Pipeline network design for gathering unconventional oil and gas production using mathematical optimization

Abstract

The optimal design of gathering networks for the unconventional oil and gas production is a relevant problem, particularly with the shale boom. In this work, we address a network design problem in which the main decisions are the location and sizing of tank batteries for oil and gas separation together with the pipeline connections. The goal is to gather the oil, gas and water production from a large number of wellpads, being also possible to install junction nodes to merge the production at some points of the field. One major challenge is due to the steep production decline curves, requiring continued connections of new wells. To address this problem, we develop a complex multiperiod formulation accounting for the varying flows over the planning horizon. We propose an optimization framework to obtain efficient solutions within reasonable computational times. To circumvent the computational burden of a large-scale, nonlinear and non-convex model due to the fluid dynamics of multiphase flows, we propose a solution algorithm based on a bi-level decomposition. Near optimal solutions are found for real-world instances, suggesting that facility planning can considerably improve the economics of unconventional projects.

Appendices

Appendix 1: Maximum flow in single phase pipelines

1.1 Liquid phase pipelines

When sizing liquid phase pipelines petroleum engineers seek to avoid corrosion, erosion and water hammer effect, among other negative phenomena associated with the flow velocity. Based on that, they usually impose a maximum mean velocity ($vma{x}^{LP}$) around 1.5 m/s (Society of Petroleum Engineers 2006). Equation 23 determines the maximum flow (in m³/s) for oil and water phases when linking tank batteries with battery junction nodes and these with central delivery points through a pipeline of diameter diam_d (in m). In other words, the maximum admissible flow is assumed to be directly proportional to the pipeline section.

$$maxflow_{d}^{oil/water} = vmax^{LP} \cdot \frac{{\uppi }}{4} \cdot \left( {diam_{d}^{2} } \right)$$

(23)

1.2 Gas phase pipelines

The compressibility of the gas phase makes the calculations of the pipeline transportation capacity more complex. Similar to Cafaro and Grossmann (2014) and Duran and Grossmann (1986), the head loss in a given pipeline segment of diameter diam_d (in m) and length l (in km) is assumed to follow the Weymouth correlation (Weymouth 1912) for gas pipelines sizing, represented by Eqs. 24 and 25. The value of the gas density ${\rho }^{GP}$ is set to 0.729 kg/m³ for shale gas in standard conditions (P₀ = 0.1013 MPa; T₀ = 288.9 K). T is the average gas temperature (in this work, equal to T₀) and the value of $\alpha$ from the Weymouth correlation is 3/16. In the most general case, input and output pressures (P_in and P_out, in MPa) are model variables, whereas in model approximations they are assumed to be known (see Appendix 4). The final Eq. 26 represents the gas flow computation (in 10⁶ m³/day).

$$dia{m}_{d}= {l}^{\alpha }{\left({P}_{in}^{2}-{P}_{out}^{2}\right)}^{-\alpha } {B}^{\alpha }$$

(24)

$$B={\rho }^{GP}T{\left[{P}_{0}/0.375{T}_{0}\right]}^{2}{[{maxflow}_{d}^{gas}]}^{2}$$

(25)

$${maxflow}_{d}^{gas}={l}^{-0.5}{\cdot diam}_{d}^{2.667}{\left\{\frac{\left({P}_{in}^{2}-{P}_{out}^{2}\right)}{\left({\rho }^{GP}T{\left[{P}_{0}/0.375{T}_{0}\right]}^{2}\right)}\right\}}^{0.5}$$

(26)

Notice that ${maxflow}_{d}^{gas}$ is proportional to: (1) the pipeline diameter to the power of 2.667, and (2) the square root of the difference of squared inlet and outlet pressures.

Appendix 2: Lockhart–Martinelli correlation

This procedure is aimed at predicting the pressure drop for fully developed horizontal gas–liquid flows, and is widely used in the Chemical Engineering field (Green and Southard 2018; Lockhart and Martinelli 1949). It is based on the calculation of the pressure drops that would be expected for each of the phases as if flowing alone through the pipeline. Then, the Lockhart–Martinelli parameter is computed as the square root of the ratio between the individual expected pressure drops. Finally, the overall two-phase pressure drop is estimated by using phase-specific multipliers that are obtained from nonlinear correlations with the LM parameter. We assume oil and water streams as part of a common liquid phase. Even though this is not entirely valid since oil and water phases are not miscible, generating particularly complex flow-patterns, the assumption is made to gain simplicity. In fact, as reviewed by Edwards et al. (2018), the pressure drop might be affected by the apparent viscosity of the liquid mixture, which depends on the water fraction. However, in our view, this assumption does not excessively affect the pressure drops computations since the gor and wor are constant terms in our model.

From Eq. 27 we obtain the ratios between the component flow rates. Equation 28 computes the superficial velocities for liquid (VS^LP) and gas (VS^GP) phases, while Eq. 29 determines the estimated Reynolds numbers for liquid and gas phases (RY^LP–RY^GP). The remaining parameters account for: diam_d, the pipeline diameter [m], $\rho$ and $\mu$, the average density [kg/m³] and viscosity [Pa.s] of each phase, respectively. The friction factor ${HFF}^{LP}$ for the liquid phase is computed through the Haaland correlation in Eq. 30, while the pressure drop gradient ${\Delta P}^{LP}/L$ is estimated by Eq. 31, as if the liquid phase were flowing alone in the pipeline. The parameter ε accounts for the pipeline internal wall roughness [m]. For the gas phase, the Weymouth equation (Weymouth 1912) estimates a pressure gradient along the pipeline, as in Eq. 32.

$$wor \cdot FP^{oil} = FP^{water} \quad gor \cdot FP^{oil} = FP^{gas}$$

(27)

$$VS_{d}^{LP} = \frac{{FP^{water} + FP^{oil} }}{{\pi diam_{d}^{2} /4}}\quad VS_{d}^{GP} = \frac{{FP^{gas} }}{{\pi diam_{d}^{2} /4}}$$

(28)

$$RY_{d}^{LP} = \frac{{\rho^{LP} \cdot VS^{LP}_{d} \cdot diam_{d} }}{{\mu^{LP} }}\quad RY_{d}^{GP} = \frac{{\rho^{GP} \cdot VS^{GP}_{d} \cdot diam_{d} }}{{\mu^{GP} }}$$

(29)

$$\frac{1}{{\left( {HFF_{d}^{LP} } \right)^{0.5} }} = - 1.8 \log \left( {\frac{\varepsilon }{{3.7 diam_{d} }} + \frac{6.9}{{RY_{d}^{LP} }}} \right)$$

(30)

$$\frac{{\Delta P_{d}^{LP} }}{L} = \frac{{HFF_{d}^{LP} }}{2}{ } \cdot \frac{{\rho^{LP} VS_{d}^{LP2} }}{{diam_{d} }}$$

(31)

$$FP^{gas} = l^{ - 0.5} \cdot diam_{d}^{2.667} \left\{ {\frac{{\left( {P_{in}^{2} - P_{out}^{2} } \right)}}{{\left( {\rho^{GP} T\left[ {P_{0} /0.375T_{0} } \right]^{2} } \right)}}} \right\}^{0.5}$$

(32)

$$\frac{{\Delta P^{GP} }}{L} = \frac{{\left( {P_{in} - P_{out} } \right)}}{l}$$

(33)

Note that in the most general case, only the pressure at the wellpads is given. As the gas flow FP^gas increases, the difference of square pressures (P²_in – P²_out) also increases, leading to a larger single-phase pressure gradient ${\Delta P}^{GP}/L$ (Eq. 33). The other parameters account for the average gas temperature T [K], the temperature T₀ [K] and pressure P₀ [MPa] in standard conditions. Also, the inlet pressure P_in has an upper limit according to the segment of the network being considered. It can be easily proved that ${\Delta P}^{LP}/L$ and ${\Delta P}^{GP}/L$ are strictly increasing functions with the oil flow rate, for fixed gor and wor.

The Lockhart–Martinelli parameter X_LM is a measure of the relation between the single-phase pressure gradients, which is computed by Eq. 34. The corresponding multipliers Y^LP and Y^GP are obtained in Eq. 35 to determine how much the single-phase pressure gradients are increased when both phases are flowing together in a multiphase flow. The latter equation follows the Wilkes approach (Wilkes, 2005) to fit the curves of the LM correlation (see Fig. 

Fig. 19

Liquid and gas pressure gradient multipliers versus Lockhart–Martinelli parameter

19). The parameter n depends on the flow regime of each phase, being equal to 4.12 if both liquid and gas phases have a turbulent regime, as usually seen in practice for unconventional streams. The overall multiphase pressure gradient $\Delta P/L$ is computed by Eq. 36 and the total pressure drop ${\Delta P}^{Total}$ is given by Eq. 37. The maximum pressure drop (${\Delta p}^{max}$) is set according to the operating conditions of the manifolds and the tank batteries. Note that the overall pressure gradient $\Delta P/L$ is also a strictly increasing function with the oil flow rate, for fixed gor and wor.

$$X_{LM} = \left( {\frac{{\Delta P^{LP} /L}}{{\Delta P^{GP} /L}}} \right)^{0.5}$$

(34)

$$Y^{GP} = \left( {1 + X_{LM}^{\frac{2}{n}} } \right) ^{n} \quad Y^{LP} = \frac{{Y^{GP} }}{{X_{LM}^{2} }}$$

(35)

$$\frac{\Delta P}{L} = Y^{LP} \cdot \frac{{\Delta P^{LP} }}{L}$$

(36)

$$\Delta P^{Total} = { }\frac{\Delta P}{L} \cdot dist \le \Delta p^{max}$$

(37)

2.1 SPE (society of petroleum engineers) guidelines

According to the SPE (Society of Petroleum Engineers 2006), some recommendations should be taken into account to determine the diameter of a multiphase pipeline. These are related to maximum and minimum velocity constraints to prevent erosion, corrosion, noise or water hammer effects. The maximum velocity recommended by the SPE is 60 ft/s to inhibit noise, and 50 ft/s for CO₂ corrosion inhibition. In turn, a minimum velocity of 10 to 15 ft/s is usually set to prevent surging and keep the line swept clear of solids.

The SPE guidelines for multiphase pipeline design and sizing are then added to the formulation. Equation 38 limits the flow in order not to exceed the maximum velocity v_max [ft/s], according to the given pipeline diameter. The parameter v_max is computed as the ratio between the constant ζ and the square root of the average density ${\rho }_{avg}$. ζ is an SPE specific constant with a value of 150 for solids-free fluids and continuous service operation. The average density is also a parameter computed as shown in Eq. 39. Moreover, z is the gas compressibility factor, r the gas/liquid ratio [ft³/bbl], T the gas/liquid flowing temperature [°R], ${P}_{in}$ the inlet pressure in PSI, sg the specific gravity of the liquid phase relative to water, and s the specific gravity of the gas, relative to air.

$$FP^{oil} + FP^{water} \le \frac{{1000 \cdot v_{max} \cdot \left( {0.0254 \,diam_{d} } \right)^{2} }}{{\left( {11.9 + \frac{z \,r \,T}{{16.7 \, P_{in} }}} \right)}}$$

(38)

$$v_{max} = \frac{\varsigma }{{\rho_{avg}^{0.5} }} ;\quad \rho_{avg} = \frac{{12409{ }\,sg{ }\,P_{in} + 2.7{ }\,r{ }\,s{ }\,P_{in} }}{{198.7\,P_{in} + z\,r\,T}}$$

(39)

Appendix 3: Multiphase maximum flow computation

These illustrative examples are developed to show the results of the NLP formulation when applied to a horizontal pipeline segment with given inlet and outlet pressures.

3.1 Illustrative example 1

To estimate the transportation capacity of a pipeline segment, assume we are given two nodes: (1) the origin, representing the outlet of a wellpad with a pressure of 250 PSI (1.72 MPa), connected by a pipeline of diameter d and length dist with (2) a junction node with a pressure of 200 PSI (1.37 MPa) (see Fig. 

Fig. 20

Setting pressures in a pipeline segment

20). The NLP formulation is solved assuming ten different segment lengths: 250, 500, 750, 1000, 1250, 1500, 2000, 3000, 4500 and 6000 m. gor and wor values are fixed at 2 and 3.5 respectively, whereas alternative pipeline diameters are 8″, 12″ and 16”.

Figure 

Fig. 21

Maximum flows in multiphase pipelines for different diameters (only oil flowrates shown for reference)

21 shows the results after solving the NLP formulation for the maximum flow determination. For every diameter, the maximum flowrate is a decreasing function with the length. Some interesting conclusions can be drawn. First, there is an important increase in the maximum flowrate with the pipeline diameter. The increase is larger for shorter pipelines. Second, for a given pipeline diameter the SPE guidelines generate a unique result, independent of the length. This indicates that, for shorter pipelines, the SPE guidelines are more restrictive than the Lockhart–Martinelli (LM) correlation (dotted lines). The red circles represent the length at which the LM and SPE determine the same maximum admissible flow. In longer pipelines, the LM procedure restricts the flow to limit the head losses, while the SPE guidelines restrict the flow in shorter pipelines to avoid damage.

3.2 Illustrative example 2

We now change the gor and wor values with the aim of studying the effects of the different phases. We analyze three conditions: (1) gor = 2 and wor = 3.5 (444 standard cubic feet of gas per barrel of liquid, i.e. oil and water), (2) gor = 3 and wor = 2 (1000 [scf/bbl]), and (3) gor = 4 and wor = 2 (1333 [scf/bbl]). Figure 

Fig. 22

Results for the Illustrative Example 2

22 shows the maximum flows for each individual component through a pipeline segment of 8″ in diameter with different lengths. Note that, for the first two scenarios, the maximum oil flowrates are similar. The increase in the gas flowrate is compensated by the increase of oil concentration in the liquid phase. However, in the third scenario, both oil and water flowrates decrease due to the greater proportion of gas in the flow.

Next, we study the behavior of the NLP model when the gor and wor move towards extreme values (gor = 10 and wor = 0.5). By comparing with the first scenario (gor = 2, wor = 3.5) it is observed that the maximum oil flowrate decreases at the expense of a larger increase in the gas flow (see Fig. 

Fig. 23

Results of the NLP (pipeline sizing) model when gor and wor move to extreme values

23). Additionally, the SPE criteria impose an upper bound on the gas flowrate that is stronger for longer segments, before changing to the LM bounds.

Appendix 4: Pressure estimations for pipeline sizing

The values in Fig. 

Fig. 24

Setting pressures at the nodes of the network

24 are the reference pressures at every node in the network when the optimization framework is performed. In the most general case, only wellpads and compressors pressures are given, while the others stand for optimization variables. We assume that the pressure at the wellheads is high enough to make oil, gas and water flow towards the tank batteries. Even though in certain cases the pressure reaches 650 PSI (4.48 MPa) a conservative value of 250 PSI (1.72 MPa) is assumed at the outlet of the wellpads. For the purpose of pipeline sizing, the inlet pressure at tank batteries has to be of at least 80 PSI (0.55 MPa), resulting in a maximum pressure drop of 170 PSI. The distance between wellpads and junctions will be generally short, leading us to arbitrarily set the maximum pressure drop in these segments to 50 PSI (0.34 MPa). From junction nodes to batteries the pipelines operate at medium–low pressures, being the maximum head loss equal to 120 PSI (0.82 MPa). After the stream flows are separated in the tank batteries, the pressure is assumed to drop to 40 PSI at the tank battery junctions, where pumps and compressors push the fluids to the CDP at high pressure conditions. Gas pressure after compression reaches 1100 PSI, which is assumed to drop to 580 PSI at the centralized delivery point. Note that, if pipelines are not used at their maximum capacities, the outlet pressures can be larger, but never smaller than the reference values.

Appendix 5: Steps of the solution algorithm

5.1 Pre-processing step

According to the maximum flowrate estimation, for both multiphase and single-phase pipeline segments, straightforward logic can be applied to reduce the size of the problem without affecting the quality of the solutions:

• For every wellpad-junction connection, the maximum flowrate over time is known in advance according to the wellpad production forecast. Hence, it is possible to minimize the pipeline costs by selecting the minimum diameter d that is able to handle the production peak of the wellpad. This is achieved by fixing the x_p,j,d’ variables to zero for all d’ ≠ d.

• For connections between junction nodes and tank batteries, the minimum flow to be transported is that coming from the less productive wellpad that might be connected to j. Then, it is possible to omit every diameter d that is not able to handle at least the production of that single wellpad.

• Also, the maximum processing capacity of a single tank battery limits the flow in the segments connecting j with b. From that, one can omit alternative diameters that are larger than required.

5.2 MILP formulation with investments at the initial time

A strong assumption is introduced in this section to reduce the size of the MILP formulation, and thereby improving its computational performance. We assume that all the facilities selected by the model are installed and made available at time zero, yielding a compact formulation with a large reduction in the model size, both in variables and equations.

5.2.1 Model reformulation

The main changes with regards to the multiperiod MILP are as follows. Regarding the binary variables, all decisions related to the installation of a facility after the first period are omitted, as stated by Eqs. 40 and 41.

$$\hat{x}_{p,j,d,t} = 0{ }\quad \forall p,j,d,t \in TS, t > 1{ };\quad \hat{y}_{j,b,d,t} = 0{ }\quad \forall j,b,d,t \in TS,t > 1;\quad \hat{u}_{b,bj,d,t}^{c} = 0\quad \forall b,c, bj, d,t \in TS, t > 1$$

(40)

$${ }\hat{w}_{bj,cdp,d,t}^{c} = 0\quad \forall bj,c,cdp,d, t \in TS, t > 1 ;\quad \hat{z}bat_{b,bz,t} = 0{ }\quad \forall b,bz,t \in TS, t > 1$$

(41)

Moreover, after solving the simplified MILP, it is important to compute the actual net present cost of every facility by postponing the capital expenditures to the time they are used for the first time, according to the network design. These computations are introduced to determine the net present cost of the solution, which is comparable with those yielded by the original MILP.

It is also interesting to note that this formulation yields feasible solutions for the multiperiod MILP model presented in Sect. 5.2. This can be easily proved by considering that: (a) every solution of the reduced MILP model is a solution in the multiperiod approach, and (b) the objective function in the multiperiod MILP is always lesser or equal to the total present cost in the reduced MILP. The implications are: (1) the simplified MILP formulation with investments at the initial time is a valid starting point for the original MILP model, and (2) it imposes a valid upper bound to the optimal solution of the MILP.

5.3 Bounds on the number of tank batteries

In this section, an iterative procedure is developed to establish the minimum and maximum number of tank batteries among which the optimal solution lies. This reduces the feasible region of the problem, limits the combination of potential locations to be activated, and also improves the search for a lower bound of the objective function.

5.3.1 Heuristic determination of the number of tank batteries

The minimum number of tank batteries required (MinBat) is determined by adding Eq. 42 to the MILP, where the parameter nTB is iteratively increased (starting from 1) until the problem becomes feasible, as shown in Fig. 

Fig. 25

Heuristic Determination of the number of tank batteries. *Adapted for feasibility search. **#TB = all potential TB locations. ***Checked during the first 2 h of CPU usage for every iteration

25. In other words, it seeks to determine the minimum number of TB at which the problem becomes feasible, regardless of the quality of the solution. Note that many constraints of the original formulation can be relaxed in order to improve this procedure. Particularly, we may relax pipeline sizing equations yielding a simple pad-battery assignment problem.

On the other hand, by imposing a minimum number of batteries to use (through the same parameter nTB) we can determine the number of tank batteries (MaxBat) above which the solution is guaranteed to be suboptimal (Eq. 43). Taking into consideration the best integer solution found so far, the heuristic procedure starts with a large value for nTB (in an extreme case, all potential tank batteries are enforced to be used), and compares the lower bound of the branch and bound search with the best solution. If the lower bound exceeds the net present cost of the best solution found, the total number of batteries has to be smaller than the imposed value, interrupting the execution, reducing the value of nTB by 1 and repeating the procedure, as depicted in Fig. 25.

$$\mathop \sum \limits_{{b, bz \in BZ_{b} }} zbat_{b,bz} \le nTB$$

(42)

$$\mathop \sum \limits_{{b, bz \in BZ_{b} }} zbat_{b,bz} \ge nTB$$

(43)

5.4 Lower bound improvement strategies

In this section, we introduce the concept of SuperBatteries and new topological constraints for cutting off suboptimal solutions. Both ideas aim to provide better optimality gaps for the overall problem.

5.4.1 SuperBatteries (SB) concept

We seek to aggregate the tank batteries capacities within the same geographical location, looking for a valid relaxation whose optimal solution has a much faster convergence. Through the synthesis of multiple batteries, the element SB replaces the tank batteries in the same site with a SuperBattery with a capacity that is an integer multiple of a single battery, which is captured by the aggregate integer variable nBat (see Fig. 

Fig. 26

Original topology (left) and SuperBattery topology (right) of the problem

26).

We prove that this aggregation provides a valid lower bound for the original problem because: (1) every solution in the original problem is also a solution of the SB relaxation (it is simply obtained by summing single flows reaching the SB); and (2) the objective function of the SB relaxation is always lesser or equal to that in the original formulation. More specifically, from the original Eq. 5, adding up the incoming flows to the potential tank batteries in every SuperBattery location SB (as in Eqs. 44 and 45), it is possible to obtain the integer variable $nba{t}_{{\varvec{s}}{\varvec{b}},bc}$ as the sum of the corresponding $zba{t}_{b,bc}$ binary variables. Clearly, the number of SuperBatteries may be significantly smaller than the tank batteries in the original formulation, which allows to: (1) reduce the model size, and (2) reduce the problem complexity by cutting off the number of potential links between junctions and tank batteries (now SuperBatteries).

$$\mathop \sum \limits_{b \in SB} \mathop \sum \limits_{{j \in J_{b} }} FPB_{j,b,t}^{c} \le \mathop \sum \limits_{b \in SB} \mathop \sum \limits_{{bz \in BZ_{sb} }} batcap_{bz}^{c} \cdot zbat_{b,bz} = \mathop \sum \limits_{bz \in BZ} batcap_{bz}^{c} \mathop \sum \limits_{b \in SB} zbat_{b,bz} \quad \forall sb,c,t \,\epsilon\, TP$$

(44)

$$\mathop \sum \limits_{{j \in J_{b} }} FPB_{j,sb,t}^{c} \le \mathop \sum \limits_{{bz \in BZ_{sb} }} batcap_{bz}^{c} \cdot nbat_{sb,bz} \quad \forall sb,c,t \,\epsilon\, TP$$

(45)

However, the relaxation gives rise to two important issues: first, it may lead to almost feasible solutions, which is illustrated in Fig. 

Fig. 27

Aggregate/disaggregate assignment of wellpads when the SuperBattery relaxation is applied

27. In that example, 11 wellpads are developed and assigned month after month to a double-capacity SuperBattery, but it turns to be impossible to re-assign the flows to the individual tank batteries without exceeding their single capacities. Secondly, it tends to consolidate flows from many wellpads into a single junction before sending them to the SB, as observed in Fig. 

Fig. 28

Infeasible gathering of the production into a single pipeline (left), which is overcome by limiting the intermediate pipeline’s capacity (right)

28 (left). That solution is infeasible in terms of the original formulation since a single flow towards the SB exceeds the capacity of a single tank battery. To overcome this issue, Eq. 46 can be added to the relaxation, which also turns out to be an effective tightening constraint. Other topological constraints to cut off infeasible solutions are developed in the next section.

$$FPB_{{j,{\varvec{sb}},t}}^{c} \le max_{bz} \{ batcap_{bz}^{c} \} \mathop \sum \limits_{d} y_{j,sb,d,t} \quad \forall {{sb}},c,j,t \,\epsilon\,TP$$

(46)

5.4.2 Topological constraints to cutoff infeasible solutions

It is worth to analyze how we can gather the production from a given cluster of wellpads in order to add special topological constraints. More specifically, we may determine if the total production from a cluster of wellpads can be sent to a single tank battery, or, in other words, determine the maximum number of wellpads from a cluster whose production can be merged in a single junction. Figure 

Fig. 29

Effects of gathering the total production from a cluster of wellpads in a single tank battery

29 shows the cumulative production of 4 wellpads of the same cluster (each of them with an initial production peak of 22,750 bbl/day of water). Note that production start dates are planned for periods 38, 41, 44 and 46, yielding an excess of 7.3% above the capacity of the largest possible tank battery.

Therefore, installing a single junction node in that cluster is an infeasible option. Two ways to avoid this are illustrated in Fig. 

Fig. 30

Logic of topological cuts

30. The first option is to install at least two junction nodes in that cluster, and the second is by sending the production of at least one of the wellpads to a junction of another cluster of wellpads. This can be generalized and applied to the models through Eq. 47. This constraint limits the number of pipelines in the cluster k connected to a junction j to be lesser or equal to the maximum number of wellpads nw_k from the same cluster k whose production can be merged and processed by a single tank battery.

$$\mathop \sum \limits_{{p \in P_{k} , d}} x_{p,j,d} \le nw_{k} \, \forall \, j \in J_{k} ,k \in K$$

(47)