SAIC-03/1040 Final Report Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Prepared by: J. W. Pritchett December 2003
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Science Applications International Corporation 10260 Campus Point Drive, Mail Stop A1 Phone 858-826-1628 Fax 858-826-1652
TABLE OF CONTENTS Section Page LIST OF FIGURES... v LIST OF TABLES... ix 1 Introduction... 1-1 2 Calculations of Microgravity Change... 2-1 3 DC Resistivity Survey Simulations... 3-1 4 Magnetotelluric Survey Simulations... 4-1 5 Calculations of Self-Potential (SP)... 5-1 6 Acknowledgement... 6-1 7 References... 7-1 iii
LIST OF FIGURES Figure Page 2.1. Vertical section through center of computational grid, showing imposed initial conditions and boundary conditions... 2-2 2.2. Computed decrease of total steam volume with time.... 2-4 2.3. Computed system state at t = 5 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 5 days (70.0% of initial steam remains). Blue: instantaneous distribution of fluid pressure.... 2-5 2.4. Computed system state at t = 10 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 10 days (42.9% of initial steam remains). Blue: instantaneous distribution of fluid pressure.... 2-6 2.5. Computed system state at t = 15 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 15 days (23.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure.... 2-7 2.6. Computed system state at t = 20 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 20 days (9.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure.... 2-8 2.7. Computed system state at t = 25 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 25 days (1.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure.... 2-9 2.8. Computed system state at t = 30 days. Red: initial volume occupied by steam (steam volume reached zero at t = 27 days). Blue: instantaneous distribution of fluid pressure... 2-10 2.9. Computed system state at t = 50 days. Red: initial volume occupied by steam (steam volume reached zero at t = 27 days). Blue: instantaneous distribution of fluid pressure... 2-11 2.10. Computed increase in earth-surface microgravity for t = 5, 10, 15, 20, 25, 30 and 50 days.... 2-12 2.11. Comparison of analytic solution (green) with microgravity change computed by postprocessor (red) for t = 50 days.... 2-13 3.1. Computational domain and spatial discretization employed for calculations with DC resistivity survey postprocessor. Blue: grid block boundaries. Upper: vertical cross-section. Lower: view from above... 3-2 3.2. Electrode arrangements adopted for Square Dipole-Dipole DC resistivity survey calculations... 3-3 v
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure Page 3.3. Electrode arrangements adopted for Linear Dipole-Dipole DC resistivity survey calculations... 3-4 3.4. Electrode arrangements adopted for Schlumberger DC resistivity survey calculations.... 3-5 3.5. Electrode arrangements adopted for Wenner DC resistivity survey calculations.... 3-6 3.6. Effects of electrode arrangement, array orientation, and spatial grid resolution on computed apparent resistivity results for a uniform 100 ohm-meter half-space earth.... 3-7 3.7. Geometry chosen for nonuniform-earth DC resistivity survey calculations. Horizontal grid block spacing = vertical spacing = 200 meters except at great distances from Wenner array (see Figure 3.1). Transition depth D separating shallow region from deep region varies from zero to 2000 meters (0 to 10 computational layers). Geometry for D = 400 m shown. Shallow region conductivity (depth < D) is either 10 or 1000 ohm-meters. At greater depths, resistivity is 100 ohm-meters.... 3-11 3.8. Comparison of analytic solution with results calculated by DC resistivity survey postprocessor for non-uniform two-layer earth... 3-12 4.1. Computational domain and problem geometry for MT survey postprocessor calculations of two-layer problem. Depth of resistivity discontinuity (D) is 800 meters. Above D, Ω = Ω S = 250 ohm-meters. Below D, Ω= Ω D = 250 ohm-meters (case A ), 125 ohm-meters (case B ) or 62.5 ohm-meters (case C )... 4-2 4.2. Comparison of apparent resistivities calculated by MT postprocessor with analytic solution of Cagniard (1953) for various frequencies... 4-4 4.3. Comparison of phase angles calculated by MT postprocessor with analytic solution of Cagniard (1953) for various frequencies.... 4-5 4.4. Geometry considered for CSAMT postprocessor calculations of two-layer problem. Frequency f is 15 Hz (cases A-1, B-1, C-1), 60 Hz (A-2, B- 2, C-2) or 240 Hz (A-3, B-3, C-3). Depth of resistivity discontinuity D is 800 meters (cases A-1, B-1, C-1), 400 m (A-2, B-2, C-2) or 200 m (A-3, B-3, C-3). Shallow resistivity Ω S is 250 ohm-meters for all cases. Deep resistivity Ω D is 250 ohm-m (A-1, A-2, A-3), 125 ohm-m (B-1, B-2, B-3) or 62.5 ohm-m (C-1, C-2, C-3). Distance between dipole signal source and survey point (R) varies from 141 to 5798 meters... 4-8 4.5. Computational domain and spatial discretization for CSAMT simulations (cases A-1, B-1 and C-1). Frequency f = 15 Hz and depth of resistivity discontinuity D = 800 meters (shallow region consists of eight 100-meter computational layers).... 4-10 vi
List of Figures Figure Page 4.6. Computational domain and spatial discretization for CSAMT simulations (cases A-2, B-2 and C-2). Frequency f = 60 Hz and depth of resistivity discontinuity D = 400 meters (shallow region consists of four 100-meter computational layers)... 4-11 4.7. Computational domain and spatial discretization for CSAMT simulations (cases A-3, B-3 and C-3). Frequency f = 240 Hz and depth of resistivity discontinuity D = 200 meters (shallow region consists of two 100-meter computational layers).... 4-12 4.8. Apparent resistivity values in survey area computed by CSAMT postprocessor as functions of S/R (skin depth distance to signal source). Black: high-resolution cases A-1, B-1 and C-1. Red: intermediate resolution cases A-2, B-2 and C-2. White: low resolution cases A-3, B-3 and C-3. Green: limiting values as R (S/R 0) from Cagniard (1953) solution... 4-13 4.9. Phase angle values in survey area computed by CSAMT postprocessor as functions of S/R (skin depth distance to signal source). Black: high-resolution cases A-1, B-1 and C-1. Red: intermediate resolution cases A-2, B-2 and C-2. White: low resolution cases A-3, B-3 and C- 3. Green: limiting values as R (S/R 0) from Cagniard (1953) solution... 4-14 4.10. Computational domain and problem geometry for MT survey postprocessor calculations of vertical resistivity contact problem. Earth resistivity is equal to 125 ohm-meters for x < 0 and 250 ohmmeters for x 0. MT surveys performed along y = 0 for 2000 m East x +2000 m East. Frequencies considered are 10, 20, 40, 80, 160, 320 and 640 Hz... 4-16 4.11. Apparent resistivities computed by MT survey postprocessor at grid block centers for vertical resistivity contact problem along y = 0 for 2000 m East x +2000 m East. Frequencies considered are 10, 20, 40, 80, 160, 320 and 640 Hz... 4-17 4.12. Computed apparent MT resistivities at grid block centers for vertical resistivity contact problem as functions of x/s.... 4-18 5.1. Computational domain and spatial discretization for SP survey postprocessor calculations, viewed from above.... 5-2 5.2. Computational domain viewed from the side. Upper: grid for STAR reservoir simulator calculations. Lower: grid for SP survey postprocessor calculations.... 5-4 vii
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure Page 5.3. Computed SP distribution at the earth surface after 4000 days of stabilization in central portion of computational domain. Red: electrical potential contours (millivolts). White: region of negative potential. Yellow: region of positive potential. Blue: grid block boundaries. Green: wellhead locations. Black: location of representative survey line A-B... 5-6 5.4. Comparison of computed 4000-day SP distribution along survey line A-B (red) with analytical solution (green)... 5-7 5.5. Correlation between change in fluid pressure and electrokinetic potential. Green: theoretical linear relationship with streaming potential = 77.44 millivolts per bar. Red: computed values at grid-block centers (all 29,040 grid blocks displayed), with pressure values from STAR reservoir simulation and potential values from SP postprocessor result.... 5-11 5.6. Magnified view of central portion of Figure 5.5... 5-12 5.7. Additional magnification of Figure 5.5.... 5-13 viii
LIST OF TABLES Table Page 2.1. Microgravity Change at 50 Days for y = 0 Meters North... 2-14 3.1. Uniform Case Computed Results for Square Dipole-Dipole Electrode Array (see Figure 3.2).... 3-8 3.2. Uniform Case Computed Results for Linear Dipole-Dipole Electrode Array (see Figure 3.3).... 3-8 3.3. Uniform Case Computed Results for 4:1 Schlumberger Electrode Array (see Figure 3.4)... 3-9 3.4. Uniform Case Computed Results for Wenner Electrode Array (see Figure 3.5).... 3-9 3.5. Computed Apparent DC Resistivity Results for Two-Layer Problem using Wenner Electrode Array (A = 1000 m = 5 x).... 3-13 4.1. Comparison of Computed MT Survey Results with Analytical Solution of Cagniard (1953) for Two-Layer Problem.... 4-6 4.2. Cases Considered for CSAMT Survey Simulations.... 4-9 5.1. Comparison of Computed Earth-Surface Self-Potential Distribution with Analytical Results... 5-8 5.2. Temporal Trends in Computed Earth-Surface SP and Comparison with Analytic Steady Solution.... 5-9 ix
1 INTRODUCTION Earth-surface survey techniques that were originally developed for geophysical exploration are attracting increasing attention in geothermal reservoir engineering because of their potential utility for monitoring temporal changes in underground conditions within and around the reservoir at moderate cost. Repeat microgravity surveys were first performed starting forty years ago at the Wairakei geothermal field in New Zealand, and the results did much to elucidate subsurface mass changes and the role of natural recharge in the reservoir (Hunt, 1977). Other promising techniques include time-lapse surveys of subsurface electrical resistivity (using either traditional DC electrode arrays or electromagnetic methods such as MT and CSAMT surveys), repeat surveys and/or continuous monitoring of electrical selfpotential (SP) at the earth surface, and repeat seismic surveying. History-matching studies are a crucial element in the development of reliable mathematical models for operating geothermal fields using numerical reservoir simulators. If the data sets that are ordinarily available for history-matching (such as records of changes in subsurface pressures and temperatures, and wellhead flow rate and enthalpy histories) could be supplemented with data collected intermittently from geophysical surveys such as those listed above, the result would presumably be more robust reservoir models and a better predictive capability, leading eventually to better reservoir management, greater efficiency, and a more competitive geothermal power industry (Pritchett et. al, 2000). Conventional numerical reservoir simulators, of course, provide a forward modeling capability for subsurface histories of such quantities as pressure, temperature and steam saturation, but do not by themselves allow the user to calculate the corresponding changes in such quantities as earth-surface gravity, electrical resistivity, or electrical potential. To provide the necessary link between measured survey results from the field and the forecasts from numerical simulation studies, postprocessors have been developed that obtain the 3-D subsurface histories of the primary quantities (pressure, temperature, etc.) from the results of conventional numerical reservoir simulations, and then perform additional calculations to predict the changes in geophysical observables that would be measured at the earth surface if the mathematical reservoir model were in fact correct. The existing suite of geophysical postprocessors, described by Pritchett (2002), was developed for use with the STAR geothermal reservoir simulator, which is used principally overseas. To promote the utilization of this technology within the U.S. geothermal community, Shook and Renner (2002) have undertaken to create an adapter to permit reservoir simulation calculations performed with the TETRAD simulator to be coupled with the STAR postprocessors. The present report describes a group of problems (computed using the STAR reservoir simulator and the STAR postprocessors) that are intended for subsequent calculation using the TETRAD implementation to verify that the adapter is performing properly. An additional purpose is to provide a suite of results that have been checked against analytical solutions to demonstrate that the postprocessors 1-1
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite themselves are reliable. The requirement for the existence of analytical solutions naturally restricts the present study to fairly simple geometries, but the test problems involve the same essential physics needed for more realistic field situations. To facilitate subsequent calculations by other investigators, a CD disk containing all input files required to run these problems using STAR (and most of the output files as well) is included as an integral part of the present report. Detailed instructions for running STAR and the postprocessors are provided by Pritchett (2002). The present calculations were performed using STAR Version 9.0 on a UNIX scientific workstation, but STAR is also available configured for execution on a personal computer using the Microsoft Windows operating system (Windows 98, NT, ME, 2000, or XP). FORTRAN and C compilers are required for a full STAR installation. At present, the STAR postprocessor suite consists of five separate programs a sixth (the seismic postprocessor) is presently under development but not yet released, and was not included in the present study. The five existing postprocessors are: The Microgravity postprocessor. This program calculates the changes in surface gravity caused by production, injection and spatial redistributions of fluid mass within the underlying reservoir volume. This is accomplished by brute-force numerical integration of Newton s Law of Gravitation for observation points located on an irregular plane representing the surface of the earth, caused by changes in fluid mass within the grid block volumes in the subsurface computational mesh used for the reservoir simulation calculation. The program is described in detail in Chapter 13 of Pritchett (2002). The DC Resistivity postprocessor. The reservoir (STAR) grid is embedded within a (usually larger) electrical grid for the calculation of voltage and electric current. The user prescribes the spatial distribution of electrical resistivity, and the effect of temporal reservoir changes (changes in temperature, fluid chemistry, steam saturation etc.) upon changes in resistivity is taken into account by interpolation from the reservoir grid to the electrical grid. Subject to the distribution of electrical resistivity, the voltage induced in each grid block by the current electrodes is calculated and evaluated at the locations of the voltage electrodes. Results are interpreted in terms of apparent resistivity. See Chapter 14 of Pritchett (2002). The MT Resistivity postprocessor. As described by Wannamaker (2001), the computational kernel within this postprocessor is based on the mathematical technique originally developed by Sasaki (1999; 2001). From the user s standpoint, the structure is similar to that of the DC Resistivity package (above), with an electrical grid overlying the reservoir grid and user-specified models for the distribution of electrical resistivity and of the influence of reservoir changes on resistivity change. Diagnostic results include spatial distributions of apparent resistivity and phase angle over the surface of the earth, and how these quantities change with time. See Pritchett (2002), Chapter 15. The CSAMT Resistivity postprocessor. This package differs from the above MT postprocessor mainly in that it allows for non-planar incident electromagnetic wavefronts, as can be important under pathological circumstances if an artificial signal source (at finite 1-2
Introduction distance) is employed instead of natural electromagnetic signals. The essential approach is outlined by Wannamaker (2002) and by Sasaki (1999; 2001); the implementation within the STAR program system is described in Chapter 16 of Pritchett (2002). The SP (Self-Potential) postprocessor. The SP program calculates electrokinetic potentials (voltages) at and below the earth surface that arise from the drag currents induced by the subterranean time-dependent flow of liquid interacting with the molecularscale electrical double layer at the fluid-solid interfaces, subject to the (temporally-varying) macroscopic spatial distribution of electrical resistivity. Such distributions of electrical potential and temporal potential change represent a powerful technique for remote sensing of subsurface fluid flow (Ishido and Pritchett, 2000). Ishido and Mizutani (1981) describe the essential mechanisms and pertinent models for the requisite electrical properties of representative geothermal rocks. The computational technique again involves an electrical finite-difference grid within which the reservoir grid is embedded, and is described (in Chapter 17) by Pritchett (2002). Sections 2 5 of the present report outline the test problems devised for the present study, display the results obtained using the computational postprocessors, and compare these results with analytic solutions. Section 2 describes a problem involving the collapse of a large spherical subsurface low-pressure steam bubble and the effects upon the earthsurface microgravity distribution above the spherical region. Section 3 provides illustrative calculations for a half-space of uniform electrical resistivity, and compares postprocessor simulations of DC resistivity surveys using various electrode array geometries and differing degrees of spatial resolution with the exact result. Then, a two-layer earth model is examined using one of the array geometries, both for a relatively resistive and a relatively conductive shallow layer, and the dependence of apparent resistivity upon the depth of the actual resistivity discontinuity is compared with analytical solutions. In Section 4, both MT and CSAMT techniques are employed to study a similar two-layer earth, for various signal frequencies and depths to the resistivity discontinuity. The MT postprocessor is also used to examine the effects of grid spatial resolution upon the computed apparent resistivity distribution over an infinite half-space within which the electrical resistivity depends on horizontal position. Finally, Section 5 examines the effects of the operation of a pair of shallow wells a production well and an injection well on the distribution of electrical selfpotential along the earth surface. 1-3
2 CALCULATIONS OF MICROGRAVITY CHANGE Figure 2.1 illustrates the geometry of the problem considered. Viewed from above, the computational domain is square and extends over 500 m East x +500 m East and 500 m North y +500 m North. Vertically, the domain is 500 m thick and extends from 100 meters depth to 600 meters depth. The spatial discretization chosen for use by the STAR numerical reservoir simulator uses x = y = z = 20 meters in the central region, with larger horizontal spacing for x or y beyond ± 250 meters. Formation properties are uniform: porosity is 0.25, absolute permeability is 10 13 m 2 (100 millidarcies), and relative permeabilities are of the straight-line type with residual water and steam saturations of 0.30 and 0.05 respectively: Water relative permeability = larger of zero, Steam relative permeability = larger of zero, 0.7 S V 0.7 SV 0.05 0.95 (2-1) (2-2) where S V is steam saturation (steam volume total fluid volume). The rock grain density is 2500 kg/m 3 (so the dry formation density is 1875 kg/m 3 and the saturated density is ~2115 kg/m 3 ), the rock grain heat capacity is 1000 J/kg- C and the thermal conductivity of the formation is 3 W/m- C. Initially, temperature is uniform and equal to 100 C. Within a 300-meter diameter spherical region centered at 350 meters depth and at x = y = 0, the pore pressure is equal to one atmosphere (1.01325 bars) and the steam saturation is equal to unity (no liquid water). Outside the spherical region, no steam is present and the pressure distribution is hydrostatic, varying between 10 bars at 100 meters depth (the top of the domain) and 57 bars at 600 meters depth (the bottom). For computational purposes, each grid block was initially examined to see if it lies (1) completely outside the sphere, (2) completely inside the sphere, or (3) partially outside and partially inside. If the block is completely outside the sphere, hydrostatic pressure was assigned to the block initially. If it is completely inside the sphere, pressure was assigned as one atmosphere and the initial steam saturation was set to unity. If the block is only partially inside the sphere, the initial pressure was again set to one atmosphere and the initial steam saturation was set equal to the fraction of the grid block volume lying within the sphere. 2-1
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.1. Vertical section through center of computational grid, showing imposed initial conditions and boundary conditions. 2-2
Calculations of Microgravity Change The lower boundary at 600 m depth is impermeable and insulated. Along the vertical sides (at x,y = ±500 meters) and the upper surface of the domain (at 100 meters depth), pressure and temperature are maintained at their initial values. The calculation was carried out to t = 50 days, using a computational time-step size of three hours. As time goes on, the steam bubble collapses and floats upwards slightly. At t = 27 days, the steam volume reaches zero, as shown in Figure 2.2. Figures 2.3 2.9 illustrate the temporal evolution of the steam-filled region and the underground distribution of pressure, at t = 5, 10, 15, 20, 25, 30, and 50 days respectively, as computed by the STAR reservoir simulator. Temperature remains constant at 100 C throughout, within a small fraction of a degree. Microgravity change (relative to the starting conditions) over the earth surface (zero depth) was then calculated using the gravity postprocessor for the same times, as indicated in Figure 2.10. Results are essentially invariant after 27 days. In Table 2.1 and Figure 2.11, comparisons are provided for t = 50 days with the analytical solution, given by: 4π gxy (, ) ( ) [ GϕDρW ρs ] = 3 2 2 2 R D + x + y 3 (2-3) where -3 2 G = universal gravitational constant (6.672 10 microgal/kg-m ), D = initial depth of center of spherical region (350 m), R = initial spherical region radius (150 m), ϕ = rock porosity (0.25), 3 ρ = liquid water density (959.9 kg/m ), and W ρ S = steam density (0.6 kg/m ). 3 As indicated in Table 2.1, the maximum absolute (0.037 microgals) and relative (1/4990; about 0.02%) deviations between the above analytical solution and the computed results appear directly above the center of the spherical region at x = y = 0, and both absolute and relative errors decline with increasing distance. The principal cause of the errors appears to be slight departures of the shape of the initial steam bubble from ideal spherical form owing to finite grid-block size, which primarily influences gravity changes at close-in locations. 2-3
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.2. Computed decrease of total steam volume with time. 2-4
Calculations of Microgravity Change Figure 2.3. Computed system state at t = 5 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 5 days (70.0% of initial steam remains). Blue: instantaneous distribution of fluid pressure. 2-5
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.4. Computed system state at t = 10 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 10 days (42.9% of initial steam remains). Blue: instantaneous distribution of fluid pressure. 2-6
Calculations of Microgravity Change Figure 2.5. Computed system state at t = 15 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 15 days (23.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure. 2-7
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.6. Computed system state at t = 20 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 20 days (9.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure. 2-8
Calculations of Microgravity Change Figure 2.7. Computed system state at t = 25 days. Red: initial volume occupied by steam. Yellow: steam-filled volume at 25 days (1.1% of initial steam remains). Blue: instantaneous distribution of fluid pressure. 2-9
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.8. Computed system state at t = 30 days. Red: initial volume occupied by steam (steam volume reached zero at t = 27 days). Blue: instantaneous distribution of fluid pressure. 2-10
Calculations of Microgravity Change Figure 2.9. Computed system state at t = 50 days. Red: initial volume occupied by steam (steam volume reached zero at t = 27 days). Blue: instantaneous distribution of fluid pressure. 2-11
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 2.10. Computed increase in earth-surface microgravity for t = 5, 10, 15, 20, 25, 30 and 50 days. 2-12
Calculations of Microgravity Change Figure 2.11. Comparison of analytic solution (green) with microgravity change computed by postprocessor (red) for t = 50 days. 2-13
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Table 2.1. Microgravity Change at 50 Days for y = 0 Meters North x (m E) Analytic (µgal) Computed (µgal) Difference (µgal) 0 184.666 184.629 0.037 100 164.160 164.314 0.026 200 120.868 120.855 0.013 300 80.826 80.821 0.005 400 52.731 52.729 0.002 500 34.826 34.825 0.001 600 23.624 23.624 0.000 700 16.517 16.517 0.000 800 11.891 11.891 0.000 2-14
3 DC RESISTIVITY SURVEY SIMULATIONS Uniform Resistivity Cases Viewed from above, the computational domain is square and extends over 7500 m East x +7500 m East and 7500 m North y +7500 m North. Vertically, the domain extends from the earth surface (a flat horizontal plane) down to 7500 meters depth. The DC resistivity postprocessor s electrical grid uses 200 meter spacing in the central region at relatively shallow depths, with coarser resolution at greater distances. The geometry of this region and of the computational grid is illustrated in Figure 3.1. The specified distribution of actual subsurface electrical resistivity is uniform and equal to 100 ohm-meters throughout this volume. The upper surface is insulated and the subsurface electrical conductivity at large distances (below and beyond the computational grid volume) is treated as infinite. Four different electrode geometries are considered at two different orientations (with a line from voltage electrode to current electrode oriented parallel and diagonal relative to the principal grid lines). Each arrangement is characterized by a distance A, equal to the minimum separation between a voltage electrode and a current electrode. Calculations were carried out for A = 200, 300, 400, 500, 600, 800 and 1000 meters (electrode separation = 1.0, 1.5, 2.0, 2.5, 3.0, 4.0 and 5.0 grid block sizes). The electrode array geometries are illustrated (for A = 5 x) in Figures 3.2 3.5. Since apparent resistivity (Ω A ) may be defined as the electrical resistivity of a uniform unbounded half-space with the same ratio of voltage to current as that observed for the same electrode array geometry in the field, it follows that in principle the results of all of these calculations should be Ω A = 100 ohm-meters, apart from the effects of numerical discretization. Results (illustrated in Figure 3.6) indicate that the calculated apparent resistivity is near 100 ohm-meters in all cases, with discrepancies decreasing as resolution improves. So long as A is equal to 5 grid blocks or more, deviations from exact results will be less than 3%. For each of the 56 cases of which the results are summarized in Figure 3.6 (four electrode geometries seven A values two orientations = 56 cases), 4096 calculations of apparent resistivity were actually performed, varying the positions of the electrodes slightly relative to the computational grid blocks. The center of the electrode array was perturbed over the rectangle [ 200 m x +200 m, 200 m y +200 m] using a 64 64 sub-grid with 6.25 meter spacing; this area corresponds to the surface area of the four central blocks in the computational grid. Results depend slightly on the exact location of the electrodes relative to the grid blocks themselves (in a manner that is spatially periodic, with wavelength one grid block size), as indicated in Tables 3.1 3.4. For each electrode arrangement, these tables list (1) the minimum value obtained from the 4096 individual apparent resistivity Continued on page 3-10 3-1
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 3.1. Computational domain and spatial discretization employed for calculations with DC resistivity survey postprocessor. Blue: grid block boundaries. Upper: vertical cross-section. Lower: view from above. 3-2
DC Resistivity Survey Simulations Figure 3.2. Electrode arrangements adopted for Square Dipole-Dipole DC resistivity survey calculations. 3-3
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 3.3. Electrode arrangements adopted for Linear Dipole-Dipole DC resistivity survey calculations. 3-4
DC Resistivity Survey Simulations Figure 3.4. Electrode arrangements adopted for Schlumberger DC resistivity survey calculations. 3-5
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 3.5. Electrode arrangements adopted for Wenner DC resistivity survey calculations. 3-6
DC Resistivity Survey Simulations Figure 3.6. Effects of electrode arrangement, array orientation, and spatial grid resolution on computed apparent resistivity results for a uniform 100 ohmmeter half-space earth. 3-7
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Table 3.1. Uniform Case Computed Results for Square Dipole-Dipole Electrode Array (see Figure 3.2). Parallel Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 75.239 75.295 75.267 ±0.015 300 1.5 99.247 99.307 99.276 ±0.016 400 2 104.843 104.881 104.862 ±0.010 500 2.5 106.221 106.250 106.236 ±0.008 600 3 105.622 105.642 105.632 ±0.005 800 4 103.869 103.881 103.876 ±0.003 1000 5 102.491 102.501 102.496 ±0.002 Diagonal Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 66.460 66.479 66.471 ±0.006 300 1.5 78.482 78.508 78.498 ±0.007 400 2 85.329 85.344 85.338 ±0.004 500 2.5 90.113 90.119 90.116 ±0.002 600 3 92.784 92.789 92.787 ±0.001 800 4 95.800 95.806 95.804 ±0.001 1000 5 97.083 97.095 97.090 ±0.002 Table 3.2. Uniform Case Computed Results for Linear Dipole-Dipole Electrode Array (see Figure 3.3). Parallel Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 67.750 67.784 67.767 ±0.009 300 1.5 89.744 89.780 89.762 ±0.010 400 2 98.808 98.834 98.821 ±0.007 500 2.5 101.634 101.654 101.645 ±0.005 600 3 102.438 102.453 102.456 ±0.004 800 4 102.213 102.221 102.217 ±0.003 1000 5 101.869 101.876 101.872 ±0.001 Diagonal Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 74.569 74.575 74.572 ±0.001 300 1.5 86.804 86.810 86.807 ±0.002 400 2 91.902 91.905 91.903 ±0.0007 500 2.5 94.981 94.984 94.982 ±0.0008 600 3 96.512 96.516 96.514 ±0.0009 800 4 98.264 98.267 98.266 ±0.0005 1000 5 99.178 99.181 99.179 ±0.0005 3-8
DC Resistivity Survey Simulations Table 3.3. Uniform Case Computed Results for 4:1 Schlumberger Electrode Array (see Figure 3.4). Parallel Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 68.365 68.387 68.375 ±0.006 300 1.5 87.891 87.923 87.907 ±0.009 400 2 97.691 97.712 97.701 ±0.006 500 2.5 100.592 100.613 100.603 ±0.006 600 3 102.009 102.022 102.015 ±0.003 800 4 101.816 101.825 101.821 ±0.002 1000 5 101.224 101.229 101.227 ±0.001 Diagonal Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 72.709 72.720 72.713 ±0.003 300 1.5 87.338 87.343 87.340 ±0.001 400 2 91.475 91.477 91.476 ±0.0003 500 2.5 94.815 94.816 94.816 ±0.0003 600 3 96.159 96.162 96.161 ±0.001 800 4 97.855 97.856 97.856 ±0.0002 1000 5 98.479 98.481 98.481 ±0.0002 Table 3.4. Uniform Case Computed Results for Wenner Electrode Array (see Figure 3.5). Parallel Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 78.160 78.190 78.175 ±0.008 300 1.5 93.912 93.940 93.926 ±0.007 400 2 99.788 99.808 99.798 ±0.005 500 2.5 101.436 101.452 101.444 ±0.004 600 3 101.777 101.788 101.783 ±0.003 800 4 101.259 101.265 101.262 ±0.002 1000 5 100.587 100.593 100.590 ±0.001 Diagonal Array Orientation: Minimum Ω A Maximum Ω A Average Ω A RMS Deviation A (m) A/ x 200 1 81.059 81.061 81.060 ±0.0004 300 1.5 90.168 90.171 90.169 ±0.0007 400 2 93.973 93.974 93.973 ±0.0003 500 2.5 96.193 96.194 96.193 ±0.0002 600 3 97.244 97.246 97.245 ±0.0004 800 4 98.296 98.298 98.297 ±0.0003 1000 5 98.586 98.588 98.587 ±0.0004 3-9
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite calculations for each situation, (2) the maximum value obtained, (3) the average of the 4096 values (this is the quantity plotted in Figure 3.6) and (4) the root-mean-square deviation of the various values from the average. The largest RMS deviation encountered was ±0.016 ohm-meters, which is insignificant compared to the 100 ohm-meter resistivity of the computational domain. Non-Uniform Resistivity Cases Additional DC resistivity survey simulations consider the same spatial domain and the same computational grid geometry. This time, the actual subsurface electrical resistivity is again taken equal to 100 ohm-meters for depths exceeding a transition depth D, but at shallower depths the resistivity is either higher (1000 ohm-meters) or lower (10 ohm-meters), as shown in Figure 3.7 for cases with D = 800 meters = 4 x. Calculations were performed for D = 0, 200, 400,, 2000 meters (0, x, 2 x, 3 x,, 10 x). A Wenner electrode arrangement was considered, at two different orientations (with the electrode line oriented parallel and diagonal to the principal grid lines, as before). The electrode spacing is 1000 meters (A = 5 x; also see Figure 3.5). For a region of unbounded lateral extent and infinite depth, this problem can be solved analytically. The apparent resistivity (Ω A ) depends on D, the actual resistivity in the shallow region (Ω = Ω S for depth D; either 10 or 1000 ohm-meters), the actual resistivity at greater depths (Ω = Ω D for depth > D; 100 ohm-meters), and the Wenner electrode spacing (A; 1000 meters): where: A S [ 1 2 1 2] Ω =Ω + (3-1) j χ 1 = (3-2) 2 j= 1 jd 1 + A 4 j χ 2 = (3-3) 2 j= 1 jd + 1 A where χ is given by: ( ΩD ΩS ) χ = (3-4) ( Ω +Ω ) D S As shown in Figure 3.8, results for apparent resistivity calculated by the DC resistivity postprocessor are in good agreement with the above analytic solution. Numerical values are provided in Table 3.5. If the shallow-layer resistivity Ω S is equal to 10 ohmmeters (conductive shallow layer) both the diagonal and parallel electrode orientation 3-10
DC Resistivity Survey Simulations Figure 3.7. Geometry chosen for nonuniform-earth DC resistivity survey calculations. Horizontal grid block spacing = vertical spacing = 200 meters except at great distances from Wenner array (see Figure 3.1). Transition depth D separating shallow region from deep region varies from zero to 2000 meters (0 to 10 computational layers). Geometry for D = 400 m shown. Shallow region conductivity (depth < D) is either 10 or 1000 ohmmeters. At greater depths, resistivity is 100 ohm-meters. 3-11
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 3.8. Comparison of analytic solution with results calculated by DC resistivity survey postprocessor for non-uniform two-layer earth. 3-12
DC Resistivity Survey Simulations Table 3.5. Computed Apparent DC Resistivity Results for Two-Layer Problem using Wenner Electrode Array (A = 1000 m = 5 x). Shallow Region Resistivity = 10 ohm-meters: Analytic Solution Computed Using Diagonal Array Computed Using Parallel Array D (m) D/ x 0 0 100.00 98.59 100.59 200 1 43.38 42.59 43.01 400 2 26.71 26.22 26.46 600 3 19.59 19.20 19.40 800 4 15.89 15.55 15.74 1000 5 13.80 13.48 13.68 1200 6 12.56 12.26 12.46 1400 7 11.79 11.50 11.70 1600 8 11.29 11.01 11.21 1800 9 10.95 10.69 10.89 2000 10 10.72 10.47 10.67 Shallow Region Resistivity = 1000 ohm-meters: Analytic Solution Computed Using Diagonal Array Computed Using Parallel Array D (m) D/ x 0 0 100 99 101 200 1 113 113 118 400 2 237 235 250 600 3 441 432 451 800 4 613 602 622 1000 5 734 722 742 1200 6 814 802 823 1400 7 867 856 876 1600 8 902 892 912 1800 9 927 916 937 2000 10 944 934 954 3-13
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite calculations tend to slightly underestimate the analytical result for apparent resistivity Ω A, typically by around 1% ( parallel ) or 2% ( diagonal ). On the other hand, if the shallow layer is resistive (Ω S = 1000 ohm-m), the diagonal calculations underestimate the analytic value slightly whereas the parallel calculations provide a slight overestimate, each typically by 1% to 2%. Presumably, improving the spatial resolution beyond A/ x = 5 would further reduce these errors. 3-14
4 MAGNETOTELLURIC SURVEY SIMULATIONS Layered Earth Calculations Similar two-layer problems are considered for illustrative calculations using the MT and CSAMT resistivity survey postprocessors. Figure 4.1 shows the general problem geometry and the computational discretization employed for these MT calculations. The volume considered extends over 5000 m East x +5000 m East and 5000 m North y +5000 m North, and from the (flat) earth surface down to 5000 meters depth. The horizontal grid block spacing is 200 meters in the central region with larger blocks near the lateral boundaries, and the vertical spatial resolution is 100 meters per layer above 1100 meters depth with thicker layers below. The total number of grid blocks is 32 32 21 = 21,504. The shallow region consists of the uppermost eight layers of grid blocks (800 meters total thickness). Three cases are considered, which differ only in the electrical resistivity assigned below depth D = 800 meters. In all cases, the shallow 800-meter-thick region above this level (the uppermost eight layers of grid blocks) has an electrical resistivity Ω S = 250 ohmmeters. Below 800 meters, cases A, B and C have electrical resistivities Ω D = 250, 125 and 62.5 ohm-meters respectively. In MT and CSAMT soundings, signal frequency plays a role that is analogous to that of electrode spacing in DC resistivity surveys. Lower frequencies (like larger DC arrays) provide greater depth of penetration into the earth, but also result in poorer spatial resolution. For a simple uniform half-space, the electromagnetic signal will decrease by a factor (1/e) after penetration of the skin depth S, given by: where 2Ω S = (4-1) µ ω Ω= electrical resistivity of the earth, 7 µ o = magnetic permeability of free space = 4π 10 Henry/meter, ω = angular frequency = 2 π f, and f = signal frequency (Hz). With the above value for µ o and for Ω expressed in ohm-meters and f in Hz, this means: o Ω S = 503.3 meters. (4-2) f 4-1
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 4.1. Computational domain and problem geometry for MT survey postprocessor calculations of two-layer problem. Depth of resistivity discontinuity (D) is 800 meters. Above D, Ω = Ω S = 250 ohm-meters. Below D, Ω= Ω D = 250 ohm-meters (case A ), 125 ohm-meters (case B ) or 62.5 ohm-meters (case C ). 4-2
Magnetotelluric Survey Simulations For each of Cases A, B and C, nine values were selected for signal frequency f resulting in a range of values for the skin depth S with Ω = Ω S = 250 ohm-m: f, Hz S, m f, Hz S, m 240 513.7 7.5 2905.8 120 726.4 3.75 4109.4 60 1027.3 1.875 5811.5 30 1452.9 0.9375 8218.7 15 2054.7 Diagnostic results from magnetotelluric surveys include apparent resistivity (Ω A ) which is analogous to the corresponding quantity obtained from DC surveys, and the phase angle (θ) between the electric and magnetic waves. These quantities are defined such that, for a plane electromagnetic wave incident on an unbounded half-space with uniform electrical resistivity Ω o, Ω A = Ω o and θ = π/4 = 45. Results obtained from the MT postprocessor for Ω A and θ are independent of the location of the monitoring station within the Figure 4.1 MT Survey Area for Cases A, B and C, and are displayed (in red) in Figures 4.2 and 4.3. An analytic solution is available for this problem, based on Cagniard (1953); the dependence of Ω A and θ upon S, D, Ω S and Ω D may be expressed by: 2 Ω A ( α exp ξ) + 2α cosξexpξ+ 1 = 2 Ω ( α exp ξ) 2α cosξexpξ+ 1 S 2 ( αexp ξ) 2αsinξexpξ 1 tanθ = 2 ( αexp ξ) + 2αsinξexpξ 1 (4-3) (4-4) where Ω D + ΩS α = (4-5) Ω Ω D S ωµ o 2D ξ = 2D = 2Ω S S (4-6) This analytic solution is also displayed (in green) in Figures 4.2 and 4.3. Quantitative comparisons between numerical calculations and analytic results are provided in Table 4.1. Agreement between computed and analytic results is essentially exact. The major difference between the MT postprocessor and the CSAMT postprocessor is that the latter takes into account the possibility that the incident wavefronts will be curved owing to the finite distance between the survey point and the artificial signal source 4-3
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 4.2. Comparison of apparent resistivities calculated by MT postprocessor with analytic solution of Cagniard (1953) for various frequencies. 4-4
Magnetotelluric Survey Simulations Figure 4.3. Comparison of phase angles calculated by MT postprocessor with analytic solution of Cagniard (1953) for various frequencies. 4-5
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Table 4.1. Comparison of Computed MT Survey Results with Analytical Solution of Cagniard (1953) for Two-Layer Problem. Apparent resistivity Ω A Phase angle θ f Ω S Ω D Analytic Computed Analytic Computed Case A 0.9375 250 250 250.000 250.000 45.000 45.000 Case A 1.875 250 250 250.000 250.000 45.000 45.000 Case A 3.75 250 250 250.000 250.000 45.000 45.000 Case A 7.5 250 250 250.000 250.000 45.000 45.000 Case A 15 250 250 250.000 250.000 45.000 45.000 Case A 30 250 250 250.000 250.000 45.000 45.000 Case A 60 250 250 250.000 250.000 45.000 45.000 Case A 120 250 250 250.000 250.000 45.000 45.000 Case A 240 250 250 250.000 250.000 45.000 45.000 Case B 0.9375 250 125 143.178 143.178 48.191 48.191 Case B 1.875 250 125 151.103 151.103 49.121 49.121 Case B 3.75 250 125 162.469 162.469 50.112 50.112 Case B 7.5 250 125 178.458 178.458 50.968 50.968 Case B 15 250 125 199.874 199.874 51.352 51.352 Case B 30 250 125 225.534 225.534 50.828 50.828 Case B 60 250 125 249.518 249.518 49.140 49.140 Case B 120 250 125 261.451 261.451 46.754 46.754 Case B 240 250 125 257.730 257.730 45.023 45.023 Case C 0.9375 250 62.5 83.183 83.183 51.549 51.549 Case C 1.875 250 62.5 92.976 92.976 53.365 53.365 Case C 3.75 250 62.5 107.761 107.791 55.241 55.241 Case C 7.5 250 62.5 129.966 129.966 56.796 56.796 Case C 15 250 62.5 162.258 162.258 57.411 57.411 Case C 30 250 62.5 204.931 204.931 56.316 56.316 Case C 60 250 62.5 249.067 249.067 53.033 53.033 Case C 120 250 62.5 272.714 272.714 48.408 48.408 Case C 240 250 62.5 265.237 265.237 45.045 45.045 where: f = frequency (Hz). Ω S = actual resistivity shallower than depth D = 800 meters. Ω D = actual resistivity deeper than depth D = 800 meters. Ω A = apparent MT resistivity. θ = MT phase angle (degrees). 4-6
Magnetotelluric Survey Simulations employed in CSAMT surveying. In actual field practice the source is ordinarily located several kilometers away from the survey area to minimize errors arising from wavefront curvature, but for the present purposes calculations were performed with a nearby source to examine these deviations from ideal far-field behavior in detail. Figure 4.4 indicates the problem geometry; we again consider a region with a horizontal surface and two underground layers, with a discontinuity in electrical resistivity located at depth D. Nine cases were considered, as summarized in Table 4.2. A single value of signal frequency f was chosen for each case, such that f D 2 is the same for all cases so that the ratio of resistivity discontinuity depth D to the skin depth S is the same for all, and equal to 0.389 as shown. The essential problem is to establish how the diagnostic results (Ω A and θ) depend on the distance between the signal source and the survey point R, and how these results for finite values of R differ from the MT results (the asymptotic result as R, given by the Cagniard (1953) solution, above). In dimensionless form, the problem is to determine how [Ω A /Ω S ] and [θ/45 ] depend upon [S/R], for [D/S] = 0.389 and for [Ω S /Ω D ] = 1.0 ( A cases), 2.0 ( B cases) and 4.0 ( C cases). The same spatial discretization scheme was employed for all nine cases; this is the same computational grid employed above for the MT calculations and is illustrated for the various cases in Figure 4.5 (high-resolution cases A-1, B-1, C-1; f = 15 Hz, D = 800 m, S = 2054.7 m), Figure 4.6 (intermediate-resolution cases A-2, B-2, C-2; f = 60 Hz, D = 400 m, S = 1027.3 m) and Figure 4.7 (low-resolution cases A-3, B-3, C-3; f = 240 Hz, D = 200 m, S = 513.7 m). In all cases the signal source is considered to be located at the earth surface, at x = 2000 m East, y = 2000 m North. Calculations were carried out for 441 survey points located at grid block centers in the high-resolution region of the spatial grid as indicated, with x = 1900, 1700,, +2100 m East and y = 1900, 1700,, +2100 m North. This means that R for the most distant survey station in each case corresponds to S/R = 0.354 (cases A-1, B-1, C-1), S/R = 0.177 (cases A-2, B-2, C-2) and S/R = 0.089 (cases A-3, B-3, C-3). Computed results for apparent resistivity Ω A and phase angle θ are displayed in Figures 4.8 and 4.9 respectively as functions of S/R. Each point displayed in these graphs represents one of the station locations at the center of a particular computational grid block. Results for each discrete value of [Ω S /Ω D ] group together as they should, although relatively low-resolution results exhibit greater scatter owing to discretization errors, particularly for large values of S/R (close to the signal source). As distance from the source increases without limit (and S/R 0), all three curves ( A, B and C ) converge to the appropriate far-field limiting values given by the Cagniard (1953) solution (equations 4-3 and 4-4 above) for both apparent resistivity Ω A and phase angle θ. Near the signal source, abnormally high values of apparent resistivity and low values of phase angle are observed, in agreement with experience (Telford et al., 1990). Also, a problem essentially equivalent to the present Case A (with Ω D = Ω S ) has been solved numerically by Zonge and Hughes (1991) for the distribution of apparent resistivity surrounding the signal source. No computed tabular data are available for direct comparison with the present results, but the graphs provided by Zonge and Hughes are consistent with the present calculations for Case A, with Ω A >> Ω S near the signal source (large values of S/R), with Ω A Ω S for S/R < ⅓ or so, and with Ω A Ω S from below as S/R 0. Continued on page 4-15 4-7
Verification and Validation Calculations Using the STAR Geophysical Postprocessor Suite Figure 4.4. Geometry considered for CSAMT postprocessor calculations of two-layer problem. Frequency f is 15 Hz (cases A-1, B-1, C-1), 60 Hz (A-2, B-2, C-2) or 240 Hz (A-3, B-3, C-3). Depth of resistivity discontinuity D is 800 meters (cases A-1, B-1, C-1), 400 m (A-2, B-2, C-2) or 200 m (A-3, B-3, C-3). Shallow resistivity Ω S is 250 ohm-meters for all cases. Deep resistivity Ω D is 250 ohm-m (A-1, A-2, A-3), 125 ohm-m (B-1, B-2, B-3) or 62.5 ohm-m (C-1, C-2, C-3). Distance between dipole signal source and survey point (R) varies from 141 to 5798 meters. 4-8
Magnetotelluric Survey Simulations Table 4.2. Cases Considered for CSAMT Survey Simulations. f (Hz) Ω S Ω D S (m) D (m) D/S D/ z Case A-1 15 250.0 250.0 2054.7 800.0 0.389 8 Case A-2 60 250.0 250.0 1027.3 400.0 0.389 4 Case A-3 240 250.0 250.0 513.7 200.0 0.389 2 Case B-1 15 250.0 125.0 2054.7 800.0 0.389 8 Case B-2 60 250.0 125.0 1027.3 400.0 0.389 4 Case B-3 240 250.0 125.0 513.7 200.0 0.389 2 Case C-1 15 250.0 62.5 2054.7 800.0 0.389 8 Case C-2 60 250.0 62.5 1027.3 400.0 0.389 4 Case C-3 240 250.0 62.5 513.7 200.0 0.389 2 where: f = frequency Ω S = earth electrical resistivity above depth D. Ω D = earth electrical resistivity below depth D. S = skin depth for frequency f, resistivity Ω S. D = depth of electrical resistivity discontinuity. z = vertical size of shallow layers in computational grid (100 m). 4-9