Robert T. Langan

Tomographic determination of velocity and depth in laterally varying media

Estimation of reflector depth and seismic velocity from seismic reflection data can be formulated as a general inverse problem. The method used to solve this problem is similar to tomographic techniques in medical diagnosis and we refer to it as seismic reflection tomography. Seismic tomography is formulated as an iterative Gauss‐Newton algorithm that produces a velocity‐depth model which minimizes the difference between traveltimes generated by tracing rays through the model and traveltimes measured from the data. The input to the process consists of traveltimes measured from selected events on unstacked seismic data and a first‐guess velocity‐depth model. Usually this first‐guess model has velocities which are laterally constant and is usually based on nearby well information and/or an analysis of the stacked section. The final model generated by the tomographic method yields traveltimes from ray tracing which differ from the measured values in recorded data by approximately 5 ms root‐mean‐square. The i...

Seismic monitoring of a CO2 flood in a carbonate reservoir: A rock physics study

A carbon dioxide (CO2) injection pilot project is underway in Section 205 of the McElroy field, West Texas. High‐resolution crosswell seismic imaging surveys were conducted before and after CO2 flooding to monitor the CO2 flood process and map the flooded zones. The velocity changes observed by these time‐lapse surveys are typically on the order of −6%, with maximum values on the order of −10% in the vicinity of the injection well. These values generally agree with laboratory measurements if the effects of changing pore pressure are included. The observed dramatic compressional (VP) and shear (VS) velocity changes are considerably greater than we had initially predicted using the Gassmann (1951) fluid substitution analysis (Nolen‐Hoeksema et al., 1995) because we had assumed reservoir pressure would not change from survey to survey. However, the post‐CO2 reservoir pore fluid pressure was substantially higher than the original pore pressure. In addition, our original petrophysical data for dry and brine‐sa...

Hybrid l 1 /l 2 minimization with applications to tomography

Least squares or l2 solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors. The lp minimization for 1 ≤ p < 2 gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions to these lp problems. We apply IRLS to a hybrid l2/l2 minimization problem that behaves like an l2 fit for small residuals and like an l1 fit for large residuals. The smooth transition from l2 to l1 behavior is controlled by a parameter that we choose using an estimate of the standard deviation of the data error. For linear problems of full rank, the hybrid objective function has a unique minimum, and IRLS can be proven to converge to it. We obtain a robust efficient method. For nonlinear problems, a version of the Gauss‐Newton algorithm can be applied. Synthetic crosswell tomography examples and a field‐data VSP tomography example demonstrate the improvement of the hybrid method over le...

High-resolution crosswell imaging of a West Texas carbonate reservoir; Part 1, Project summary and interpretation

A carbon dioxide flood pilot is being conducted in a section of Chevron’s McElroy field in Crane County, west Texas. Prior to CO2 injection, two high‐frequency crosswell seismic profiles were recorded to investigate the use of seismic profiling for high‐resolution reservoir delineation and CO2 monitoring. These preinjection profiles provide the baseline for time‐lapse monitoring. Profile #1 was recorded between an injector well and an offset observation well at a nominal well‐to‐well distance of 184 ft (56 m). Profile #2 was recorded between a producing well and the observation well at a nominal distance of 600 ft (183 m). The combination of traveltime tomography and stacked CDP reflection amplitudes demonstrates how high‐frequency crosswell seismic data can be used to image both large and small scale heterogeneity between wells: Transmission traveltime tomography is used to image the large scale velocity variations; CDP reflection imaging is then used to image smaller scale impedance heterogeneities. The...

Tracing of rays through heterogeneous media: An accurate and efficient procedure

We present an accurate and computationally efficient method for tracing rays through a medium of heterogeneous velocity. Efficiency is obtained through approximations, while accuracy is maintained by using higher-order terms in the approximations when necessary. The method was developed as part of a general seismic inversion technique for traveltime information where the medium is discretized into rectangular cells, each of which is characterized by a velocity gradient. However, as a stand-alone tool for forward modeling this method may be applied to other cell geometries and the velocity field need not be a linear function of position in each cell. We use path length as the independent variable rather than position or orientation. The position, orientation, and traveltime of the ray are algebraic expressions of path length, so that trigonometric functions are avoided. If either position or orientation is used as the independent variable, trigonometric expressions invariably arise in evaluating the dependent variables. Such transcendental functions are computationally inefficient in large problems.

Theoretical and numerical issues in the determination of reflector depths in seismic reflection tomography

Seismic reflection tomography obtains an estimate of the subsurface slowness field and the location of strong reflectors by minimizing the difference between measured travel times from seismic reflection events and the corresponding travel times computed from a model of the subsurface. We present some theoretical results for the undiscretized problem regarding the possible ambiguity between slowness and depth. These results indicate that the depths of the reflectors are determined in theory except for edge effects, but a sufficiently large aperture at the reflector is necessary to resolve this ambiguity in practice. The slowness field, however, does have some undetermined features. These results have strong implications for how the tomography problem should be discretized and regularized to compute solutions which are accurate in the features of the model which are well determined from the travel time data. In particular, the slowness model should not be discretized much more coarsely than the reflectors as a way of regularizing the problem because that may force the computed reflector depths to try to match aspects of the travel time data which are caused by features in the slowness field.

Thermal Evolution of the Earth: Some Recent Developments

Publisher Summary This chapter reviews the recently developed methods for thermal modeling of the Earth and illustrates that the Earth has been cooling over the last 3 billion years. The mechanism of core formation is examined. The assumption that total melting and eruption of mantle material may have cooled the Earth after core formation and that the temperature increased again during the Archean from radioactive heating is examined. The future of thermal modeling is discussed. In parameterized convection calculations, the global spreading rate and the residence time are obtained from the physics of convective flow. A derivation is presented based on the force balance of plate tectonics. In “parameterized convection,” calculations for thermal history models, the radioactive heat generation is assumed and the dependence of surface heat flow on temperature is obtained from the physics of convection. It is suggested that the Earth has been cooling since Archean time and a substantial amount of the Earths heat flow comes from cooling rather than radioactivity. The chapter reveals that the rate of plate motions in Archean and early Proterozoic times is poorly constrained. Kinematic models using 40 Ar degassing give rates similar to present rates.

High-resolution crosswell imaging of a West Texas carbonate reservoir; Part 5, Core analysis

We conducted a core analysis program to provide supporting data to a series of crosswell field experiments being carried out in McElroy Field by Stanford University’s Seismic Tomography Project. The objective of these experiments is to demonstrate the use of crosswell seismic profiling for reservoir characterization and for monitoring CO2 flooding. For these west Texas carbonates, we estimate that CO2 saturation causes P‐wave velocity to change by −1.9% (pooled average, range = −6.3 to +0.1%), S‐wave velocity by +0.6% (range = 0 to 2.7%), and the P‐to‐S velocity ratio by −2.4% (range = −6.4 to −0.3%). When we compare these results to the precisions we can expect from traveltime tomography (about ±1% for P‐ and S‐wave velocity and about ±2% for the P‐to‐S velocity ratio), we conclude that time‐lapse traveltime tomography is sensitive enough to resolve changes in the P‐wave velocity, S‐wave velocity, and P‐to‐S velocity ratio that result from CO2 saturation. We concentrated here on the potential for CO2 sat...

A continuation approach to regularization of ill-posed problems with application to crosswell-traveltime tomography

In most geometries in which seismic-traveltime tomography is applied (e.g., crosswell, surface-reflection, and VSP), determination of the slowness field using only traveltimes is not a well-conditioned problem. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in measured traveltimes can cause large errors in the computed slowness field. A priori information often is available — well logs, initial rough estimates of slowness from structural geology, etc. — and can be incorporated into a traveltime-inversion algorithm by using penalty terms. To further regularize the problem, smoothing constraints also can be incorporated using penalty terms by penalizing derivatives of the slowness field. What weights to use on the penalty terms is a major decision, particularly the smoothing-penalty weights. We use a continuation approach in selecting the smoothing-penalty weights. Instead of using fixed smoothing-penalty weights, we decrease them step by step, using the slowness...

A continuation approach to regularization for traveltime tomography

In most geometries in which seismic traveltime tomography is applied (e.g., surface reflection, cross-well, or VSP), the slowness field is not well-determined from traveltimes alone. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in the measured traveltimes can lead to large errors in the computed slowness field. A priori information is often availablewell-logs, initial rough estimates of the slowness from structural geology, etc. This a priori information can be incorporated into a traveltime inversion algorithm using penalty terms. To further regularize the problem, smoothing constraints can also be incorporated using penalty terms by penalizing derivatives of the slowness field. A major decision to be made is the of the weights on the penalty terms, particularly the smoothing penalty weights. We use a continuation approach for selecting the smoothing penalty weights. Instead of fixing the smoothing penalty weights; we decrease the smoothing penalty weights in a step-by-step fashion, using the slowness model computed using the previous (larger) weights as the initial slowness model for the next step using the new (smaller) weights. A surprising outcome in synthetic problems is that the model error continues to decrease as we continue to decrease the smoothing penalty weights even after the data error has leveled off at the noise level. This continuation approach can solve synthetic problems more accurately than with fixed smoothing penalty weights, and appears to yield more features of interest in real-data applications of traveltime tomography.