Research on DC resistivity for an arbitrarily anisotropic earth using circular scanning measurement

Transcription

1 Reearch on DC reitivity for an arbitrarily aniotroic earth uing circular canning meaurement Zhilong Yang* Changchun Yin Xiuyan Ren Changkai Qiu Xiaoyue Cao Jilin Univerity Jilin Univerity Jilin/RMIT Univerity Jilin Univerity Jilin Univerity Changchun, China Changchun, China Melbourne, Autralia Changchun, China Changchun, China SUMMARY Thi aer reent a 3D forward modelling algorithm with an adative finite-element method baed on untructured grid that when ued with a circular canning DC meaurement technique can rovide an indication of aniotroy in a layered earth. The accuracy of thi algorithm i checked againt 1D emi-analytical olution for an arbitrarily aniotroic earth. Thi aniotroy roduce a meaurement aradox, reult are deendent on the hyical relationhi between DC tranmitter and receiver. To reolve thi aradox, a circular canning meaurement technique ha been uggeted. Through analying the reone of tyical aniotroic model, we tudy the characteritic of aarent reitivity related to the electrically aniotroic media and identification of underground electrical aniotroy. The reult of numerical exeriment how the effectivene of our algorithm and the analyi will be helful to the interretation of aniotroic DC reitivity data. Key word: numerical modelling, untructured grid, local adative finite-element method, electrical aniotroy, circular canning meaurement INTRODUCTION The direct current (DC) reitivity method ha been widely ued in the exloration of uburface mineral, hydrogeology urvey and environmental & engineering invetigation for year (Xu et al, 1994). The method can redict the reitivity ditribution of the uburface tructure through the ditribution of the aarent reitivity oberved on the ground (Pain et al, 2003). The electrically iotroic model are widely ued in exiting forward and inverion algorithm. However, uing an iotroic model to interret the data from an aniotroic earth will caue big mitake (Yin & Weidelt, 1999), eecially when there are ditinct tratification or fracture, or directional tructure in the earth. Therefore, in area with ditinct aniotroy the DC reitivity modelling hould take into account the electrical aniotroy (Yin & Weidelt, 1999; Yin, 2000). The analytical olution of the reitivity reone for a tranverely iotroic uniform half-ace are known (Habberjam, 1975), a are olution for aniotroic layered media (Wait, 1990; Li & Uren, 1997; Yin & Maurer, 2001). For comlicated aniotroic media, the reent numerical algorithm include the volume integral method (Li & Uren, 1998), the finite-difference method (Wang & Fang, 2001), the finite-element method (Wang & Wu, 2013), and the ectral element method (Zhou et al, 2009). Li et al (1997) modelled the arbitrarily aniotroic uburface uing integral equation method, and Wang et al (2013) modelled the reitivity reone of aniotroic bodie embedded by iotroic rock uing a finite-element method. In traditional DC reitivity roecting, the obervation data only contain one-directional reitivity information, the reitivity in other direction i neglected. Therefore the interretation of the obervation data cann t accurately redict the uburface tructure becaue of the exitence of an aniotroic aradox in the determination of the aarent reitivity (Yin & Weidelt, 1999). In order to olve the meaurement roblem, Yin et al. (2016) rooed a circular canning meaurement technique and validated the method uing a one-dimenional layered model of an aniotroic medium. By lotting aarent reitivity a a olar diagram, any uburface aniotroy could be recognized eaily. In thi aer, we reent an adative finite-element method baed on untructured grid to model the arbitrarily aniotroic media. To guarantee the comutational accuracy in the region near the ource and with high reitivity contrat, we adoted the local adative grid refinement technique baed on the gradient recovery theory. Thi technique ha a maller memory requirement but fat comutational eed than the global adative refinement algorithm in IAG ( Zhengyong Ren, 2010). It can imrove the meh adative flexibility by etting deviation threhold for different area. Furthermore, a circular canning meaurement ha been uggeted in the imulation of the 3-D uburface tructure. We et the tranmitter in the centre of a detecting region, and the urvey oint are oitioned circularly with ame ditance to the tranmitter. By changing the ditance to the tranmitter, we can get more information on the reitivity ditribution and aniotroy at different deth of the earth. METHOD AND RESULTS The boundary value roblem For the arbitrarily aniotroic uburface model, the conductivity tenor = -1 or reitivity vector can be exreed by a ymmetric tenor with 6 individual comonent: AEGC 2018: Sydney, Autralia 1

2 xx xy xz yx yy yz. (1) zx zy zz The relationhi between the conductivity tenor and the rincial conductivityc tenor i (Yin, 1999) T R cr, (2) where T rereent the matrix tranoe, R=RxRyRz i the Euler rotation matrix. In the 3-D domain, the otential u=u +u (u i the econdary otential and u i the rimary otential) induced by the oint current ource atifie the Poion equation, the otential on the ground boundary 0 atifie the Newman boundary condition and the otential in the infinite atify the hybrid boundary condition. Therefore, the boundary value roblem for the econdary otential u can be written a u u, r Ω. (3) where B 0 u n u n, r Γ0. (4) n r The analytical olution of rimary ingular field i P -1 0 T r n r u n u u n u, B B I π 0 BP r, 0 i the conductivity tenor of the urrounding rock. r Γ. (5) u, (6) We et a tet function N in the Hilbert domain H 1 () and obtain the following equation with the Galerkin finite-element method: 0 u NdΩ 0 Ω u. (7) The modelling domain i then dicretized into a et of tetrahedral element with the oen-ource meh generator Tetgen (Si. 2006). We combine the boundary condition with equation 9 and imlify it via the Green identitie and obtain KU b, (8) where K i the global coefficient matrix, b i the ource term, U i the econdary otential vector to be olved. In thi way, we can obtain the econdary otential from the large are linear equation by uing the arallel direct olver MUMPS and get the aarent reitivity from the total field otential that i the um of the rimary otential and the econdary otential. Local adative meh refinement In 1987, Zienkiewicz rooed a oteriori error etimation algorithm (which i called z-z method) baed on gradient recovery technique. The method which i indeendent in ecific roblem i imle and efficient, o it i eay to tranlant it to untructured grid. In the L2 norm, the element relative error of the 3-D tatic electromagnetic field in the finite-element imulation i a follow: F T F e du - u du - u dω i 1, 2, 3, n. i, (9) Ω i where i i the number of the element, n i the total number of the element dicretized in the imulating region, i i the real ace of each element, u F i the otential gradient. The element recovery gradient du i obtained by the uer-convergent atch recovery technique that can be written a where T 1 2 d u Ga, (10) 1 a0x a0 y a0z x G ; a1x a1 y a1z a. (11) y a 2x a2 y a2z z a3x a3y a3z Where a i a matrix of unknown arameter ( Zhengyong Ren, 2010). And the element gradient recovery can be obtained by a in the element atch that contain all the element adjoining the element domain i. After that, we can obtain the element relative error e i through the equation 9. In the adative meh refinement trategy, we firt et the maximum error threhold for each target region: e j, j 1, 2, k., (12) where k i the number of the target region. The volume of each element can be adjuted by the element relative error and the maximum threhold. The adjuted element volume i AEGC 2018: Sydney, Autralia 2

3 e j Vi Vi, (13) e where Vi * i the element volume before each iteration. After obtaining the new volume of each element, we can regenerate the new grid by Tetgen. We reeat the above te until the element relative error i maller than the threhold. Accuracy verification We will demontrate the accuracy of our algorithm by imulating a two-layer aniotroic model hown in Figure 1. The rincial reitivity and the Euler rotation angle of the firt layer are reectively x1 =L1=100 m, y1=l1=100 m, z1=t1=400 m, 1=0 o,1=90 o, 1=0 o. For the econd layer, they are x2=l2=10 m, y2=l2=10 m, z2=t2=40 m, 2=0 o, 2=90 o, 2=0 o. The accuracy verification i howed in Figure 2. By analying the reult we can ee: 1) the algorithm of thi aer ha a high modelling accuracy. 2) When the offet r between the tranmitter and the receiver tart at 0, the lot in Figure 2 a and c mainly how the reitivity of the firt layer. However, the value in Figure 2 a doe not equal c. Figure 2 a how the aarent reitivity in x-direction with a value equal to L1=100 m, which i the reitivity along the tratification of the firt layer. Figure 2 a how the aarent reitivity in y- direction and the value equal to =200 m that i the geometric mean of the reitivity along the tratification of the M1 L1 T1 firt layer L1 and the reitivity erendicular to the tratification of the firt layer T1. When the offet r become large enough, the value of Figure 2 a and c tay invariant that how the reitivity of the econd layer. The reult are imilar to the firt layer: the aarent reitivity in x-direction i L2=10 m while that in y-direction i =20 m. Thee reult tyify the meaurement aradox that occur when meauring aarent reitivity in an aniotroic layered earth; the aarent reitivity i erendicular to the ditribution of the true earth reitivity. i M2 L2 T2 Figure 1: A two-layer model AEGC 2018: Sydney, Autralia 3

4 Figure 2: Comarion of the reult comuted by thi aer with emi-analytical olution and the relative deviation a,b i around the x direction; c,d i around the y direction, r i the offet. Modelling of an arbitrarily aniotroic half-ace Figure 3 how the aarent reitivity ditribution of an arbitrarily aniotroic half-ace whoe rincial reitivity i x0/y0/z0=100/400/100 m, and the reult are calculated for a circular canning urvey (refer to Figure 4). In Figure 3 a, the rincial reitivity i rotated around the z axi. From the olar lot we can ee that the hae of the aarent reitivity i imilar to an ellie: it i tretched in the direction of the low rincial reitivity, and the value (ellitic long axi) equal to the geometric mean of the rincial reitivity in the x and the y direction; while it i comreed in the direction of the high rincial reitivity. The value (ellitic hort axi) equal to the rincial reitivity in the x direction. Thi reult correond to the above dicued aniotroic aradox. Thi imlied that via the aniotroic aradox of aarent reitivity we can ditinguih the aniotroic characteritic of the underground media. In Figure 3 b, the rincial reitivity i rotated around the x axi. We can ee that before the rotation (=0 o ), the aarent reitivity of the aniotroic half-ace i imilar to the above aarent reitivity ditribution. When the aniotroic halface i rotated around the x axi, the ellitic hort axi turn to be tretched while the ellitic long axi tay invariant. When rotated 90 o around the x axi, the model turn to be horizontal iotroic (at thi oint the rincial reitivity in the x direction i the ame a in the y direction). The olar lot become a circle with radiu equal to the geometric mean M L T x z y z =200 m. Thu we can conclude that the aniotroic characteritic and the inclination of the trata of an aniotroic medium can be identified from the characteritic of the olar lot. AEGC 2018: Sydney, Autralia 4

5 Figure 3: Polar lot of aarent reitivity for an aniotroic half-ace a. The rincial reitivity of the half-ace medium i rotated 0 o, 45 o, 90 o, 135 o around the z axi; b. the rincial reitivity of the half-ace medium i rotated 0 o, 45 o, 90 o, 135 o around the x axi. Modelling of an arbitrarily aniotroic anomaly embedded by arbitrarily aniotroic rock We will tudy the characteritic of the aarent reitivity for an arbitrarily aniotroic anomalou body embedded by arbitrarily aniotroic rock. Figure 4 how the detail of our model, the comutational region i 2000 m2000 m1000 m and the extended region i m10000 m10000 m. The rincial reitivity of the anomalou body i x1=10 m, y1=40 m, z1=10 m, while the rincial reitivity of the urrounding rock i x1=100 m, y1=400 m, z1=100 m. Figure 4: Circular canning meaurement baed on bi-ole configuration over a 3D model We et 600 m600 m600 m area a the comutational region which contain the anomalou body and et the element relative error a 5% to refine the meh. To get a high accuracy of the econdary otential, we further et the element relative error a 1% for two region that include the anomalou body and a mall region containing the tranmitter. To how clearly the reult of the local refinement, in Figure 5 we only how the y-z lane of a region which i 600 m600 m300 m. Comaring the three ma in Figure 5, we can ee that after the third refinement of the volume element, the element relative error become maller than the threhold in all target region. The about 3% relative element error hown near the urface of the anomaly i caued by a large (1000%) reitivity contrat in thi region. AEGC 2018: Sydney, Autralia 5

6 Figure 5: Local adative meh refinement for 3-D model Figure 6: Polar lot of the aarent reitivity for an aniotroic abnormal body embedded in arbitrarily aniotroic urrounding rock The rincial reitivity of the urrounding rock i x0/y0/z0=100/400/100 m, and the rincial reitivity of the anomaly i x1/y1/z1=10/40/10 m. The rotation angel of the urrounding rock and the anomalou body in each ma are: a. AEGC 2018: Sydney, Autralia 6

7 0/0/0=0 o /0 o /0 o, 1/1/1=0 o /0 o /0 o ; b. 0/0/0=0 o /0 o /45 o, 1/1/1=135 o /0 o /0 o ; c. 0/0/0=0 o /0 o /90 o, 1/1/1=90 o /0 o /0 o ; d. 0/0/0=45 o /0 o /0 o, 1/1/1=0 o /0 o /90 o ; e. 0/0/0=90 o /0 o /0 o, 1/1/1=0 o /0 o /45 o ; f. 0/0/0=135 o /0 o /0 o, 1/1/1=45 o /0 o /0 o. Figure 6 how the modelling reult for the aniotroic anomalou body embedded by homogeneou aniotroic rock. Each figure in Figure 6 ha two different olar lot reectively for offet of 50 m (dotted line) and 140 m (olid line). From the figure, one ee that: 1) when the urrounding rock are not diing aniotroic, the ditribution of aarent reitivity i ymmetric, however, the aniotroic aradox till exit; 2) when the urrounding rock are diing aniotroic, the ditribution of aarent reitivity i unymmetric, with the maller ellioid ointing toward the inclination of the tratification; 3) the aarent reitivity ditribution for the abnormal body how imilar characteritic, only the anomaly i largely overwhelmed by urrounding rock; 4) the aniotroic characteritic of the abnormal body and the urrounding can be ditinguihed from the aarent reitivity ditribution for different offet between tranmitter and receiver. CONCLUSIONS In thi aer, we conduct the 3-D DC reitivity imulation for an arbitrarily aniotroic model uing finite-element method baed on untructured grid. The algorithm ha a high accuracy when checked againt 1D emi-analytical olution for an arbitrarily aniotroic earth. We have modelled an arbitrarily aniotroic half-ace and an arbitrarily aniotroic anomalou body embedded in an arbitrarily aniotroic rock. Through the analyi of the everal numerical reult, we come to concluion a following: 1) The circular canning meaurement ha advantage to recognize the characteritic of aniotroy. Comaring to the traditional meaurement, the reult of circular canning meaurement can clearly how the direction of aniotroic axi and aarent reitivity arameter. 2) There are aarent reitivity aniotroic aradoxe in the reult of the circular canning meaurement. The aniotroic ditribution characteritic of the urrounding rock and the anomalou body can be ditinguihed by changing the offet. ACKNOWLEDGMENTS Thi aer i uorted by National Nature Science Foundation of China ( , ) and Key National Reearch Project of China (2016YFC , 2017YFC ). REFERENCES Habberjam, G., 1975, Aarent Reitivity, Aniotroy and Strike Meaurement: Geohyical Proecting, 23, Jin, Jianming., 2014, The Finite Element Method in Electromagnetic: Xi an Univerity Pre. Li, P., Uren, N., 1997, Analytical Solution for the Point Source Potential in an Aniotroic 3-D Half-Sace Ι: Two-Horizontal-Layer Cae: Mathematical and Comuter Modelling, 26, Li, P., Uren, N., 1998, The Modelling of Direct Current Electric Potential in an Arbitrarily Aniotroic Half-Sace Containing a Conductive 3-D Body: Journal of Alied Geohyic, 38, Pain, C. C., et al., 2003, Aniotroic Reitivity Inverion: Invere Problem, 19, Ren, Z., Tang, J., 2010, 3D direct current reitivity modeling with untructured meh by adative finite-element method: Geohyic, 75(1), Ren, Z., Tang, J., 2014, A Goal-Oriented Adative Finite-Element Aroach for Multi-Electrode Reitivity Sytem: Geohyical Journal International, 199, Si, H., 2006, A Quality Tetrahedral Meh Generator and Three-Dimenional Delaunay Triangulator: Ph.D. Thei, Weiertra Intitute for Alied Analyi and Stochatic. Wait, J. R., 1990, Current Flow into a Three-Dimenionally Aniotroic Conductor: Radio Science, 25, Wang, Feiyan., 2009, 2.5-D DC Reitivity Modeling by the Adative Finite-Element Method with Untructured Triangulation: M.S. Thei, Central South Univerity. Wang, W., et al, 2013, Three-Dimenional DC Aniotroic Reitivity Modelling Uing Finite Element on Untructured Grid: Geohyical Journal International, 193, Wang, T., Fang, S., 2001, 3-D Electromagnetic Aniotroy Modelling Uing Finite Difference: Geohyic, 66, Xu, Shizhe, et al., 1994, The Finite Element Method for Solving Anomalou Potential for Reitivity Survey: Chinee Journal of Geohyic, 37, Yin, C., Weidelt, P., 1999, Geoelectrical Field in a Layered Earth with Arbitrary Aniotroy: Geohyic, 64, Yin, C., 2000, Geoelectrical Inverion for a One-Dimenional Aniotroic Model and Inherent Non-Uniquene: Geohyical Journal International, 140, Yin, C., Maurer, H. M., 2001, Electromagnetic Induction in a Layered Earth with Arbitrary Aniotroy: Geohyic, 66, Yin, C., et al., 2016, Forward Modelling of Marine DC Reitivity Method for a Layered Aniotroic Earth: Alied Geohyic, 13, Zhou, B., Greenhalgh, M., & Greenhalgh, S., 2009, 2.5-d/3-d reitivity modelling in aniotroic media uing Gauian quadrature grid: Geohyical Journal International, 176, Zienkiewicz, O. C., Zhu, J. Z., 1992, Suer-Convergent Patch Recovery and a Poteriori Error Etimate: Part1: The Recovery Technique: Internatioanl Journal for Numerical Method in Engineering, 33, AEGC 2018: Sydney, Autralia 7