Schemes for improving efficiency of pixel-based inversion algorithms for electromagnetic loggingwhile-drilling measurements
Y. Lin, A. Abubakar, T. M. Habashy, G. Pan, M. Li and V. Druskin, Schlumberger-Doll Research, USA, and L. Knizhnerman, Central Geophysical Expedition, Russia
SUMMARY
We present an application of the two-and-half dimensional (2.5D) multiplicative-regularized Gauss-Newton inversion method for the interpretation of the directional resistivity
measurements [electromagnetic logging-while-drilling (LWD)
measurements]. In this work, an inversion algorithm is employed for obtaining a two-dimensional (2D) resistivity distribution (pixel-based) of the subsurface. Several modifications on both forward and inversion algorithms have been
implemented to minimize the computational time and memory usage as well as to improve the accuracy of the algorithms. Numerical examples demonstrate the feasibility of using this 2.5D pixel-based inversion algorithm for electromagnetic LWD data.
algorithm, such as separating the forward and inverse grid, the
implementation of a near-offset correction scheme, and the use
of a sliding window scheme. All these modifications are discussed in this paper, and we show some test examples. From
those numerical results, we observe that the 2.5D pixel-based
inversion algorithm has a potential for improving the resistivity LWD data interpretation.
FORMULATION
Inversion algorithm
The inversion algorithm is based on the regularized GaussNewton method. A multiplicative regularization is used to construct the cost function:
Φn (m) = ϕ d (m) × ϕnm (m) ,
INTRODUCTION
LWD technology allows us to acquire a variety of measurements during the drilling of a well. These measurements can
help in optimizing the well positioning to minimize the drilling
cost and to characterize the reservoir. We concentrate on the
interpretation of the resistivity LWD data. Typical LWD resistivity tools are operated at multiple frequencies, with multiple
source and receiver arrays. Some of the receivers are tilted
with respect to the tool axis. The rotation of the tool provides
nearly triaxial (three orthogonal magnetic field components)
measurements. The source-receiver offset is restricted by the
size of the tool; the smallest offset is on the order of a few centimeters. On the other hand, the total investigation domain is
on the order of hundreds of meters. Hence, we are dealing with
a multiscale problem. This multiscale problem raises difficulties in applying modeling and inversion algorithms. Therefore,
most of the current two-dimensional (2D) or three-dimensional
(3D) pixel-based inversion algorithms such as Abubakar and
van den Berg (2000), Alumbaugh and Wilt (2001);Tartaras and
Zhdanov (2004);Abubakar et al. (2006, 2009), and Abubakar
and Habashy (2010) will not be efficient for inverting resistivity LWD data. There are also few works on parametric 2D
and 3D nonlinear inversion of resistivity LWD data. The twoand-half dimensional (2.5D) finite-difference and 3D finiteelement forward solvers have been applied to model the resistivity LWD data, and a ”trial-and-error” method is employed
for the interpretation, (see Omeragic et al. (2009)). A onedimensional (1D) parametric inversion algorithm for this application can be found in Omeragic et al. (2005).
In this work, we apply the 2.5D pixel-based inversion algorithm developed in Abubakar et al. (2008) to resistivity LWD
data for obtaining a curtain image (a 2D pixel-based image)
of the subsurface. To efficiently and accurately invert the resistivity LWD data, several improvements were made to the
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SEG Las Vegas 2012 Annual Meeting
(1)
where ϕ d is the data misfit cost function, measuring the difference between the measurement data d and the simulated data
s(m), where m is the vector of unknown parameters (pixelby-pixel conductivity). The simulated data are calculated by
solving the Maxwell’s equations for the 2D geometry and 3D
field (2.5D problem). The data misfit cost function is given by
ϕ d (m) =
1
∥Wd [d − s(m)]∥2 ,
2
(2)
where Wd is a data weighting matrix, which is a real-valued
diagonal matrix.
The function ϕnm (m) is the regularization cost function at the
n-th iteration. The multiplicative regularization poses a self adjusting regularization term. More details on the data weighting matrix and the regularization cost function can be found
in Abubakar et al. (2008) and Abubakar et al. (2009).
A Gauss-Newton minimization approach is employed to minimize the cost function in equation 1. At the n-th iteration, a
linear system of equations must be solved to obtain the step
vector pn ,
Hn pn = −gn ,
(3)
where Hn is the Hessian matrix, and can be approximated as
follows:
d
T
Hn ≈ JH
n Wd Wd Jn + ϕ (mn )L(mn ) ,
(4)
where the superscript H denotes a matrix conjugate transposition and the superscript T denotes a matrix transposition.
In equation 4, Jn is the Jacobian matrix and L is the second
derivative of ϕnm with respect to the unknown parameter m.
The gradient vector gn is given by
T
d
gn = −JH
n Wd Wd [d − s(mn )] + ϕn (mn ) L(mn )mn .
(5)
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After obtaining the step vector p, the unknown model parameter m is updated as follows:
mn+1 = mn + νn pn ,
(6)
where νn is the step length, which is obtained through a linesearch procedure described in Habashy and Abubakar (2004).
This procedure guarantees the decrease of the cost function
for each iteration. In the inversion process, the inverted model
parameters are enforced to lie within their physical bounds by
using a constrained minimization as described in Habashy and
Abubakar (2004). The iterative process stops when one of the
error criteria in Habashy and Abubakar (2004) is satisfied.
The Jacobian matrix is computed using the adjoint approach
(see Abubakar et al. (2008)) as follows:
∫
∂ hi, j,k
(rr R , r S ) =
eSi,k (rr , r S ) · eRj,k (rr , r R ) dV , (7)
∂ σp
Ωp
where hi, j,k is the simulated field at the j-th receiver, i-th source,
and k-th frequency, respectively. Ω p is the domain of the pth pixel whose conductivity is σ p ; eSi,k and eRj,k are the electric
fields excited by sources located and oriented as the i-th source
and j-th receiver respectively, both at the k-th frequency.
To invert the resistivity LWD data, the 2.5D pixel-based inversion algorithm in Abubakar et al. (2008) must be modified in
several aspects, first the calculation of Jacobian matrix need to
be more accurate especially for the near region, second the efficiency of the inversion need to be improved. Next we discuss
all improvements that were made to make the algorithm more
suitable for resistivity LWD data inversions.
Figure 1: Schematic of nonconforming forward and inversion
grids.
problem. By using these two forward responses, we calculate
the corrected fields h2D,corrected for each data point according
to the following formula:
1D,analytical
1D,FD
h2D,corrected
= h2D,FD
i, j,k
i, j,k − hi, j,k + hi, j,k
.
(8)
The additional cost for using the near-offset correction scheme
is only to solve a 1D layer model analytically and a 1D layer
model by using the 2.5D algorithm. Since the 1D model is
generated based on the initial model in the inversion, these extra calculations were only done once. We did not calculate 1D
model responses in each Gauss-Newton iteration. Although
this correction scheme is very simple and its computational
overhead is negligible, it provides a good improvement on the
inversion results.
Inversion using uniform inversion grids and nonuniform
forward-modeling grids
A near-offset correction scheme
The 2.5D finite-difference frequency domain (FDFD) forwardmodeling algorithm is not accurate for calculating electromagnetic fields near the source location because of the well-known
singularity problem. For a typical resistivity LWD tool, where
the tool spacing is from 0.4 to 2.5m, the electromagnetic fields
of some source-receiver pairs are in the near-field region. In
addition, the direct field is significantly larger for short-offset
fields. Hence, a small inaccuracy of the forward-modeling may
significantly affect the inversion results.
To improve the accuracy of the simulated field close to the
source, we implemented the so-called near-offset correction
scheme. The idea of the near-offset correction scheme is to
use an analytical solution of a one-dimensional (1D) layered
model to correct fields close to the source location (Our correction model is usually generated from initial model. In cases
we choose homogeneous initial model as initial model, the
1D layer models become 0D homogeneous model ). From a
given 2D medium, we can generate an approximate 1D layered model. Then, the 1D model can be solved analytically, to
obtain h1D,analytical . The same 1D model is also solved by using 2.5D forward-modeling algorithm, to obtain h1D,FD . These
two forward responses should have only large differences for
fields close to the source location. The difference between
these two forward responses represents the inaccuracy of the
2.5D forward-modeling algorithm because of the singularity
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From our numerical study, we observe that a very fine spatial
discretization is needed for the forward modeling to obtain accurate electromagnetic fields. This is because some of sourcereceiver offsets are small. If a uniform grid is used for the
whole computation domain, the finite-difference grids are too
fine for regions where we do not have source or receiver. A solution to this problem is to use nonuniform grids, which have
a high density near the source and receiver and a low density
for other regions. On the other hand, for inversion we prefer to
use uniform grids, as there is no a priori information about the
formation. Hence different grid sets must be employed for forward and inversion algorithms. In this way, the inversion grids
are uniform, while the forward grids are dense in regions close
to sources and receivers.This reduces the cost of the forward
simulation. Since the fields on the forward grids are needed
for calculating the Jacobian matrix according to equation 7, an
interpolation procedure is needed for mapping the fields from
the forward to inverse grids.
The schematic of nonconforming forward and inversion grids
is shown in Figure 1. The red box is an inversion grid cell. The
blue boxes are four forward grid cells, which have shared area
with the inversion cell. S1, S2, S3, and S4 correspond to the
shared area of the forward grids (F1, F2, F3, and F4) and the
inversion cell (I1). The calculation of derivative by the adjoint
method as in equation 7 needs to be done on the inverse grid
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Figure 2: A sliding window scheme.
I1 as follows:
∫
∂ hi, j,k
(rr R , r S ) =
eSi,k (rr I1 , r S ) · eRj,k (rr I1 , r R )dV
∂ σI1
ΩI1
NF ∫
∑
eSi,k (rr Fα , r S ) · eRj,k (rr Fα , r R )dV ,
=
α =1 ΩFα ∩ΩI1
(9)
where NF is the number of forward cells that share an inversion
cell.
After solving for the step vector p in equation 3 for the next
Gauss-Newton iteration, the material properties of the forward
grids need to be updated. When we updated the material property of the forward-modeling grid, we assumed the material
property to be homogeneous inside a forward cell Fα . So if
Fα cell coincides with NI inverse cells, we use the area of the
corresponding shared cells as the interpolation coefficients as
follows:
mFα =
NI
∑
Sαβ
β =1
Sα
mIβ ,
(10)
where Sαβ is area of forward cell α sheared with inversion cell
β , and Sα is the area of the forward cell α .
Sliding window scheme
The inversion domain usually has a size of several hundred
meters, while the typical skin depth of deep directional EM or
resistivity LWD measurements is from 0.1 meters to tens of
meters. On the other hand, the longest source-receiver offset
is only several meters. The spatial discretization in the FDFD
forward algorithm needs to be fine enough to get accurate simulated fields and to avoid singularity problems. A sufficient
discretization size is about 0.2 m for the tool we studied, and
therefore the number of unknowns will easily become more
than tens of thousands if we use the same forward and inverse
grid for a typical LWD inversion problem. Thus, we use a sliding window scheme to reduce the size of the inversion problem.
A sliding window scheme is used to divide the inversion domain into several smaller inversion domains, as shown in Figure 2. We use a sliding window scheme with an overlapping
area. Every subdomain has an overlapping area with previous and subsequent subdomains. We invert every subdomain
with the same homogeneous initial model. After obtaining inversion results for all subdomains, we construct the final conductivity image from the inversion results from those nonoverlapping regions. Each subdomain is chosen to be sufficiently
much smaller than the original whole problem, thus computational cost for solving a single subdomain is much smaller
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Figure 3: The resistivity distribution of the model based on the
Alaska North Slope field (scale on right is the logarithm of the
resistivity, in ohm-m).
than solving the original whole problem. Furthermore, since
the same homogeneous initial model is used for each subdomain inversion, there is no dependency on different subdomain
inversions, so it is very easy to simultaneously run different
subdomain inversions on parallel computing resources.
NUMERICAL EXAMPLES
As an example, we constructed a model based on the Alaska
North Slope field Omeragic et al. (2009). In that paper, the
formation is anisotropic, in this work, the model is limited to
isotropic. The size of this model is 200 m by 20 m. The model
consists of a fault and several dipping layers with resistivity
varying from 3 ohm-m to 50 ohm-m. The true resistivity distribution is shown in Figure 3. In this figure, the resistivity
is given in terms of the logarithm of the resistivity. In the inversion we use synthetic data corresponding to 30 log points
located inside the oil layer. For every log point, 13 channels
were used in the inversion at two frequencies: 100 kHz and
400 kHz. The green line in Figure 3 shows the trajectory of
the logging. Our objective is to identify the fault region and
the dipping layer boundaries using resistivity LWD measurements.The size of the finite-different grid is chosen to be 0.2
m; hence in total, we have 1000 x 100 grid cells. A homogeneous initial model with mean value of the true model is used
for all inversions.
First, we invert the whole domain. The inversion results from
the Gauss-Newton method after 30 iterations (all tests in this
paper reach the maximum prescribed number of iteration) are
shown in Figure 4. The inverted result shows that the algorithm
can accurately find the location of layer boundaries and can, to
some extent, find the location of fault. However, we observe
that there is a pinchout in the low-resistivity layer close to the
fault region.
Next, we invert the same data set using nonuniform forward
grids. The setup of the nonuniform grid is as follows: In the
z-direction from 5 m to 15 m, the forward grids are uniform
with a cell size of 0.2 m. For z < 5 m and z > 15 m the cell
size is 0.4 m. The cell size in the x-direction is 0.2 m. As
the inversion grids we use uniform grid with cell size of 0.2 x
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Figure 4: The inverted resistivity model using the GaussNewton method.
Figure 6: The inverted resistivity model using the GaussNewton method with the near-offset correction scheme.
Figure 5: The inverted resistivity model using the GaussNewton method with a nonuniform forward grid setup.
Figure 7: The inverted resistivity model using the GaussNewton method with the sliding window scheme and the nearoffset correction scheme.
0.2 m. The number of forward grids is reduced from 100,000
to 76,000, and the CPU time for the forward simulation using
the nonuniform grid is about 68% of the one using a uniform
grid. The inversion results using the nonuniform forward grid
setup after 30 iterations are shown in Figure 5. We note that
we obtain a similar result as in Figure 4 while we reduce the
computational time to some extent.
In the next test, we use the near-offset correction scheme. The
inversion results after 30 iterations are shown in Figure 6. The
result shows that we obtain a more accurate location of the
dip of the fault. Furthermore, there is no pinchout in the lowresistivity layer as in the inversion results without using the
near-offset correction scheme.
In this final test, we use the sliding window scheme in the inversion. Every inversion subdomain is extended by 10 m on
each horizontal side except for the first and last subdomain.
After all inversions of all subdomains are obtained, we discard
the results for extended parts, and we combine all the inversion
results to obtain the final result, which is shown in Figure 7.
By using the extended inversion subdomain, we can reduce
the discontinuity problem, which may exist in a nonoverlapping sliding window scheme. The one-domain Gauss-Newton
inversion cost 440,479 s for 30 iterations. The largest CPU
time for all extended inversion subdomains in this example is
41,527 s for 30 iterations. However, if we use a nonoverlapping sliding window, the largest CPU time for all extended
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SEG Las Vegas 2012 Annual Meeting
inversion subdomains in this example is further reduced to
18,290 s. The inversions are performed on a cluster with Xeon
R
⃝E5410
2.33 GHz processors and 8 cores are used for all calculations. By using this sliding window scheme, we can solve
much larger problems while using less CPU time.
CONCLUSION
We discussed the application of 2.5D pixel-based inversion approach for resistivity LWD data interpretation. We employed
several schemes, such as the near-offset correction, an independent forward and inversion grid scheme and the sliding
window scheme, to improve the accuracy and efficiency of
the approach, especially for resistivity LWD applications. We
show that this pixel-based inversion algorithm has a promising
potential for improving resistivity LWD data interpretation.
ACKNOWLEDGMENTS
The authors would like to thank Dr. Jianguo Liu for his contribution on the development of the 2.5D pixel-based inversion
code.
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EDITED REFERENCES
Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012
SEG Technical Program Expanded Abstracts have been copy edited so t hat references provided with the online metadata for
each paper will achieve a high degree of linking to cited sources that appear on the Web.
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SEG Las Vegas 2012 Annual Meeting
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