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  GIMMC WORKSHOP - RESISTIVITY WELL LOGGING 1 Resistivity Well Logging Maria Inez C. Goncalves, Jeff P. Grossman, Ying Han, Gerald H.W. Lim, Zheng Meng,Benjamin W.L. Ong, Yongji Tan. Abstract  — This article is a result of group collaborationduring a GIMMC 2002 modelling workshop. We model re-sistivity well-logging, and develop an algorithm to solve aninverse problem. Keywords —Resisitivity Well Logging, Finite Elements, to-tal flux boundary condition, inverse problem. I. Introduction In petroleum exploration, resistivity well-logging is per-formed in order to estimate oil saturation levels in porousmedia. The method employs a log tool consisting of 3 dis-charging and 20 measuring electrodes, which is lowered intoa drilled well. Current is supplied by the discharging elec-trodes to generate a static electric field, and the resultingelectric potential is sampled at the measuring electrodes. Fig. 1. Domain IdealizationM. Goncalves is with the University of Victoria, Department of Math and Statistics. E-mail: inez@math.uvic.caJ.P. Grosmann is with the University of Calgary, Department of Geology and Geophysics. E-mail: grossman@geo.ucalgary.caY. Han is with McGill University, Department of Math and Statis-tics. E-mail: hua@math.mcgill.caG. Lim is with Simon Fraser University, Department of Physics.E-mail: ghlim@sfu.caZ. Meng is with Rice University, Department of Math and Statistics.E-mail: zhmenh@rice.eduB. Ong is with Simon Fraser Unviersity, Department of Math. E-mail: bwo@sfu.caY. Tan was the mentor for the group. He is with Fudan University,Department of Math.Fig. 2. Log Tool The goal is to infer the conductivity of the domain; thisis achieved through an iterative scheme - There are twomethods to solve the forward problem (i.e., find the poten-tial given conductivities and boundary conditions): impos-ing the total boundary flux condition and re-expressing theproblem in terms of classical PDEs. II. Mathematical Model A. Problem Formulation  Given that the ã  governing Laplace equation for electric potential  u  is ∇· ( k ( x ) ∇ u ( x )) = 0 (1)where  k ( x ) is the unknown conductivity that is spatiallydependent ã  The current at the discharging electrodes satisfy    A i k∂u∂ndS   =  I  i  (known constant)  ,  (2)where  A i  is the surface of the  i -th electrode, and  I  i  isthe current discharged by it. ã  The electric potentials at the measuring electrodes,¯ u j , j  = 1 ,...,J   (3)where J is the number of measuring electrodes.We wish to find the conductivity  k ( x ). Unfortunately, theproblem as posed is impossible to solve.  GIMMC WORKSHOP - RESISTIVITY WELL LOGGING 2 B. Simplification  We asssume cylindrical symmetry and discretize our do-main into several sub-regions, assuming that k is piece-wiseconstant in each sub-region. A natural discretization allowsfor a top rock layer, a bottom rock layer, an oil-saturatedsandstone layer, a mud filled well bore and a contaminatedregion within the saturated layer. (see fig 1) Fig. 3. Numerical Domain We note that although this discretization may appearsomewhat simplistic, it nevertheless represents the essentialgeophysical features in a reasonable way. One can alwaysrefine the discretization to P subdomains provided  P   ≤  J  (the number of measuring electrodes). In addition, we caninfer the following boundary conditions. C. Physical Boundary Conditions  ã  On the rubber insulating surface,  ∂u∂n  = 0 ã  By symmetry,  ∂u∂n  = 0 on the cylindrical axis. ã  On the measuring electrodes I   =    M  i k 1 ∂u∂ndS   = 0or equivalently,  ∂u∂n  = 0. (The above three boundariesform Γ 2 ã  For far fields,  u  →  0 as distance  → ∞  (Γ 1 ) D. Interface conditions  To obey the laws of physics, we require ã  Continuity of electric potential  u | −  =  u | + . ã  Continuity of current,  k − ∂u∂n  − =  k + ∂u∂n  + across all interfaces. E. The Inverse Problem  This inverse problem cannot be solved in a straightfor-ward manner. We adopt an iterative approach that en-forces (3) by minimizing   = J   j =1  u i ( z j ) −  ¯ u j   ,  (4)so that the iterative solution at the measuring electrodes, u i ( z j ), converge to ¯ u j . The outline of the iterative algo-rithm is as follows:Algorithm:1. Guess an initial set of conductivities,  k i = [ k i 1 ,k i 2 ,k i 3 ,k i 4 , 1 ,k i 4 , 2 ] for  i  = 0.2. Solve (1) for  u i excluding condition (3).3. Calculate    using (4)4. If    <  , where    is a user-defined tolerance, the   k i isaccepted; otherwise, it is modified to   k i +1 =  f  (  k i ) by asimplex search algorithm in matlab.5. Return to step 2, using   k i +1 , and repeat steps 2 to 4until the condition   <   is satisfied. III. Variational Principle and Finite ElementMethod A. Formulation  The resistivity well-logging problem can be re-expressedby incorporating (2) (the total flux boundary condition) ina variational formulation. This is equivalent to finding theminimum of the energy functional: J  ( v ) = 12     Ω k |∇ v | 2 dxdydz − L  i =1 I  i v | Γ i 3 ,  (5)where  L  is the number of discharging electrodes and theminimum is taken over  V   0  =  { v ( x ) | v  ∈  H  1 (Ω) ,v | Γ 1  =0 ,v | Γ i 3 = const. ( i  = 1 ,...,L ) }  , i.e., we wish to find  u  ∈  V   0 such that  J  ( u ) = min v ∈ V   0 J  ( v ).An equivalent formulation using the variational principle isto find  u  ∈  V   0  such that     Ω k ∇ u ·∇ Φ dxdydz  = L  i =1 I  i Φ | Γ i 3 ,  ∀ Φ  ∈  V   0 .  (6) B. Implementation  Using the finite element method, the natural boundary con-ditions are automatically satisfied. Neverthless, some spe-cial treatment of the total flux boundary condition is re-quired. Without loss of generality, we simplify our problemas follows: ã  We consider the 50m  ×  50m cross section Ω 4  split intotwo sub-regions with piecewise constant conductivities k 1  and  k 2 . ã  Reduce problem to one discharging electrode and eightmeasuring electrodes.  GIMMC WORKSHOP - RESISTIVITY WELL LOGGING 3 J  ( u ) simplifies to J  ( u ) = 12 L  i =1    Ω i 4 k i |∇ v | 2 drdz  −  I  2 πu | Γ 3 .  (7)We can now form the stiffness matrix and treat the equi-valued boundary value condition explicitly (details of thistechnique can be found in [Surv. Math. Ind. (1995): 133-167]. We use linear triangular elements based on the per-pendicular network. The mesh is denser near the discharg-ing electrode because the electric potential changes rapidlythere. Fig. 4. Sample Mesh 0 0.05 0.1 0.15 0.2 0.25 0.3 0.3500.20.40.60.8r        z Fig. 5. Simplified Domain 0 5 10 15 20 25 30 35 40 45 5005101520253035404550 Ω 4,1 (k 1 ) Ω 4,2  (k 2 ) In our numerical experiment, we set  I   = 1 on the discharg-ing electrode, set   k  = ( k 1 ,k 2 ) = (5 , 0 . 05) and we computedthe potential  u  over Ω 4 . The solution is plotted on thefollowing page. Fig. 6. Electric Potential with  k 1  = 5,  k 2  = 0 . 05 0 5 10 15 20 25 30 35 40 4505101520253035404550 Observe that the potential  u  decreases rapidly from thetool to the far field boundaries. Also note the effect of current continuity at  r  ≈  2 C. Inverse Problem  Suppose we take eight potentials on the log tool (fromthe above computation) as measured data. Given anothertwo resistivities  k 01  and  k 02  as initial values, can we recover  k  = (5 , 0 . 05)? Yes! Our final solution converges to ourgiven data. The deviation    is  o (10 − 12 ), and we recover  k  = [5 . 00003424142100 , 0 . 04999434491546]. D. Stability  We tested the stability of our method by introduc-ing 0.5% random error noise into our measured data.We find that the conductivities now converge to   k  =[3 . 35348 , 0 . 03354] with    =  o (10 − 10 ). This indicates thatour method is not stable, and that some regularization of the parameters or some added weights is required in ourfuture work, e.g., replacing (4) by   = min k 1 ,k 2 8  i =1 k i ( u −  ¯ u i ) 2 (8) IV. Classical Method A. Formulation  We can also reformulate our boundary conditions to amore familiar form: Dirichlet and Neumann bondary con-ditions, namelyreplace    Γ i 3 k 1 ∂u∂n  =  I  i  by  u | Γ 23 = 1 (9)In fact, through linearity of (1), if we just solve the PDEin Section II with (2) replaced by u | Γ j 3 = 1 , u | Γ k 3 = 0 , (  j   =  k ) .  GIMMC WORKSHOP - RESISTIVITY WELL LOGGING 4 Our solution is then a linear combination u  = L  j =1 β  j u j We can then solve for  β  j  by solving a linear system of equa-tions. For example, the case with 3 discharging electrodesresults in    Γ 13 β  1 k 1 ∂u 1 ∂n  +    Γ 13 β  2 k 1 ∂u 2 ∂n  +    Γ 13 β  3 k 1 ∂u 3 ∂n  =  I  1    Γ 23 β  1 k 1 ∂u 1 ∂n  +    Γ 23 β  2 k 1 ∂u 2 ∂n  +    Γ 23 β  3 k 1 ∂u 3 ∂n  =  I  2    Γ 33 β  1 k 1 ∂u 1 ∂n  +    Γ 33 β  2 k 1 ∂u 2 ∂n  +    Γ 33 β  3 k 1 ∂u 3 ∂n  =  I  3 B. Implementation  We used a finite element package (FEMLAB) to solvethe (full) problem posed in section 2.2. Only 4 regions arevisible in the following plot because the computation wasdone to scale. (The mud region is not visible) ã  the depth of the well is 100m ã  the electric field affects a radius of approximately 50m, ã  the drill has a 10 cm radius resulting in a mud regionwith a 10cm radius ã  the tool has a length of 8 meters ã  the single discharging electrode has a height of 0.5m. ã  the contaminated region is approximately 1 meter wideWhen I = 0.645 and   k  = [ k 1 ,k 2 ,k 3 ,k 41 ,k 42 ] =[10 , 1 , 1 , 0 . 5 , 0 . 01],Notice that the same features are present in the more com-plex subdomain structure - the continuity of current pro-vides a jump in  ∂u∂ n .A more detailed plot of the mesh is shown: Fig. 7. Femlab Mmesh 0 5 10 15 20 25 30 35 40 45 500102030405060708090100 C. Inverse Problem  As in Chapter 3, the search direction for   k  was com-puted using  fminsearch  , a routine found in the matlaboptimization toolbox. We had problems converging tothe correct solution for the 5 parameter problem,   k  =[ k 1 ,k 2 ,k 3 ,k 41 ,k 42 ]. It appeared that there were local min-imums in this problem. It is also likely that scaling prob-lems due to the different region dimensions would play arole. The problem did converge however when  k 1 ,k 2 & k 3 was kept fixed, and   k  = [ k 41 ,k 42 ] was optimized.Finding a good optimization routine is of course the keyto this problem. Due to time constraints, only  fminsearch  was attempted.  fminsearch   took some undesirable steps:the conductivities became negative, which is not physicallypossible in this problem. Given time, one could probablycode a more suitable optimization routine which involvestrust regions and appropriate constraints. V. Conclusions Given the time restrictions of a modeling camp, we werequite pleased with the results that we obtained. We thinkthat our formulation of the problem models the problempresented, and our finite element solvers adequately solvedthe Laplace equation with boundary conditions. We man-aged to solve the inverse problem using  fminsearch   for asimple test case, but it would have been nice to code ourown optimization routine.
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