production rate during fracking process at shale site i in time period t. wrfi is the recovery ratio of water for hydraulic fracturing process at shale site i. wdi denotes the average drilling water usage at shale site i. NNi,t stands for the number of wells drilled at shale site i in time period t.
The wastewater production rate during the fracking process is proportional to the total shale gas production rate at a shale site, and the coefficient is estimated based on real data.[124]
, ,
WP =cc SP⋅ ∀i t (A29)
where cci is the correlation coefficient for shale gas production and wastewater production of a shale well at shale site i. SPi,t is the shale gas production rate at shale site i in time period t.
At each shale site, the total amount of wastewater, including the wastewater from drilling, hydraulic fracturing, and completion, should equal to the total amount of water treated by different water management options, including CWT and onsite treatment.
, , , , , , ,
where wrdi denotes the recovery ratio for drilling process at shale site i. WTCi,t denotes the amount of wastewater transported from shale site i to CWT facilities in time period t.
The total amount of shale gas produced at any existing shale site i at any time period t can be calculated by,
, ,, ,
i t i i t e
SP =ne spp⋅ ∀ ∈i I t (A31)
where nei denotes the number of existing shale wells drilled at shale site i. sppi,t denotes the shale gas production of a shale well of age t at shale site i.
The total shale gas production rate at a shale site equals the summation of that of different wells.
where sppi,t-t’ denotes the shale gas production profile of a shale well drilled at time period t’ at shale site i in time period t. Thus, the age of this well would be t- t’. We use
this time-dependent parameter to describe the decreasing feature of the shale gas production profile of a certain well.
The shale gas production at each shale site is then transported to different processing plants.
where STPi,p,t denotes the amount of shale gas transported from shale site i to processing plant p in time period t.
The methane and NGLs are separated at processing plants, and their corresponding amounts are dependent on the processing efficiency and the composition of raw shale gas.
where pef denotes the NGL recovery efficiency at processing plants. mci denotes the methane composition in shale gas at shale site i. lci denotes the NGL composition at shale site i. SPMp,t is the amount of natural gas produced at processing plant p in time period t. lci is the average NGLs composition in shale gas at site i. SPLp,t stands for the amount of NGLs produced at processing plant p in time period t.
The total amount of natural gas separated at a processing plant equals the summation of natural gas transported from the processing plant to different power plants.
, , ,, ,
where STPGp,g,t denotes the amount of natural gas transported from processing plant p to power plant g in time period t.
The total amount of electricity generation at a power plant in each time period is proportional to the amount of natural gas transported to the power plant from processing plants.
where GEg,t denotes the amount of electricity generated at power plant g in time period t. ue denotes the amount of electricity generated from unit natural gas input.
3.5.2.4 Capacity constraints
The total amount of wastewater from different shale sites treated by each CWT facility cannot exceed its capacity, given by,
, ,
where ccat denotes the capacity for wastewater treatment at CWT facility in time period t.
If a certain onsite treatment technology is applied at a shale site, the amount of wastewater treated onsite should be bounded by its capacity; otherwise, the amount of wastewater treated onsite should be zero. This relationship can be modeled by the following inequality,
, , , , , , ,
o i o i o t o i o
ocl YO⋅ ≤WTO ≤ocu YO⋅ ∀i o t (A39)
where oclo and ocuo denote the minimum and maximum treatment capacities for onsite treatment technology o, respectively. YOi,o is a binary variable that equals 1 if onsite treatment technology o is applied at shale site i.
The total amount of shale gas processed at a processing plant cannot exceed its capacity,
, , , ,
where PCp denotes the capacity of processing plant p.
The total amount of shale gas transported from a shale site to a processing plant is constrained by the transportation capacity of corresponding gathering pipeline.
, , , , , , ,
where TCPi,p,r denotes the capacity of pipeline from shale site i to processing plant p with capacity level r.
Similarly, the amount of natural gas transported from processing plant p to power plant g is bounded by the capacity of corresponding pipeline,
, , , , , , ,
where TCPGp,g,r denotes the capacity of pipeline from processing plant p to power plant g with capacity level r.
The total electric power generation is constrained by the lower bound and upper bound of local demand.
, , ,, ,
g t g t g t
dgl ≤GE ≤dgup ∀g t (A43)
where dglg,t and dgupg,t denote the minimum demand and maximum demand of electricity at power plant g in time period t, respectively.