Article pubs.acs.org/IECR
Well Placement, Infrastructure Design, Facility Allocation, and Production Planning in Multireservoir Oil Fields with Surface Facility Networks M. S. Tavallali,† I. A. Karimi,*,† A. Halim,† D. Baxendale,‡ and K. M. Teo§ †
Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore, Singapore 117585 ‡ RPS Energy Limited (Singapore Branch), Unit #03-01 China House, 19 China Street, Far East Square, Singapore, Singapore 049561 § Department of Industrial & Systems Engineering, National University of Singapore, 1 Engineering Drive 2, Singapore, Singapore 117576 S Supporting Information *
ABSTRACT: Managing oil fields with multiple reservoirs and shared surface networks optimally is a challenging practical problem. The majority of studies on optimal well placement have focused on either the interactions within the surface network facilities or the dynamic states of the subsurface environment. This work holistically integrates both surface and subsurface sections and addresses well placement, surface network design and allocation, and production/injection planning in a field with multiple irregular-shaped reservoirs. We consider the dynamic, economic, and operational interdependencies of the reservoirs, shared surface network, and common oil market through a complex, nonconvex, and deterministic mixed integer nonlinear programming model. Our approach determines the optimal number and locations of wells, design or retrofit of surface network, connections/allocations of wells with surface facilities, and optimal injection/production planning in water-drive reservoirs. We illustrate the complex interplay of well operations and throughputs using a literature example.
1. INTRODUCTION The world population has increased from almost 5.4 billion in 1990 to about 7.3 billion in 2012. The global energy demand has thus been constantly increasing.1 With the advent of new drilling technologies such as horizontal drilling and hydraulic fracturing and discoveries of shale gas,2 fossil fuels such as oil and gas will continue to be the major energy resources for the world. Thus, it is critical to exploit these limited resources (both existing and new) in a wise, efficient, and cost-effective manner. Cost-effective production of oil requires optimal design and planning of the field’s infrastructure. Well drilling is an expensive part that poses a considerable financial risk; up to 60% or more of the capital expenditure (CAPEX) on an exploitation project might be due to well drilling and associated activities alone. Yet, once a well is drilled, connected to the production infrastructure, and operated successfully, it can provide huge revenue and return the initial investment. Therefore, optimal well placement, infrastructure design, and planning are critical steps which must consider the integration of different elements in the infrastructure. 1.1. Oil Field and Production Challenges. Figure 1 shows a schematic for a multireservoir field. It has three main components: (1) the porous subsurface formation, (2) the well strings, and (3) the surface network. The surface network includes wellheads, manifolds (headers) that collect/mix the oil/gas flows from the wellheads, processing centers that receive the commingled flows from the manifolds, a maze of valves and flow lines that interconnect the wellheads, manifolds, and processing centers, and finally long pipelines that supply the fluids to the market after pressurization. The interaction between the above components strongly affects production. In practice, fields often © 2014 American Chemical Society
have multiple reservoirs that share common surface infrastructure and production facilities. These facilities strongly interlink the operations of these reservoirs. Since each reservoir can no longer be studied separately, well placement and production planning become even more challenging for various reasons. First, the surface settings would depend on the conditions of the connected wells from multiple reservoirs. Production variations at some wells due to changes in subsurface conditions can alter production at other wells connected to the same flow path. When new producer wells are drilled to increase/ sustain the production rates and connected to the surface network, the conditions of the surface network may change drastically. These variations may even necessitate changes in the design and operation of the entire surface network. Second, each reservoir may have different geological characteristics and production mechanisms. Treating these separate reservoirs as one aggregated reservoir with several inactive portions is very inefficient.3 In fact, facility design studies such as well placement4 even in a single reservoir5 are already complex and difficult problems. Lastly, the complex multiphase flow regimes in the subsurface, wells, and surface network add even more complexities. Clearly, addressing well placement jointly with the design and planning of surface facilities at the field levelrather than the individual reservoir levelsis of paramount importance and poses special challenges. Received: Revised: Accepted: Published: 11033
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approximations15 of reservoir pressure and gas-oil-ratio (GOR) versus cumulative oil production, and linear pressure drop approximation for the multiphase flow in the surface flow lines.12,16 Despite these simplifications, these contributions are important as they address a broad range of surface design applications. However, to our best knowledge, no solid study that considers the dynamics of both subsurface and surface networks integrally with infrastructure planning and production exists today. Studies on short-term scheduling decisions over days or weeks ignore capital cost considerations and reservoir dynamics and focus on surface operation, production scheduling, and flow-line routing. These models usually integrate multiphase flow, starting from a wellbore and ending in a separation unit. Since the reservoir dynamics is not included, two key elements for comparing these contributions are (a) the way they model the multiphase flow in the flow lines and (b) the algorithm used for optimization. Different piecewise linear approximations for the momentum balances17 and pressure drops18 have been on the top list of the former. The latter includes decomposition methods (into smaller subproblems and using branch and price framework,18b Lagrange relaxation and Dantzig−Wolfe18a) and the outer approximation algorithm17 using heuristics such as branching priories18c and prescreening. Codas et al.18c provided a good industrial overview on the surface-oriented and short-term scheduling studies. In brief, most existing studies either have not used rigorous multiphase flow reservoir models or have not included the surface facility network effects. Moreover, in most cases this problem is either not addressed at the field level or addressed with substantial compromise in the governing equations. The industry has already identified the importance of such integrated studies, and packages such as AVOCET by Schlumberger19 are working in that direction. The complexity and curse of dimensionality are the biggest obstacles in solving these problems. In our recent work5 we addressed well placement and production planning jointly in a single 2D rectangular reservoir using a rigorous reservoir model and (multiphase) well flow up to the well head. Although we did not consider the surface network dynamics, the problem was already a challenging mixed-integer dynamic optimization (MIDO) problem requiring a spatiotemporal and dynamic nonconvex MINLP model. We developed an effective modification of the outer approximation algorithm by Grossmann and co-workers20 for its solution and obtained interesting results. Our work provides a good basis for the work in this paper. In this work, we aim to extend our previous study5 to address well placement/surface network design (location and allocation) and production/injection planning in a field with multiple irregular-shaped reservoirs supplying to a shared surface production−network facility. In this sense, it represents a significant and novel advance over our previous study. We simultaneously address all the dynamic, economic, and operational interdependencies of the entire field and its reservoirs. Our deterministic model holistically includes financial considerations, market demand, dynamic, and structural constraints in a surface network of wells−manifolds−separators and provides drilling/ network design decisions and detailed production/injection plans simultaneously. We first define and describe the scope of our problem. After that we discuss our modeling and solution strategy. Then, we present a comprehensive case study to assess our approach and conclude.
Figure 1. Schematic diagram of the subsurface and surface components of a hydrocarbon field. Thick black and blue lines represent the production and injection wells, respectively, r1, r2, r3, and r4 are four reservoirs with possibly different production mechanisms.
1.2. Literature Review. A typical petroleum field may have a life span of 15−20 years, which will involve a variety of shortterm and long-term planning decisions. One must assess capital investments for infrastructure design and well placement years in advance, whereas one must plan well production, gas lifting, flow routing from wells to manifolds, etc., on a daily and/or weekly basis.6 An ideal design and operation must integrally consider both horizons.7 Hence, we review studies on both domains, i.e., long-term planning (well placement and infrastructure design) and short-term scheduling studies (flow routing). Several studies exist in the literature on well placement, but many have focused on the reservoir only, without due regard to the dynamics of the surface network facilities. Additionally, production planning is simply left to the reservoir simulator using some heuristics based on limited production/injection rates or bounded BHP/THP. Most of these studies have linked the reservoir simulator or its proxy (under aforementioned heuristic conditions) to evolutionary and direct search,8 and relatively few have used gradient-based optimization9 to search for drilling sites. Some of these studies have employed the reservoir’s static data for well screening,10 and this naturally ignores the dynamics. However, the above trend is changing. More studies are now paying attention to well placement jointly with production planning.11 ECLIPSE now offers a reservoir coupling and network option based on a master−slave heuristic. Some others have addressed well placement, infrastructure design, and production planning in a surface network.12 However, they have grossly approximated the subsurface flows, which may be inappropriate for well placement studies.13 In such cases, no meaningful distinction exists between single-reservoir and multireservoir (field) subsurface models. Moreover, they have not considered fluid injection, which is a common field activity. Some of these empirical approximations include influence function and superposition,14 piecewise linear12 and nonlinear 11034
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2. PROBLEM STATEMENT It is desired to initiate or increase oil production from a multireservoir hydrocarbon field (Figure 1) by drilling new/infill vertical producer wells and installing the required surface network (manifolds, processing centers, valves, flow lines, etc.). If the field is already producing then it may have some existing wells, manifolds, and processing centers. In this case, the surface network may need changes to accommodate new wells and production. The well placement problem is then as follows. Given 1. Geological information for each reservoir such as location, dimensions from seismic studies or history matching, porosity, and permeability. 2. Pressure and saturation profiles in each reservoir at time zero. 3. Pressure−volume−temperature (PVT) related information such as viscosity, density, compressibility, surface tension, and formation volume factor (from core samples or previous production data) for each reservoir. 4. Existing wells (if any), their locations, types (producer vs injector), diameters, lengths, tubing roughness, etc. 5. Potential/existing manifolds, their locations, capacities, maximum numbers of connections, and existing/potential well-to-manifold and manifold-to-separation center allocations. 6. Operational data such as the required inlet pressures at the separation centers, separation-center capacity for each phase, water-cut limits, and incremental capacity expansion plans for surface facilities. 7. Field production horizon of H years. 8. Relevant economic data such as drilling budget and costs, manifold installation cost, costs per well-to-manifold and manifold-to-center connections, injection costs, oil/gas revenue forecasts, discount rate, demand curve, etc. Obtain 1. Number and locations of new producer wells (and hence the reservoirs to be exploited) and their production profiles. 2. Number and location of manifolds and processing centers and incremental capacity expansion plan for the surface processing centers. 3. Potential well-to-manifold, well-to-surface-center, and manifold-to-surface-center allocations. 4. Throughput profiles for all producer/injector wells, flow lines, manifolds, and processing centers. 5. Dynamic pressure profiles along the network at processing centers, manifolds, wellheads, well bore holes, and corresponding valve settings. 6. Dynamic pressure and saturation profiles for each reservoir. Aiming to maximize the net present value (NPV) of oil/gas production over the planning horizon. Assuming 1. The reservoirs are horizontal and planar. They may overlap, but they are disconnected. Field surface elevation may vary from point to point. 2. Wells are vertical and can pass through multiple reservoirs but can be perforated to access only one reservoir. 3. A wellhead may be connected to one or more manifolds/ centers. 4. The manifold capacities remain constant over time.
5. Each well (existing or potential) is preallocated to some manifolds/centers (existing or potential) based on some criteria such as distance, from which the best allocations will be selected. This can be relaxed but will increase computation time.12 6. Capillary pressure in the reservoir is negligible, and the reservoir is undersaturated (i.e., its pressure exceeds bubble point pressure). 7. All phases (oil, water, and gas) are compressible. The driving mechanisms in the field can be water-drive injection and/or primary expansion. Each reservoir might have different pressure and saturation distribution; however, all have the same fluid. Following 1. All new wells, manifolds, and surface centers begin operations simultaneously at time zero. Furthermore, each well must be beyond some minimum distance from all other wells. 2. A well that hits its water-cut limit is shut in. Allowing 1. Reservoirs may have arbitrary and irregular shapes. 2. Existing manifolds and centers can make/receive new connections. 3. Processing centers can receive fluids from wells directly or through manifolds. 4. Central water processing units supply water to injector wells directly or via shared manifolds.
3. MATHEMATICAL FORMULATION Let R (r = 1, 2, ..., R) denote the number of reservoirs, C (c = 1, 2, ..., C) denote the number of processing centers (existing/ potential), and M (m = 1, 2, ..., M)) denote the number of manifolds (existing/potential). Of these we assume that the first Me manifolds (1 ≤ m ≤ Me) and first Ce processing centers (1 ≤ c ≤ Ce) already exist. There are two types of manifolds and processing centers. The ones associated with processing oil/gas are called production manifolds or oil production centers, and the ones associated with treating and injecting water are called injector manifolds and water treatment centers. To differentiate among these different types of manifolds and centers we define PM = {m|m is an existing/potential producer manifold} PC = {c|c is an existing/potential oil processing center}
Thus, if m ∉ PM then m is an injector manifold, and if c ∉ PC then c is a water treatment center. 3.1. Reservoir Dynamics and Spatial Discretization. Following Tavallali et al.,5 we discretize the spatial coordinates. To this end we use the field’s geological map to define the tightest possible rectangle around each r. Then we discretize each rectangle r by defining Ir cells of arbitrary lengths Δxjr (i = 1, 2, ..., Ir) in the x direction and Jr cells of arbitrary lengths Δyjr (j = 1, 2, ..., Jr) in the y direction. Then we name each cell in a reservoir by a single index, n = i + (j − 1) × Ir, and define three sets for each reservoir as follows IPr = {n|cell n belongs to reservoir r }
IX r = {n|n ∈ IPr , (n + 1) ∈ IPr , and i < Ir } IYr = {n|n ∈ IPr , (n + Ir ) ∈ IPr , and j < Jr } 11035
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transmissibilities in x and y directions, Mtf,(x+)r, Mtf,(x−)r, Mtf,(y+)r, and Mtf,(y−)r are upstream weighted mobilities, which are taken as those of the neighboring cells with higher pressures. The last four terms within the big parentheses in both equations represent the convective flow terms along the four faces of cell n. The transmissibilities, upstream weight mobility terms, and variable accumulation multipliers are as in eqs 3, 4, 7−10, and A1−A4 of Tavallali et al.5 Mto,nr and Mtw,nr are the oil and water mobilities defined as (n ∈ IPr)
IPr excludes from the subscribing rectangle the cells that do not belong to reservoir r, IXr excludes the reservoir’s border cells in the x direction, and IYr excludes them in the y direction. A typical water-drive reservoir will have two types of wells. The ones producing fluids from the reservoirs are called producer wells, and the others that inject water into the reservoirs are called injector wells. PWr = {n|n ∈ IPr is an existing producer well} IWr = {n|n ∈ IPr is an existing injector well}
Mot , nr
NWr = {n|n ∈ IPr is a potential location for a new producer
=
well}
M wt , nr
=IPr − PWr − IWr − {n ∈ IPr |a new producer well is infeasible at cell n}
kro0
(1)
+
+
(
−
n ∈ IPr
)
+ {Mot ,(y+)r Tnry[Pnrt − P(tn + Ir)r ]}n ∈ IYr = 0
To manage well−manifold, well−center, and manifold−well connections, we define following sets and variables: (2)
M nr = {m|cell n of reservoir r is or can be connected to manifold m}
(Vnr /dht ){dwt ,1, nr[Pnrt − Pnrt− 1] + dwt ,2, nr[Snrt − Snrt− 1]}
+
r
(
1 ≤ r ≤ R , n ∈ IPr
)
{M wt ,(x−)r T(xn − 1)r[Pnrt
(5)
krw0
⎧1 if a well should exist at cell n of reservoir r ynr = ⎨ ⎩ 0 otherwise
P(tn − 1)r ]}(n − 1) ∈ IX r
+ {Mot ,(y−)r T(yn − Ir)r[Pnrt − P(tn − Ir)r ]}(n − I ) ∈ IYr
(
⎝ 1 − Sr , wr
⎧1 if manifold m should exist zm = ⎨ ⎩ 0 otherwise
+ {Mot ,(x+)r Tnrx[Pnrt − P(tn + 1)r ]}n ∈ IX r
+ qnt , r − qot, nr ∉ IW
⎞b ⎟⎟ − Sr , or ⎠
Snrt − Sr , wr
⎧1 if center c should exist uc = ⎨ ⎩ 0 otherwise
(Vnr /dht ){dot ,1, nr[Pnrt − Pnrt− 1] + dot ,2, nr[Snrt − Snrt− 1]} {Mot ,(x−)r T(xn − 1)r[Pnrt
=
(4)
where and are the end-point relative permeabilities of oil and water, Sr,or and Sr,wr are the residual oil and water saturations, respectively, a and b are the exponents in Corey’s correlation, (μr,Bo,1, μr,Bo,2) are regression parameters for the product (μo,r × Bo,r) of the viscosity and formation volume factor of oil, Cw,r is the water compressibility factor, B0w,r is the formation volume factor for water at reference pressure Pr, and μw is the water viscosity. The nomenclature provides a comprehensive list of sets, parameters, and variables. 3.2. Drilling and Infrastructure Design Decisions. We define the following binary variables for drilling and locating the surface facilities
where ε is the porosity, Bf is the formation volume factor of phase f (f = o for oil and f = w for water), Sf is the saturation, Pf is the pressure, and qf is the well flow from (+ve for out, −ve for in) the reservoir, krf is the relative permeability, μf is the viscosity, ρf is the density, and K is the absolute permeability tensor. It is important to note that all flow rates in eq 1 are at the processing center (versus in situ) conditions. The formation volume factors (Bo and Bw) map back the center flows to flows at the reservoir conditions. Tavallali et al.5 discretized the planning horizon H into T intervals of arbitrary lengths Δht (t = 1, 2, ..., T) and employed backward finite difference approximation for the derivatives to obtain the following equations for oil and water phases (n ∈ IPr) qot, nr ∉ IW r
⎛
kr0w⎜⎜
× [1 + Cw , r(Pnrt − Pr )]/Bw0 , r μw , r
The dynamics multiphase flow in a reservoir can be modeled through the following partial differential equation21 ⎡ kr ⎛ ⎤ g ⎞ ∂ ⎡⎢ Sf ⎤⎥ f ε + q f − ∇⎢ K⎜⎜∇Pf − ρf ∇z⎟⎟⎥ = 0 ⎢⎣ μf Bf ⎝ ∂t ⎢⎣ Bf ⎥⎦ gc ⎠⎥⎦
⎛ 1 − Snrt − Sr , or ⎞a ⎟⎟ /(μr , Bo ,1Pnrt + μr , Bo ,2 ) 1 − S − S ⎝ r , wr r , or ⎠
kr0o⎜⎜
Cnr = {c|cell n of reservoir r is or can be connected to manifold c}
− P(tn − 1)r ]}(n − 1) ∈ IX r
1 ≤ r ≤ R , n ∈ IPr
+ {M wt ,(x+)r Tnrx[Pnrt − P(tn + 1)r ]}n ∈ IX r + {M wt ,(y−)r T(yn − Ir)r[Pnrt − P(tn − Ir)r ]}(n − I ) ∈ IYr
)
+ {M wt ,(y+)r Tnry[Pnrt − P(tn + Ir)r ]}n ∈ IYr = 0
Cm = {c|manifold m is or can be connected to center c} 1 ≤ r ≤ R , n ∈ IPr (3)
Unless stated otherwise, all variables and constraints involving index t (defined later), r, m, and c are to be written for all their valid values, i.e., 1 ≤ t ≤ T, 1 ≤ r < R, 1 ≤ m < M, 1 ≤ c < C. To select from the above connection options we define the following binary variables
where Stnr and Ptnr are saturation and pressure at the end of interval t, Sonr, and Ponr are the initial saturation and pressure at time zero, dtf,1,nr and dtf 2,nr are the variable accumulation multipliers, qtnr and dto,nr are liquid (oil + water) and oil flow rates, Txnr and Tynr are 11036
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⎧1 if a well at cell n of reservoir r ⎪ =⎨ should be connected to manifold m ⎪ ⎩ 0 otherwise
R
CMWcU uc ≥
zunc = ⎧1 if manifold m should be connected to ⎪ ⎨ center c ⎪ ⎩ 0 otherwise
Q c(t + 1) * = dQ ct * + Q ct *
QC t * =
If any facility or connection from the above already exists then we set its binary to 1. Thus, we set uc = 1 for 1 ≤ c ≤ Ce, zm = 1 for 1 ≤ m ≤ Me, and ynr = 1 for n ∈ PWr ∪ IWr. Similarly, if a well (n, r) is already connected to manifold m then yznrm = 1. All corresponding installation and drilling costs are set to zero. For preventing wells (injectors or producers) from being adjacent, Tavallali et al. 5 proposed the following constraints ynr + y(n + 1)r ≤ 1
n ∈ IX r
(6)
ynr + y(n + I )r ≤ 1
n ∈ IYr
(7)
ynr + y(n + I + 1)r ≤ 1
(n + Ir ) ∈ IX r
r
∑
yznrm +
m ∈ M nr
∑
QCwt* =
∑
yznrm ≥ zm
(8)
∑
zumc ≥ zm
(16)
∑
Q ct *
(17)
n ∈ IPr
(18)
QMmt ≤ QMmt, U zm
(9)
(19)
QCct ≤ QCct , U uc
(20)
where δnr = −1 for n ∈ IWr and 1 otherwise and and QCt,U c are reasonable upper bounds. Similar constraints apply for the various connections qt,U nr ,
t t ,U qwmnrm ≤ qwmnrm yznrm
t qwmot , nrm ≤ qwmnrm
t t ,U qwcnrc ≤ qwcnrc yunrc t qwcot , nrc ≤ qwcnrc
(10)
t qmcot , mc ≤ qmcmc
QMt,U m ,
n ∈ IPr , m ∈ M nr
(21)
n ∈ IPr − IWr , m ∈ M nr
(22)
n ∈ IPr , c ∈ Cnr
(23)
n ∈ IPr − IWr , c ∈ Cnr
t qmcmc ≤ QMmt, U zumc
c ∈ Cm c ∈ Cm
(24) (25) (26)
where and are the total and oil flows through the connection between well n and manifold m, qwctnrc and qwcto,nrc are the total and oil flows through the connection between well n and center c, and qmctmc is the total flow through the connection between manifold m and center c. Note that we do not write eqs 22, 24, and 26 for injector wells as their oil flows are zero. qwmtnrm
1≤m≤M (11)
c ∈ Cm
Q ct *
δnrqnrt ≤ qnrt, U ynr
r = 1 n ∈ IPr ∍ m ∈ M nr
MCmU zm ≥
(15)
where is the initial center capacity and f r is the maximum possible expansion fraction from each available capacity. 3.3. Well and Surface Network Flow Management. Figure 1 depicts the various liquid (oil + water) flow rates from wells, manifolds, and processing centers. Nonexistent wells, manifolds, and centers cannot send/receive any flows at any time. If QMtm and QCtc are the total fluid flows through manifold m and processing center c during interval t then we have
c ∈ Cnr
∑
(t < T )
c ∉ PC
R
MW mU zm ≥
∑
(14)
Q0c *
yunrc ≥ ynr
1 ≤ r ≤ R , n ∈ IPr
(13)
c ∈ PC
On the basis of the above constraints, we exclude the cells adjacent to the existing injector and producer wells from NWr and fix their ynr to zero as discussed earlier. Additionally, we imposed the Clique cuts. Suppose that a well n ∈ IPr can have at most WMCUnr manifold/ center connections, a manifold m can have at most MWUm well connections and at most MCUm center connections and a center c can have at most CMWUc well/manifold connections. Similarly, if a well, manifold, or center should exist then it must have at least one connection. Then we have WMCnrU ynr ≥
yunrc ≥ uc
0≤t
