Identification of three-dimensional electric conductivity changes from time-lapse electromagnetic observations

https://doi.org/10.1016/j.jcp.2011.02.015Get rights and content

Abstract

We present a novel solution algorithm for 3D parameter identification based on low frequency electromagnetic data. With focus on large-scale applications such as monitoring of subsea oil production, CO2 sequestration, and geothermal systems, the proposed solution algorithm is designed to meet challenges related to low parameter sensitivity, nonuniqueness of the inverse solutions, nonlinearity in the mapping from the data to the parameter space, and costly numerical simulations. Motivated by earlier investigations on the relation between sensitivity, nonlinearity and scale, the proposed solution approach is based on a reduced, composite parameter representation. Though a reduced representation restricts the solution space, flexibility with respect to which parameter functions that can be represented is obtained by facilitating the estimation of the structure and smoothness of the representation itself. Moreover, the resolution of the parameter function is detached from the computational grid and determined as part of the estimation. The performance of the proposed solution algorithm is illustrated through numerical examples for identification of underground electric conductivity changes from time-lapse electromagnetic observations.

Introduction

In many geophysical applications, like subsea petroleum production, CO2 sequestration, and geothermal systems, there is a need to monitor fluid flows to be able to control the process. As an example, consider a petroleum reservoir with several production wells under water-assisted production. If fluid-flow monitoring indicates that injected water (brine) is rapidly approaching one particular production well, it will be beneficial to reduce the production rate in that well for some period of time. While it is crucial for decision making that the applied monitoring technique is able to identify a coarse-scale approximation to the real shape of the brine front, it is of little importance to identify all details of the shape.

Use of time-lapse seismics for monitoring is now quite common for subsea petroleum reservoirs. Seismic signals are sensitive to elastic properties of a porous medium, which depend on elastic properties of its fluid content. A porous rock saturated with gas is significantly more compressible than if the same rock was saturated with liquid. Although the elastic properties of oil-saturated rocks differ somewhat from those of brine-saturated rocks, it is more difficult to use time-lapse seismics to discriminate between oil and brine than between gas and liquid. An additional difficulty is that elastic properties are also sensitive to fluid pressure in addition to fluid saturation.

Electromagnetic (EM) signals are sensitive to the electric conductivity of a medium. The electric conductivity of a saturated porous medium depends strongly on the electric conductivity of its fluid content, while the dependence on fluid pressure is considered negligible. Furthermore, the electric conductivity of brine-saturated rocks is significantly (10–100 times) higher than that of oil- or gas-saturated rocks. Hence, use of time-lapse EM signals will be complimentary to use of time-lapse seismics for fluid-flow monitoring.

Due to increased damping of the EM signal with increasing frequency, very low frequencies (0.1–10 Hz) are typically applied. Hence, one can not expect a high resolution of the brine front from such signals. Since typical applications do not require a high resolution from the monitoring to be useful for decision making, time-lapse EM might still provide valuable information. The focus of this paper is to present a methodology for approximately identifying the change in electric conductivity in three spatial dimensions from time-lapse EM data in a stable manner.

Due to the restricted resolution power of the data, the inversion is usually under-determined, leading to nonuniqueness of the inverse solutions. Moreover, possible high noise levels make the inversion highly unstable. Additional difficulties are caused by the nonlinearity of the mapping from electric conductivity to electromagnetic data increasing the risk of getting trapped in a local minimum when applying gradient based optimisation algorithms.

Regularisation is a means to overcome these challenges and make the inverse problem better posed. Different approaches to regularisation exist. The most common approach is to apply a pixel based inversion, incorporating some a priori assumptions about the solution properties and penalise deviations from these. Recent progress in 3D inversion of controlled source electromagnetic (CSEM) data along these lines includes [1], [2], [3], [4].

In the current work, we address the problem of over-parametrisation directly, and apply regularisation by reduced representations. Previous work on electromagnetic data inversion utilising reduced representations includes [5], [6], [7], [8], [9]. In these works, the representation of the unknown parameter function is determined a priori, whereas the actual resolution power of the available data will not, in general, be known prior to the estimation. We consider solving the extended inverse problem, where one seeks to determine both a representation warranted by the available information, and associated coefficients in the representation [10]. The current work is a further development of a general framework for parameter estimation developed for 2D estimation of fluid conductivity in porous media [11], [12].

The solution algorithm is based on a composite, nonlinear parameter representation. The composite representation is designed to be flexible with respect to the parameter structures it can represent, allowing for discontinuous as well as smooth structures. In particular, optimisation with respect to the smoothness of the transitions between regions of different conductivity values is possible. That is, the representation facilitates identification also of the smoothness in the parameter function itself. In the limit, the chosen representation is reduced to a zonation with an implicit representation of the interior boundaries that is equivalent to a level-set representation [13], [14].

The composite, nonlinear reparametrisation is detached from the computational grid. This enables a multi-level estimation strategy, where the resolution in the parameter representation is found as part of the solution algorithm. Our parameter identification strategy starts at a coarse resolution, which then is adaptively refined, giving a gradually finer parameter representation. Advantages of an adaptive refinement strategy are reduced risk of getting trapped in a local minimum, and, also, a reduction in the number of repeated solves of the electromagnetic forward model. Examples of adaptive refinement strategies are given in [10], [15], [16], [17], [18], [19]. To our knowledge, this work is the first attempt of introducing adaptive strategies for determining both dimensionality and structure to subsurface parameter identification in three space dimensions.

The paper is organised as follows: the forward problem is stated in Section 2, and the corresponding inverse problem is presented in Section 3. The nonlinear representation of the parameter function is outlined in Section 4, while parameter identification, including the adaptive multi-level estimation strategy, is discussed in Section 5. In Section 6, the performance of the proposed solution strategy is shown for identification of the evolution in the electric conductivity from time-lapse CSEM data with application to subsurface oil-production monitoring.

Section snippets

Forward model equations

Electromagnetic phenomena are generally modelled by Maxwell’s equations. Assuming time variation e−iωt, no free electric charges, and neglecting displacement currents (due to the low frequencies), Maxwell’s equations in the frequency domain are×E=iωμH,×H=σE+Je,·(μH)=0,·(ϵE)=0,where E denotes the electric field, H denotes the magnetic field, ω denotes the angular frequency, μ denotes the magnetic permeability, ϵ denotes the permittivity, σ denotes the electric conductivity (the sub-sediments

Inverse problem

We aim at monitoring changes in fluid saturation utilising information from time-lapse CSEM observations.

The relation between fluid saturation and the electric conductivity can be found through empirical rock physics models such as, e.g., Archie’s law [28], where saturation is given as a function of the electric conductivity S = S(σ). Hence, by identifying the electric conductivity, we can derive information about the corresponding saturation profile.

To monitor the evolution of fluid saturation

Representation of the electric conductivity

High degree of flexibility is obtained by choosing a composite parameter representation of the formp(r)=c1E1(I(r))+c2E2(I(r)),where I is denoted the interior function, and Ej is denoted the exterior function [11], [12]. The coefficients c1 and c2 are denoted exterior coefficients. While {Ej}j=12 are fixed functions, the function I(r) will be given by a parametric representation, resulting in a nonlinear representation of the electric conductivity with respect to the coefficients in the

Parameter identification

To obtain a well-posed problem by regularisation with reduced representations, it is crucial to select an appropriate resolution in the representation of the unknown parameter function. Selecting a very coarse representation is tempting as this can stabilise the estimation by increasing parameter sensitivities. Moreover, for a broad class of inverse problems, a coarse representation is shown to reduce model nonlinearity [33], [34], [35], [12]. On the other hand, a representation with too little

Numerical examples

Marine CSEM surveying is currently in commercial use for hydrocarbon exploration [40], and lately there has been a growing interest for marine CSEM data as a potential new tool for offshore oil production monitoring. Motivated by earlier feasibility studies on marine CSEM data for subsea oil production monitoring [31], [32], we present a series of synthetic examples where we consider the identification of time-lapse conductivity changes corresponding to saturation changes within oil reservoir

Summary and discussion

We have presented a methodology for parameter identification in 3D from data with limited information content, utilising a reduced representation of the sought parameter function. The methodology is based on a composite, nonlinear representation, where both the dimensionality and structure of the parameter function are subject for estimation. The degree of freedom in the reduced representation is determined using a novel adaptive multi-level strategy, where the parameter representation is

Acknowledgments

The second and third authors want to acknowledge the financial support of the Norwegian Research Council (PETROMAKS), Rocksource, Statoil and Total to perform this study.

References (40)

  • Y. Zhang, A. Abubakar, T. Habashy, A model-based inversion algorithm for controlled-source electromagnetic data, in:...
  • M. Commer et al.

    New advances in three-dimensional controlled-source electromagnetic inversion

    Geophys. J. Int.

    (2008)
  • M. Commer et al.

    Three-dimensional controlled-source electromagnetic and magnetotelluric joint inversion

    Geophys. J. Int.

    (2009)
  • N.-Z. Sun, A.Y. Sun, Parameter identification of environmental systems, in: Environmental Fluid Mechanics, ASCE, 2002...
  • I. Berre et al.

    Estimation of a Piecewise Constant Function Using Reparameterized Level-Set Functions

  • O. Dorn et al.

    Level set methods for inverse scattering some recent developments

    Inverse Problems

    (2009)
  • H. Ben Ameur et al.

    Refinement and coarsening indicators for adaptive parameterization: application to the estimation of hydraulic transmissivities

    Inverse Problems

    (2002)
  • F.T.-C. Tsai et al.

    Global-local optimization for parameter structure identification in three-dimensional ground water modeling

    Water Resour. Res.

    (2003)
  • A.-A. Grimstad et al.

    Adaptive multiscale permeability estimation

    Comput. Geosci.

    (2003)
  • P.E. Wannamaker et al.

    Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations

    Geophysics

    (1984)
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