B44 Adapted Nonlinear Optimization Method for Production Data and 4D Seismic Inversion

International audienceIntegrated inversion of production history data and 4D seismic data for reservoir model characterization leads to a nonlinear inverse problem that is usually cumbersome to solve : the associated forward problem based, on one hand, on fluid flow simulation in the reservoir for production data modeling, and on the other hand, on a petro-elastic model for 4D time lapse seismic data modeling, is usually computationally time consuming, the number of measurements to be inverted is large (up to 500 000), the number of model parameters to be determined is up to 100. Moreover, all the derivatives of the modeled data with respect to those parameters are usually not available. We propose an optimization method based on a Sequential Quadratic Programming algorithm which uses gradient approximation coupled with a BFGS approximation of the Hessian. In addition, the proposed method allows to handle equality and inequality nonlinear constraints. Some realistic applications are presented to illustrate the efficiency of the method

Uncertainty Analysis in Prestack Stratigraphic Inversion: a first insight

International audienceStratigraphic inversion of prestack seismic data allows the determination of subsurface elastic parameters (density, P and S-impedances). Based on a Bayesian approach, the problem is formulated as a non-linear least-squares local optimization problem. The objective function to be minimized is composed of two terms, the first one measures the mismatch between the synthetic seismic data (computed via a forward operator) and the observed seismic data, the second one models geological a priori information on the subsurface model. It is crucial to estimate the a posteriori uncertainties because the solution model of the inversion is only one solution among the range of admissible models that fit the data and the a priori information. The goal of this paper is to propose an optimized deterministic method to estimate a posteriori uncertainties in stratigraphic inversion. The proposed method is based on the hypothesis that the covariance matrices describing the uncertainties on the data and on the model are laterally uncorrelated (no cross correlation among parameters of different traces). Moreover, the covariance matrix on the data is also supposed laterally stationary. Application on 2D synthetic PP data illustrates the performances of the method. Extension and limitations of the method are discussed

How Topological Rearrangements and Liquid Fraction Control Liquid Foam Stability

International audienceThe stability of foam is investigated experimentally through coalescence events. Instability (coalescence) occurs when the system is submitted to external perturbations (T1) and when the liquid amount in the film network is below a critical value. Microscopically, transient thick films are observed during film rearrangements. Film rupture, with coalescence and eventual collapse of the foam, occurs when the available local liquid amount is too small for transient films to be formed. Similar experiments and results are shown in the two-bubble case

A dedicated constrained optimization method for 3D reflexion tomography

International audienceSeismic reflection tomography is a method for the determination of a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a nonlinear least-squares function measuring the mismatch between observed traveltimes and those calculated by raytracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as constraints in the optimization problem. This constrained optimization is based on a Gauss-Newton Sequential Quadratic Programming approach. At each Gauss-Newton step, a solution to a strictly convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian relaxation method. Our choice for this optimization method is motivated and its original aspects are described. The efficiency of the method is shown on applications on a 2D OBC real data set and on a 3D real data set: the introduction of constraints coming both from well logs and from geological knowledge allows to reduce the under-determination of the 2 inverse problems. Introduction Reflection tomography allows to determine a velocity model from the traveltimes of seismic waves reflecting on geological interfaces. This inverse problem is formulated as a nonlinear least-squares function which measures the mismatch between observed traveltimes and traveltimes computed by ray tracing method. This method has been successfully applied to real data sets (Ehinger et al, 2001, Broto et al, 2003). Nevertheless, the under-determination of the inverse problem generally requires the introduction of additional information on the model to reduce the number of admissible models. Penalization terms modelling this information can be added to the seismic terms in the objective functions but the tuning of the penalization weights may be tricky. In this paper, we propose to handle the a priori information by the introduction of equality and inequality constraints in the optimization process. This approach allows to introduce lot of constraints of different types, provided we have at our disposal an adequate constrained optimization method. We developed an original method designed for the tomographic inverse problem which presents the following characteristics: it is a large scale problem (10000-50000 unknowns), the forward operator is nonlinear and its computation may be expensive (large number of source-receiver couples, up to 500000), the problem is ill-conditioned. In the first part of this paper, the chosen method is motivated and its original aspects are shortly described (for further details, refer to Delbos et al, 2004). Applications on a 2D PP/PS real data set and on a 3D PP real data set are presented in a second part

Constrained nonlinear optimization for extreme scenarii evaluation in reservoir characterization.

International audienceThe goal of reservoir characterization is the estimation of unknown reservoir parameters (the history matching problem), by integrating available data in order to take decisions for production scheme and to predict the oil production of the field in the future (the forecast problem). The reservoir parameters could be classified in two classes: • those related to the geological modeling (spatial distribution of porosity, permeability, faults), • and those related to the fluid flow modeling (relative permeability curves, productivity index of the wells). Those parameters could not be directly determined by measurements (or only locally using well logs), this is the reason why this parameter estimation problem is formulated as an inverse problem with forward simulators that compute synthetic measurable data from those parameters. Observed data are well data acquired at production/injection wells (bottom-hole pressure, gas-oil ratio, oil rate) at different calendar times during the production of the field. The main contribution of this work is the integration of nonlinear optimization methodology to predict the oil production of a field and to give a confidence interval on this prediction. We believe that applying non linear optimization methods will increase accuracy and then give more reliable production forecast than approaches with simplified models of forward operators (linear approximations or response surfaces). The first and second sections of this paper are respectively dedicated to the history matching problem and to the forecast problem. In the third section, we described the optimization methods used to solve both problems. Then, in the last section the previous methodology is applied to a 3D synthetic reservoir application (the PUNQ test case)

Constrained optimization in seismic reflection tomography: a Gauss-Newton augmented Lagrangian approach

International audienceS U M M A R Y Seismic reflection tomography is a method for determining a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint , the problem consists in minimizing a non-linear least-squares function measuring the mismatch between observed traveltimes and those calculated by ray tracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as inequality constraints in the optimization problem. This technique is based on a Gauss-Newton (GN) sequential quadratic programming approach. At each GN step, a solution to a convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian algorithm. Our choice for this optimization method is motivated and its original aspects are described. First applications on real data sets are presented to illustrate the potential of the approach in practical use of reflection tomography

A Derivative Free Optimization method for reservoir characterization inverse problem

International audienceReservoir characterization inverse problem aims at building reservoir models consistent with available production and seismic data for better forecasting of the production of a field. These observed data (pressures, oil/water/gas rates at the wells and 4D seismic data) are compared with simulated data to determine unknown petrophysical properties of the reservoir. The underlying optimization problem is usually formulated as the minimization of a least-squares objective function composed of two terms : the production data and the seismic data mismatch. In practice, this problem is often solved by nonlinear optimization methods, such as Sequential Quadratic Programming methods with derivatives approximated by finite differences. In applications involving 4D seismic data, the use of the classical Gauss-Newton algorithm is often infeasible because the computation of the Jacobian matrix is CPU time consuming and its storage is impossible for large datasets like seismic-related ones. Consequently, this optimization problem requires dedicated techniques: derivatives are not available, the associated forward problems are CPU time consuming and some constraints may be introduced to handle a priori information. We propose a derivative free optimization method under constraints based on trust region approach coupled with local quadratic interpolating models of the cost function and of non linear constraints. Results obtained with this method on a synthetic reservoir application with the joint inversion of production data and 4D seismic data are presented. Its performance is compared with a classical SQP method (quasi-Newton approach based on classical BFGS approximation of the Hessian of the objective function with derivatives approximated by finite differences) in terms of number of simulations of the forward problem