Multi-period Optimization for Long-Term Oilfield Production Planning

Abstract

In this work, a multi-period nonlinear programming formulation is presented to obtain the optimal long-term oilfield production planning, based on a two-phase, one-dimensional, and Cartesian-coordinated phenomenological reservoir model. The phenomenological model contains a set of second-order partial differential equations, which are approximated by a collocation on finite element method. This CFE method prevents mathematical stability limitations due to stiffness problems, resulting in an algebraic equation system added as an optimization set of constraints. This is a significant and innovating approach as there are only a handful of similar studies in the literature that integrate phenomenological models as mathematical constraints in the optimization problem. However, these works do not solve the model using long-term production planning coupled with a simultaneous strategy. Also, formulation applied to two study cases allowed solving the optimization problem within an adequate time without requiring a high-performance computing platform. Results show the economic impact of simultaneously considering the constraints and the state variables evolution throughout the reservoir’s life span to obtain the optimal long-term production planning.

Appendix A

Before implementing CFE method in the phenomenological model, the terms that contain the second-order spatial derivative must be reduced to Eqs. 32 and 33:

$$ \frac{\partial }{\partial x}\left[ {A\frac{{K\dot{K}}}{{\dot{\mu }}} \frac{\partial P}{{\partial x}}} \right] = \frac{{\partial \dot{U}}}{\partial x} $$

(32)

$$ \frac{\partial }{\partial x}\left[ {A\frac{{K\overline{K}}}{{\overline{\mu }}} \frac{\partial P}{{\partial x}}} \right] = \frac{{\partial \overline{U}}}{\partial x} $$

(33)

where

$$ \dot{U} = 0.001127{*}A\frac{{K\dot{K}}}{{\dot{\mu }}} \frac{\partial P}{{\partial x}} $$

(34)

$$ \overline{U} = 0.001127{*}A\frac{{K\overline{K}}}{{\overline{\mu }}} \frac{\partial P}{{\partial x}} $$

(35)

When the collocation method is applied:

$$ \left. {\frac{{\partial \dot{U}}}{\partial x}} \right| _{y,e} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1}^{{\dot{e}}} \overline{M}_{j,y} *\dot{U}_{j,e} , \quad \forall \;y \in Y,\; e \in E $$

(36)

$$ \left. {\frac{{\partial \overline{U}}}{\partial x}} \right| _{y,e} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1}^{{\dot{e}}} \overline{M}x_{j,y} *\overline{U}_{j,e} , \quad \forall \;y \in Y, \;e \in E $$

(37)

Replacing Eqs. 36 and 37 in Eqs. 32 and 33, respectively,

$$ \left. {\frac{\partial }{\partial x}\left[ {A\frac{{K\dot{K}}}{{\dot{\mu }}} \frac{\partial P}{{\partial x}}} \right]} \right| _{y,e} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1}^{{\dot{e}}} \overline{M}_{j,y} *\left( {A\frac{{K\dot{K}}}{{\dot{\mu }}} } \right)_{j,e} *\left( {\frac{\partial P}{{\partial x}} } \right)_{j,e} , \quad \forall \;y \in Y, \;e \in E $$

(38)

$$ \left. {\frac{\partial }{\partial x}\left[ {A\frac{{K\overline{K}}}{{\overline{\mu }}} \frac{\partial P}{{\partial x}}} \right]} \right| _{y,e} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1}^{{\dot{e}}} \overline{M}_{j,y} *\left( {A\frac{{K\overline{K}}}{{\overline{\mu }}} } \right)_{j,e} *\left( {\frac{\partial P}{{\partial x}} } \right)_{j,e} , \quad \forall \;y \in Y, \;e \in E $$

(39)

The term $\left( {\frac{\partial P}{{\partial x}} } \right)_{j,e}$ is approximated using the same method. The position in which this derivative is located should be considered to solve the generated system adequately, as shown from Eqs. (11–15). Terms $\left( {A\frac{{K\dot{K}}}{{\dot{\mu }}} } \right)_{j,e}$ and $\left( {A\frac{{K\overline{K}}}{{\overline{\mu }}} } \right)_{j,e}$, also known as $\dot{T}_{j,e}$ and $\overline{T}_{j,e}$, refer to the fluid transmissibility and they are estimated as established by [14].

On the other hand, the phenomenological model has a first-order temporal derivative term that is discretized as done in Eq. 40.

$$ \left. {\frac{\partial P}{{\partial t}}} \right|_{y,e}^{r,d} = \frac{1}{{\Delta t_{d} }}\mathop \sum \limits_{w = 1}^{{\dot{r}}} \dot{M}_{w,r} *P_{y,e}^{w,d} , \quad \forall \;y \in Y,\; e \in E, \;r \in R, \;d \in D $$

(40)

Including time dimensions for the Eqs. 38 and 39, and assuming the terms $\dot{T}_{j,e}$ and $\overline{T}_{j,e}$ remains constant through each temporal element, the resulting expression are shown in Eqs. 41 and 42.

$$ \left. {\frac{\partial }{\partial x}\left[ {A\frac{{K\dot{K}}}{{\dot{\mu }}} \frac{\partial P}{{\partial x}}} \right]} \right|_{y,e}^{r,d} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1} \left[ {\overline{M}_{j,y} *\dot{T}_{j,e} { } *\left( {\frac{\partial P}{{\partial x}} } \right)_{j,e} } \right] ,\quad \forall \;y \in Y, \;e \in E, \;r \in R, \;d \in D $$

(41)

$$ \left. {\frac{\partial }{\partial x}\left[ {A\frac{{K\overline{K}}}{{\overline{\mu }}} \frac{\partial P}{{\partial x}}} \right]} \right|_{y,e}^{r,d} = \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1} \left[ {\overline{M}_{j,y} *{ }\overline{T}_{j,e} *\left( {\frac{\partial P}{{\partial x}} } \right)_{j,e} } \right] ,\quad \forall \;y \in Y,\; e \in E, \;r \in R,\; d \in D $$

(42)

After replacing Eqs. 42, 41 and 40 in 3, the obtained result is presented in 12. In addition, the IBC for spatial elements (see Eq. 43) are also required to ensure the solution function continuity.

$$ \frac{1}{{\Delta x_{e} }}\mathop \sum \limits_{j = 1} \left[ {\overline{M}_{{j,\dot{y}}} *P_{j,e}^{r,d} } \right] = \frac{1}{{\Delta x_{e + 1} }}\mathop \sum \limits_{j = 1} \left[ {\overline{M}_{{j,\dot{y}}} *P_{j,e + 1}^{r,d} } \right] , \quad \forall \;e \in E| e < \dot{z},\;r \in R,\;d \in D $$

(43)