Abstract
A sequential inversion methodology for combining geophysical data types of different resolutions is developed and applied to monitoring of large-scale CO2 injection. The methodology is a two-step approach within the Bayesian framework where lower resolution data are inverted first, and subsequently used in the generation of the prior model for inversion of the higher resolution data. For the application of CO2 monitoring, the first step is done with either controlled source electromagnetic (CSEM) or gravimetric data, while the second step is done with seismic amplitude-versus-offset (AVO) data. The Bayesian inverse problems are solved by sampling the posterior probability distributions using either the ensemble Kalman filter or ensemble smoother with multiple data assimilation. A model-based parameterization is used to represent the unknown geophysical parameters: electric conductivity, density, and seismic velocity. The parameterization is well suited for identification of CO2 plume location and variation of geophysical parameters within the regions corresponding to inside and outside of the plume. The inversion methodology is applied to a synthetic monitoring test case where geophysical data are made from fluid-flow simulation of large-scale CO2 sequestration in the Skade formation. The numerical experiments show that seismic AVO inversion results are improved with the sequential inversion methodology using prior information from either CSEM or gravimetric inversion.
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Open Access funding provided by NORCE Norwegian Research Centre AS.
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The authors are grateful for the financial support from Research council of Norway (RCN), Octio, DEA, DONG, ConocoPhilips, Store Norske Spitsbergen Kulkompani, and Statoil through the SUCCESS project (grant 193825/S60). The first and second authors are also grateful for the financial support from RCN, DEA, and Total through the PROTECT project (grant 233736).
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Appendices
Appendix 1. Reduced, smoothed level-set representation
Recalling the notation introduced in Section 3.1, let \(\left \{\phi _{i}\right \}_{i=1}^{N_{\phi }}\) denote a set of real-valued, continuous functions on Ω—the level-set (LS) functions. Utilizing this set to construct \(\left \{ {\Omega }_{j} \right \}_{j=1}^{N_{c}}\) in a particular manner will render (6) a LS representation. With Nc> 2, alternative LS representations (LSR)s exist [18, 56, 59, 80] which are able to represent between Nϕ + 1 and \(2^{N_{\phi }}\) subregions using Nϕ LS functions. For detailed expositions of the LSRs proposed by [80] and [59] in the context of modelling of geophysical exploration problems, we refer to [76] and [75], respectively. We will, however, only require the case where Nc = 2, in which case the LSR is unique and only a single LS function, ϕ, is applied.
To arrive at the LS representation from (6) with Nc = 2 inserted, we first replace the explicit dependence of χ1 and χ2 on x and a by an implicit dependence through the LS function,
Next, we select Ω1 as the part of Ω where \(\phi \left (\mathbf {x}; \mathbf {a} \right ) > 0\). Since Ω2 = Ω ∖Ω1, we obtain the LSR in standard notation,
where H denotes the Heaviside function (indicator function for the positive real axis). There are few restrictions on ϕ. Hence, the LSR is a very flexible way to represent subregions in Ω, as illustrated in Fig. 17. The shapes of Ω1 and Ω2 are governed by the LS function, whose spatial variation is controlled by the parameters in a.
The LSR has been extended [18] to incorporate arbitrary spatial variation within each zone by replacing (28) with
where \(\mathbf {c}_1 \in {\mathbb R}^{N_{c_1}}\), \(\mathbf {c}_2 \in {\mathbb R}^{N_{c_2}}\), and \(N_{c_1} + N_{c_2} = N_{c}\). Both (28) and (29) will be applied in numerical examples, where relevant quantities, such as \(N_{c_1}\) and \(N_{c_2}\), will be specified. To complete the general description of the LSR, the dependency of ϕ on x and a must be specified. When applying (29), also the dependencies of k1 on x and c1 and k2 on x and c2 must be specified. We will apply the same type of representation for the LS function, ϕ, as for the coefficient functions, k1 and k2.
1.1 A.1 Reduced parameterization of level-set and coefficient functions
Let ψ represent either of the functions ϕ, k1, or k2, and correspondingly, let b represent either a, c1, or c2. We express the dependency of ψ on x and b by [6]
The basis functions \(\left \{\xi _{k}\right \}_{k=1}^{N_{b}}\) are defined on a rectangular parameter grid that is not attached to, and much coarser than, the forward model grid (Fig. 18a). Hence, Nb ≪Ng, and our parameterization is therefore significantly reduced with respect to a pixel parameterization. There will, however, still be sufficient flexibility to approximately represent the large-scale structures that we aim to estimate.
While alternative representations are viable, we represent ψ in a finite-element fashion [6], and let ξu be a normalized piecewise bilinear function with support on the four rectangular elements adjacent to node u (arbitrary) (Fig. 18b). Its value is unity in node u and zero in all other nodes. Figure 18c shows node u and three of its adjacent nodes, v, r, and s, and the supports of the basis functions associated with these four nodes. Figure 18d shows the element where ξu, ξv, ξr, and ξs have common support. The projections of ξu, ξv, ξr, and ξs onto this element are normalized bilinear functions, so whenever ψ is to be evaluated at a forward model grid point, its value is calculated using bilinear interpolation.
1.2 A.2 Smoothed level-set representation
We replace H in the LSR by a smoothed approximation,
resulting in \(q \left (\mathbf {x}; \mathbf {m} \right )\) no longer being a zonation since \(\widetilde {H}\) will have global support in Ω. Introducing smoothness in q can be beneficial since the nonlinearity in the mapping \(\mathbf {a} \rightarrow q\) will decease with increasing smoothness [53]. This consideration should, however, be balanced by the desire to keep a relatively sharp transition between subregions where \(q \left (\mathbf {x}; \mathbf {m} \right ) \approx c_{1}\) (\(q \left (\mathbf {x}; \mathbf {m} \right ) \approx k_{1} \left (\mathbf {x}; \mathbf {c} \right )\) if (29) is applied) and subregions where \(q \left (\mathbf {x}; \mathbf {m} \right ) \approx c_{2}\) (\(q \left (\mathbf {x}; \mathbf {m} \right ) \approx k_{2} \left (\mathbf {x}; \mathbf {c} \right )\) if (29) is applied). The width of the transition region is decided by the behaviour of ϕ in the vicinity of its zero-level set, ζ. Let n be a unit normal vector to ζ. A sharp transition in q over ζ then corresponds to large values of |∇ϕ ⋅n|. Figure 19 illustrates the difference between a LSR and a smoothed approximation to a LSR when (28) is applied.
Appendix 2. Initial ensemble generation
The ensemble-based inversion methodologies described in Section 3.2 require generation of an initial ensemble. The initial ensemble is generated from the prior PDF, p(m0), which is chosen to be Gaussian,
Standard Cholesky decomposition method can thus be used to generate realizations from p(m0),
where \(\mathbf {z}\sim \mathcal {N}(0, 1)\) and \(\mathbf {L}\mathbf {L}^{T}=\mathbf {C}_{m^0}\), with L being a lower triangular matrix. Based on knowledge of the CO2 plume, e.g. from previous time-lapse vintage data, suitable values for \(\bar {\mathbf {m}}^0=((\bar {\mathbf {c}}^0)^{T},(\bar {\mathbf {a}}^0)^{T})^{T}\) can be generated. To generate \(\mathbf {C}_{m^0}\), it is assumed that a and c are not correlated, and, moreover, it is assumed that c1 is not correlated with c2. Hence,
where \(\mathbf {C}_{c^0_i}\) and \(\mathbf {C}_{a^0}\) denote covariance matrices for ci, i = 1, 2, and a, respectively. Note that if (28) is applied, the covariance matrix \(\mathbf {C}_{c^0_i}\) reduces to a scalar variance, βi.
To generate \(\mathbf {C}_{c^0_i}\) and \(\mathbf {C}_{a^0}\), a spherical covariance function [14],
is applied. Here, h denotes spatial distance between two nodes in the parameter grid (confer Section A.1), and α denotes the correlation length. The covariance matrix can thus be generated as
where the subscript ‘ * ’ denotes either a0 or \({c_{i}^{0}}\) which leads to ‘ ‡ ’ being either Na or \(N_{c_i}\), respectively.
The covariance matrices \(\mathbf {C}_{c^0_1}\), \(\mathbf {C}_{c^0_2}\)m and \(\mathbf {C}_{a^0}\) can be non-diagonal, to allow for anisotropic correlations. The anisotropy will be specified trough the angle, γ, from the z-axis to the principal axis corresponding to the largest eigenvalue, and the anisotropy ratio, δ. Numerical values for α, β, γ, and δ will be given in Section 4.2.
For an in-depth description of the EnKF applied to a geophysical method (CSEM) and generation of the initial ensemble with the reduced, model-based representation, with examples, see [75].
Appendix 3. Sample mean and covariance matrix
Let \(\mathbf {Y} = \left (\mathbf {y}_{1}, \mathbf {y}_{2}, \ldots , \mathbf {y}_{N_e} \right )\) denote an arbitrary ensemble matrix, and let u denote an Ne vector where all entries equal unity. The sample (empirical) mean may then be written as
Furthermore, let \(\MakeUppercase {\mathbf {u}} = \left (\mathbf {u}, \mathbf {u}, \ldots , \mathbf {u} \right )\) (i.e. with Ne columns), and define the sample mean matrix as \(\widetilde {\mathbf {Y}} = \frac {1}{N_e} \mathbf {Y} \MakeUppercase {\mathbf {u}}\). The sample cross-covariance matrix between two arbitrary random vectors, y and z, is then given as
The sample auto covariance matrix, \(\widetilde {\mathbf {C}}_{y}\), is given by (38) with Z = Y.
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Tveit, S., Mannseth, T., Park, J. et al. Combining CSEM or gravity inversion with seismic AVO inversion, with application to monitoring of large-scale CO2 injection. Comput Geosci 24, 1201–1220 (2020). https://doi.org/10.1007/s10596-020-09934-9
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DOI: https://doi.org/10.1007/s10596-020-09934-9