Parameter Estimation in Reservoir Engineering Models via Data Assimilation Techniques

Parameter Estimation in Reservoir Engineering Models via Data Assimilation Techniques Mariya V. Krymskaya TU Delft July 6, 2007

Ensemble Kalman Filter (EnKF) Iterative Ensemble Kalman Filter (IEnKF) State Vector Feasibility Re-scaling state vector Experimental Setup EnKF IEnKF

Structure of Reservoir Engineering Reservoir Engineering Production Engineering Reservoir Simulation approaches Analogical Experimental Mathematical

A Reservoir General information > 40, 000 oil fields in the world 300 m to 0 km below the surface 2 500 million years old Ghawar oil field the biggest among discovered Location: Saudi Arabia Recovery: since 95 Size: 280 30 km Age: 320 million years old Production: 5 million barrels (800, 000 m 3 ) of oil per day

Oil Recovery Primary 20% extracted

Oil Recovery Primary 20% extracted Secondary (water flooding) 25% to 35% extracted

Oil Recovery Primary 20% extracted Secondary (water flooding) 25% to 35% extracted Tertiary 50% left

Reservoir Properties Rock properties Porosity (Absolute) permeability Rock compressibility

Reservoir Properties Rock properties Porosity (Absolute) permeability Rock compressibility Fluid properties Fluid compressibility Fluid density Fluid viscosity

Reservoir Properties Rock properties Porosity (Absolute) permeability Rock compressibility Fluid properties Fluid compressibility Fluid density Fluid viscosity Fluid-rock properties Fluid saturation Capillary pressure Relative permeability (Corey-type model)

Mass balance equation for each phase Darcy s law for each phase Capillary pressure equation Relative permeability equations (Corey-type model) Equations of state

Mass balance equation for each phase Darcy s law for each phase Capillary pressure equation Relative permeability equations (Corey-type model) Equations of state Initial / boundary conditions

Mass balance equation for each phase Darcy s law for each phase Capillary pressure equation Relative permeability equations (Corey-type model) Equations of state Initial / boundary conditions Well model

Model Discretization Ê(X) Ẋ Â(X) X ˆB(X) U = 0 accumulation matrix system matrix state vector input matrix input vector X = [ p S ]

History Matching Process Historical Match Manual Automatic approaches Traditional Kalman Filtering

Data Assimilation Problem Statement Data Assimilation Problem Statement Ensemble Kalman Filter (EnKF) Iterative Ensemble Kalman Filter (IEnKF) System X k+ = F (X k, U k, m) + W k, Z k+ = MX k + V k Uncertainties X 0 N (X 0, P 0 ) uncertain initial state, W k N (0, Q) model noise, V k N (0, R) measurement noise, Independency assumption State conditional pdf X 0 W k V k (X k Z,..., Z l ) N (mean, cov)

Outline Ensemble Kalman Filter (EnKF) Data Assimilation Problem Statement Ensemble Kalman Filter (EnKF) Iterative Ensemble Kalman Filter (IEnKF) Initial condition X 0, P 0 Data Z k Ensemble generation (ξ i ) 0, X 0, P 0 Forward model integration Forecasted ensemble and state (ξ i ) f k, X f k, P f k Data assimilation Analyzed ensemble and state (ξ i ) a k, X a k, P a k

Parameter Estimation via EnKF Data Assimilation Problem Statement Ensemble Kalman Filter (EnKF) Iterative Ensemble Kalman Filter (IEnKF) Augmented state vector X = [ p S ] X = log k} } = m p = Y S d

Outline Data Assimilation Problem Statement Ensemble Kalman Filter (EnKF) Iterative Ensemble Kalman Filter (IEnKF) Iterative Ensemble Kalman Filter (IEnKF) [m 0, Y 0] T, P 0 m a k Analyzed ensemble Initial condition X 0, P EnKF and state 0 (ξ i ) a k, X a k,p a k Estimated model parameter m a k

Outline State Vector Feasibility State Vector Feasibility Re-scaling state vector Experimental Setup Ensemble generation New static parameters Forward model integration Data assimilation Yk a m a k Re-initialized ensemble and state (ξ i ) a k, X a k, P a k Forward model integration from k to k Confirmation step Confirmed ensemble and state (ξ i ) c k, X c k, P c k

Re-scaling state vector State Vector Feasibility Re-scaling state vector Experimental Setup A B = X = 0 7 0 log k p S Y, 0 7 0 3 AML BL

Re-scaling state vector State Vector Feasibility Re-scaling state vector Experimental Setup Kalman gain K = N ( ) LL T M T N MLLT M T + R

Re-scaling state vector State Vector Feasibility Re-scaling state vector Experimental Setup Kalman gain K = ( B B ) LL T M ( T AA ) ( ) ( N N MLLT M T + R A A )

Re-scaling state vector State Vector Feasibility Re-scaling state vector Experimental Setup Kalman gain K = ( B B ) LL T M ( T AA ) ( ) ( N N MLLT M T + R A A ) ( ) = B (BL) (AML)T N N AML (AML)T + ARA A }{{} K Ensemble update (ξ i ) a k = (ξ i) f k + K ( Z M(ξ i ) f k + Vi) = (ξ i ) f k + B K A ( Z M(ξ i ) f k + Vi) = (ξ i ) f k + B ( K ( AZ AM(ξi ) f k + AVi))

Experimental setup State Vector Feasibility Re-scaling state vector Experimental Setup Twin experiment Initialization True permeability field (log(m 2 )) 28 Mean of permeability fields ensemble (log(m 2 )) 2 4 28.5 2 4 28.5 6 29 6 28.6 8 0 2 29.5 30 8 0 2 28.7 28.8 4 6 30.5 3 4 6 28.9 8 20 5 0 5 20 3.5 32 8 20 5 0 5 20 29 29.

RMS Error in Model Parameter EnKF IEnKF 2.8 Ensemble mean Ensemble members.6 RMS error (log(m 2 )).4.2 0.8 0.6 0.4 0 00 200 300 400 500 600 Time (days)

Estimated Permeability Field EnKF IEnKF Mean of initial ensemble (log(m 2 )) Variance of initial ensemble ((log(m 2 )) 2 ) 28 5 0 29 30 5 0 0.8 0.6 2 4 6 8 0 2 4 6 8 20 True permeability field (log(m 2 )) 5 0 5 20 28 28.5 29 29.5 30 30.5 3 3.5 32 5 20 5 0 5 20 3 32 5 20 5 0 5 20 Estimated permeability field (log(m 2 )) Variance of estimated ensemble ((log(m 2 )) 2 ) 28 5 0 29 30 5 0 0.4 0.2 0 0.8 0.6 5 20 5 0 5 20 3 32 5 20 5 0 5 20 0.4 0.2 0

EnKF IEnKF Forecasted Reservoir Performance Water saturation S w at well (,) ( ) Water saturation S w at well (,2)( ) 0.8 0.6 0.4 0.2 0 500 000 500 Time t ( days) 0.8 0.6 0.4 0.2 0 500 000 500 Time t ( days) Water saturation S w at well (2,)( ) Water saturation S w at well (2,2)( ) 0.8 0.6 0.4 0.2 0 500 000 500 Time t ( days) 0.8 0.6 0.4 0.2 0 500 000 500 Time t ( days)

RMS Error in Model Parameter EnKF IEnKF 2.8 Ensemble mean Ensemble members 2.8 Ensemble mean Ensemble members.6.6 RMS error (log(m 2 )).4.2 RMS error (log(m 2 )).4.2 0.8 0.8 0.6 0.6 0.4 0 00 200 300 400 500 600 Time (days) 0.4 0 00 200 300 400 500 600 Time (days)

RMS Error in Model Parameter EnKF IEnKF 2.2 2 Ensemble mean Ensemble members 2.2 2 Ensemble mean Ensemble members.8.8 RMS error (log(m 2 )).6.4.2 RMS error (log(m 2 )).6.4.2 0.8 0.8 0.6 0 00 200 300 400 500 Time (days) 0.6 0 00 200 300 400 500 Time (days)

Estimated Permeability Field EnKF IEnKF Mean of initial ensemble (log(m 2 )) Variance of initial ensemble ((log(m 2 )) 2 ) 28 5 0 5 20 5 0 5 20 29 30 3 32 5 0 5 20 5 0 5 20 Estimated permeability field (log(m 2 )) Variance of estimated ensemble ((log(m 2 )) 2 ) 28 5 0 5 20 5 0 5 20 29 30 3 32 5 0 5 20 5 0 5 20 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 Mean of initial ensemble (log(m 2 )) Variance of initial ensemble ((log(m 2 )) 2 ) 28 5 0 5 20 5 0 5 20 29 30 3 32 5 0 5 20 5 0 5 20 Estimated permeability field (log(m 2 )) Variance of estimated ensemble ((log(m 2 )) 2 ) 28 5 0 5 20 5 0 5 20 29 30 3 32 5 0 5 20 5 0 5 20 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0

Model calibration is essential EnKF provides reasonable parameter estimation There are cases at which IEnKF is superior to EnKF Further investigations on IEnKF sensitivities are required

Outline