Short-term steady-state production optimization of offshore oil platforms: wells with dual completion (gas-lift and ESP) and flow assurance

Abstract

Research on short-term steady-sate production optimization of oilfields led to the development of models and solution methods, several of which have found their way into practice. Early models considered satellite wells that operate with a fixed topside pressure and gas-lift injection, while recent approaches address distinct types of artificial lifting, pressure control, and processing equipment. By integrating existing approaches, this work presents a flexible model for production optimization that considers new features, including flow assurance constraints and smart selection of artificial lifting operating modes (gas-lift with multiple valves, electrical submersible pumping, and dual completion). Given that the proposed model is conceptual, piecewise-linear functions are obtained from field and simulation process data to approximate nonlinear relations. This way, the methodology decides the best combinations of routing and operation modes to maximize production gains. Simulated results are reported considering a representative asset that illustrates complex behavior.

Appendix A: Piecewise-linear approximation

The piecewise-linear approximations are achieved by studying the behavior of these relations through tests or simulation runs, which aim to define breakpoints between which the relation is considered to be linear. Several binary variables and SOS2 constraints are employed to define the operational region for each such constraint—i.e., they define the active boundaries for a region in the state space in which the function approximation is linear. Informally, this can be thought of as a continuous region of the piecewise linearization.

SOS2 stands for specially ordered sets of type 2 (Beale and Tomlin 1970), which means that at most two elements of the ordered set can assume nonzero values, in which case they must be consecutive in the order.

In order for us to obtain a practical model, relations must be properly defined for the initial model to express oil production (${\widehat{q}}_{\mathrm{o, GL}}^{\,n}$, ${\widehat{q}}_{\mathrm{o, ESP}}^{\,n}$ and ${\widehat{q}}_{\mathrm{o, surg}}^{\,n}$), lift-gas injection (${\widehat{q}}_{\mathrm{gl}}^{\,n}$), pressure drop ($\widehat{\varDelta {p}}_{\mathrm{fl}}^{m,h}$), downthrust and upthrust bounds for ESP pumps (${\widehat{q}}^{\min , n}_{\mathrm{l}}$, ${\widehat{q}}^{\max , n}_{\mathrm{l}}$), hydrate envelope (${\widehat{\tau }}_{\mathrm{env-hyd}}^{n}$) and pressure and temperature along the pipeline ($p_{\mathrm{hyd, gl}}$, $p_{\mathrm{hyd, esp}}$, $\tau _{\mathrm{hyd, gl}}$, $\tau _{\mathrm{hyd, esp}}$).

Analytical functions that accurately model these relations are either unknown or very complex, resulting in non-linear formulations. Fortunately, values for these relations can be obtained for a given combination of inputs through experimentation (either by performing simulations or field tests) (Gunnerud and Foss 2010). This process can be repeated for several different inputs, for each well and manifold. As a result, for each such function, we can obtain a set of points in the domain for which the value of the function is known. These points can then be used to define a piecewise-linear approximation by assuming that the function is linear in each polytope defined by the adjacent known points.

Among the various formalisms that could be used to express a piecewise-linear function (Vielma et al. 2010), we chose to use a SOS2 model (Beale 1980). SOS2 constraints are natively supported by several commercial solvers and have some properties that allow them to be handled more efficiently by a branch-and-bound procedure.

Generically, let $f:{\mathcal {D}} \rightarrow {\mathbb {R}}$ be a piecewise function that maps from domain ${\mathcal {D}}$ to the real numbers, and let ${\mathcal {D}}' \subset {\mathcal {D}}$ be an ordered set of points $d \in {\mathcal {D}}'$ for which f(d) is known, i.e., $d \in {\mathcal {D}}'$ are the vertices of the function domain, which we call the breakpoints of f.

A weighting variable $\lambda ^{d}$ can be assigned to each d, in order to express how much that breakpoint contributes to the value of the function for a given input. For any given $x \in {\mathcal {D}}$, $x = \sum _{d \in {\mathcal {D}}'} d \cdot \lambda ^{d}$ and $f(x) = \sum _{d \in {\mathcal {D}}'} f(d) \cdot \lambda ^{d}$, with $\lambda ^{d}$ being SOS2 variables, i.e., only the breakpoints corresponding to the vertices of the polytope that contains x should be active. Formally:

• $\lambda ^{d} \ge 0, \ \forall \ d \in {\mathcal {D}}'$;

• $\sum \nolimits _{d \in {\mathcal {D}}'} \lambda ^{d} = 1$;

• Only two adjacent variables $\lambda ^{d}$ can be nonzero.

SOS2 constraints can be extended to multidimensional piecewise-linear functions (Beale 1980) by limiting convex combinations to be inside a hypercube of the function domain and using a linked chain of SOS2 constraints, one for each dimension of the domain (Williams 2013).

Due to space considerations, the piecewise-linear model is not entirely shown. Instead, parts of the model are presented to highlight how it can be linearized. Following, we show how SOS2 constraints can be used for a piecewise-linear approximation of the oil production $q_{\mathrm{o}}^n$ of satellite well n, operated with ESPs.

1.1 A.1 Oil production in satellite ESP wells

Recall Eq. (6a). For simplicity, let us consider only satellite wells that produce using ESP or naturally, namely wells $n\in {\mathcal {N}}_{\mathrm{ESP}}^{\star }$. In this case, Eq. (6a) reduces to:

$$\begin{aligned} q_{\mathrm{o}}^{n} = {\widehat{q}}_{\mathrm{o, ESP}}^{\,n}(w_{\mathrm{f}}^n,p_{\mathrm{wh}}^n) \cdot t^{n}_{\mathrm{esp}} + {\widehat{q}}_{\mathrm{o, surg}}^{\,n}(p_{\mathrm{wh}}^n) \cdot t^{n}_{\mathrm{surg}} \end{aligned}$$

(20)

We model the relation expressed by (20) with the following piecewise-linear approximation, accounting for the possibility of multiple production lines (which is not explicitly stated in the initial model):

$$\begin{aligned} q_{\mathrm{o}}^{n}= & {} \sum _{c \in {\mathcal {C}}} q_{\mathrm{o}}^{n,c} \end{aligned}$$

(21a)

$$\begin{aligned} q_{\mathrm{o}}^{n,c}= & {} \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}} \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \theta ^{n, c}_{k, j} \cdot q^{n, k, j}_{\mathrm{o}} \nonumber \\&+ \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \theta ^{n, c}_{j} \cdot q^{n, j}_{\mathrm{o}}, \ \forall c \in C \end{aligned}$$

(21b)

$$\begin{aligned} \eta _{j}^{n}= & {} \sum _{c\in {\mathcal {C}}} \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}} \theta ^{n,c}_{k,j},\,\forall j \in {\mathcal {J}}_{\mathrm{ESP}} \end{aligned}$$

(21c)

$$\begin{aligned} \eta _{k}^{n}= & {} \sum _{c\in {\mathcal {C}}} \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \theta ^{n,c}_{k,j},\,\forall k \in {\mathcal {K}}_{\mathrm{ESP}} \end{aligned}$$

(21d)

$$\begin{aligned} \nu _j^{n}= & {} \sum _{c\in {\mathcal {C}}} \theta ^{n,c}_{j}, \, \forall j \in {\mathcal {J}}_{\mathrm{ESP}} \end{aligned}$$

(21e)

$$\begin{aligned} t^{n,c}_{\mathrm{esp}}= & {} \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}} \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \theta ^{n, c}_{k, j} \end{aligned}$$

(21f)

$$\begin{aligned} t^{n,c}_{\mathrm{surg}}= & {} \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \theta ^{n, c}_{j} \end{aligned}$$

(21g)

$$\begin{aligned} t^{n}_{\mathrm{esp}}= & {} \sum _{c \in C} t^{n,c}_{\mathrm{esp}} \end{aligned}$$

(21h)

$$\begin{aligned} t^{n}_{\mathrm{surg}}= & {} \sum _{c \in C} t^{n,c}_{\mathrm{surg}} \end{aligned}$$

(21i)

$$\begin{aligned} t^{n}= & {} t^{n}_{\mathrm{surg}} + t^{n}_{\mathrm{esp}} \end{aligned}$$

(21j)

$$\begin{aligned}&\left\{ \eta _{j}^{n} \right\} _{j \in {\mathcal {J}}_{\mathrm{ESP}}},\, \left\{ \eta _{k}^{n} \right\} _{k \in {\mathcal {K}}_{\mathrm{ESP}}}, \, \left\{ \nu ^{n}_{j} \right\} _{j \in {\mathcal {J}}_{\mathrm{ESP}}}\text { are SOS2} \end{aligned}$$

(21k)

where:

• $q_{\mathrm{o}}^{n,c}$ is the oil production of well n routed to production line c.

• ${\mathcal {K}}_{\mathrm{ESP}}$ is the set of breakpoints for ESP frequency, for wells operating with ESP.

• ${\mathcal {J}}_{\mathrm{ESP}}$ is the set of breakpoints for wellhead pressure, for wells operating with ESP.

• $\theta ^{n, c}_{k, j}$ is the convex combination variable for oil production of a well operating with ESP, for well n, production line c, ESP frequency k and wellhead pressure j.

• $\theta ^{n, c}_{ j}$ is the convex combination variable for oil production of a well operating with surgence, for well n, production line c and wellhead pressure j.

• $q^{n, k, j}_{\mathrm{o}}$ is the value of oil production for a well operating with ESP, corresponding to breakpoint (n, k, j), for any production line.

• $q^{n, j}_{\mathrm{o}}$ is the value of oil production for a well operating without artificial lifting corresponding to breakpoint (n, j), for any production line.

• $t^{n,c}_{\mathrm{esp}}$ is a binary variable that assumes the value 1 if and only if well n is active, routed to production line c, and producing using an ESP.

• $t^{n,c}_{\mathrm{surg}}$ is a binary variable that assumes the value 1 if and only if well n is active, routed to production line c, and producing by surgence.

• $\eta _{j}^{n}$, $\eta _{k}^{n}$ and $\nu _{j}^{n}$ are SOS2 variables used to model the condition that only one polytope of the piecewise linearization can be active.

Equation (21a) states that the total oil produced by a well equals the sum of the oil routed from that well to each production line. In practice, a well will be routed to at most one production line, so $q_{\mathrm{o}}^{n,c}$ will be nonzero for at most one value of c. Equation (21b) approximates oil production with a piecewise linear function. Oil produced depends on the weight variables $\theta ^{n, c}_{k, j}$ for ESP (or $\theta ^{n, c}_{j}$ for a surging well) and the corresponding production values $q^{n, k, j}_{\mathrm{o}}$ (or $q^{n, j}_{\mathrm{o}}$ for a surging well) obtained experimentally. Equations (21c) to (21e) associate an array of SOS2 variables to each dimension of the weighting variables. This guarantees that at most one polytope of the piecewise-linear approximation has the associated weights greater than zero. Equations (21f) and (21g) enforce the sum of the weighting variables corresponding to well n and production line c to assume value 1 if well n is active and routed to production line c; and 0 otherwise.

Equations (21h) to (21j) guarantee that a well can be routed to only one production line and be active in at most one operation mode (in this case, either ESP or surgency).

1.2 A.2 Liquid flow at ESP wells

Recall Eq. (6i), which states that the total liquid flow in wells operated by ESP is bounded by a minimum and a maximum flow, which in turn are functions of the ESP frequency. These frequency-dependent bounds can be approximated by piecewise-linear functions for all $n \in {\mathcal {N}}_{\mathrm{ESP}}$ as follows:

$$\begin{aligned}&q_{\mathrm{l},\min }^{n} \le \left( \frac{1}{1-bsw^n}\right) \cdot \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}} \sum _{j \in {\mathcal {J}}_{\mathrm{ESP}}} \sum _{c\in {\mathcal {C}}} \theta ^{n,c}_{k,j} \cdot q_{\mathrm{o}}^{n,k,j} \le q_{\mathrm{l},\max}^{n}, \end{aligned}$$

(22a)

$$\begin{aligned}&q_{\mathrm{l},\min }^{n} = \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}}\lambda ^{n,k} \cdot q_{\mathrm{l},\min }^{n,k}, \end{aligned}$$

(22b)

$$\begin{aligned}&q_{\mathrm{l},\max }^{n} = \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}}\lambda ^{n,k} \cdot q_{\mathrm{l},\max }^{n,k}, \end{aligned}$$

(22c)

$$\begin{aligned}&w_{\mathrm{f}}^{n} = \sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}}\lambda ^{n,k} \cdot w_{\mathrm{f}}^{n,k}, \end{aligned}$$

(22d)

$$\begin{aligned}&\sum _{k \in {\mathcal {K}}_{\mathrm{ESP}}}\lambda ^{n,k} = t^n_{\mathrm{esp}}, \end{aligned}$$

(22e)

$$\begin{aligned}&\left\{ \lambda ^{n,k} \right\} _{k \in {\mathcal {K}}_{\mathrm{ESP}}}\text { is SOS2} \end{aligned}$$

(22f)

where $q_{\mathrm{l},\min }^{n}$ and $q_{\mathrm{l},\max }^{n}$ are the minimum and maximum bounds, respectively, for liquid flow from well n as a function of the ESP frequency, and $\lambda ^{n,k}$ is a weighting variable for the convex combination of frequencies. The set of $\lambda ^{n,k}$ variables is SOS2.

Equation (22a) is the piecewise-linear approximation of Eq. (6i) that imposes the limits on the liquid flow. Equations (22b) and (22c) define the bounds as a linear combinations of weight coefficients $\lambda ^{n,k}$, with breakpoints $(q_{\mathrm{l},\min }^{n,k},$ $q_{\mathrm{l},\max }^{n,k}$) obtained by simulation. Eqs. (22d) and (22e) ensure that the operating frequency given by this linear combination is the same as the one used in other equations of the model and that this frequency is nonzero only if the corresponding well is active and operating with ESP.