SAGD Optimization under Uncertainty

Transcription

1 WHOC SAGD Optimization under Uncertainty J. GOSSUIN & P. NACCACHE Schlumberger SIS, Abingdon, UK W. BAILEY & B.COUËT Schlumberger-Doll Research, Cambridge, MA, USA This paper has been selected for presentation and/or publication in the proceedings for the 2011 World Heavy Oil Congress [WHOC11]. The authors of this material have been cleared by all interested companies/employers/clients to authorize dmg events (Canada) Ltd., the congress producer, to make this material available to the attendees of WHOC11 and other relevant industry personnel. Abstract A means to optimize a SAGD process in the presence of reservoir uncertainty is presented. While various methods for optimizing a SAGD model have been presented, these all applied to discrete models where uncertainty was not explicitly considered in the course of the analysis. Studies utilizing a sensitivity analysis on reservoir simulations have been used but these mostly only identify which parameters impact the desired outcome, an assumption that does not consider their distribution as an explicit part of the computation. This work presents a consistent and operationally realizable method for SAGD optimization that can consider any number of uncertainties (and their distribution) subject to sufficient computational capacity. We present results on a SAGD model where 3 different (and equi-probable) geological models are possible. We map the results into a graphical construct based on the well-known Efficient Frontier that permits the decision maker to make an operational decision given their own (often very specific) risk tolerance (their degree of risk aversion). The Efficient Frontier provides the necessary operational input to allow the asset holder to maximize their recovery (or NPV) when faced with genuine and real reservoir uncertainty. Finally, we show how we can obtain better strategies, with confidence, and the potential of such a workflow to help select a production pattern. Introduction The substantial amount of energy required to produce the necessary steam in Steam Assisted Gravity Drainage (SAGD) results in it being a costly process, thereby motivating the need to examine methods to make the most efficient use of resources and improve subsequent project economics. Numerous studies aimed at optimizing a SAGD process have been published (Yang et al., 2007, Piña et al., 2008, among many others), most of which involve some form of sensitivity analysis where one parameter is varied while others are kept constant. The best value for this parameter is then kept fixed while the next variable is investigated, and so on. This method is not only time consuming but is liable to miss optimal combinations of parameters due to their likely interaction. To improve upon such an analysis, dedicated optimization algorithms have been successfully applied to a single SAGD simulation model (Card et al., 2006, Gossuin et al., 2010). However, such schemas are deterministic in nature, as they are bound by the fundamental assumption that the model being optimized is actually the right one. This might not be necessarily applicable to fields with heterogeneities, which have been shown, particularly with the presence of shale close to either well bore, to drastically influence SAGD production (Chen et al., 2007). Some authors (e.g., Yeten et al., 2002) have considered optimization under uncertainty but did so using deterministic optimizations. In essence, the "optimum" result presented was the mean of the optima obtained from individual deterministic runs. The problem with this approach is that the final result becomes unrealizable as one cannot tell which combination of unique control variables is to be used to obtain this mean value. In other words, it is not practically possible to actually realize this optimum. The scheme proposed here corrects this limitation by furnishing the actual unique combination of controls necessary to achieve the stated optimum - and to do so with an associated degree of confidence. The aim of this article is to 1

2 explain this in detail and to demonstrate an optimization with geological uncertainty such that the results have real operational meaning and can be applied in practical operational settings. The uncertainty in our example is in the form of three different but equi-probable geological realizations. As will be discussed in more detail later, while we still need to run simulations for each of these realizations it is in the processing of the results from these runs that we can extract meaningful results and the analysis provides a summary that can incorporate the decision makers own particular value (or degree) of risk tolerance. We first outline preparatory work necessary to identify the key control variables. This is followed by a description of the proposed optimization under uncertainty scheme and its application to an example. To demonstrate its utility, we compare and discuss the results against their deterministic counterparts to validate the proposed approach. Problem Description The overarching objective of this work is to maximize the net present value (NPV) of a SAGD process at the end of ten years of production. The well pair is 500m each and completed in a 25m thick sandstone reservoir unit. Reservoir Grid: The volume modeled is a m rectangular domain. To enable a more rapid solution, only half of the reservoir (divided along the plane containing the well) is simulated, i.e., we assume symmetrical property distributions either side of the wells. The grid itself consists of cells, each 50m long (in the direction of the wells), 5m wide (perpendicular to the wells) and 1m thick. These dimensions were selected to balance run time and accuracy and were also based on prior work by Gossuin et al. (2010). Three equi-probable geological realizations of porosity and permeability were geostatistically generated with average porosity of around 26% and permeability of 5500 md. Their properties are described in Table 1 and illustrated in Figures 1, 2 and 3. Grid 1 (shown in Figure 1) has a large thickness pay zone situated in the bottom part of the reservoir with quality generally decreasing as depth increases. Grid 2 (shown in Figure 2) has more homogeneous properties with some variations of quality along the well direction. Grid 3 (shown in Figure 3) has a large thickness pay zone roughly in the middle of the reservoir with slightly decreased quality at the top and bottom of the reservoir. We need to stress that these geological representations were deemed reasonable for the problem at hand. If a problem requires a larger number of realizations, the optimization framework proposed here can accommodate such requirements seamlessly. However, one must be mindful that the more uncertainties one introduces into a problem, the longer the computation time required per trial will be. Reservoir & Fluids: The oil is an Orinoco heavy oil with a viscosity of 6450 cp at reservoir conditions (51.2 C, 70 bars) and summarized in Table 2. A viscosity versus temperature curve is shown in Figure 4. Economic Parameters: The objective function, F, defined for our optimization is the after-tax NPV of the field at the end of a 10-year production period. The suitability of this singlevalued objective function for our optimization was demonstrated by Gossuin et al. (2010). The economic parameters used are outlined in Table 3. Costs are divided into energy- and non-energy-dependent terms with discounting performed continuously and an allowance made for oil and gas price drift in time. Preparatory Optimization Steps Before embarking on any major optimization (with uncertainty or otherwise), it is highly beneficial to conduct a set of preparatory tasks (steps 0 to 4) to properly identify the most important control variables and to ensure the most efficient and effective optimization possible. Such preparatory tasks associated with a deterministic optimization have been covered in detail by Gossuin et al., A rapid proxy is used to identify the most important (impactful) set of control parameters and their practical lowerand upper bounds. For this study, the proxy was chosen to be a 2D model representing the volume directly perpendicular to the wells. The 2D model porosity and permeabilities are chosen to be homogeneous and equal to the average of the 3D three grids properties. However, the inclusion of uncertainty requires the introduction of a new, initial step, denoted as "Step 0". Our modified preparatory steps are as follows: Step 0: Identify Key Uncertainties. This may be simple (both the nature of the uncertainty and its distributional form are known) or much more involved, necessitating considerable analysis to properly identify and enumerate (i.e., describe its distribution, its probability density function). Once these uncertainties have been identified (and their distributions agreed upon), they should be applied in step 3 below to better understand how uncertainty impacts our control variables. In our case, we have established 3 equi-probable grids, each with symmetric heterogeneity. Step 1: List the possible control variable (CV) candidates to focus on practical, operational variables that can be realistically optimized for. These should be accorded a crude ranking based on engineering judgment and experience. This step identified the following initial set of CVs: Wells positions in the vertical plane (k-index) The injector bottom hole pressure (BHPINJ) The BHP of the producer (BHPPROD): Steam trap temperature (STEMP) Furthermore, we sub-divide the 10-year forecast period into a number of separate time periods (not necessarily of equal duration). Step 2: Proxy sensitivity. This step utilizes a straightforward sensitivity analysis to provide a preliminary ranking of the control variables according to their impact, being mindful that the more CVs one selects, the longer the optimization will take to converge to a solution. From an initial parameter set (see Table 4), we chose to subdivide the forecast into five fixed time periods to cover the 10- year (3650-day) production, resulting in a new initial set comprising 17 CVs. These are: 2, for each well position, 5 for the BHP of the injector (one for each time period), 5 for the BHP of the producer (again, one per time period) and 5 separate values for the stream trap temperature (again, one per period). Time periods at the beginning are chosen to be shorter while the steam chamber is being established. We then conducted a sensitivity analysis to establish an approximate ranking of the 17 initial CVs. We should stress here that the criticism leveled earlier against single-parameter sensitivity analysis applies only to its use as an optimization tool. It has a valid and important role in this parameter reduction step. The difference between the lowest and highest NPV obtained from this simple sensitivity analysis is shown as a tornado plot in Figure 5, which illustrates parameters by decreasing order of influence. Note that the producer BHP (BHPPROD) has little effect on the final result (probably because the producer well is under steam trap control for most of the ten years). This lack of impact was confirmed later in our 3D validation analysis. 2

3 Step 3: 3D sensitivity: this step is recommended, but can be optional if computer resources or time are limited. Information forthcoming from step 2 are now validated on the full 3D model to validate our candidate CVs. To achieve this, similar sensitivities were run on each of the 3D grids. In terms of their relative ranking, the results obtained are very similar and are shown in Figure 6. Well location and injector BHP (BHPINJ) remain the most influential parameters. While 3D grids and 2D proxy share similar behavior, there does appear to be a major difference in the impact of well position. The best values from a sensitivity analysis for a 2D homogeneous model suggest that the wells should be separated as much possible with the injector in the upper layer and the producer at the bottom. However, the introduction of heterogeneity in the 3D model results in a much more complex response, as shown in Figure 7 (injector location) and Figure 8 (producer location). These two figures demonstrate the substantial impact on NPV not only from well location but also between the 3 grids, due to the 3D reservoir heterogeneity. Step 4: Control Variable Reduction. We must exercise caution in eliminating control parameters without some additional confirmation otherwise we may lose fidelity. The ranking obtained from the sensitivity analysis is used to guide parameter reduction, not specify it. A limited set of control variables is selected by testing different optimizations of the proxy. Note that we tested the best number of time sub-divisions using our proxy simulations, being mindful that the more subdivisions there are, the more CVs will be created. In other words, we needed to strike a balance between increased resolution (and hopefully better optima) and run time. The results are shown in Table 5, which confirms that the best balance is obtained with a 5-period sub-division (as was originally suggested by Gossuin et al., 2010). With the producer BHP having little effect, as mentioned above, and with the steam injection rate (STRATE) being ultimately dominated by BHPINJ, as seen in Figures 5 and 6, the final selection of CVs, confirmed using our 3D model, is shown in Table 6(a) with 11 controls being carried through into our full 3D optimization study. Note that we are now letting the duration of the time periods to become variable. Table 6(b) provides their initial values and their lower and upper bounds. As stated above, Period 5 is simply the time remaining between 3650 days and the sum of periods 1 to 4. The initial guesses stated in Table 6(b) are the resulting optimum values obtained from optimizing the 2D proxy with the selected control variables. They will be used in the following part as the initial (starting) values of any 3D optimization. The idea is to provide the full 3D model with the best starting value given the state of knowledge at the time. Optimization Under uncertainty Our discussion so far has been focused solely on the preparatory work and some sensitivity studies and validation. The following covers the treatment of uncertainty. To do this in a systematic manner, we first outline the nature of the objective function under investigation - which is pivotal to our proper understanding and interpretation of the results. We follow by results of our optimization runs for the SAGD model considered. This includes a description of the Efficient Frontier and how it pertains to an operational strategy, given the definition of the chosen uncertainty. We also include a discussion on the results obtained. We should also stress that the following discussion does not apply to just our problem (defined by 3 equi-probable uncertainties). Rather, the nature of the solution allows any number of uncertainties to be incorporated into the analysis given that there is sufficient computational resource to solve them in a timely manner. Objective Function When optimizing under uncertainty, it is necessary in each trial to simulate the three geological grid realizations separately - but each with identical operating conditions. The results from each realization are combined into a single representative objective function to be optimized. One could, of course, simply consider the average over the three results for the objective function. However, taking the mean of these three separate simulations is just one of a spectrum of possible outcomes. Rather, a more general form of the objective function, F, can be used that takes into account the decision makers own particular risk-aversion, which we define as a risk aversion factor, λ. This concept was originally described by Couët et al. (2004) and is written as: F = μ λσ, where μ is the average of the cumulative NPV obtained from the three runs and σ their standard deviation. The higher the value of μ, the higher the objective function will be, but accordingly, the higher the dispersion σ between the results, the lower the objective function will be. One can relate the riskaversion factor, λ, to the confidence levels typically used in standard probability distributions. It is possible to compute the probability that a realized gain, X, will be greater than F, i.e., to say with what confidence (in percent) one can be certain that X will be greater than F. If we assume that the individual gains in NPV of the various individual scenarios follow a normal (Gaussian) distribution, selecting λ = 0, 1 or 2 means that the corresponding NPV gain is to be maximized at a 50%, 84% or 98% confidence level, respectively, or that one has 50%, 84% or 98% certainty to obtain more than the corresponding F. A different optimization process will be run for each of these three values of λ. In other words, if we wish to optimize about the mean (i.e., λ = 0) then this corresponds to a risk-neutral stance towards the objective function, i.e., there is a 50% chance that the actual value realized will be higher or lower than the value of F found from optimization (assuming, of course, that the reservoir model is a fair and representative proxy of the actual asset). If λ = 1.0, then one is taking a more risk-averse position towards the outcome as this translates to an 84% likelihood that the actual realized value will be higher than that provided by the optimization of F. As the value of F computed is correspondingly smaller (by a value of 1 σ), the degree of confidence attributed to the result is higher. The larger the given value of λ, the smaller will be our value of F but with a corresponding increase in likelihood that that value or more will actually be realized. Our optimization process involves a sequential optimization starting with risk-aversion constant, λ = 0 (i.e., an optimization about the mean, our risk-neutral scenario). Using the best values of this run as the starting vector, a second optimization is conducted, this time with λ = 1. This, in turn, is followed by a third (and final) optimization run with λ = 2, this time starting with the optimum values obtained from the previous run. The reader should note that separate optimizations for each value of λ are necessary as λ determines the objective function F being considered. There is no strict requirement to restart the next λ run with the best (optimum) values from the previous. However, 3

4 doing this accelerates convergence as initial control parameters are likely to be closer to the optimum than using initial estimate starting values. Results are described and discussed in the following section. Results and Discussion Before describing the results of optimizing with uncertainty, we first present the results of deterministic optimizations performed individually over each grid in turn, hence separate results for each grid. Mean of the Optima (Unrealizable): We conducted three separate deterministic optimizations, one separate, distinct optimization for each grid, and the results are shown in Table 7. This shows a significant percentage gain for Grid 3, while Grids 1 & 2 have gains of around 19% over their respective baselines. The baselines are defined as the results of the 3D cases ran with the strategy obtained by proxy optimization. Using the approach suggested by some authors (e.g., Yeten et al., 2002), the average over these optima is around $13 million. The problem with such a value, however, is that it is not realizable as there is no single unique operational policy to achieve this value. Indeed, what are the corresponding values of the control variables associated with this mean value? The answer does not come out of the analysis. The mean of optima value is merely an indicator of potential upside with no associated operating control settings. In other words, such a value is not physically realizable but rather an indicator that upside exists. Optimum of the Mean (Realizable): To address this practical consideration of no corresponding control parameters, we conduct our optimization for the objective function defined earlier over a set of risk aversion factors (λ = 0, 1 and 2). The results shown in Table 8 are presented according to each risk aversion value (λ) and its associated degree of confidence (C.I.). Table 8 may be referred to as a 'decision table' as the decision maker can simply look-up their personal preference for risk and read-off the corresponding value of F (for which we have an associated set of controls). Recall that the objective function in the optimization is defined as F = μ λσ, meaning that as we increase our risk aversion (i.e., λ increases) the value of F correspondingly reduces, as can be seen in Table 8. While the value of F may reduce as λ increases, the degree of confidence associated with that value also increases. Thus, for λ = 2, we can translate the optimum value as "there is a 98% certainty that we will get $ (or more)" if we implemented the operational policies suggested. The gain stated in Table 8 relates to F and its corresponding initial value set for that particular λ. The initial set of control variables for a λ run is chosen to be the best set from the previous run and demonstrate the practical benefit of starting from a good initial value as the number of iterations is also reduced. Also note from Table 8 how the gain for successive λ reduces - once again due to the improved initial value. For λ = 0, the initial set of control variables was chosen as the optimum set from the 2D proxy optimization. So what are these policies? Recall from the earlier discussion that each trial comprises 3 separate simulations (realizations) - one for each of our uncertain grids but with identical control variables. Thus each and every value of F computed during the optimization is associated with a unique set of values for the control variables. We can represent all possible outcomes (both optimal and sub-optimal) in the form of the Efficient Frontier. The Efficient Frontier Representing the results for all trials in a succinct and meaningful manner is achieved through an Efficient Frontier (EF) (Couët et al and Bailey et al. 2005). The EF is the leftmost limit (shown as a green line) made from the various iterates in the optimization on the plot of μ against σ, shown in Figure 9. This plot provides all the information decision makers will need to adopt an operational policy that is in concert with their own degree of risk tolerance. Each point on the plot (labeled "Sample Point" with a blue circular marker) corresponds to a unique set of control variables. A sub-optimal operation would be one involving a point that is not on the frontier. This is because any such point corresponds to a nonoptimal combination of standard deviation (risk) and its mean value. Points on the frontier itself represent the optimum combination of risk (as defined by standard deviation) and reward (the mean value). No set of points can lie above the frontier as this space is non-realizable. A rational decision maker will first select their own value of risk aversion - say 84% certainty of achieving an outcome (i.e., λ = 1). Then, using the EF, locate the point on the frontier corresponding to that particular value of risk aversion. This point will have associated with it a unique set of corresponding control variables that represents the policy the decision maker should adopt in order to obtain their desired value. For this example (λ = 1), the optimum value of F is $ (see Table 8) that is attainable with a prescribed set of control variables. Thus we can say that we have an 84% likelihood of achieving $ or more from our asset given that the uncertainty is reasonably defined by the 3 equi-probable grids (discussed earlier). The EF is a flexible construct in that any number of uncertainties may be folded into its representation. For example, if we have 20 separate geostatistical grids then - subject to a lot more computation - we may still represent the results of all these in this simple 2D plot. If we have more than one uncertainty, say grids and PVT properties, these too may be folded seamlessly into the construct. One is really only restricted by the almost exponential computational cost required to perform the computation as the number of realizations per trial increases by the product of their sampling points. For example, say we have 3 separate uncertainties, each sampled with 3 points, then each trial of our optimization will require 27 separate simulation runs (27 realizations), which could, however, be evaluated concurrently on a computer cluster, for example. Obtaining better strategies with confidence For a risk-tolerating user (λ = 0), cumulative NPV has been improved for each grid by between 64% and 115% if we compare with our original SAGD cases (see Table 9). These large benefits are due to the fact that cumulative oil produced has increased between 23% and 57% while steam oil ratio has been divided by two. Using different time periods was instrumental into obtaining such a strategy: it enables the creation and evaluation of flexible combinations. The fact that the duration of the periods is itself a variable gives the optimizer more latitude to achieve higher values. Yet, this kind of approach greatly complicates the problem, making it even more complex to interpret by successive sensitivity alone. Using the approach described above enables us to tackle a complex problem with numerous control variables while at the same time obtaining a unified solution valid over the uncertainty studied. Had we used a deterministic approach for each of the grids, we 4

5 would have risked obtaining inconsistent results and proposing a sub-optimal policy. Referring again to the Efficient Frontier in Figure 9, we know each and every value for BHPINJ, PERIOD, POSINJ and POSPROD for every point indicated. For deterministic runs, we do not have a single set of values for the three grids and it is only with proper dealing of the uncertainty that we may make the optimal choice. In other words, when faced with real and quantified uncertainty, performing the optimization in the manner discussed here will produce results that are indeed optimal, given the uncertainty, but with the caveat that the objective function is not a single number but it also includes a percentile describing the confidence of its value. Table 11 shows the values for all control variables for our optimizations with uncertainty (various λs) and without (the 3 individual grids). Figure 10 presents the results for the BHPINJs. The left-hand plot (bounded in green) shows the results using optimization under uncertainty for each λ. The right-hand plot (bounded in red) was generated from individual deterministic optimizations for each grid. Although it appears that the results of optimizing individual grid produce almost identical values of BHPINJ per time period, when combined under one uncertainty, one can definitely conclude that the BHPINJ takes alternatively a high value of 80 and 20 bars from the first to the fourth period and stays low in the last period. The similarity between deterministic and uncertainty runs illustrates the fact that the uncertainty chosen might not have much impact on the optimization for the BHPINJ. Figure 11 shows the duration of each forecast time period, denoted 'TIME'). We notice that the left-hand (green bounded) plots (optimization with uncertainty) are markedly different from those established from individual optimizations. This clearly indicates the dangers of drawing conclusions from deterministic results alone. These periods relate to alternating injection and non-injection of steam, the duration of which is clearly sensitive to the presence of our uncertainty. Finally, Figure 12 shows producer and injector completion locations (POSPROD and POSINJ, respectively). There is good agreement at all confidence levels between these locations for the optimization with uncertainty results (the green-bounded, left-hand plot). The core operational strategy behind the improvement obtained relies on a discontinuous steam injection pattern: following a short period of injection, injection is stopped for several years and depressurization ensues. Injection starts afterwards again at a rate higher than what is achieved in the continuous steam injection case. The injector BHP limit is either at its lowest 20 bars or at its maximum (82 bars) for the 3D simulations. Why is this strategy of alternation of aggressive injection and non injection more efficient than continuous injection? The original problem shows the steam channels reaching the producer quickly, which leads the necessary steam trap control to choke the wells. Shutting the injector lets the steam chamber dissipate partially inside the reservoir, spreading its heat. The producer well switches then from being constrained by the steam presence to a bottom hole pressure control and is able to produce much more, as displayed in Figure 13. The ensuing depressurization is what enables the injector well to reach such higher rates of steam injection afterwards, as showed in Figure 14. At the end of this injection period, more steam has been introduced inside the reservoir than in the base case, as showed in Figure 15. The cost of the steam injected reduces temporarily the overall economics of the field, but the investment is recovered during the remaining time, as displayed in Figure 16. Great care was taken to ensure that this strategy was not a direct consequence of the approach chosen. Two optimizations under uncertainties, described in Table 10, were run: one to test the possible improvements with a tighter control of the producer well, and another one for the five period constraints. In that last case, the injector well is either injecting at 82 bars or shut and the duration of its injection or soaking period is optimized. A similar strategy of two successions of injection/depressurization periods was obtained in both cases with only marginal benefits. The robustness provided by the uncertainty handling combined with the ability of optimization to explore complex problems makes it possible to really assess the potential of production processes. Compared to cold production, using a base SAGD strategy would improve oil recovery by more than 100%. Yet, the strategy we obtained after optimization is one where we interrupt steam injection. That leaves opened the possibility that another thermal pattern like Cyclic Steam Stimulation may be more adapted for such a field and economic conditions. Conclusion Optimization with uncertainty has been applied successfully to a SAGD process. The uncertainty in this example was fairly benign, comprising of three heterogeneous and equi-probable grids. However, the scheme can accommodate any number of uncertainties subject to sufficient computational power. Optimization is performed on a simple utility function (involving the mean, standard deviation and risk aversion factor) that considers the specific risk tolerance of the decision maker. A set of preparatory steps necessary to reduce the number of control variables is also suggested and resulted in a reduction from 17 to 11 for the number of controls deemed significant to the example discussed here. The optimum operating strategy forthcoming prevents premature steam channels that can choke production by alternating periods of depressurization and injection. Using a mathematical optimization approach enables one to handle more complex problems that simple sensitivity analysis can address. Trying to tackle such problem manually could force, for example, the imposition of constant operating parameters over the life of the field. References Bailey, W.J., Couët, B., and Wilkinson, D., Framework for Field Optimization to Maximize Asset Value, SPE PA, SPE Reservoir Engineering J., 8, 1, February 2005, pp Card, C., Chakrabarty, N., Gates, I., and Kovscek, A. R. Automated Global Optimization of Commercial SAGD Operations, Petroleum Society Chen, Q., Gerritsen, M. G., and Kovscek, A. R. Effects of Reservoir Heterogeneities on the Steam-Assisted Gravity-Drainage Process, SPE Couët, B., Bailey, W.J., and Wilkinson, D., Reservoir Optimization Tool for Risk and Decision Analysis, Proceedings of the 9 th European Conference on the Mathematics of Oil Recovery, Cannes, France, August 30 - September 2, Gossuin, J., Bailey, W.J., Couët, B., and Naccache, P., Steam-Assisted Gravity Drainage Optimization for Extra Heavy Oil, Proceedings of the 12 th European Conference on the Mathematics of Oil Recovery, Oxford, UK, September 6-8,

6 Piña, J.A.R., Bashbush, J.L. and Fernandez, E.A.: "Applicability and Optimization of SAGD in the Eastern Venezuelan Reservoirs. SPE MS Yang, C., Card, C. and Nghiem, L. Economic Optimization and Uncertainty Assessment of Commercial SAGD Operations, paper , Canadian International Petroleum Conference, Calgary, Alberta, June 12-14, Yeten, B., Durlofsky, L.J., and Aziz, K.: Optimization of Smart Control, SPE

7 Table 1 Properties of the three grids selected for porosity, permeability and fluid in place. Grid 1 Grid 2 Grid 3 Average Porosity (%) Min. Porosity (%) Max. Porosity (%) Porosity, std. Dev (%) Average Permeability (md) Min. Permeability (md) Max. Permeability (md) Permeability, std. Dev (md) Fluid in place (Mrm3) Table 2 Reservoir and fluid properties Reservoir temperature 51.2 C Reservoir Pressure 70 bars Viscosity 6300 cp API degree 10.1 Reservoir Thickness 25 m Reservoir Width 225 Molar percentage light component 0.18 Table 3 Economic Parameters Parameter Value Non Energy-Dependent Terms Costs Water treatment $0.28 /stb (water) Water injection cost $2.43 /stb (water) CO2 cost $0.43 /stb (water injected) Oil production costs $4.13 /stb (oil) Oil upgrade costs $3.07 /stb (oil) Electricity $0.05 /kw hr Prices Gas Price $10.0 /Mscf Oil Price $75.0 /stb (oil) Drift Oil price 4.0% per annum Cost drift 2.0% per annum Discount rate 4.0% per annum Energy-Dependent Terms Water injection cost $0.29 /stb (water) Oil production costs $1.18 /stb (oil) Oil upgrade costs $0.60 /stb (oil) Taxes and Royalties Oil Royalties 5.6% Tax rate 33.8% Table 4 Values used during sensitivity and optimization. POSINJ is the location of the injector (from the top of the reservoir). POSPROD is the corresponding location of the producer. With the last digit representing the specific production periods, TIME, BHPINJ, BHPPROD, and STRATE indicate the duration of the forecast, the injector and the producer bottom hole pressures, and the steam injection rate, respectively. Note that TIME5 is computed such that the total production period equals 10 years. Base Min. Max. value Value Value Units POSINJ m POSPROD m Production period 1 TIME Days BHPINJ Bars BHPPROD Bars STRATE Sm3/day Production period 2 TIME Days BHPINJ Bars BHPPROD Bars STRATE Sm3/day Production period 3 TIME Days BHPINJ Bars BHPPROD Bars STRATE Sm3/day Production period 4 TIME Days BHPINJ Bars BHPPROD Bars STRATE Sm3/day Production period 5 TIME Days BHPINJ Bars BHPPROD Bars STRATE Sm3/day Table 5 Results of optimization of the proxy for different numbers of time periods being used. Number of periods Number of CVs Iterations Final NPV ($millions) 3 Flexible Periods Flexible Periods Flexible Periods Flexible Periods Fixed periods

8 Table 6 Table (a) (top) defines the control parameters identified. Table (b) (bottom) shows their initial guesses and bounds. CV Type Notation Units 1 Position of Producer POSPROD k-layer 2 Position of injector POSINJ k-layer 3 Period 1 Duration TIME1 Days 4 Period 2 Duration TIME2 Days 5 Period 3 Duration TIME3 Days 6 Period 4 Duration TIME4 Days 7 Injector BHP (for Period 1) BHPINJ1 Bars 8 Injector BHP (for Period 2) BHPINJ2 Bars 9 Injector BHP (for Period 3) BHPINJ3 Bars 10 Injector BHP (for Period 4) BHPINJ4 Bars 11 Injector BHP (for Period 5) BHPINJ5 Bars Notation Initial Guess Lower Upper POSPROD POSINJ TIME TIME TIME TIME BHPINJ BHPINJ BHPINJ BHPINJ BHPINJ Figure 1: Grid 1 porosity. High quality area in the bottom, but the upper part of the reservoir has lower porosity. Figure 2: Grid 2 porosity. More homogeneous overall, with the presence of a reduced permeability layer. Table 7 Results of deterministic optimization for the three equi probable grids. The baseline for 3D optimizations is not the original SAGD strategy, but the optimal strategy obtained from the proxy model. Grid 1 Grid 2 Grid 3 Baseline NPV ( 10 6 ) $9.698 $12.63 $8.793 Optimum NPV ( 10 6 ) $11.7 $14.95 $12.6 Gain (over baseline) 19.7% 18.4% 41.4% Trials Mean of Optima $ Figure 3: Grid 3 porosity. Quality of the reservoir is lower at the bottom and at the top but very good in the middle. Table 8 Results of optimization with uncertainty. The percentage gain shown relates to the gain in F against the initial value set for that λ optimization. For λ = 0, the initial set is chosen as the optimum of the 2D optimization. For subsequent λs, the initial set is the optimum of the previous λ optimization. Gain Trials λ C.I. F μ σ US$ 10 6 US$ 10 6 US$ % $12.81 $12.81 $ % % $11.65 $12.81 $ % % $10.54 $12.69 $ % 26 Figure 4: Viscosity vs. temperature of the fluid considered using a semi logarithmic scale. Viscosity can be reduced by several orders of magnitude by increasing the temperature. 8

9 Figure 5: 2D sensitivity results: what is displayed here is the difference between the lowest and highest cumulative NPV after ten years for each parameter. Figure 8: Cumulative NPV vs. producer position for the 2D proxy (purple) and the three 3D grids. Figure 6: Tornado plot obtained for a 3D model (Grid 2). The ranking is very similar to the 2D model with well positions and injector BHP being the most influential parameters. Figure 9: Efficient Frontier. Each blue point represents the outcome of a unique set of control parameters applied equally to all 3 grids. The frontier (shown in green) represents the best possible pair of (μ,σ) attainable and for each pair corresponds a set of control variables. The value of λ increases from right to left along the frontier. Points below the frontier are sub optimal. Figure 7: Cumulative NPV vs. injector position for the 2D proxy (purple) and the three 3D grids. The 2D proxy exhibits a somewhat linear behavior favoring the top positions, whereas heterogeneous grids have more complex variations. 9

10 Figure 10: Results of BHPINJ (injector bottom hole pressure) for the 5 periods of the forecast. The left hand plot (bounded in green) shows the values obtained from optimization under uncertainty for each of the three λ values. The right hand plot (bounded in red) shows optimum BHPINJ, but obtained with deterministic optimization (each grid optimized separately). The legend applies to both plots. Table 11 tabulates all the results shown here. Figure 11: Results of Period. The left plot (bounded in green) shows the values obtained from optimization under uncertainty for each of the three values of λ while the right hand plot (bounded in red) shows the same parameter but with deterministic optimization (each grid optimized separately). The legend applies to both plots. Table 11 tabulates all the results shown here. Figure 12: Position of Producer (blue) and injector (red). Left plot (bounded in green) shows the values obtained from optimization under uncertainty for each of the three values of λ while the right hand plot (bounded in red) shows the same parameter but with deterministic optimization (each grid optimized separately). The legend applies to both plots. Table 11 tabulates all the results shown here. 10

11 Table 9 Summary of main optimization results comparing the economics of the base SAGD case (constant operating pressure) for each grid versus the optimized results (chosen with λ = 0), followed by the cumulative oil produced and the final cumulative steam oil ratio. Cumulative NPV ($millions) Cumulative oil produced (Msm3) Steam oil ratio Grid 1 Base SAGD Final result Grid 2 Base SAGD Final result Grid 3 Base SAGD Final result Table 10 Cumulative NPV ($ millions) for the different optimization approaches: In the second row, the producer BHP limit is setup as the BHP of the injector minus a variable difference of pressure. In the third one, the control variables chosen are the well position and the duration of the periods where the injector is opened or shut. Only marginal additional benefits are obtained. Figure 14: Steam injection rate for base case (green) vs. optimized case (red) of the Grid 3. The optimized case can reach a much higher injection case than the base SAGD case. Grid1 Grid2 Grid3 Original approach (lambda=0) Pressure differential approach Cyclic (Inj, soak, Inj, soak, Inj ) Figure 15: Cumulative steam injected vs. simulated time for the optimized case (red) and the base case (Green), Grid 1. Figure 9: Cumulative oil produced of Grid 1 for the base SAGD case (green), the optimized case (red) and a cold production case (black). 11

12 Figure 16: Cumulative NPV ($) of Grid 2 for the base SAGD case (green), the optimized case (red) and a cold production case (black). Table 11 Control variable results for optimization with and without uncertainty. Optimization under uncertainty results (shown under the group heading "With Uncertainty") are shown for their respective values of risk aversion (λ). Deterministic optimization results (shown under the group heading "Deterministic Optima") are shown for each of the 3 grids. We observe that while there is little difference in values for bottom hole injection well pressure, much more pronounced differences exist for the durations of each period and also for well completion location. With Uncertainty Deterministic Optima λ=0 λ=1 λ=2 Grid 1 Grid 2 Grid 3 Injector Bottom Hole Pressure (bars) BHPINJ1 (applied over TIME1) BHPINJ2 (applied over TIME2) BHPINJ3 (applied over TIME3) BHPINJ4 (applied over TIME4) BHPINJ5 (applied over TIME5) Duration of Each Period (days) TIME TIME TIME TIME TIME5 (derived) Completion Layer of Well (k-index) POSPROD (production well location) POSINJ (injection well location)