GLE 594: An introduction to applied geophysics

Transcription

1 GLE 594: An intoduction to applied geophsics Electical Resistivit Methods Fall 4 Theo and Measuements Reading: Toda: -3 Net Lectue: 3-5

2 Two Cuent Electodes: Souce and Sink Wh un an electode to infinit when we can use it? souce sink souce P souce sink i π souce sin k i π sink Total oltage at P: p souce sin k i π souce sink Measuement Pacticalities Can t measue potential at single point unless the othe end of ou volt mete is at infinit. This is inconvenient. It is easie to measue potential diffeence (). This lead to use of fou electode aa fo each measuement. Resulting measuement given as Can be ewitten G I π whee G*/π is sometimes efeed as the Geometical Facto P * P I π 3 4

3 Cuent densit and equipotential lines fo a cuent dipole d faction total cuent i f π tan d i f.5 at i f.7 at d d Wide spacing Deepe cuents Appaent Resistivit Pevious epession can be eaanged in tems of esistivit: (/I) (π/g). This can be done even when medium is inhomogeneous. Result is then efeed to as Appaent Resistivit. Definition:Resistivit of a fictitious homogenous subsuface that would ield the same voltages as the eath ove which measuements wee actuall made. 3

4 Geometical Factos Aa advantages and disadvantages Aa Wenne Schlumbege Dipole-Dipole Advantages. Eas to calculate a in the field. Less demand on instument sensivit. Fewe electodes to move each sounding. Needs shote potential cables. Cables can be shote fo deep soundings Disadvantages. All electodes moved each sounding. Sensitive to local shallow vaiations 3. Long cables fo lage depths. Can be confusing in the field. Requies moe sensitive equipment 3. Long Cuent cables. Requies lage cuent. Requies sensitive instuments 4

5 5 Govening Equation Continuit: What goes in must comes out i A Cuent Densit (like hdo q): Appling Ohm s Law: ; ; s equation LaPlace' Govening Equation and, sin, cos o using θ θ

6 Govening Equation - Solution The Laplace s equation is a homogeneous, patial second ode diffeential equation Solution: Eact solutions: onl fo simple geometies Gaphical solutions: Flow nets, maste chats Numeical solutions: finite diffeence and finite elements solutions Appoimate solutions: methods of fagments Phsical analogies (electical, hdaulic and heat flow) Geo-electic Laeing Often the eath can be simplified within the egion of ou measuement as consisting of a seies of hoiontal beds that ae infinite in etent. Goal of the esistivit suve is then to detemine thickness and esistivit of the laes. Longitudinal conductance (one lae): Tansvese esistance (one lae): Longitudinal esistivit (one lae): Tansvese esistivit (one lae): S L h/hσ Th L h/s T T/h Longitudinal conductance (one lae): S L Σ(h i / i ) Tansvese esistance (one lae): TΣ(h i i ) 6

7 oltage and Flow in Laes Tangent Law: The electical cuent is bent at a bounda i a b θ θ c d dl i Relations: Cuent: i i oltage: d d Resistivit: > dl d tanθ tanθ If < then the cuent lines will be efacted awa fom the nomal If > then the cuent lines will be efacted close to the nomal oltage and Flow in Laes Method of electical image S oltages at points P and Q: 3 P I P 4π k S Q Q I k 4π 3 whee k 7

8 Solving the diffeential equation fo two laes and a souce and sink Govening Equation C a P int h Bounda Conditions. i. at inteface 3. at 4. π i ( ) inteface at, No cuent at suface oltage is continuous Nomal cuent densit is continous Paticula solution Lae Calculations Can use fo image theo fo multiple boundaies. Fo two lae case: k I p π I π whee n k n k k It obviousl gets much moe difficult with moe laes. n n ( nh) k... n n... 8

9 Lae Calculations (cont.) I Integal method: ) p K( λ) J ( λ dλ π J is the Bessel function of eo ode. T ( λ) K(λ) given b elationship K ( λ) T i (λ) solved fo ecusivel upwad fom bottom lae to lae using: [ Ti i tanh( λhi )] Ti ( λ) [ T ] i tanh( λhi ) / i whee and e tanh( λhi) e T n (λ) λhi λhi n Solutions fo a Wenne Aa fo two laes C P P C k 9

10 etical Electic Sounding When ting to pobe how esistivit changes with depth, need multiple measuements that each give a diffeent depth sensitivit. This is accomplished though esistivit sounding whee geate electode sepaation gives geate depth sensitivit. ES Data Plotting Convention Plot appaent esistivit as a function of the log of some measue of electode sepaation. Wenne a spacing Schlumbege AB/ Dipole-Dipole n spacing Asmptotes: Shot spacings << h, a. Long spacings >> total thickness of oveling laes, a n To get a tue fo intemediate laes, lae must be thick elative to depth.

11 Equivalence: seveal models poduce the same esults Ambiguit in phsics of D intepetation such that diffeent laeed models basicall ield the same esponse. Diffeent Scenaios: Conductive laes between two esistos, whee lateal conductance (σ h ) is the same. Resistive lae between two conductos with same tansvese esistance ( h ). Equivalence: seveal models poduce the same esults Although ER cannot detemine unique paametes, can detemine ange of values. Also eists in D and 3D, but much moe difficult to quantif. In these multidimensional cases simpl efeed to as non-uniqueness.

12 Pinciple of suppession: Thin laes of small esistivit contast with espect to backgound will be missed. Thin laes of geate esistivit contast will be detectable, but equivalence limits esolution of bounda depths, etc. Suppession