Halliburton Energy Services' Dual Laterolog Logging. Tool (DLLT-BTM) to reproduce measurements acquired in Middle-East carbonate reservoirs. Numerical.
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
INTERPRETATION OF FREQUENCY-DEPENDENT DUALLATEROLOG MEASUREMENTS ACQUIRED IN MIDDLE-EAST CARBONATE RESERVOIRS USING A SECOND-ORDER FINITEELEMENT METHOD Wei Yang, and Carlos Torres-Verdín, The University of Texas at Austin; Ridvan Akkurt, Saleh Al-Dossari, and Abdullah Al-Towijri, Saudi Aramco; Haluk Ersoz, Halliburton Copyright 2007, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors.
INTRODUCTION
This paper was prepared for presentation at the SPWLA 48th Annual Logging Symposium held in Austin, Texas, United States, June 3-6, 2007.
Laterolog measurements use a galvanic conduction principle to excite electrical conduction in rock formations penetrated by a borehole. While essentially DC in nature (Lacour-Gayet, 1981), a strictly zerofrequency laterolog measurement is impractical due to contact-impedance noise at electrode locations. Laterolog measurements are commonly acquired in the frequency range from 10 Hz to 2 KHz (Anderson, 2001). The application of non-zero values of frequency often complicates the interpretation of laterolog measurements in the presence of large contrasts of electrical resistivity, including the conspicuous examples of the so-called Delaware and Groningen effects (Anderson, 2001, Lacour-Gayet, 1981, Lovell, 1993, Trouiller et al., 1978, Woodehouse, 1978).
ABSTRACT Laterolog tools operate at low frequencies because of prevalent contact-impedance noise at electrode locations. However, most existing laterolog modeling codes are based on zero-frequency (DC) electricalpotential formulations. In this paper, we develop a new second-order finite-element algorithm to simulate the frequency-dependent laterolog response of axiallysymmetric, invaded and anisotropic formations. When compared to first-order (linear) DC finite-element solutions, the new algorithm provides enhanced accuracy due to the implementation of second-order shape functions. In addition, numerical results indicate that the new algorithm can accurately simulate cases of extreme contrast in electrical resistivity such as those arising in the presence of steel casing, air, or anhydrite layers.
Most of the available laterolog modeling codes are based on voltage potential formulations (Li et al., 1995, Yang et al., 1997, Zhang, 1986). Such simulation methods are strictly accurate only at DC (Lovell, 1993). Lovell (1993) and Zhang (1986) proposed a simulation method based on the solution of the partial differential equation of the current potential. The latter method can simulate finite-frequency measurements and enables the efficient calculation of electric current lines. Chen et al. (1998) utilized the same method for the case of DC simulations and described the corresponding spatial distribution of electric current lines. Lovell (1993) applied a similar simulation method for the non-zero frequency (AC) case but did not document simulation results for the case of dual laterolog measurements.
To benchmark the reliability, accuracy, and applicability of the new simulation algorithm, we consider the specific electrode configuration of Halliburton Energy Services’ Dual Laterolog Logging Tool (DLLT-BTM) to reproduce measurements acquired in Middle-East carbonate reservoirs. Numerical simulations incorporate the tool electrode and insulator dimensions as well as the operating modes of the deep(LLD) and shallow-sensing (LLS) measurements at their respective frequencies. Our simulations indicate that non-DC measurements are affected by the presence of steel casing. We quantify the influence of anhydrite layers of varying thickness located immediately below the casing shoe on measurements acquired across porous and permeable carbonate reservoirs. Simulations show that laterolog apparent resistivities acquired across low-resistivity carbonate reservoirs shouldered by anhydrite beds could exhibit a slight bias and also give a false indication of invasion. In such complex environments, reliable assessment of hydrocarbon saturation can only be accomplished with accurate simulations of laterolog measurements.
The linear (i.e., first order) finite-element method (FEM) is typically used to simulate laterolog measurements (Li et al., 1995, Lovell, 1993, Yang et al., 1997, Zhang, 1986). Experience shows that the accuracy of the first-order FEM is acceptable in the presence of low to moderate contrasts of electrical resistivity. However, in extreme contrast situations, such as those that involve steel casing (resistivity approximately equal to 2e-7 Ω-m), air, halite, and anhydrite layers (all of them exhibiting practically infinite electrical resistivity) the accuracy of the first1
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
electrical conductivities, respectively. Equation (2) can be written as
order FEM is not adequate to reproduce the measurements. In this paper, we develop a second-order FEM method based on the solution of the current potential to simulate frequency-dependent dual laterolog measurements in invaded and anisotropic formations. First, we introduce the mathematical formulation, the associated boundary conditions, and the second-order FEM variational formulation. We consider the specific electrode configuration of Halliburton Energy Services’ Dual Laterolog Logging Tool (DLLT-BTM) to perform the numerical simulations. Subsequently, we discuss several benchmarking examples and draw conclusions about the accuracy and reliability of our simulation method. Additional simulation results are discussed based on laterolog measurements acquired in MiddleEast carbonate formations that include hydrocarbonbearing carbonate formations invaded with water-base mud and shouldered by anhydrite beds. The objective of the latter studies is to assess whether shallow- and deep-sensing laterolog measurements across porous and permeable carbonate layers remain affected by the concomitant presence of casing and anhydrite beds.
−
With the definition J = 2πρ Hφ , we rewrite Eq. (4) as −
∂ ⎛ 1 ∂J ⎞ ∂ ⎛ 1 ∂J ⎞ iωμ J = 2π M φ . ⎟+ ⎜ ⎟− ⎜ ∂ρ ⎝ ρσ z ∂ρ ⎠ ∂z ⎜⎝ ρσ ρ ∂z ⎟⎠ ρ
(5)
Physically, the connecting lines of J are exactly the electric current lines excited in the formation by the impressed galvanic source. In fact, the excitation term 2π M φ can be imposed with the boundary conditions shown in Figure 1. Finally, the energy functional used in our finite-element (FE) simulation algorithm is given by F (J ) =
1 2π
⎡ 1 ⎛ ∂J ⎞ 2 1 ⎛ ∂J ⎞ 2 ⎤ , (6) 2 1 ∫Ω ⎢⎢ σ z ⎜⎝ ∂ρ ⎟⎠ + σ ρ ⎜⎝ ∂z ⎟⎠ + iωμ J ⎥⎥ ρ d ρ dz ⎣ ⎦
where Ω is the spatial domain of the simulations.
FORMULATION
BOUNDARY CONDITIONS AND TOOL DESCRIPTION
Assuming a time harmonic excitation of the form e−iωt , Maxwell’s equations are given by (Lovell, 1993)
∇ × E = iωμ H , ∇ × H = σ E + J ,
Figure 1 describes the computational domains ΩLLD and ΩLLS for the deep-and shallow-sensing laterolog modes, respectively. The domain termination boundary is denoted as m1. Figure 2 shows the configuration of the laterolog tool adopted in this paper for the deep- and shallow-sensing modes. Such a configuration corresponds to Halliburton Energy Services’ Dual Laterolog Logging Tool (DLLT-BTM), which operates at 131.25 Hz for the deep-sensing mode and at 1050 Hz for the shallow-sensing mode. In Figure 1, the letter B identifies the current return electrode, placed 33 m away from the A2 electrode. Electrodes are denoted as mi , i = 1,...,11 starting from the electrode placed at infinity and proceeding counterclockwise. The insulator adjacent to the electrode mi is denoted as mi′ . In the
(1)
where E and H are the electric and magnetic field vectors, respectively, μ is the magnetic permeability, σ is the complex-valued anisotropic conductivity tensor, J is the impressed current density, σ E is the induced current density, i = − 1 , ω is radian frequency, and t is time. For the case of transverse-magnetic (TM) excitation in cylindrical coordinates (ρ,φ,z), the only non-zero components of the electric and magnetic fields are H φ , Eρ and Ez . Thus, from Eq. (1) it follows that (Jin et al., 1999)
φˆ ⋅∇× (σ −1∇ × Hφφˆ) + iωμ Hφ = M φ ,
(2)
voltage potential method, homogeneous Neumann boundary conditions are enforced on insulators (these conditions are automatically satisfied in finite-element formulations). We enforce Dirichlet and equipotential surface boundary conductions at electrode locations, with the number of boundary conditions equal to the number of electrodes. Because of the duality of the voltage potential method, the current potential method requires the enforcement of boundary conditions on the insulators, with the number of enforced boundary conditions equal to the number of insulators. These
where φˆ is the unit vector in the azimuthal direction, and M φ is the magnetic current density in the azimuthal direction, defined as ∂ ⎛J M φ = φˆ ⋅∇ × (σ −1 ⋅ J ) = ⎜ ρ ∂z ⎝ σ z
∂ ⎛ 1 ∂ ( ρ H φ ) ⎞ ∂ ⎛ 1 ∂H φ ⎞ ⎟ + iωμ H φ = M φ . (4) ⎜ ⎟− ⎜ ∂ρ ⎝ ρσ z ∂ρ ⎠ ∂z ⎜⎝ σ ρ ∂z ⎟⎠
⎞ ∂ ⎛ Jz ⎜⎜ ⎟− ⎠ ∂ρ ⎝ σ ρ
⎞, ⎟⎟ ⎠
(3)
where σ ρ and σ z are the horizontal and vertical 2
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
boundary conditions are given by (Chen et al., 1998): 1.
Deep Laterolog J m' = 0
2.
Vm' = ?
Vm' = ?
3.
Vm' = 0
Vm' = −Vm'
4.
Vm' = −Vm' = −1 + Vm'
1
1
2
2
3
9
4
9
5.
Vm' = 0
Vm' = 0
6.
Vm' = ?
Vm' = ?
5
10
7.
Vm' = −Vm'
Vm' = −Vm'
Vm' = 0
Vm' = 0
9. 10. 11.
10
11
7
6
7
8
9
6
Deep Laterolog J m ' − J m ' + J m' − J m ' = 0
8
1.
Vm' = 1 − Vm' − Vm'
Vm' = 1 − Vm'
11
9
Vm' = 0 10
10
9
11
11
8
5
Shallow Laterolog J m ' − J m ' + J m ' − J m' = 0
4
9
J m' − J m ' + J m ' − J m ' = 0
J m' − J m' + J m' − J m' = 0
Vm' = ?
J m' = 0
J m' = 0
8
7
6
5
8
11
i
J m' = 0 2
the total current on M 1 and M 1′ be 0, the third equation ensures that the current flow at infinity be 0, and the fourth equation ensures that the current at electrode B be 0 for the case of the shallow-sensing laterolog mode. In similar fashion to the field superposition technique used in the voltage potential method, one can solve for the current potential using the principle of superposition. In so doing, one divides the original problem into four partial problems. Let J (1) ( ρ , z ) be the solution to the energy functional (6) with Vm' = −1, Vm' = 1, else = 0 ; J (2) ( ρ , z ) the solution to the
2
parameter which needs to be determined by the enforcement of constraint conditions. Condition 3 ensues because the electrode A2 is always connected to A2′ . There is an unknown parameter in this equation for the case of the shallow-sensing mode. However, for the case of the deep-sensing mode, the voltage difference between A2 and A1 is 0. Condition 4 ensues
4
9
energy functional (6) with Vm' = 1, Vm' = −1, else = 0 ; 6
because the electrode A1 is always connected to A1′ . In Condition 5, the voltage difference between the electrodes M 2 and M 1 is 0. Condition 6 defines Vm'
7
J ( ρ , z ) the solution to the energy functional (6) with (3)
Vm' = 1, else = 0 ; J (4) ( ρ , z ) the solution to the energy 2
functional (6) with Vm' = 1, Vm' = −1, Vm' = 1, else = 0 ;
6
as an unknown parameter. Condition 7 ensues because the electrode M 1 is always connected to M 1′ .
11
and J with
Condition 8 enforces that the voltage difference between M 1′ and M 2′ be 0. Condition 9 is based on the fact that the voltage difference between M 2′ and infinity is 1; i.e., Vm' + Vm' + Vm' = 1 . Condition 10 10
5
where the first condition ensures that the total current on M 2 and M 2′ be 0, the second conditions ensures that
above conditions are often referred to as equipotential surface conditions which have to be enforced by a series of superposition operations on the finite-element stiffness matrix (Li et al., 1995, Zhang, 1986). Conditions on the current potential are of the Dirichlet type and can be enforced directly. In Condition 1, the current potential J at the top insulator m1' is defined as zero. Condition 2 designates Vm' as an unknown
10
6
11
4.
where Vm' is the voltage decay on insulator m . The
7
, (8)
4
2.
' i
9
5
3.
11
identifies Vm'
8
Vm' = ? 10
Vm' = ?
11
constraint conditions for the deep-sensing laterolog mode and four conditions for the shallow-sensing laterolog mode. These conditions are expressed as
6
8.
6
Vm' and Vm' . Therefore, one needs to include three
5
6
11
2
10
Vm' = −Vm' = −1 + Vm' + Vm'
11
6
sensing laterolog mode, the unknowns are Vm' , Vm' ,
(7)
2
3
4
In the above boundary conditions, there are three unknowns for the case of the deep-sensing laterolog mode: Vm' , Vm' and Vm' . For the case of the shallow-
Shallow Laterolog J m' = 0
(5)
9
4
( ρ , z ) the solution to the energy functional (6) Vm' = 1, Vm' = 1, Vm' = −1, Vm' = −1, else = 0 . 10
4
9
3
Accordingly, the solution for the deep- and shallowsensing dual laterolog currents must be a linear combination of J (i ) ( ρ , z ), i = 1,...5 , respectively given by
11
as the zero voltage difference between
J Deep = J (1) + α J (2) + β J (3) + γ J (4)
electrodes A1′ and A2′ (deep-sensing mode) or else as an unknown parameter (shallow-sensing mode). Condition 11 designates Vm' as the unknown voltage
J Shallow = J (1) + α J (2) + β J (3) + γ J (4) + η J (5)
,
(9)
where the constants α , β , γ ,η are determined from the solution of Eq. (8). Moreover, because the voltage can be specified for M2, it follows that
11
difference between A2′ and infinity. 3
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
Vm' + Vm' + Vm' = 1 . 9
10
two solutions agree very well in the presence of low to moderate contrasts of electrical resistivity. However, significant differences between the two methods are observed in the presence of steel casing. The following section describes benchmarking exercises undertaken to appraise the accuracy of the second-order FEM developed in this paper. We also describe examples intended to study the sensitivity of the simulated laterolog apparent resistivities to several extreme conditions of measurement acquisition.
(10)
11
The corresponding apparent resistivities are given by RDeep = kdeep
(J
1 (R)
m7'
(R)
− J m'
6
)
, RShallow = k shallow deep
(J
1 (R)
m7'
(R)
− J m'
6
)
, (11) shallow
where kdeep , k shallow are the tool constants for deep- and shallow-sensing laterolog modes, respectively, at 0 Hz. These constants can be determined from simulations in a homogenous medium where the measured apparent resistivities RDeep , RShallow are equal to the actual resistivity
of
the
(R)
(R)
7
6
probed
medium.
NUMERICAL RESULTS Code Validation. We first verified the accuracy and reliability of the new second-order simulation method against (a) analytical solutions of point sources in a homogeneous whole space (Zhang, 1986), (b) DC numerical mode-matching solutions (Liu et al., 1994) in layered and invaded formations including a borehole, and (c) existing DC first-order laterolog algorithms based on the solution of the electric potential (Li et al., 1995, Zhang, 1986) for layered and invaded formations that included a borehole. All the simulations were performed on a Dell Dimension 8400 personal computer. For the sake of conciseness, we omit graphical results obtained from the above comparisons. We note, however, that all the comparisons conclusively indicated that the accuracy of the new second-order finite-element algorithm was better than 1% even in cases of large contrasts of electrical resistivity. In what follows, we focus our attention to specific comparisons performed between first- and second-order finite-element solutions. All the simulations are performed specifically for the laterolog configuration described in Figure 2.
The
variables J m' , J m' identify the real part of the current flowing on m7' and m6' , respectively. (R)
(R)
7
6
Therefore,
J m' − J m' identifies the real part of the current at the
electrode A0. A similar approach can be used to calculate out-of-phase apparent resistivities. SECOND-ORDER FE SOLUTION
To simulate the response of dual laterolog measurements in the presence of large contrasts of electrical conductivity, we make use of second-order shape functions in the finite-element formulation. The shape function, shown schematically in Figure 3, is an 8-node quadrilateral element, with the corresponding two-dimensional interpolation function given by u = α1 + α 2 x + α3 y + α 4 xy + α 5 x 2 + α 6 x 2 + α 7 x 2 y + α8 xy 2 ,
(12)
Figure 4 shows results from the first comparison example at DC. The formation model consists of three layers with a borehole. Mud resistivity is 1 Ω-m and borehole diameter is 0.2m One of the layers is invaded. Layer resistivities are 1, 50, and 1 Ω-m, with the resistivity of the invasion zone in the invaded layer equal to 10 Ω-m (thickness equal to 2 m and radial length of invasion equal to 0.2 m). Solid blue (solid triangles) and dashed blue (open triangles) lines identify deep laterolog measurements simulated with the first- and second-order finite-element methods, respectively. Solid red (solid circles) and dashed red (open circles) lines identify shallow laterolog measurements simulated with the first- and secondorder finite-element methods, respectively. The dotted purple line describes the actual value of model resistivity, Rt. We observe that the two sets of simulations agree very well with each other.
where x and y are the free variables and the subscripted α values designate arbitrary real-valued constants. Nodal values of shape function are then given by N i = 1/ 4(1 + ξiξ )(1 + ηiη )(ξiξ + ηiη − 1) N i = 1/ 2(1 − ξ )(1 + ηiη )
i = 1, 2,3, 4
(13)
i = 5, 7
2
N i = 1/ 2(1 − η )(1 + ξiξ )
i = 6,8
2
[ξi ] = [ −1,1,1, −1, 0,1, 0, −1] , [ηi ] = [ −1, −1,1,1, −1, 0,1, 0] T
T
Figure 4 compares simulation results obtained with first- and second-order finite-element methods for the case of a single-layer invaded isotropic formation. Figure 5 shows the second model used to compare the two solution methods. It consists of a single-interface formation and includes the presence of a borehole and vertically truncated steel casing. Electrical resistivity contrasts considered in this model are extremely large. Simulation results, shown in Figure 6, indicate that the 4
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
Figure 5 shows the model constructed to perform the second set of numerical comparisons. It consists of a single layer interface and includes both a borehole and steel casing. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. Layer resistivities are 1 and 10000 Ω-m, with their interface located at the relative vertical location of -1 m. Casing thickness, electrical resistivity, and relative magnetic permeability are equal to 0.01 m, 2e-7 Ω-m, and 1, respectively. The lower termination boundary of casing is located at the relative vertical location of 1 m and the spatial domain of the simulation is terminated at the relative vertical location of 1000 m. Figure 6 describes the simulation results obtained with the first- and second-order finite-element algorithms at DC. Simulations agree very well with each other along depth segments with low contrasts of electrical resistivity. However, we note a significant difference between the two simulation results along the depth range occupied by steel casing. Further testing with a numerical mode-matching code indicated that the relative error of the second-order finite-element solution was below 1%, thereby confirming the reliability of the new simulation algorithm.
shallow laterolog measurements, respectively, simulated for the isotropic case. Dashed blue (open triangles) and red (open circles) lines identify deep- and shallow-sensing laterolog measurements, respectively, simulated for the anisotropic case. The dotted purple and dashed cyan lines identify the actual horizontal (Rh) and vertical formation resistivities (Rv), respectively. Simulations confirm that neither the shallow-sensing nor the deep-sensing laterolog measurements possess significant sensitivity to the presence of electrical anisotropy (this results is consistent with the so-called anisotropy paradox of laterolog measurements). However, we observe that shallow-sensing laterolog measurements exhibit a marginal sensitivity to electrical anisotropy. Effect of Non-Zero Probing Frequencies. Figure 10 compares simulated in-phase laterolog measurements for both deep- and shallow-sensing modes at two operating frequencies. The formation model includes a borehole and two layers. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. Layer resistivities are 1 and 10000 Ω-m. Solid blue (solid triangles) and dashed blue (open triangles) lines identify deep-sensing measurements simulated at 0 Hz and 131.25 Hz, respectively. Solid red (solid circles) and dashed red (open circles) lines identify shallow-sensing measurements simulated at 0 Hz and 1050 Hz, respectively. The dotted purple line identifies Rt. Simulations indicate that frequency has no appreciable effect on laterolog measurements. Figure 11 shows the simulated out-of-phase (quadrature) laterolog measurements for both deep- and shallow-sensing modes at different frequencies. The corresponding formation model is the same as that shown in Figure 10. Solid blue (solid circles) and dashed red (open circles) lines identify deep-sensing measurements simulated at 131.25 Hz and shallow-sensing measurements simulated at 1050 Hz, respectively. The dotted purple line identifies Rt. Simulations indicate that out-of-phase laterolog measurements exhibit a similar behavior to that of the in-phase measurements. However, we note that the simulated out-of-phase apparent resistivities require a different normalization constant to that of the in-phase measurements to properly reproduce the actual layer resistivities when the resistivity contrast is high.
Effect of Electrical Anisotropy. The objective of this simulation exercise is to shed light to the influence of resistivity anisotropy on the electrical current lines enforced within the formation probed by the deep- and shallow-sensing laterolog modes. We consider a formation with horizontal and vertical resistivities equal to 1 Ω-m and 2 Ω-m, respectively. Figures 7 and 8 show the electric current lines within the probed formation calculated for the deep- and shallow-sensing laterolog modes, respectively. Thick solid lines and thin dashed lines identify electric current lines for the cases of isotropic and anisotropic homogeneous formations, respectively. Electric current lines for the anisotropic formation have a slightly enhanced tendency to flow in the horizontal direction compared to those of the isotropic formation because the assumed vertical resistivity is larger than the vertical resistivity. In addition, Figure 8 indicates that electric current lines for the shallow-sensing laterolog mode are slightly more sensitive to the presence of electrical anisotropy than those of the deep-sensing laterolog mode.
Figure 9 compares simulated laterolog measurements for the cases of isotropic and anisotropic layered formations. The formation model consists of three layers (thickness of the center layer is 2 m) and includes a borehole. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. For the isotropic case, layer resistivities are 1, 10, and 1 Ω-m, whereas for the anisotropic case, vertical resistivities in the three layers are 10, 100, and 10 Ω-m, respectively. Solid blue (solid triangles) and red (solid circles) lines identify deep and
Multiple Layers in the Presence of Steel Casing. Figure 12 shows an eight-layer model that includes steel casing. The origin of coordinates is assumed located at the center of the Limestone C layer, with the casing shoe located at 600 ft (183 m). Mud resistivity is 0.061 Ω-m and borehole diameter is equal to 8.5 inches (0.2159 m). Casing resistivity, relative magnetic permeability, and thickness are equal to 1e-6 Ω-m, 1, and 0.025 m, respectively. Layer interfaces from top to
5
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
bottom are located at 200, 160, 140, 100, 50, -50, and 250 ft (or 61, 48.8, 42.7, 30.5, 15.25, -15.25, and -76.25 m), respectively. Formation resistivities from top to bottom are 1e4, 0.625, 1e4, 0.4, 1e4, 0.625, 1e4, and 1.11 Ω-m, respectively. This model was constructed based on typical Middle East carbonate reservoirs with alternating layers of anhydrite and porous limestone. The objective of the simulations is to ascertain whether laterolog apparent resistivities are indicative of actual resistivities across porous and permeable limestone layers because of the concomitant presence of anhydrite layers (high electrical resistivity) and steel casing (low electrical resistivity).
across porous and permeable limestone layers. Comparison of Figures 14 through 17 indicates that the simulated in-phase apparent resistivities remain sensitive to the presence of invasion regardless of the presence of both casing and shouldering anhydrite beds. In particular, the simulated shallow-sensing apparent resistivities exhibit an almost linear sensitivity to the corresponding perturbation of invaded-zone resistivity regardless of the presence of both casing and anhydrite shouldering beds. CONCLUSIONS
We have formulated, implemented, and successfully tested a new second-order finite-elemnt method to simulate axially-symmetric laterolog measurements based on the variational formulation of the frequencydependent current potential. Benchmarking exercises confirmed the reliability and accuracy of the simulation method in the presence of large contrasts of electrical resistivity, electrical anisotropy, and invasion. Specific simulations conducted for the case of Halliburton Energy Services’ Dual Laterolog Logging Tool (DLLTBTM) configuration indicate that the placement of the return current electrode, N, at infinity has a negligible effect on non-zero frequency measurements. We found that steel casing has a significant effect on shallow- and deep-sensing laterolog measurements. The effect of steel casing on both in-phase and out-phase laterolog measurements depends on the specific value of frequency. Moreover, simulations indicate that the effect of steel casing remains confined to the spatial neighborhood of casing (within 100 m) with marginal effect on apparent resistivities measured tens of feet away from the casing shoe, even in the extreme case of presence of highly resistive layers of anhydrite. Our simulations indicate measurable differences between true layer resistivities and laterolog apparent resistivities across porous and permeable limestone layers shouldered by anhydrite beds. Simulations also revealed separation between shallow- and deep-sensing laterolog measurements across porous and permeable limestone layers shouldered by anhydrite beds that could give a false indication of invasion. In cases of significant variations of layer resistivities, the work presented in this paper suggests that the petrophysical interpretation of laterolog apparent resistivities should be guided by numerical simulation.
Figures 13 shows the simulated laterolog measurements at DC for the formation model described in Figure 12. Figures 14 and 15 show the corresponding simulated inphase and out-of-phase apparent resistivities, respectively. Simulations indicate that measurements across the various layers are, in general, consistent with true formation resistivities. However, we note that apparent resistivities across low-resistivity layers (corresponding to porous and permeable limestone formations) are slightly different from the corresponding layer resistivities. The difference between actual layer resistivities and in-phase apparent resistivities across limestone formations depends on both frequency and layer thickness. Moreover, we observe that the simulated deep- and shallow-sensing laterolog apparent resistivities do not overlap across low-resistivity limestone layers, thereby opening the possibility of erroneous interpretations about the presence and radial extent of mud-filtrate invasion. As expected, frequency has a marked effect on the simulated apparent resistivities across steel casing. The simulated out-of-phase apparent resistivities are substantially different from actual layer resistivities, possibly due to the fact that the normalizing geometrical constant should be corrected for frequency. We also observe anomalous “horns” in the simulated out-of-phase apparent resistivities across the upper interfaces of low-resistivity limestone layers. Figures 16 through 18 show the results of an additional sensitivity study performed to assess the combined influence of steel casing and shouldering anhydrite beds on the diagnosis and quantification of invasion. For this study, invasion was included only in the porous and permeable limestone layers with a single piston-like invasion front of radial length equal to 1 ft and with invaded-zone resistivity (Rxo) equal to 0.5 times the corresponding value of true (uninvaded) formation resistivity. Comparison of Figures 14 and 16 indicates that the effect of casing on the simulated in-phase apparent resistivities remains only within 100 m of the casing shoe and does not affect measurements acquired
ACKNOWLEDGEMENTS
The work reported in this paper was funded by University of Texas at Austin Research Consortium on Formation Evaluation, jointly sponsored by Anadarko, Aramco, Baker Atlas, BP, British Gas, ConocoPhilips, Chevron, ENI E&P, ExxonMobil, Halliburton Energy 6
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
Services, Hydro, Marathon Oil Corporation, Mexican Institute for Petroleum, Occidental Petroleum Corporation, Petrobras, Schlumberger, Shell International E&P, Statoil, TOTAL, and Weatherford.
Geosystems Engineering of The University of Texas at Austin. Since December 2006, he has been with WesternGeco Electromagnetics of Schlumberger as Senior Research Scientist. He conducts research on marine CSEM and MT. He published over 30 papers and holds a Chinese patent.
REFERENCES
Carlos Torres-Verdín received a Ph.D. degree in Engineering Geoscience from the University of California, Berkeley, in 1991. During 1991–1997 he held the position of Research Scientist with Schlumberger-Doll Research. From 1997–1999, he was Reservoir Specialist and Technology Champion with YPF (Buenos Aires, Argentina). Since 1999, he has been with the Department of Petroleum and Geosystems Engineering of The University of Texas at Austin, where he currently holds the position of Associate Professor. He conducts research on borehole geophysics, well logging, formation evaluation, and integrated reservoir characterization. Torres-Verdín has served as Guest Editor for Radio Science, and is currently a member of the Editorial Board of the Journal of Electromagnetic Waves and Applications, and an associate editor for Petrophysics (SPWLA) and the SPE Journal. He is co-recipient of the 2003 and 2004 Best Paper Award by Petrophysics, and is recipient of SPWLA’s 2006 Distinguished Technical Achievement Award.
Anderson, B. I., 2001, Modeling and inversion methods for the interpretation of resistivity logging tool response: PhD Thesis, Delft University of Technology. Chen Y. H., Chew W. C. and Zhang G. J., 1998, A novel array laterolog method: The Log Analyst, v. 39, no. 5, pp. 23-33. Jin J. M., M. Zunoubi, K. C. Donepudi, and W. C. Chew, 1999, Frequency-domain and time-domain finite-element solution of Maxwell’s equations using the spectral Lanczos decomposition method: Computer Methods in Applied Mechanics and Engineering, v. 169, no. 1999, pp. 279-296. Lacour-Gayet, P., 1981, The Groningen effect, causes and a partial remedy: Schlumberger Technical Review, v. 29, no. 1, pp. 37-47. Li, T. T., and Tan, Y.J., 1995, Mathematical problems and methods in resistivity well-logging: Surveys Math. Industry, v. 5, pp 133-167. Liu, Q. H., Anderson, B., and Chew, W.C., 1994, Modeling low-frequency electrode-type resistivity tools in invaded thin beds: IEEE Transactions on Geoscience and Remote Sensing, v. 32, no. 3. Lovell, J. R., 1993, Finite element methods in resistivity logging: PhD Thesis, Delft University of Technology. Trouiller J. C., and Dubourg I., 1978, A better deep laterolog compensated for Groningen and reference effects: in Transactions of the 35th SPLWA Symposium, pp. 1-16, Paper VV. Woodehouse, R., 1978, The laterolog Groningen phantom can cost you money: in Transactions of the 19th SPLWA Symposium, pp. 1-17, Paper R. Yang F., and Nie Z. P., 1997, A precise numerical simulation of DLL logging response: Chinese Well Logging Technology, v. 21, no. 4. Zhang, G. J., 1986, Electrical well logging (II) (in Chinese): The Petroleum Industry Press, Beijing. ABOUT THE AUTHORS
Ridvan Akkurt is a Petroleum Engineering Consultant at Aramco. Ridvan started his oilfield career in 1983 in Africa as a wireline field engineer for Schlumberger, then worked for GSI in the Middle East as a field seismologist, for Schlumberger-Doll Research on NMR research, for Shell as a geophysicist, and for NUMAR as a senior research scientist. He founded NMRPLUS in late 1997, and consulted for major oil and service companies on various aspects of NMR logging until joining Aramco in 2005. Ridvan has a B.Sc. degree in Electrical Engineering from the Massachusetts Institute of Technology and a Ph.D. degree in Geophysics from the Colorado School of Mines. He has several publications and patents in the area of NMR logging, is recipient of best paper award by SPWLA in 1995, teaches industrial courses on NMR logging, and has served as a Distinguished Lecturer for SPE during 1998-1999, and for SPWLA during 1995-1996.
Wei Yang received a Ph.D. degree in General Mechanics from Harbin Institute of Technology, China, in 1995. During 1995-1997 he was a postdoctoral researcher at the University of Petroleum, Beijing, China. During 1997-2005, he held the position of Associate Professor with the Borehole Research Center of the same university. From 2005 to 2006, he was a Research Fellow with the Department of Petroleum and
Haluk Ersoz is a Technical Advisor for Halliburton Saudi Arabia. He holds a M.Sc. degree in Nuclear Physics and a B.Sc. degree in Mechanical Engineering from University of Manchester, U.K. He has over 20 years experience in log interpretation and formation evaluation with Schlumberger and Halliburton as field engineer, field service coordinator, field service quality coordinator and technical advisor.
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SPWLA 48th Annual Logging Symposium, June 3-6, 2007
m1’
m1
B
m2
A2
m3 m3’ m4 m4’ m 5 m5’ m 6 m6’ m7 I0 m7’ m8 m8’ m9 m9’ m10 m10’ m11 m11’
A1 M3 M1 A0 M1’ M3’ A1’ A2’
m2’vvv
V0 ΩLLD
m1
m1 (a)
m1’ B
m2
A2
m3 m3’ m4 m4’ m 5 m5’ m 6 m6’ m7 I0 m7’ m8 m8’ m9 m9’ m10 m10’ m11 m11’
A1 M2 M1 A0 M1’ M2’ A1’ A2’
(a)
m1 m2’
V0 ΩLLS
m1
m1 (b) (b)
Figure 1. Computational domain (Ω) for dual laterolog measurements. (a) Computational domain for the deepsensing laterolog mode. (b) Computational domain for the shallow-sensing laterolog model. The variables mi , i = 1,...,11 identify electrode locations, where m1
Figure 2. Configuration of the dual laterolog tool assumed in this paper. (a) Deep-sensing laterolog mode: it includes 10 electrodes: the main electrode A0, the bucking electrodes A1, A1’, A2 and A2’, the monitoring electrodes M1, M1’, M3, and M3’, and the current return electrode B. (b) Shallow-sensing laterolog mode: It includes 9 electrodes: the main electrode A0, the bucking electrodes A1, A1’, A2 and A2’, and the monitoring electrodes M1, M1’, M2, and M2’. Distances are given in meters.
identifies the domain boundary, and the variables
mi′ , i = 1,...,11 identify insulators. Dashed purple lines describe the main current I0 enforced by the main electrode A0 . Dashed green lines describe the electrical current enforced by the focusing electrodes A1 and A2. 8
SPWLA 48th Annual Logging Symposium, June 3-6, 2007
Figure 3. Description of the 8-node quadrilateral element used in the second-order finite-element formulation.
Figure 5. Single-interface formation model with borehole and steel casing. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. Layer resistivities are 1 and 10000 Ω-m, with their interface located at the relative vertical location of -1 m. Casing thickness, electrical resistivity, and relative magnetic permeability are equal to 0.01 m, 2e-7 Ω-m, and 1, respectively. The lower termination boundary of casing is located at a the relative vertical location of 1 m and the spatial domain of the simulation is terminated at the relative vertical location of 1000 m.
FIRST ORDER-LLD SECOND ORDER-LLD FIRST ORDER-LLS SECOND ORDER-LLS Rt
-2
0
1
0.2m 50Ω-m 2 m 10Ω-m
2
1Ω-m 1
10
Apparent Resistivity (Ω-m)
FIRST ORDER-LLD SECOND ORDER-LLD FIRST ORDER-LLS SECOND ORDER-LLS Rt
2
1Ω-m
0.2m
1
1Ω-m
Depth (m)
Depth (m)
-1
100
0
-1
Figure 4. Comparison of first- and second-order finiteelement solutions. Figure 2 shows the assumed laterolog tool configuration operating at 0 Hz. The formation model includes three layers with a borehole. Mud resistivity is 1 Ω-m and borehole diameter is 0.2m One of the layers is invaded. Layer resistivities are 1, 50, and 1 Ω-m, with the resistivity of the invasion zone in the invaded layer equal to 10 Ω-m (thickness equal to 2 m and radial length of invasion equal to 0.2 m). Solid blue (solid triangles) and dashed blue (open triangles) lines identify deep laterolog measurements simulated with the first- and second-order finite-element methods, respectively. Solid red (solid circles) and dashed red (open circles) lines identify shallow laterolog measurements simulated with the first- and secondorder finite-element methods, respectively. The dotted purple line identifies Rt.
-2 -3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
Apparent Resistivity (Ω-m)
4
10
5
10
Figure 6. Comparison of first- and second-order finiteelement solutions for the formation model described in Figure 5. Figure 2 shows the assumed laterolog tool configuration operating at 0 Hz. Solid blue (solid triangles) and dashed blue (open triangles) lines identify deep laterolog measurements simulated with the first- and second-order finite-element methods, respectively. Solid red (solid circles) and dashed red (open circles) lines identify shallow laterolog measurements simulated with the first- and secondorder finite-element methods, respectively. The dotted purple line identifies Rt.
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SPWLA 48th Annual Logging Symposium, June 3-6, 2007
p
p
p
p
2.5
LLD_Iso LLS_Iso Rt
-2
2
-1
1
Depth (m)
Vertical Coordinate (m)
1.5
0.5 0 -0.5
0
0.2m 1Ω-m
1
-1
10Ω-m
2m
-1.5 -2
2
1Ω-m
1Ω-m
-2.5 0.5
1
1.5
2
2.5
3
3.5
4
4.5
1
5
Radial Coordinate (m)
Figure 7. Spatial distribution of electric current lines for deep-sensing laterolog measurements simulated across isotropic and anisotropic homogeneous formations. The horizontal and vertical resistivities of the formation are 1 Ω-m and 2 Ω-m, respectively. Thick solid lines and thin dashed lines identify electric current lines for the cases of isotropic and anisotropic homogeneous formations, respectively.
10
Apparent Resistivity (Ω -m)
(a) p LLD_Ani LLS_Ani Rh Rv
-2
Depth (m)
-1
0
1
2.5
Rh=1,Rv=10Ω-m 0.2m
2
Vertical Coordinate (m)
1.5
Rh=10,Rv=100Ω-m 2m
2
1Ω-m
1
1 0.5 0
10
Rh=1,Rv=10Ω-m
Apparent Resistivity (Ω -m)
100
(b)
-0.5
Figure 9. Simulation of dual laterolog measurements in (a) isotropic and (b) anisotropic inhomogeneous formations. The formation model consists of three layers (thickness of center layer is 2 m) and includes a borehole. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. For the isotropic case, layer resistivities are 1, 10, and 1 Ω-m, whereas for the anisotropic case, vertical resistivities in the three layers are 10, 100, and 10 Ω-m, respectively. Figure 2 shows the assumed tool configuration; the operating frequency is 0 Hz. Solid blue (solid triangles) and red (solid circles) lines identify deep and shallow laterolog measurements, respectively, simulated for the isotropic case. Dashed blue (open triangles) and red (open circles) lines identify deep and shallow laterolog measurements, respectively, simulated for the anisotropic case. The dotted purple and dashed cyan lines identify the horizontal (Rh) and vertical resistivities (Rv), respectively.
-1 -1.5 -2 -2.5 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Radial Coordinate (m)
Figure 8. Spatial distribution of electric current lines for shallow-sensing laterolog measurements simulated across isotropic and anisotropic homogeneous formations. The horizontal and vertical resistivities of the formation are 1 Ω-m and 2 Ω-m, respectively. Thick solid lines and thin dashed lines identify electric current lines for the cases of isotropic and anisotropic homogeneous formations, respectively.
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SPWLA 48th Annual Logging Symposium, June 3-6, 2007
p
50
Depth (m)
0.2 m
LLD-0 Hz LLD-131.25 HZ LLS-0 HZ LLS-1050 HZ Rt
0
Casing
1 Ω-m 100 m 1 Ω-m
1E4 Ω-m
Anhydrite
100
BS=8.5 inches
400 ft
Rm=0.061 Ohm-M @ 223 F 150
200
Limestone A 10
0
1
2
10
10
10
3
40 ft
4
10
Apparent resistivity (Ω-m)
Anhydrite
20 ft 40 ft
Limestone B
Figure 10. Comparison of simulated in-phase deep- and shallow-sensing laterolog measurements acquired at different frequencies with the tool configuration described in Figure 2. The formation model includes the borehole and two layers. Mud resistivity is 1 Ω-m and borehole diameter is 0.2 m. Layer resistivities are 1 and 10000 Ω-m. Solid blue (solid triangles) and dashed blue (open triangles) lines identify deep-sensing measurements simulated at 0 Hz and 131.25 Hz, respectively. Solid red (solid circles) and dashed red (open circles) lines identify shallow-sensing measurements simulated at 0 Hz and 1050 Hz, respectively. The dotted purple line identifies Rt.
Limestone C
Anhydrite
100 m
Depth (m)
1E4 Ω-m 1 Ω-m
100
LLD-131.25 Hz LLS-1050 HZ Rt
150
200 -1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
200 ft
Figure 12. Eight-layer formation model that includes a borehole and truncated steel casing. Mud resistivity is 0.061 Ω-m and borehole diameter is 8.5 inches (0.2159 m). Casing thickness, electrical resistivity, and relative magnetic permeability are equal to 0.025 m, 1e-6 Ω-m, and 1, respectively. Depths are measured with respect to the center of the Limestone C layer. The casing shoe is located at 600 ft (182.88 m), and layer interfaces (from top to bottom) are located at 200, 160, 140, 100, 50, -50, and -250 ft (or 60.96, 48.768, 42.672, 30.48, 15.24, -15.24, and -76.20 m). Layer resistivities (from top to bottom) are 1e4, 0.625, 1e4, 0.4, 1e4, 0.625, 1e4, and 1.11 Ω-m.
1 Ω-m 50
100 ft
Limestone D
0.2 m
0
50 ft
Anhydrite
8
10
Apparent resistivity (Ω-m)
Figure 11. Simulated out-of-phase (quadrature) deepand shallow-sensing laterolog measurements acquired with the tool configuration described in Figure 2. The formation model is the same as that described in Figure 10. Layer resistivities are 1 and 10000 Ω-m. Solid blue (solid circles) and dashed red (open circles) lines identify deep-sensing measurements simulated at 131.25 Hz and shallow-sensing measurements simulated at 1050 Hz, respectively. The dotted purple line identifies Rt.
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SPWLA 48th Annual Logging Symposium, June 3-6, 2007
200
200
Depth (m)
Depth (m)
100
50
0
100
50
0
-50
-50
-100
-100 -1
10
0
10
Rt LLD-131.25 Hz LLS-1050 Hz
150
Rt LLD-0 Hz LLS-0 Hz
150
1
10
2
10
3
10
-2
10
4
10
-1
10
10
0
1
2
10
3
10
10
4
10
5
10
6
10
7
10
8
Figure 15. Out-of-phase apparent resistivities of deepand shallow-sensing laterolog measurements simulated for the formation model shown in Figure 12 with the tool configuration described in Figure 2. The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- (131.25 Hz) and shallow-sensing (1050 Hz) laterolog measurements.
Figure 13. Deep- and shallow-sensing laterolog measurements simulated at DC for the formation model shown in Figure 12 and with the tool configuration described in Figure 2. The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- and shallow-sensing sensing laterolog measurements, respectively.
200
200
Depth (m)
100
50
0
100
50
0
-50
-50
-100
-100 -1
10
0
10
1
10
Rt LLD-131.25 Hz LLS-1050 Hz
150
Rt LLD-131.25 Hz LLS-1050 Hz
150
Depth (m)
10
Resistivity (Ω-m)
Resistivity (Ω-m)
2
10
3
10
10
4
10
-1
10
0
10
1
10
2
10
3
4
10
Resistivity (Ω-m)
Resistivity (Ω-m)
Figure 16. In-phase apparent resistivities of deep- and shallow-sensing laterolog measurements simulated for the formation model shown in Figure 12 with the tool configuration described in Figure 2. Casing was not included in the simulations. The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- (131.25 Hz) and shallow-sensing (1050 Hz) laterolog measurements (cf. Figs. 14, 17, and 18).
Figure 14. In-phase apparent resistivities of deep- and shallow-sensing laterolog measurements simulated for the formation model shown in Figure 12 with the tool configuration described in Figure 2. The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- (131.25 Hz) and shallow-sensing (1050 Hz) laterolog measurements.
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SPWLA 48th Annual Logging Symposium, June 3-6, 2007
200
200
Rt LLD-131.25 Hz LLS-1050 Hz
100
50
0
100
50
0
-50
-50
-100
-100
10
-1
0
10
1
10
Rt LLD-131.25 Hz LLS-1050 Hz
150
Depth (m)
Depth (m)
150
10
2
10
3
10
4
-1
10
Resistivity (Ω-m)
0
10
10
1
10
2
3
10
10
4
Resistivity (Ω-m)
Figure 17. In-phase apparent resistivities of deep- and shallow-sensing laterolog measurements simulated for the formation model shown in Figure 12 with the tool configuration described in Figure 2. Casing was not included in the simulations whereas invasion was only included in the low-resistivity limestone beds (Rxo = Rt /2 and radial length of invasion equal to 1 ft). The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- (131.25 Hz) and shallow-sensing (1050 Hz) laterolog measurements (cf. Figs. 14, 16, and 18).
Figure 18. In-phase apparent resistivities of deep- and shallow-sensing laterolog measurements simulated for the formation model shown in Figure 12 with the tool configuration described in Figure 2. Both casing and invasion were included in the simulations. Invasion was only included in the low-resistivity limestone beds (Rxo= Rt /2 and radial length of invasion equal to 1 ft). The solid black line identifies values of true (uninvaded) formation resistivity. Solid blue (solid circles) and dashed red (open circles) lines identify the simulated deep- (131.25 Hz) and shallow-sensing (1050 Hz) laterolog measurements (cf. Figs. 14, 16, and 17).
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