Optimized production profile using a coupled reservoirnetwork model
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Abstract
This paper presents an integrated approach for scheduling and forecasting oil and gas production by integrating models of the entire value chain, from the reservoirs to the sales points. The methodology ensures maximum oil production at each time step of the reservoir simulator while honoring all operational constraints of the system. The proposed method is applied to a small North Sea offshore field consisting of two oil reservoirs with API gravities of 37 and 39. 3 gaslifted wells are producing in each reservoir. They are arranged in a production network connected to a surface process. Control variables include individual well choke opening (early stage) and gas lift injection rate (later stage). The system is subject to numerous operational constraints (e.g., maximum field liquid production, maximum gas lift injection rate). The proposed solution is built in a commercial IAM platform that connects the models and orchestrates the software execution and optimization. The optimization problem is formulated as a Mixed Integer Linear Program. The well and flowline performance curves are approximated with piecewise linear functions. Results show that such an integrated approach can significantly affect the production profile (up to 15% difference against traditional “silo” approach). The proposed integrated solution is twotothree times faster than traditional nonlinear optimization methods, guarantees convergence towards the global maximum and it represents with an appropriate level of accuracy the original blackbox model. This allows to run a lot of different scenarios making it a suitable tool for field development and planning optimization. The proposed method is used to optimize the field design and schedule. Optimal surface capacities are determined by brute force exploration of net present value function.
Keywords
Production forecast Field development and planning Artificial lift Optimization Mixed Integer Linear Programming (MILP)List of symbols
 \(\varDelta N_\mathrm{p}\)
Incremental oil production (stb)
 \(\varDelta t\)
Reservoir time step (day or months)
 \(E_\mathrm {cap}\)
Capital expenditure (USD)
 \(E_\mathrm {op}\)
Operating expenditure (USD)
 \(f_\mathrm {w}\)
Water cut (%)
 i
Interest rate (\(\%\))
 J
Well productivity index (stb/day/psi)
 \(N_\mathrm {years}\)
Duration of the operator license (years)
 \(P_\mathrm {o}\)
Price of oil (USD/stb)
 \(p_\mathrm{R}\)
Reservoir pressure (psi)
 \(R_\mathrm{p}\)
Gas oil ratio (scf/stb)
 \(R_y\)
Revenue of year y (USD)
 \(S_{\pi }\)
Saturation of phase \(\pi \in \{o,g,w\}\) (%)
 \(V_{\mathrm {o}y}\)
Volume of oil produced during year y (stb)
 GOR
Gas oil ratio (scf/stb)
 ID
Inner diameter
 IPR
Inflow performance relationship
 MD
Measured depth (ft)
 MILP
Mixed Integer Linear Programming
 NPV
Net present value (USD)
 SOS2
Special ordered set of type 2
 SQP
Sequential Quadratic Programming
 TVD
True vertical depth (ft)
 WC
Water cut (%)
Introduction
The forecasting of oil and gas production rates is a critical activity typically performed during the field development stage and when the field is already producing. In the field development phase, revenue streams are calculated from the hydrocarbon production rates and are further used in economic evaluations [e.g., net present value (NPV)] of relevant development alternatives (Jahn et al. 2008).
Production scheduling consists of defining the production rates of wells and field with time (production profiles). Although to produce as much as possible as early as possible would appear to be the best alternative to early mitigate the development expenses, this is not always true (Haldorsen 1996). Higher production flow rates require bigger processing capacities of production fluids (oil, gas, and water) and injection (e.g., gas injection, water injection, etc.). Bigger processing capacities translate into higher capital expenditures (due to the increase in size and weight of the processing equipment) that might reduce the NPV of the project. This is especially relevant for standalone offshore developments, where capacity of topside facilities directly affect the design, size, and cost of the supporting structures such as platforms and floating vessels.
There are other problems that high production rates could potentially cause in the reservoir, e.g., gas and water coning or cusping, excessive sand production, etc. These problems reduce the ultimate recovery factor of the field, affecting negatively the project value.
Production profiles are usually the result of an iterative process between several disciplines within the company. Initial profiles are generated using reservoir models and taking into consideration factors like drainage area, recovery factor, well productivity, well placement, gas or water coning and sand production, among others. The reservoir models employed have none or very simple models for pressure drop in wells and surface network. Production rates are then validated or corrected by production engineers to account for the pressure drop in wells, artificial lift design, surface network, among others. Facilities engineers perform a predesign of the processing system, map the requirements and operational constraints. The corrections and modifications are communicated back to the custodian of the reservoir model and the process is repeated.
This procedure is usually labor intensive, time consuming, and performed manually; therefore, it does not normally allow for an exhaustive evaluation of all development alternatives nor a probabilistic and robust assessment of uncertainty. This often leads to unoptimized production scenarios, lower revenues, and suboptimal decision making.
The use of integrated models (i.e., reservoir, wells, network, and facilities) is an alternative to obtain more realistic production profiles. However, it is not easy to implement, primarily because models are built in different tools or simulators, the custodians of the models are usually in separate business units, and the layout and characteristics of the production system are not defined in early development stages. Another important challenge is that the subsurface uncertainty is usually very high.
Coupling of reservoir and surface network models is a topic that has been researched extensively in the past. Barroux et al. (2000) presents a comprehensive review of commonly used coupling methods and approaches and their advantages and disadvantages. This study and others (e.g., AlShaalan et al. 2002) also discuss that coupled models provide in general production profiles that are more realistic than those obtained using standalone reservoir simulators. This is especially important in cases, where the backpressure on the well sand face is significant (e.g., deep offshore projects) or where there is a complex surface network.
Coupling these models, however, can be time consuming and challenging. Some examples of integration are given by Dempsey et al. (1971), Fang and Lo (1996), Hepguler et al. (1997), and Valbuena et al. (2015). Some of the challenges are due to the complexity and nonlinearity of the fundamental equations used to describe flow in porous media and in pipes and equipment and the solving strategies employed in each model. Explicit integration strategies limit the exchange of data between the models to a minimum and require fewer modifications to the individual solving algorithms. However, these strategies often exhibit stability problems and oscillations in the solutions (see Zapata et al. 2001). Implicit integration strategies are stable but require significant data transfer and modifications in the solving algorithms of the models thus making them more difficult to maintain and upgrade in the future.
Several papers address the issue of surface network optimization within the coupled model, see Hepguler et al. (1997) and Stanko and Venstad (2016). One recurring issue with network optimization lies in the combination of strongly nonlinear behavior (well and flowline performances) and integer variables (routing or disjunctive constraints). When coupled with a reservoir simulation, additional constraints appear with runtime and stability, the network optimization being run at each time step with different reservoir conditions (reservoir pressure, GOR, WC).
In this paper, this issue was addressed using an MILP formulation of the network optimization problem. The nonlinear well and flowline performances are approximated using SOS2 piecewise linear models. This approach is similar to what other authors have presented in the past, e.g., Codas et al. (2012), Codas and Camponogara (2012), Silva and Camponogara (2014), Hulse and Camponogara (2017), and Silva et al. (2015). Hoffmann et al. (2016) successfully apply this technique to optimize downhole diluent injection for an offshore heavy oil field and forecast field production. The work of Kosmidis et al. (2004) is the first one to employ piecewise linear approximations for nonlinear functions in the petroleum engineering production problem. Piecewise linear tables are generated using sequentially the existing blackbox model for different flow conditions. The formulation of the optimization problem takes considerably more effort and time to set up, but the running time is considerably lower than the traditional nonlinear approach. The other advantage lies in the robust handling of integer variables.
This paper proposes an integrated methodology to schedule and forecast production. The approach integrates all elements of the production chain (wells, network and process) into a single integrated model to predict, within a single run, an optimized production profile that (1) honors all constraints throughout the production system and (2) ensures the highest production at each time step. The integrated model can then be used to assess the economical feasibility of different development alternatives.
Production system modeling and optimization
Coupling strategy
The proposed coupling strategy is presented in Fig. 2. In any given time step \(t_i\), reservoir pressure \(p_{\rm R_{i}}\), producing water cut (WC) \(f_{\rm w_{i}}\) and gas oil ratio (GOR) \(R_{\rm p_{i}}\) are transferred to the surface network model as wellboundary conditions. The network model is optimized, and the optimal well rates, choke openings, and gas lift rates are found. The cumulative production for step \(t_i\) is found by summing the rate of all wells and multiplying by the length of the time step \(\varDelta t\). The incremental oil production \(\varDelta N_\mathrm{p}\), the reservoir conditions of the previous time step (\(p_{\rm R_{i}}\) and gas, oil, and water saturations: \(S_{\rm g_{i}}\), \(S_{\rm o_{i}}\), \(S_{\rm w_{i}}\)) and the material balance model are used to calculate reservoir pressure and producing WC and GOR of step \(t_{i+1}\). The process is repeated.
Description of the base case
The field studied in this paper is a small offshore field located in the North Sea.
Reservoir models
Reservoir properties
Reservoir A  Reservoir B  

Initial pressure (psia)  4650  3905 
Oil API gravity (\(^\circ \hbox {API}\))  37  39 
Initial GOR (scf/stb)  800  500 
Reservoir temperature (\(^\circ \hbox {F}\))  210  250 
Bubble point (psia)  3150  2300 
Well performance model
A simple IPR model is used for each well. Above the bubble point, a straight line IPR is used, while the Vogel IPR is used below the bubble point.
Well performance is modeled with vertical lift performance tables that depict required bottomhole pressure as a function of oil rate, wellhead pressure, gas lift rate, and WC. Wells are naturally flowing in the early stage of the field and are boosted with gas lift injection in the later phase. Vertical lift performance (VLP) models are built in a commercial blackoil steadystate simulator.
Well characteristics, layout, and configuration are provided in Appendix A.
Network model
In the surface network, the pipeline between manifolds A and B is 7.5 km long and has an inner diameter of \(8^{\prime \prime }\). The pipeline between manifold B and the production separator is 12 km long and has an inner diameter of \(12^{\prime \prime }\). The flowlines that connect the wellhead to manifolds A and B are short and have an inner diameter of \(8^{\prime \prime }\).
The network model is built in a commercial steadystate simulator.
Surface process
The production separator is operated at a constant pressure of 200 psia.
Network optimization
Nonlinear formulation
Linear formulation
The optimization problem has been reformulated as a mixed integer linear problem, to speed up the optimization runtime. To do so, all dependencies have to be explicitly expressed. This is a major difference compared to traditional nonlinear blackbox optimization, where part of the calculation (e.g., the well performance) are totally invisible to the solver. In particular, the network has to be modeled and solved within the optimization formulation. This is achievable by fist splitting the network model into smaller independent elements. An independent element is defined by an inlet and an outlet with a constant GOR, WC, and mass rate.
Well modeling
Flowline modeling
Network equilibrium modeling
SOS2 piecewise linear approximation
Note that in an MILP optimization formulation, the weighting factors \((\lambda _i)_{1\le i \le N}\) are variables and Eqs. (11), (12) and (13) are constraints.
The difference between f(x) and its piecewise approximation \({\widetilde{f}}(x)\) can be minimized by increasing the number of breakpoints \(x_i\). More breakpoints mean more weighting factors \(\lambda _i\); therefore, more variables in the optimization problem and ultimately longer optimization runtime. Consequently, there is a balance to find between accuracy and runtime.
Resulting mixed integer linear program
The nonlinear optimization problem can be reformulated into a Mixed Integer Linear Program (MILP) by piecewise linearising all performance functions using SOS2 models. The integer variables (in fact, binary variables) are consequences of the usage of SOS2 models. Appendices B and C give the complete details of MILP formulation of the problem.
In this paper, we use the simplex with the branch and cut algorithm implemented in a commercial solver. Default solver settings are used.
Results
Production forecast
Parameters used in the reference case
Parameter  Value 

Field liquid capacity  20,000 stb/day 
Gas lift startup  After 4 years 
Gas lift injection capacity  10 MMscf/day 
Duration of the operations  20 years max. 
Time step size  2 months 
 1.
Silo approach: material balance simulators with capacity constraints and minimum WHP as boundary conditions (400 psia for wells of reservoir A and 300 psia for wells of reservoir B);
 2.
IAM without gas lift;
 3.
IAM with gas lift optimization at each time step.
Economical analysis and field development
Economical data
 1.The revenue of a given year y is given bywhere \(P_\mathrm {o}\) is the current oil price, \(V_{\mathrm {o}y}\) is the volume of oil produced during year y, and \(E_\mathrm {op}\) is the operational expenditure. We assume here that gas does not generate any revenue (flared, reinjected or used as gas lift).$$\begin{aligned} R_y = P_\mathrm {o} \cdot V_{\mathrm {o}y}  E_\mathrm {op}, \end{aligned}$$(16)
 2.
The CAPEX \(E_\mathrm {cap}^\mathrm {init}\) depends on the initial surface installations (e.g., treatment capacity).
 3.
The CAPEX \(E_\mathrm {cap}^\mathrm {gl}\) depends on the surface installation size related to gas lift.
Numerical values of key field development parameters
Parameter  Value 

Oil price \(P_\mathrm {o}\)  70 USD/stb 
Interest rate i  8% 
Application to the reference case
Main economical results for the reference case
KPI  Value 

Project NPV  893.5 million USD (of year 0) 
Initial CAPEX  1.52 billion USD 
Secondary CAPEX  12.1 million USD 
Duration of operations  17 years 
Field development optimization
NPV (in 1E8 USD) as a function of liquid rate and gas lift capacities
Liquid capacity (\(\hbox {sm}^3/\hbox {day}\))  

5000  10,000  15,000  20,000  25,000  30,000  35,000  40,000  
Gas lift capacity (MMscf/day)  0  \(\,9.32\)  − 0.46  3.48  5.35  5.98  5.78  5.55  5.34 
5  \(\,9.51\)  0.80  6.28  8.63  9.24  9.02  8.78  8.55  
10  \(\,9.62\)  0.73  6.30  8.94  9.67  9.44  9.20  8.96  
15  \(\,9.71\)  0.67  6.24  8.91  9.69  9.47  9.22  8.98  
20  \(\,9.79\)  0.58  6.15  8.83  9.64  9.42  9.17  8.94 
Conclusions

A methodology to perform field production forecasting and scheduling is presented and discussed. The methodology successfully maximizes oil production in each depletion step by varying gas lift rate and honoring multiple operational constraints.

The running times of the proposed methodology are low and might be suitable for early field development studies that require to perform multiple sensitivity studies and uncertainty analysis.

The methodology was then used to optimize field design. Optimum ranges for liquid processing capacity and gas lift capacity were determined by brute force exploration of NPV function. The difference between the minimum and maximum NPV in the explored space is substantial.

Optimal gas lift startup was also determined in the same manner. It seems it is best to start gas lift from field production startup.
Further work
The methodology presented in this paper should be applied to other use cases with more wells. Good candidates are systems with routing issues (e.g., HP/LP lines or routing between platforms). This work has employed a material balance model to represent the reservoir. It should be extended for cases with a 3D reservoir simulation.
Field  Conversion  S.I. 

1 psi  = 6.894757  Pa 
1 ft  = 0.3048  m 
1 in  = 0.0254  m 
1 bbl  = 0.1589873  m\(^3\) 
1 cf  = 0.028316846592  m\(^3\) 
1 BTU  = 1055.06  J 
1 lb  = 0.453592  kg 
\(^\circ \hbox {API}\)  \(\rho = 141.5/(131.5 + \gamma _\mathrm {API})\)  g/cm\(^3\) 
\(^\circ \hbox {F}\)  \(^\circ \text {C} = (^\circ \text {F}  32) / 1.8\)  \(^\circ \hbox {C}\) 
Notes
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