Assessing uncertainties on production forecasting based on production Profile reconstruction from a few Dynamic simulations Gaétan Bardy – PhD Student Total S.A. – Lorraine University

Introduction Assessing static uncertainties Facies NTG Permeability … Well data … … Geo-statistic Seismic data Thousands of models can be produced Computation of: V, Voil, Vconnected, HuPhiSo … algorithms … … Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

introduction Assessing dynamic uncertainties: Parameters’ influence Well placement Production forecast Green field Brown field Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

introduction Assessing dynamic uncertainties: Parameter influence Well placement Production forecast Green field Brown field (History matching) Two main applications for proxies Try to reproduce the true dynamic simulation Select representative models Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

introduction [Scheidt and Caers, 2007] Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

introduction How to use proxy information to improve the quantile computation and assess the uncertainty on it ? [Scheidt and Caers, 2007] Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Proxy Full physic Proposed Workflow Simulation & Trend estimation [N x N] Simulation & distance matrix Proxy Trend estimation Distances reconstruction [m x m] Simulation & distance matrix Full physic Profiles reconstruction Confidence intervals on quantiles determination Quantiles computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Proxy Full physic Proposed Workflow Simulation & Trend estimation [N x N] Simulation & distance matrix Proxy Trend estimation Distances reconstruction [m x m] Simulation & distance matrix Full physic Profiles reconstruction Confidence intervals on quantiles determination Quantiles computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance REgression Challenge: predict “true” dynamic distance from: All proxy distances Some true distances from few simulated models Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance regression Plot: true distances ( ) vs. proxy distances ( ) between the same pairs of models (for the few selected models) m models give : m(m-1) / 2 distances; Here 10 models give 45 distances Dynamic distance Proxy distance Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance regression Domain is split in regions of identical number of samples m models give : m(m-1) / 2 distances; Here 10 models give 45 distances Dynamic distance Proxy distance Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance regression Domain is split in regions of identical number of samples Linear regressions are computed in each region m models give : m(m-1) / 2 distances; Here 10 models give 45 distances Dynamic distance Proxy distance Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance regression Domain is split in regions of identical number of samples Linear regressions are computed in each region Residual statistics are determined in each region m models give : m(m-1) / 2 distances; Here 10 models give 45 distances Dynamic distance Proxy distance Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Dynamic distance regression Missing pairs are drawn according to the region they belong Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Proxy Full physic Proposed Workflow Simulation & Trend estimation [N x N] Simulation & distance matrix Proxy Trend estimation Distances reconstruction [m x m] Simulation & distance matrix Full physic Profiles reconstruction Confidence intervals on quantiles determination Quantiles computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction For each missing profile (full physics), the distance to the simulated curves can be: computed ( ) estimated from previous method ( ) Time step FOPT (m3) Observed profiles Current reconstructed profile Creation of an objective function to minimize ^d is the estimated distance D* is the distance computed between the mathamatical model and a simulated model. Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction Work on FOPT profile Represent the cumulative oil production of the field Common part for all profiles First regular phase limited only by the infrastructures; Second part corresponding to progressive water arrival; The proposed model should respect these physical constraints Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction Choice of a mathematical model that can fit our profiles: Linear part for the field first period; Hermite cubic spline for the field decrease; t Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction Choice of a mathematical model that can fit our profiles: Linear part for the field first period; Hermite cubic spline for the field decrease; t parameters Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction Choice of a mathematical model that can fit our profiles: Linear part for the field first period; Hermite cubic spline for the field decrease; Fit try on the simulated profiles: True profile Fitted profile Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Missing profiles reconstruction Best profile is found using a minimization algorithm Constraints Increasing curve cumulative production profile Decreasing tangent daily production decrease Simulated profile Minimized profile ^d is the estimated distance D* is the distance computed between the mathamatical model and a simulated model. Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Curves correction Creation of an error map Use difference between analytical model and simulated profile FOPT Time step Dynamic profile Analytical model Model #1 Model #2 Time step Error FOPT Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Curves correction Creation of an error map Use difference between analytical model and dynamic profile Use interpolation to create the map Error map FOPT Time Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Curves correction Creation of an error map Use difference between analytical model and dynamic profile Use interpolation to create the map Each curve is corrected according to the map correction Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Proxy Full physic Proposed Workflow Simulation & Trend estimation [N x N] Simulation & distance matrix Proxy Trend estimation Distances reconstruction [m x m] Simulation & distance matrix Full physic Profiles reconstruction Confidence intervals on quantiles determination Quantiles computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Quantile computation Corresponds to low, base and high cases Quantiles are computed at each time step Provide a synthetic profile True quantiles Heterogeneous set based quantiles Selected model based quantile Improvement of quantile computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Proxy Full physic Proposed Workflow Simulation & Trend estimation [N x N] Simulation & distance matrix Proxy Trend estimation Distances reconstruction [m x m] Simulation & distance matrix Full physic Profiles reconstruction Confidence intervals on quantiles determination Quantiles computation Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Confidence interval computation Use the residuals’ laws to create multiple realizations Redoing minimization procedure Drawing new distances Minimizing missing profiles Computing quantiles Using rejection sampling Parameters’ bounds are determined from initial fits Acceptance criteria: Positive and monotonic curve Computed distance with the simulated curves respecting the law of residuals Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Application Methodology apply on real data field. 793 000 cells (130x100x61); 3 injectors and 4 producers; 162 models with slightly different geometry Possibility to benchmark the proxies ?: Width of the intervals gives information on the proxy accuracy Illustration with 2 different proxies: Upscaling based STOOIP based Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Application – preliminary results Distance estimation Rejection Sampling With minimization Upscaling based proxy STOOIP based proxy Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Conclusion & Further work Improvement of the production profile quantile computation Benchmark of proxies Not finalized yet Direct application Improve Accumulation/reserve cross plot Error map For rejection sampling Choice of interpolation parameters Test on another application field Work with water profiles Manage two development phases Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Thank you for your attention ! Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Gradual upscaling of static properties: ACTNUM ORIGINAL LAYERS: 16 17 18 UPSCALED: 9 9 6 Coarse cell is active in and only if it contains at least one active fine cell (in other words, maximum upscaling method is used). As the main advantage of this approach, we do not lose the volume of active cells (in contrast to other upscaling techniques, such as choosing the most probable value of the cell or the nearest to the cell center value). For example, using the nearest to the cell center value for ACTNUM leads to 30% loss in hydrocarbons in place volume. On the other hand, our approach might lead to increase of hydrocarbons in place volume and to avoid this we carefully process the NTG property. 2x2x2 3x3x2 3x3x3 SUMMARY: Coarse cell is active in and only if it contains at least one active fine cell (maximum method). The volume of active cells is not lost (in contrast to most probable and nearest to center upscaling methods). To avoid increasing the volume of hydrocarbons in place and initial water saturation we correct NTG. Coarsening the stratigraphic grid and upscaling the fine grid properties

Gradual upscaling of static properties: NTG ORIGINAL LAYERS: 16 17 18 UPSCALED: 9 9 6 Generally, for upscaling NTG we use arithmetic averaging with volume weighting. To avoid increase of the volume of hydrocarbons in place, a preprocessing procedure is used – substitution of the NDV values with the 0 NTG (that helps to avoid 3% bias in volume). That explains the amount of dark green color on the uspcaled layers. 2x2x2 3x3x2 3x3x3 SUMMARY: Volume weighted arithmetic average is used. A preprocessing procedure (substitution of NDV by 0) is used to avoid increasing of the net rock volume. Coarsening the stratigraphic grid and upscaling the fine grid properties

Gradual upscaling of static properties: POROSITY ORIGINAL LAYERS: 16 17 18 UPSCALED: 9 9 6 For porosity we use arithmetic averaging with volume and NTG weighting. On the upscaled layers black color corresponds to 0 NTG. 2x2x2 3x3x2 3x3x3 SUMMARY: Volume and NTG weighted arithmetic average is used. PORO is 0 if NTG is 0 Coarsening the stratigraphic grid and upscaling the fine grid properties

Gradual upscaling of static properties: Water saturation ORIGINAL LAYERS: 16 17 18 UPSCALED: 9 9 6 For SWL we use arithmetic average with volume, NTG and PORO weighting. On the upscaled layers, white color means 1 SWL and corresponds to 0 NTG. 2x2x2 3x3x2 3x3x3 SUMMARY: Volume, NTG and porosity weighted arithmetic average is used. SWL is 1 if NTG is 0 Coarsening the stratigraphic grid and upscaling the fine grid properties

Gradual upscaling of static properties: ROCK TYPES ORIGINAL LAYERS: 16 17 18 RT0 NDV UPSCALED: 9 9 6 We use the nearest to the coarse cell center value of the fine cell. That enables us to preserve the small proportions of different rock types. A preprocessing procedure is also used. RT0 is substituted with NDV, because we are not allowed to prescribe RT0 value to an active cell (and that is correct from the other point of view: a cell on the fine grid is active only if its rock type is not NDV and RT0, so an active coarse cell contains inside other values than RT0). 2x2x2 3x3x2 3x3x3 SUMMARY: Nearest to the coarse cell center value of fine cell is used. Preserves the proportions of rock types. Preprocessing: substitution of RT0 with NDV (to avoid 0 rock type in an active cell) NDV Coarsening the stratigraphic grid and upscaling the fine grid properties

Confidence interval computation (minimization based) Drawing new distances Minimizing missing profiles Computing quantiles … Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Confidence interval computation (Rejection sampling based) Parameters’ bounds are determined from initial fits Parameters are drawn successively inside bounds Acceptance criteria: Positive and monotonic curve Computed distance with the simulated curves respecting the law of residuals Example with 200 samples for each curve Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014

Accu/Reserve crossplot Stanford Center for Reservoir Forecasting Annual Affiliates Meeting – May 8th , 2014