How Do We Perform Stochastic Reservoir Optimization?*

How Do We Perform Stochastic Reservoir Optimization?* Benoît Couët Schlumberger-Doll Research Ridgefield, CT * Work in collaboration with B. Raghuraman, P. Savundararaj, and R. Burridge

Asset Team Reservoir Characterization with uncertainty Forward Modeling - simulation Financial risk (e.g., NPV) linked to uncertainty Analysis & Decision Valuation of new information Reservoir Risk Management Workflow

Valuation under Uncertainty • Given uncertainty in the reservoir parameters, risk analysis can bring many benefits: • Decide if the gain from implementing technology covers the investment, i.e., risk-reward of using technology • Choose optimum operating strategy (e.g. flow rates) to achieve gain, even before reducing uncertainty with more measurements • Get at the value of the sensors used to reduce uncertainty in the reservoir properties

When to Perform Reservoir Optimization Reservoir time line Production Phase (real-time) - Sense changes - Compute (real-time strategy) - Act on reservoir Planning Phase - Cost/Benefit analysis - Design tool

Reservoir Optimization • Extract maximum fraction of oil • Maximize Net Present Value of oil produced • Minimize water production • Account for economic constraints (stimulation cost, sensors) and physical constraints (pumps/valves limitations) Given reservoir parameters (e.g. # wells, geological uncertainty), there are control mechanisms (e.g. pumping rates) to optimize for to achieve these objective functions. The presence of reservoir and financial uncertainty turns the optimization problem into a risk management problem.

Thought Process Given a reservoir description: • What is the optimal production strategy under reservoir uncertainty? • What about financial uncertainty? • How confident (probability) are we to actually achieve the optimum under uncertainty, i.e., what is the downside risk?

Example 1: Horizontal well with ACS 10,000 bpd f1 f2 f3 low kv low kv 100 ft high kv 500 mD 10 mD aquifer 6000 ft Objective: Maximize gain in NPV due to control Control variable: 3 flow rates Constraint: total flow= 10,000 bpd P> bubble pressure time= 2 years

Sample Cost Function • Total cost (per unit of time): K=LKql+ TKqo+ PKqw+ FK • PK Process cost (applied to water) • LK Lifting costs (applied to liquid) • TKTransport cost (applied to oil) • FK Fixed costs (per unit of time)

f1 f2 f3 low kv low kv high kv aquifer Valuation of ACS : Deterministic case No control NPV= $ 43.2 MM Optimum NPV = $ 49.5 MM Gain over 2 years= $ 6.3 MM Cost of ACS= $ 2.25 MM Benefit > Cost Algorithm convergence time ~ 15 min

f1 f2 f3 low kv low kv Example 1: Stochastic reservoir • Uncertainty • width ? • Questions: • what is the gain at 50% confidence; 95% confidence? • what are optimum flow rates? • what is value of monitoring here?

Treatment of uncertainty • Variables • Normal • Lognormal • Upper/lower bounds • Sampling for weighted mean and standard deviation • Avoid Monte Carlo • Equi-probable points • Quadrature formulae

Equi-probable points • n points from variable distribution • m uncertain variables - uncorrelated • nm equi-probable scenarios population mean population standard deviation

Objective function gi = gain of a scenario = f (flow rates) weight factor of scenario Mean Std. deviation Objective function

Results of stochastic optimizationsingle uncertainty *assuming normal distribution ACS Cost =$ 2.25 MM

Optimization at different risk aversion factors

Added value through monitoring Value of perfect information = $ 2.2 MM

Example 1: Two uncertainties f1 f2 f3 low kv low kv aquifer PI ? ? • Two uncertainties • width • aquifer PI

Results of stochastic optimization *assuming normal distribution ACS Cost =$ 2.25 MM

Optimization for different risk aversion factors Scenarios are equi-probable combinations of width and aquifer PI

Added value through monitoring Value of perfect information = $ 1.8 MM

Which uncertainty to resolve? vary aquifer PI constant width=700 ft vary width constant aquifer PI=4500

Example 1: Three uncertainties f1 f2 f3 low kv K? low kv aquifer PI ? ? • Three uncertainties • width • aquifer PI • permeability

Results of stochastic optimization *normal distribution should be relaxed ACS Cost =$ 2.25 MM

Issues • Optimization • Multi-stage • Number and/or position of drainage points • Algorithms • Uncertainty • Efficiency

Optimization Algorithms • FSQP (“Feasible Sequential Quadratic Programming”) • Gradient-based package for solving constrained nonlinear local optimization with bounds on the continuous variables. • Downhill Simplex method • Non-gradient local optimization modified to handle bounds on continuous variables.

Optimization Algorithms • OptQuest • Scatter-search (à la SA or GA) for solving constrained nonlinear global optimization with bounds on the discrete or continuous variables. Neural network extension. Parallel version available • Others • Simulated annealing • Etc.

Uncertainty • How to treat skewed objective function distributions? • Ranking – optimizing by percentile • Semi-variance to minimize downside risk • Global uncertainty / heterogeneity • Better integration of physical and financial uncertainty

Efficiency • Coarse grid cells for reservoirs • Material Balance • Neural Network as a proxy/surrogate for simulator

Neural Network • Can be used (every few iterations) to forecast the objective value rather than having to run a lengthy simulation. • Tested by running optimization and adding a new train data point every iteration. Half-way through optimization, train neural net for few iterations (minimal cost). Finish optimization using neural net (no cost).

Eclipse iterations: 410 seconds Training neural net: 55 seconds Neural net iterations: ~ 1 second Final answer

Eclipse only: 765 seconds Final answer error = 0.03%

How Do We Perform Stochastic Reservoir Optimization?*