We apply a rock physics analysis to well log data from the North-East Gulf of Mexico

Rock Physics for Fluid and Porosity Mapping in NE GoM JACK DVORKIN, Stanford University and Rock Solid Images TIM FASNACHT, Anadarko Petroleum Corporation RICHARD UDEN, MAGGIE SMITH, NAUM DERZHI, AND JOEL WALLS, Rock Solid Images May 07, 2003 We apply a rock physics analysis to well log data from the North-East Gulf of Mexico to establish an effective-medium transform between the acoustic and elastic impedance on the one hand and lithology, pore fluid, and porosity on the other hand. These transforms are upscaled and applied to acoustic and elastic impedance inversion volumes to map lithology and porosity. INTRODUCTION. The ultimate goal of the geoscientist is to determine the reservoir properties (lithology and porosity) and conditions (pore fluid and pressure) from seismic data. This goal can be achieved by applying rock physics transforms to a volume of seismically-derived elastic properties. The basis for establishing rock physics transforms are controlled experiments where the reservoir and elastic sediment properties are measured on the same sample of earth at the same conditions. Such experimental data come from well logs and cores. The rock physics diagnostic is aimed at finding a deterministic effective-medium model appropriate for the rock under examination, calibrating it to the data, and then using it as the ultimate link (transform) between various rock properties. We select this rational deterministic approach, as opposed to pure statistics, in order to (a) understand and generalize trends seen in the data and use them outside of ranges available in the data; and (b) determine the domains of applicability of the trends. Below, the rock physics diagnostic is applied to well log data from the NE GoM and then used to map fluid and quantify porosity from acoustic and elastic impedance inversion volumes. WELL LOG DATA. The interval under examination includes shales and several high- 1

porosity sand packages (Figure 1). The upper sands are water-saturated while the lower sand contains hydrocarbons. The hydrocarbon-filled sand layer includes high-impedance calcite streaks. The S-wave data are not available in the well. One goal of this study is to quantify the porosity of the reservoir from seismicallyderived elastic parameters. This is why in Figure 1 we display an impedance-porosity cross-plot together with the depth plots of gamma-ray (GR), saturation, porosity, and impedance. The shale and wet sand form two separate branches in the impedanceporosity frame. In the same range of the total porosity, the impedance in the shale is smaller than in the wet sand. In both lithologies, evident are distinctive impedanceporosity trends. The impedance in the high-porosity hydrocarbon sand is smaller than in the wet sand of comparable porosity mostly due to the effect of the pore fluid on the bulk modulus. This is why the impedance-porosity trend present in the hydrocarbon reservoir is different from that observed in the wet sand located above. ROCK PHYSICS DIAGNOSTIC. In order to find an effective-medium model that will provide a porosity-lithology-impedance transform, we first theoretically saturate, using the V p -only fluid substitution, the entire interval under examination with the same pore fluid, namely, the formation brine. This pore-fluid equalization (we also call this procedure common fluid denominator ) allows us to concentrate purely on the effects of lithology and porosity on the elastic properties. The impedance in the brine-substituted interval is plotted versus porosity in Figure 2, top. It is color-coded by GR and the in-situ water saturation. The model-based impedance-porosity curves from the uncemented sand/shale model are superimposed upon the data points in the same figure. The sand data points are bound by the curves drawn for pure quartz sand and sand with 25% clay content. Model-based impedance-porosity curves are also drawn for rock with 75% and 100% clay content. These curves serve at lower bounds for the shale data. Most of the shale data lie above the 75% clay content curve which simply means that the shale is not pure clay. 2

Figure 1. Well log curves (from left to right, GR, water saturation, the total porosity, and impedance) and the impedance-porosity cross-plot. In the second, third, and fourth rows, we highlight (in red) the shale, wet sand, and hydrocarbon sand, respectively. Depth is in fictitious units. The original (not fluid-substituted) impedance is plotted versus porosity in Figure 2, 3

bottom. The uncemented sand/shale curve drawn for sand with 10% clay and for the insitu reservoir fluid is superimposed on the data. It can serve as an average impedanceporosity transform in the reservoir sand. The corresponding best-fit equations are: I p =12.45-39.62f + 47.23f 2, f = 0.741-0.117I p + 0.0051I p 2, (1) where the impedance ( I p ) is in km/s g/cc and porosity (f) is in fraction. Figure 2. Impedance versus total porosity color-coded by GR (left) and water saturation (right). Top: Impedance calculated for the brine-saturated interval, including the reservoir sand. The model curves come from the uncemented sand/shale model and are calculated for brine-saturated conditions. The curves are (top to bottom) for zero, 25%, 75%, and 100% clay content, respectively. Bottom: The insitu impedance. The yellow curve is for 10% clay content and the in-situ reservoir fluid. A comparison between the in-situ velocity and impedance data and the velocity and 4

impedance calculated using the uncemented sand/shale model with the in-situ clay content, porosity, and saturation as input (Figure 3) indicates that the selected rock physics transform is appropriate for the interval under examination. Figure 3. Comparison between sonic data and modeled data. Well log curves (from left to right, GR, water saturation, P-wave velocity, and impedance). In the velocity and impedance frames, the black curves represent the well log data while the red curves are computed from the uncemented sand/shale model using the clay content, porosity, and saturation curves as input. Depth is in fictitious units. In Figure 4 we compare the porosity measured in the pay zone with that predicted from the measured impedance using the transform from the second Equation (1). The prediction is very close to the data. Figure 4. Comparison between porosity data and prediction in the pay zone. In the porosity frame, the black curve is measured porosity while the red curve is porosity calculated from the measured impedance using the transform from Equation (1). Based on these results, we adopt the uncemented sand/shale model as appropriate for 5

the well under examination. As we have already seen, this model can serve to predict porosity from impedance. It can also be used to calculate the S-wave velocity and Poisson s Ratio (PR) which are not available at the well. Our calculations indicate that, as expected, PR in the reservoir sand is low (about 0.13); PR in the wet sand is about 0.28; and PR in the shale is about 0.35. UPSCALING. Elastic waves in the seismic frequency range usually sample large portions of the subsurface whose dimensions are on the order of the wavelength. As a result, the elastic structure of the subsurface extracted from seismic data can only represent the elastic properties averaged over relatively large intervals. The details that are apparent at the well log scale cannot be recovered from seismic data. To illustrate this effect, consider synthetic seismic traces modeled at the well under examination using a 50 Hz and 25 Hz Ricker wavelet and the velocity (V p and the modelpredicted V s ) and bulk density at the well (Figure 5). This synthetic seismic modeling is based on the convolution of the wavelet with the reflectivity series derived at the well. The reflectivity R pp (q) at an angle of incidence q is calculated according to Hilterman s approximation to the Zoeppritz equations: R pp (q) = R pp (0)cos 2 q + 2.25Dn sin 2 q, R pp (0) = D(0.5ln I p ), (2) where n is PR and D indicates the difference between the lower and upper half-space at the reflecting interface. The pay zone is manifested by a positive reflection whose amplitude decreases with offset. The positive normal reflection is due to the gradual impedance increase from the shale above the pay zone, through the pay zone, and to the underlying hard shale. The AVO effect is due to the small Poisson s ratio characteristic of porous sand with hydrocarbon. Also in Figure 5 shown are the pseudo-impedance ( I pp ) and pseudo-pr (n p ) curves which were calculated from the synthetic traces by using Hilterman s AVO approximation the pseudo-impedance is simply the cumulative sum of the normalincidence trace while pseudo-pr is a linear combination of the cumulative sum of the far- 6

offset trace and the pseudo-impedance to produce Poisson s ratio reflectivity: I pp = t Ú T 0 (t)dt, n p = [cos -2 q Ú T q (t)dt - I pp ]/(2.25tan 2 q), (3) 0 0 t where T 0 is the normal-incidence trace; T q is the trace at angle q ; and t is the two-way travel time. Figure 5. Synthetic seismic at the well. From left to right: Synthetic gather (black) and stack (red); impedance and PR at the well; and pseudo-impedance and pseudo-pr calculated from the synthetic traces. Travel time is in fictitious units. The pseudo-impedance and pseudo-pr curves presented in Figure 5 capture the 7

character of the impedance and PR at the well but, at the same time, omit the details and, effectively, smooth the well log curves. Because of this smoothing effect, it is not clear whether the impedance-porosity transform expressed in Equation (1) can be useful at the seismic scale, how it has to be corrected (upscaled), and, finally, how to interpret porosity estimates that are based on the seismic-scale impedance values. To investigate the effects of upscaling on the elastic properties and also on rock physics transforms, we use the Backus average which estimates the seismic-scale response by harmonically averaging of the elastic moduli at the well. The bulk density is upscaled by means of arithmetic averaging. Then the upscaled impedance and PR are calculated from the Backus-upscaled elastic moduli and arithmetically-upscaled density according to the appropriate theory-of-elasticity equations. The upscaled elastic curves as well as the arithmetically-averaged porosity curve are shown in Figure 6. This display confirms what we already learned from the synthetic seismic traces: the log-scale details of the interval will be omitted in seismic data. Figure 6. Original (black) and upscaled (bold cyan) porosity, impedance, and PR curves at the well. Depth is in fictitious units. UPSCALED IMPEDANCE-POROSITY TRANSFORM. A question to address is whether the transform from the impedance to porosity derived from well log data -- 2 f = 0.741-0.117I p + 0.0051I p -- is valid at the seismic scale. To investigate this problem, let us plot the porosity versus impedance at the log scale and seismic scale (Figure 7). The upscaled trend generally follows the original porosity-impedance trend but slightly overestimates the porosity at a given impedance value. The best-linear-fit 8

transform between the upscaled impedance and averaged porosity is f u = 0.205-0.037(I pu - 6.5), (4) where the impedance is in km/s g/cc and porosity is in fraction, and the subscript u refers to the upscaled values. This equation is site-specific and should not be generalized because the results of upscaling may be strongly influenced by the size and shape of the reservoir and surrounding strata. In general, the validity of log-scale rock physics transforms at the seismic scale improves as the reservoir becomes thicker and the variations of the rock properties with depth around the reservoir become smoother. Figure 7. Log-scale porosity versus log-scale impedance (black); log-scale porosity versus upscaled impedance (blue); and upscaled porosity versus upscaled impedance (red). The yellow line is from Equation (4). INFLUENCE OF HIGH-IMPEDANCE LAYERS ON REFLECTION. Let us investigate is whether the high-impedance layers that bound the reservoir affect the seismic reflection and, eventually, porosity estimates from the seismic impedance. To address this question, we alter the elastic curves at the well by reducing the impedance in the three high-impedance layers to the surrounding impedance values (Figure 8). Then we calculate the Backus-upscaled impedance curves. 9

Figure 8. The impedance curve at the reservoir. Black is for the measured impedance; red is for the impedance artificially reduced within the high-impedance layers; cyan is for the upscaled original impedance curve; and yellow is for the upscaled reduced impedance curve. Depth is in fictitious units. The upscaling results indicate that the influence of the high-impedance layers on the upscaled impedance values is small. To further support this conclusion, we generate synthetic traces using the impedance curves at the well with and without the highimpedance layers (Figure 9). The reflections appear to be virtually unaffected by the high-impedance layers. This apparently counter-intuitive result is due to the fact that the averaging of the elastic properties in the subsurface is harmonic and as such gives larger weights to the softer parts of the interval. POROSITY FROM UPSCALED IMPEDANCE. Next we apply Equation (4) to the upscaled impedance curve to assess the accuracy of porosity estimates from the seismicscale impedance. The estimated porosity is compared to the log-scale porosity and also to the arithmetically-averaged porosity in Figure 10. The porosity curve derived from the upscaled impedance is essentially the same as the arithmetically averaged porosity curve. Both curves give a reasonably accurate average estimate to the log-scale porosity. 10

Figure 9. Synthetic traces with (top) and without (bottom) high-impedance layers in the reservoir zone. The display is the same as in Figure 5. 11

Figure 10. Left: Water saturation curve in the reservoir zone. Right: the log-scale porosity curve (black); arithmetically upscaled porosity curve (red); and porosity derived from the upscaled impedance (yellow). Depth is in fictitious units. POROSITY FROM SEISMIC IMPEDANCE INVERSION. Next we use acoustic and elastic impedance inversion sections obtained from real seismic data to identify the reservoir and estimate the total porosity within the reservoir. In the section displayed in Figure 11, the reservoir is identified by a low-value-pr strip, as is expected from the above rock physics analysis and pseudo-impedance and pseudo-pr inversion curves shown in Figures 5 and 9. The minimum-value strip in the PR inversion section does not precisely coincide in space with the maximum-value strip in the impedance-inversion section. This is expected from the impedance structure of the well log profile where the maximum of the impedance is located slightly below the reservoir while the minimum of the PR is located precisely at the reservoir. The reservoir in the inversion section is identified as the low-pr strip. The porosity in the reservoir is directly calculated from the impedance inversion using Equation (4). The calculated porosity section is displayed in Figure 11. The same approach as used in the vertical section is employed to create a seismic porosity cube from 3D acoustic and elastic impedance inversion (Figure 12). The dipping sand layer filled with hydrocarbon is apparent in the volume. The curved section of the sand by a constant-time horizon visible in the 3D display is an apparent indication of a three-way closure. 12

Figure 11. Left: PR inversion from real seismic. Middle: Acoustic impedance inversion from real seismic. Right: Seismic porosity derived from the impedance inversion in the reservoir zone. Depth and cross-line distance are in fictitious units. The vertical bars indicate the location of the well. The well log curves of saturation (left); GR (middle); and porosity (right) are superimposed upon the vertical sections. Figure 12. Mapping fluid and porosity from 3D acoustic and elastic impedance inversion. From left to right: PR inversion; acoustic impedance inversion; and total porosity produced from the acoustic impedance according to Equation (4). CUMULATIVE POROSITY. An important desired reservoir characterization parameter is the cumulative porosity ( C f ) also known as the total pore volume in the reservoir, or net-to-gross. This quantity is often expressed as the product of the porosity (f) and reservoir thickness ( h): C f = fh. Strictly speaking, it is the integral of the porosity with respect to depth, calculated within the reservoir: C f = z B Ú f(z)dz, (5) z T 13

where the integration is between the top ( z T ) and bottom ( z B ) depths of the reservoir. Moreover, only the accumulated porosity within the pay zone is of immediate interest. To calculate the cumulative pay-zone porosity ( C fp ), Equation (5) should be modified as: C fp = z B Ú f(z) Sw <1 dz, (6) z T where condition S w <1 simply means that during the integration, porosity outside of the pay zone is not counted. The cumulative pay-zone porosity is plotted versus depth in Figure 13 for the original (log-scale) curve as well as for the upscaled porosity. The two curves are very close to each other which means that the seismically derived C fp and the log-scale C fp are essentially the same. Figure 13. Cumulative porosity (left); cumulative impedance (middle); and cumulative inverse impedance (right) versus depth. Black curves are for the log scale values while yellow curves are for the upscaled values. The cumulative porosity is in fraction times meter; the cumulative impedance is in km/s g/cc times meter; and the cumulative inverse impedance is in s/km cc/g times meter. CUMULATIVE ATTRIBUTES. As we have already established, porosity can be related to the impedance. Next question is which seismic attribute can be used to estimate the cumulative porosity. One possibility is that such a cumulative attribute (CATT) is the cumulative impedance in the pay zone (C I p P ) which can be defined in the same manner as C fp : 14

C I p P = z B Ú I p (z) Sw <1 dz. (7) z T C I p P is plotted versus depth in Figure 13 (middle) for the original log-scale curve as well as for the upscaled curve. Once again, the two curves are very close to each other which means that the proposed integration operator will produce approximately the same results at the seismic scale as at the log scale. The next question is whether the proposed CATT can be uniquely translated into the cumulative porosity within a seismic volume. To investigate this possibility, let us crossplot C fp versus C I p P at the log and seismic scale. The result shown in Figure 14 (left) indicates that the cross-plot curves at both scales are close to each other which means that the proposed CATT (C I p P ) can be used to calculate the cumulative porosity from seismic. Figure 14. Cumulative porosity versus cumulative impedance (left); and versus cumulative inverse impedance (right). Black curves are for the log scale values while yellow curves are for the upscaled values. Another possible CATT may be the inverse cumulative impedance (C I p -1 P ): C I p -1 P = z B Ú I -1 p (z) dz. Sw <1 (8) z T This CATT is plotted versus depth in Figure 13 (right). The cumulative porosity is cross-plotted versus C I in Figure 14, right. In this cross-plot, the match between the -1 p P log-scale and seismic-scale values is even better than for the cumulative impedance on 15

the left. This is likely due to the fact that the elastic moduli upscale by means of harmonic averaging and, in this particular case, such averaging rule may be approximately valid for the impedance. The approach proposed here essentially introduces a new class of seismic attributes which are calculated from a seismic trace by repeated integration -- the seismic impedance is due to a one-time integration of a trace which is approximately treated as the reflectivity series and the cumulative impedance is due to one-time integration of the seismic impedance or repeated integration of the trace. CAVEAT -- DELINEATING PAY ZONE. An important condition for calculating C fp from a CATT was to integrate the latter within the pay zone. To meet this condition we will have to employ an additional attribute, such as the seismic PR, to locate the pay zone. Finding a threshold for this attribute to accurately delineate a pay is important for a successful use of CATTS. CONCLUSION. The rock physics diagnostic approach enables us to establish a sitespecific effective-medium model to transform the elastic rock properties into lithology, fluid, and porosity. An effective-medium model allows us to systematically and consistently perturb rock properties and conditions to predict the elastic response away from a well. A model established for log-scale data has to be upscaled to the seismic scale in order to be used with seismic inversion cubes. Such upscaling may call for corrections in the model, especially so in a heterogeneous environment with large elastic contrasts. ACKNOWLEDGEMENT. Uwe Strecker helped with stratigraphic and geological understanding of the example presented in this study. SUGGESTED READING Dvorkin, J., and Nur, A., 1996, Elasticity of High-Porosity Sandstones: Theory for Two North Sea Datasets, Geophysics, 61, 1363-1370. Mavko, G., Chan, C., and Mukerji, T., 1995, Fluid substitution: Estimating changes in Vp without knowing Vs, Geophysics, 60, 1750-1755. Backus, G.F., 1962, Long-wave elastic anisotropy produced by horizontal layering, JGR, 67, 4427-4441. 16