th
Proceedings of the 20 International Conference on Automation & Computing, Cranfield University, Bedfordshire, UK, 12-13 September 2014
Optimal Feedback Control for Reservoir Waterflooding A. S. Grema and Y. Cao School of Engineering, Cranfield University Cranfield, Bedford, U.K
[email protected] Abstract— Waterflooding is a process where water is injected into an oil reservoir to supplement its natural pressure for increment in productivity. The reservoir properties are highly heterogeneous; its states change as production progresses which require varying injection and production settings for economic recovery. As water is injected into the reservoir, more oil is expected to be produced. There is also likelihood that water is produced in association with the oil. The worst case is when the injected water meanders through the reservoir; it bypasses pools of oil and gets produced. Therefore, any effort geared toward finding the optimal settings to maximize the value of this venture can never be over emphasized. Traditional optimal control is an open-loop solution, hence cannot cope with various uncertainties inevitably existing in any practical systems. Reservoir models are highly uncertain. Its properties are known with some degrees of certainty near the well-bore region only. In this work, a novel data-driven approach for control variable (CV) selection was developed and applied to reservoir waterflooding process for a feedback strategy resulting in optimal or near optimal control. The results indicated that the feedback control method was close to optimal in the absence of uncertainty. The loss recorded in the value of performance index, net present value (NPV) was only 0.26%. Furthermore, the new strategy performs better than the open-loop optimal control solution when system/model mismatch was considered. The performance depends on the scale of the uncertainty introduced. A gain in NPV as high as 30.04% was obtained. Keywords- intelligent wells; optimal control theory; feedback control; geological uncertainty, control variable
I.
INTRODUCTION
The quest for an efficient oil recovery from aging reservoirs led to optimization studies involving smart wells control for waterflooding operation. Waterflooding is a secondary recovery mechanism where water is injected into a reservoir through an injection well with the aim to sweep the oil toward a production well for an increased output. A lot of work has been reported in the area [1] in recent years. The optimization involves determining optimum injection and production settings in order to maximize the profitability of the venture using some performance indices such as net present value or total oil recovery. In the work of [2], the optimization studies are model-based and performed using open-loop configuration. Open-loop control assumed that the reservoir models are perfect and capture all reservoir characteristics. Unfortunately, oil reservoirs are highly heterogeneous and their properties are known with some
certainties near the well region only. The reservoir geometry is usually deduced from seismic data. As a result, its boundaries are highly uncertain [3]. Some properties such as thin, high-permeability zones may not be captured within the given model resolution. Similarly, productions can be dominated by some near-well effects for example, cusping or coning which are rarely captured well in simulation models [4]. Traditional dynamic optimization techniques based on optimal control theory can only provide open-loop solutions, hence, [5] lacks robustness to handle uncertainty, the sensitivities of which can lead the optimal strategy to be suboptimal, or even non-optimal at all. To deal with these uncertainties several studies have been reported to use robust optimization technique, [6]. This involves the use of a set of reservoir realizations to account for geological uncertainty within optimization framework [6]. The main assumption to this procedure is that, the realizations capture all possible reservoir characteristics and production behaviors; which however, is not possible in reality. The usual practice in oil and gas industries is to determine the injection and production strategies using one of the methods mentioned above. When new data such as production data, well logs, seismic data and data from core analysis become available, the reservoir model(s) is/are updated using a procedure called history matching which are performed periodically on a campaign basis, then new optimized strategies are obtained based on the updated models [7]. Unfortunately even history-matched reservoir models may fail to predict reality correctly [8]. Other optimal control solutions existing in the literature were for batch process optimization problems based on repeated learning algorithms [9] but unfortunately waterflooding process is not repeatable. In view of this, several authors are of the opinion that there should be a shift from present practice of periodic updating of model and strategies for every history matching activity to a more efficient utilization of production measurements where control strategies are implemented in a closed-loop fashion [3; 7]. Several works have been reported in literature to this regard [10]. In most of these studies, the control actions were determined using ad hoc models. Recently, Dilib and Jackson [4], designed their feedback configuration by optimizing feedback control relationship between measured data and inflow control settings. Typically, the relationship was constructed from measured water-cut
and valve openings of smart wells. Parameter values of the relationship were obtained via model-based optimization that assumes perfect prediction. The benefit of a direct feedback strategy is that control performance is insensitive to model errors as these are inevitable in any real system. The most important and fundamental task that has not been addressed in oil and gas industries is, determining a controlled variable in the feedback configuration which is not sensitive to uncertainty so that when it is maintained constant, the system is near optimal. This type of problem is receiving a great attention for continuous process through a series of studies on a concept known as selfoptimizing control (SOC). This involves selection of controlled variables (CVs) among available measurements [11] so that when CVs are kept constant at their set point through feedback control the operation of the plant will be optimal or near optimal [12]. Several methods have been reported by different authors for CV selection using SOC over the years [13]. These methods require process models and linearization of nonlinear systems around a nominal point leading to a local solution. To avoid the weakness of a local solution, Ye et al. [14] developed a method to select CVs to approximate necessary condition of optimality (NCO) globally. However, this approach requires a process model to evaluate the NCO. We recently developed a regressionbased data driven method that approximates the NCO or compressed reduced gradient from either operation or simulated data [15]. Aforementioned SOC for continuous processes is static. Recently, extension of SOC to batch processes, hence dynamic SOC has also been reported in [16-19]. Unfortunately, these approaches are either local, or too complicated to be applicable to any practical applications, such as the waterflooding problem. This inspired the present work to extend the method proposed in [15] to dynamical systems. This paper presents a novel approach to determine CVs for dynamic systems. The proposed approach is then applied to the waterflooding process with uncertainties. The method involves use of measurement data to design a CV function offline and implementing this CV in a closeloop fashion. This was compared to open-loop optimal solutions obtained by optimal control theory (OC) through simulations with some geological uncertainties. The contribution of this work is in two folds. First, this is the only work to report the use of SOC principles to waterflooding process. Secondly, an entirely new and practically applicable approach to dynamic SOC was developed. II.
APPROACH
(
,
,
(
,
, )=
,
(2)
where xk is the reservoir states vector. A change in , at time k will not only affect Jk directly but will affect the states according to (2). The states will in turn influence the outputs, , through the measurement equations as (
,
)=
,
(3)
To develop SOC methodology for the above dynamic system, the following procedures are followed: 1.
,
A control sequence
,………….
,
is defined, the reservoir model (2) is solved to obtain a solution sequence , , ……. , a measurement sequence , , ……. and cost Ji.
2.
A perturbation is applied to the control sequence where we have , ,…. ,
the reservoir model (2) is then solved again to get perturbed solutions
measurements,
3.
,
,
…….
,
, , ……. , and cost Ji+1. Taylor series expansion is used to approximate the gradient of the objective function with respect to the control given as −
=∑
−
(4)
In the above formulation, superscript, k denote time step, subscript, i refers to a trajectory and G is the gradient (CV). The aim of dynamic SOC is at all time-steps to maintain the gradient at zero. Therefore, the gradient in (4) as a target CV can be replaced by a measurement function with n past histories, as shown in (5) ∑ ,vector , , −through (5) where −is a= parameter to, …be determined regression.
A. Dynamic Data-Driven SOC for Waterflooding In reservoir waterflooding, the objective function to be maximized can be written in the following form for a total number of time steps N and single input. = ∑
The contribution to J in each time step is given by Jk, where uk, yk, and dk are controls, measurements and disturbances respectively at time k. The reservoir models can be written in a discretized form as
)
Petroleum Technology Development Fund (PTDF), Abuja
(1)
B. Reservoir Configurations and Geological Uncertainties MATLAB Reservoir Simulation Toolbox, MRST [20] developed by SINTEF was used for reservoir simulation. Four different cases were considered in this study with
differing reservoir uncertainty. For cases 1-3, same reservoir geometry was adopted which is 20x20x5 grids with Cartesian coordinate system. Each grid has a dimension of 1 m. One each of vertical injection and production wells are placed at the two opposite corners of the reservoir. These wells are perforated at each layer. Both wells are rate constrained. Case 1 was considered as the nominal case with uniform permeability and porosity of 100 mD and 0.3 respectively and served as a basis for comparison with the other cases. In Case 2, permeability in each layer is log-normally distributed and independent to each layer. The five geological layers have mean permeability values of 200 mD, 500 mD, 350 mD, 700 mD and 250 mD from top to bottom. An example of this field is shown in Fig. 1 (left). In addition to permeability variation, porosity was also changed from nominal value of 0.3 to 0.45 in Case 3 to further increase the system/model mismatch. The permeability distribution was similar to that of Case 2 with same mean values. Another form of geological uncertainty was introduced in Case 4. Here, different reservoir size and geometry was considered with possibility of presence of a sloping fault which can transmit fluid if the pressure drop across it is sufficient [7], see Fig. 1 (right). A corner-point gridding system was used that enabled us to model the reservoir with a wavy top and bottom surfaces. The model has 30x3x1 cells with a total reservoir size of 225x22.5x1 m. The width of the fault was set to 0.12 m. C. Data Collection and Regression Data used for regression to determine CV parameters, in (5) were collected from simulations of Case 1. As mentioned earlier, the two wells are rate-constrained for all cases. The manipulative variables (MVs) are injection and total production rates. A voidage replacement assumption was made, that is, the total injection must equal to the total production at all time-steps. Therefore, the system has only one MV (that is, one degree of freedom, DOF). Two measurements were taken which are oil production rate, y and water production rate, y in addition to the MV, uw, that is, the water injection rate. The measurement vector can therefore be represented as = [
]
(6)
The objective function used here is net present value, NPV of the venture [1], which is given in (7). It is computed with each available measurements set. =
∑
,
∑
(
)
,
,
∆
(7)
where rwi, rwp and ro are water injection and production costs, and oil price respectively. uwi, yw and yo are water injection and production rates, and oil production rate respectively. Ninj and Nprod are number of injection and production wells respectively. b is a discount factor, ∆tk is time-step size, tk is the actual time period for which NPV is computed while τ is the time unit. In this work, b is set to zero, rwi and rwp taken as $10/bbl while ro as $100/bbl.
Fig. 1: Permeability Distribution (left) and Faulted Reservoir (right)
With the measurements set given in (6), the CV function in (5) is chosen to be a linear time-series of the form given in (8). Two past histories were used (n = 2). The total number of coefficients to be determined is therefore 2(n +1) +1 = 7. =
(
,
+ )
,
+
,
+
+⋯
,
(
)
,
+
(8)
Regressions are performed by minimizing the square of the residual given in (9). min ∑
(
− )−
(9)
where q represents the right-hand side of (5). A feedback control law can be obtained from (8) by setting it to zero (NCO) which can be written as (for n = 2) ,
=
[
+
+
+
]
+
+
(10)
For Case 1, 500 solution trajectories were used for data collection. At each trajectory, the reservoir flooding process was simulated for a period of 2 years with time step size of 1 day. Actual optimal injection rates were used for this purpose which were slightly perturbed at each time step. So, with this set up, a 500x730 data matrix was obtained which was used to obtain the CV via regression. This CV was first implemented to the nominal case study and then to the other three cases with various degrees of uncertainty. Bench mark (BM) results were also obtained from Cases 2-4 by solving the optimal control problem directly on the models using OC. Losses recorded by applications of SOC and OC strategies on the uncertain reservoir models are computed by =
/
× 100%
(11)
Similarly, increased NPV obtained by application of SOC in comparison to OC on the uncertain models is calculated as gain using (12). =
× 100%
(12)
Sensitivities to objective function of some of the considered uncertain properties were investigated before carrying out the numerical experiment. In particular,
about 50 permeability fields similar to that in Case 2 were generated. The nominal optimal solution in Case 1 was used to simulate the reservoir with these fields. III.
RESULTS AND DISCUSSIONS
C. Case 3:Varying Porosity and Permeability Similarly, the CV and optimal solutions obtained in Case 1 were implemented to Case 3 where both permeability and porosity differed from the nominal case. This is to test the capability of the SOC method in hand-
The feedback control law obtained is given as ,
= (−1.2203 × 10 ) [0.0000 + 0.2245 + 1.2211 + 0.0000 + 0.0008 + ] × 10 0.9968 (13)
The R-squared value is 0.9912, so no higher or more sophisticated model is required. A. Case 1: Nominal Parameters The results obtained for this case are shown in Figs. 2 and 3. After a period of 2 years of production the NPV obtained using OC is $182,775 while that generated using SOC is $182,298 which represents a loss of only 0.26% (Fig. 2). The value of the gradient fluctuates between 9.54x10-7 and 9.54x10-7 which indicates a good performance. It can be observed from Fig. 3 that relatively high water injection rates which average at 0.8 m3/day in the case of OC was applied right from beginning of production. This enables higher production rates from the inception. In the case of SOC, the water injection rates steadily increases from a fixed value of 0.67 m3/day to a peak value of 0.78 m3/day and then slightly drops to around 0.75 m3/day to maintain the reservoir pressure as oil is being depleted. In summary, the total oil produced for the SOC and OC techniques are respectively 360.34 m3 and 370.69 m3. Water productions are 173.19 m3 and 215.91 m3 for SOC and OC respectively. The increased oil production in the case of OC is largely due to high water injection at the beginning of production which is associated with higher water production in comparison to SOC as well as early water break-through (difference of about 30 days).
Fig. 2: NPV Case 1
Fig. 3: Production and Injection Rates for Case 1
B. Case 2: Log-Normal Layered Permeability Reaization The sensitivities of permeability are shown in Fig. 4. It can be seen that variations in permeability affect greatly the generated NPV. To test the robustness of the feedback strategy using SOC against uncertainty in permeability, the obtained feedback control law (13) was implemented to the layered reservoir. The open-loop optimal solution obtained by OC in case 1 was also used to simulate this uncertain reservoir. Furthermore, a BM was established directly from this realization by solving the optimization problem using OC. This will give the highest possible NPV since the model is assumed to be perfect. Here, SOC out performed OC with a gain of 0.21% when uncertainty is considered. With reference to the BM, the loss by SOC and OC are 0.19% and 0.39% respectively (Fig. 5). It can be seen from the figure that NPV generated by SOC slightly surpassed those generated by OC.
Fig. 4: Permeability Sensitivity to NPV
ling unexpected reservoir behaviors that might have not been captured during data acquisition. A BM solution was also obtained. For this system/model mismatch, a higher gain in NPV of 3.16% was recorded in favor of SOC. The loss based on BM for SOC is 4.66% and 7.67% in the case of OC. This is shown in Fig. 6.
TABLE 1: SUMMARY OF THE OPTIMIZATION RESULTS
Case I
Case II
D. Case 4: Faulted Reservoir After implementing the CV to Case 4 and simulating it with the nominal optimal solutions, NPVs obtained are shown in Fig. 7. Because of the very large model/system mismatch introduced in this case, a very good performance by SOC can be observed. The nominal solu-
Case III
Case IV
OC SOC BM OC SOC BM OC SOC BM OC SOC
NPV($) 182,775.00 182,297.70 159,723.50 159,096.40 159,428.90 239,271.50 220,918.20 228,116.4 487,520.10 333,904.70 477,309.60
% Gain 0.21 3.16 30.04
% Loss 0.39 0.19 7.67 4.66 31.51 2.09
Gain is with respect to SOC over OC. Loss is based on BM
tion turned to be completely non optimal in this case. An extremely high gain of up to 30.04% was recorded in favor of SOC. Based on the BM scenario the losses are 2.09% and 31.51% for SOC and OC respectively. These are summarized in Table 1.
IV.
Fig. 5: NPV Case 2
CONCLUSIONS AND RECOMMENDATIONS
A feedback-controlled reservoir waterflooding study has been carried out. The CV was formulated using dynamic data-driven SOC. Several geological uncertainties were considered to test the robustness of the method. The developed feedback strategy was compared with the traditional open-loop OC. The developed CV was found to be insensitive to the uncertainties considered. In the absence of system/model mismatch, although the OC was seen to have a better performance as expected, the SOC results in a loss of only 0.26%. However, when uncertainty was introduced, the feedback approach performed better. The relative performance has increased with complexity in uncertainties. For instance, when uncertainty was considered in only permeability in Case 2, the gain in NPV was 0.21%. Introducing more mismatches in the form of reservoir size, geometry and structure in Case 4, a gain of up to 30.04% was recorded. Comparing this with the BM case, a loss of only 2.09% was seen for the case of SOC as against 31.51% for OC. We recommend exploring other uncertainties with this approach especially uncertainties in fluid properties.
Fig. 6: NPV Case 3
ACKNOWLEDGMENT We acknowledge SINTEF for providing free licence of the software, MATLAB Reservoir Simulation Toolbox (MRST).
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Fig. 7: NPV Case 4
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