GLE 594: An introduction to applied geophysics Electrical Resistivity Methods Fall 2004 Theory and Measurements Reading: Today: 210-223 Next Lecture: 223-251 1 Two Current Electrodes: Source and Sink • Why run an electrode to infinity when we can use it? source sink rsource P rsink Vsource = Total Voltage at P: iρ 2πrsource Vsin k = Vp = Vsource − Vsin k = iρ 2 πrsink iρ ⎛ 1 1 ⎞ ⎟ ⎜⎜ − 2 π ⎝ rsource rsink ⎟⎠ Measurement Practicalities Can’t measure potential at single point unless the other end of our volt meter is at infinity. This is inconvenient. It is easier to measure potential difference (∆V). This lead to use of four electrode array for each measurement. ρ Resulting measurement given as ∆V = VP1 − VP 2 = Can be rewritten ∆V = ρI G* 2π ρI ⎛ 1 1 1 1 ⎞ ⎜ − − + ⎟ 2 π ⎜⎝ r1 r2 r3 r4 ⎟⎠ where G*/2π is sometimes referred as the Geometrical Factor 2 Current density and equipotential lines for a current dipole d fraction total current ⎛ 2z ⎞ 2 i f = tan−1⎜ ⎟ ⎝d⎠ π if=0.5 at z= d 2 if=0.7 at z = d Wider spacing → Deeper currents Apparent Resistivity Previous expression can be rearranged in terms of resistivity: ρ=(∆V/I) (2π/G). This can be done even when medium is inhomogeneous. Result is then referred to as Apparent Resistivity. ρ1 ρ2 Definition:Resistivity of a fictitious homogenous subsurface that would yield the same voltages as the earth over which measurements were actually made. 3 Geometrical Factors Array advantages and disadvantages Array Advantages Disadvantages Wenner 1. Easy to calculate ρa in the 1. All electrodes moved each sounding field 2. Sensitive to local shallow 2. Less demand on variations instrument sensivity 3. Long cables for large depths Schlumberger 1. Fewer electrodes to move 1. Can be confusing in the field each sounding 2. Requires more sensitive 2. Needs shorter potential equipment cables 3. Long Current cables Dipole-Dipole 1. Cables can be shorter for deep soundings 1. Requires large current 2. Requires sensitive instruments 4 Governing Equation Continuity: What goes in must comes out ∂jy ⎞ ⎛ ∂j ∂j ⎛ ⎞ ⎛ ⎞ ∆y − jy ⎟⎟ + ⎜ jz − z ∆z − jz ⎟ = 0 ⎜ jx − x ∆x − jx ⎟ + ⎜⎜ jy − ∂x ∂y ∂z ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ ∂j ∂j ∂j − x ∆x − y ∆y − z ∆z = 0 ∂x ∂y ∂z z jz + ∂jz ∆y ∂y jx ∆z jy + jy jx + x ∆y ∂jx ∆y ∂y ∆x ∂j y ∆y ∂y Current Density (like hydro q): r r i j= A y jz Governing Equation Applying Ohm’s Law: jx = − 1 ∂V 1 ∂V 1 ∂V ; jz = − ; jy = − ρ ∂x ρ ∂y ρ ∂z ∂ ⎛ 1 ∂V ⎞ ∂ ⎛ 1 ∂V ⎞ ∂ ⎛ 1 ∂V ⎞ ⎜ ⎟+ ⎜ ⎟+ ⎜ ⎟=0 ∂x ⎜⎝ ρ ∂x ⎟⎠ ∂y ⎜⎝ ρ ∂y ⎟⎠ ∂z ⎜⎝ ρ ∂z ⎟⎠ or using x = r cos θ, y = r sin θ, and x 2 + y 2 = r 2 ⎫ ∂ 2V ∂ 2V ∂ 2V + 2 + 2 =0⎪ 2 ∂x ∂y ∂z ⎪ 2 ⎬∇ V = 0 ⇒ LaPlace' s equation 2 2 ∂ V 1 ∂V ∂ V ⎪ + = 0⎪ + ∂r 2 r ∂r ∂z 2 ⎭ 5 Governing Equation - Solution • The Laplace’s equation is a homogeneous, partial second order differential equation • Solution: – Exact solutions: only for simple geometries – Graphical solutions: Flow nets, master charts – Numerical solutions: finite difference and finite elements solutions – Approximate solutions: methods of fragments – Physical analogies (electrical, hydraulic and heat flow) Geo-electric Layering • Often the earth can be simplified within the region of our measurement as consisting of a series of horizontal beds that are infinite in extent. • Goal of the resistivity survey is then to determine thickness and resistivity of the layers. Longitudinal conductance (one layer): Transverse resistance (one layer): Longitudinal resistivity (one layer): Transverse resistivity (one layer): SL=h/ρ=hσ T=hρ ρL=h/S ρT=T/h Longitudinal conductance (one layer): Transverse resistance (one layer): SL=Σ(hi/ρi) T=Σ(hiρi) 6 Voltage and Flow in Layers Tangent Law: The electrical current is bent at a boundary i1 ρ1 a dl1 θ1 b ρ2 θ2 c Relations: Current: Voltage: Resistivity: dV1 i2 i1=i2 dV1=dV2 ρ1>ρ2 ρ 2 tan θ1 = ρ1 tan θ 2 dl2 dV2 If ρ2<ρ1 then the current lines will be refracted away from the normal If ρ2>ρ1 then the current lines will be refracted closer to the normal Voltage and Flow in Layers Method of electrical image S Voltages at points P and Q: r1 P r3 ρ1 ρ2 r2 Q VP = Iρ1 ⎛ 1 k ⎞ ⎜ + ⎟ 4 π ⎜⎝ r1 r2 ⎟⎠ VQ = Iρ 2 ⎛ 1 + k ⎞ ⎜ ⎟ 4 π ⎜⎝ r3 ⎟⎠ S’ where k = ρ 2 − ρ1 ρ 2 + ρ1 7 Solving the differential equation for two layers and a source and sink C1 Governing Equation ∂ 2 V 1 ∂V ∂ 2 V + + =0 ∂r 2 r ∂r ∂z 2 Boundary Conditions a P1 ρ1 zint = h ρ2 1. i z = 0 z =0 No current at surface 2. V1 = V2 at z = z interface Voltage is continuous 3. 1 ∂V1 1 ∂V2 at z = z interface = ρ1 ∂z ρ 2 ∂z 4. V = ( iρ1 2π r 2 + z 2 ) 1 2 at r = 0, z = 0 Normal current density is continous Particular solution Layer Calculations • Can use for image theory for multiple boundaries. For two layer case: Vp = = k= ρ 2 − ρ1 ρ 2 + ρ1 ⎞ Iρ1 ⎛ 1 2k 2 k 2 2k n ⎜⎜ + + + ..... + + .... ⎟⎟ 2π ⎝ r r1 r2 rn ⎠ ∞ Iρ1 ⎛ 1 kn ⎞ ⎜⎜ + 2∑ ⎟⎟ 2π ⎝ r n =1 rn ⎠ where rn = r 2 + (2 nh ) 2 • It obviously gets much more difficult with more layers. 8 Layer Calculations (cont.) ∞ Iρ • Integral method: V p = 1 ∫ K ( λ ) J 0 (λr )dλ 2π 0 • J0 is the Bessel function of zero order. – K(λ) given by relationship K (λ ) = T1 ( λ ) ρ1 • Ti(λ) solved for recursively upward from bottom layer to layer 1 using: Ti ( λ ) = where [Ti +1 + ρi tanh(λhi )] [1 + Ti +1 tanh( λhi ) / ρi ] tanh(λh i ) = ⋅ e 2 λh i − 1 ⋅ e 2 λh i + 1 and Tn ( λ ) = ρ n ⋅ Solutions for a Wenner Array for two layers C1 P1 P2 C2 k= ρ 2 − ρ1 ρ 2 + ρ1 9 Vertical Electric Sounding • When trying to probe how resistivity changes with depth, need multiple measurements that each give a different depth sensitivity. • This is accomplished through resistivity sounding where greater electrode separation gives greater depth sensitivity. VES Data Plotting Convention • Plot apparent resistivity as a function of the log of some measure of electrode separation. • Wenner – a spacing • Schlumberger – AB/2 • Dipole-Dipole – n spacing • Asymptotes: • Short spacings << h1, ρa=ρ1. • Long spacings >> total thickness of overlying layers, ρa=ρn • To get ρa=ρtrue for intermediate layers, layer must be thick relative to depth. 10 Equivalence: several models produce the same results • Ambiguity in physics of 1D interpretation such that different layered models basically yield the same response. • Different Scenarios: • Conductive layers between two resistors, where lateral conductance (σh) is the same. • Resistive layer between two conductors with same transverse resistance (ρh). Equivalence: several models produce the same results • Although ER cannot determine unique parameters, can determine range of values. • Also exists in 2D and 3D, but much more difficult to quantify. In these multidimensional cases simply referred to as non-uniqueness. 11 Suppression • Principle of suppression: Thin layers of small resistivity contrast with respect to background will be missed. • Thin layers of greater resistivity contrast will be detectable, but equivalence limits resolution of boundary depths, etc. 12