Methodology

Technical approach and mathematical formulation

Problem Formulation Gaussian Process Models QKP Formulation Low-Rank Approximation Optimization Algorithm

Problem Formulation

The well placement optimization problem involves selecting the best locations to drill new wells from a set of potential locations to maximize the expected Net Present Value (NPV) while minimizing risk.

Mathematical Formulation

We formulate the well placement problem as a mean-variance portfolio optimization:

\[ \max_{\mathbf{x}} \left\{ \mathbf{r}^T \mathbf{x} - \lambda \mathbf{x}^T \mathbf{Q} \mathbf{x} \right\} \]

subject to:

\[ \sum_{i=1}^{n} x_i = N \]

\[ x_i \in \{0,1\} \quad \forall i \in \{1,2,\ldots,n\} \]

where:

  • \(\mathbf{x}\) is a binary decision vector indicating which wells to drill
  • \(\mathbf{r}\) is the vector of expected NPVs for each potential well
  • \(\mathbf{Q}\) is the covariance matrix capturing the uncertainty and correlation between wells
  • \(\lambda\) is the risk aversion parameter
  • \(N\) is the number of wells to select

Gaussian Process Models

We use Gaussian Process (GP) regression to model the spatial distribution of well productivity across the reservoir. This allows us to capture the correlation structure between different locations.

Variogram Models

Our framework supports multiple variogram models to capture spatial correlation:

  • RBF (Radial Basis Function): General-purpose kernel with smooth properties
  • Exponential: For modeling sharper transitions in reservoir properties
  • Matérn: Provides greater flexibility for modeling spatial heterogeneity

RBF Kernel:

\[ k(x_i, x_j) = \sigma^2 \exp\left(-\frac{||x_i - x_j||^2}{2l^2}\right) \]

Exponential Kernel:

\[ k(x_i, x_j) = \sigma^2 \exp\left(-\frac{||x_i - x_j||}{l}\right) \]

The GP model provides the mean vector \(\mathbf{r}\) and covariance matrix \(\mathbf{Q}\) needed for the optimization problem.

QKP Formulation

The mean-variance portfolio optimization is formulated as a Quadratic Knapsack Problem (QKP), which is an extension of the classic knapsack problem with quadratic terms in the objective function.

QKP Model
import gurobipy as gp
from gurobipy import GRB

def solve_qkp(mean_vector, cov_matrix, n_select, lambda_value):
    n = len(mean_vector)
    
    # Create model
    model = gp.Model("well_placement_qkp")
    
    # Add binary decision variables
    x = model.addVars(n, vtype=GRB.BINARY, name="x")
    
    # Set objective: maximize mean - lambda * variance
    objective = gp.quicksum(mean_vector[i] * x[i] for i in range(n))
    
    # Add risk term
    for i in range(n):
        for j in range(n):
            objective -= lambda_value * cov_matrix[i,j] * x[i] * x[j]
    
    model.setObjective(objective, GRB.MAXIMIZE)
    
    # Add constraint on number of wells to select
    model.addConstr(gp.quicksum(x[i] for i in range(n)) == n_select)
    
    # Solve
    model.optimize()
    
    # Return selected indices
    if model.status == GRB.OPTIMAL:
        selected = [i for i in range(n) if x[i].X > 0.5]
        return selected
    else:
        return None

Low-Rank Approximation

For large-scale problems, we employ Randomized Singular Value Decomposition (RSVD) to create a low-rank approximation of the covariance matrix, significantly reducing computational requirements.

The full covariance matrix \(\mathbf{Q}\) is approximated as:

\[ \mathbf{Q} \approx \mathbf{U}_r \mathbf{\Sigma}_r \mathbf{U}_r^T \]

where \(\mathbf{U}_r\) contains the first \(r\) left singular vectors and \(\mathbf{\Sigma}_r\) is a diagonal matrix with the largest \(r\) singular values.

This allows us to reformulate the quadratic term as:

\[ \mathbf{x}^T \mathbf{Q} \mathbf{x} \approx \mathbf{x}^T \mathbf{U}_r \mathbf{\Sigma}_r \mathbf{U}_r^T \mathbf{x} = \mathbf{y}^T \mathbf{\Sigma}_r \mathbf{y} \]

where \(\mathbf{y} = \mathbf{U}_r^T \mathbf{x}\).

SVD Approximation Error vs. Rank

Visualization of approximation error

Optimization Algorithm

The complete optimization workflow involves the following steps:

  1. Generate multiple well scenarios using Gaussian Process models
  2. Compute the mean vector and covariance matrix
  3. Optional: Apply RSVD for low-rank approximation
  4. Formulate and solve the QKP using Gurobi
  5. Analyze results across different risk aversion parameters
Parameter Tuning

Key parameters to tune in the optimization process:

Parameter Description Impact
λ (lambda) Risk aversion parameter Higher values prioritize risk reduction over expected returns
r (RSVD rank) Target rank for approximation Lower ranks improve computational efficiency at the cost of accuracy
Variogram parameters Correlation length, sill, etc. Affects the spatial correlation structure and uncertainty quantification