Correction to: A novel methodological approach for land subsidence prediction through data assimilation techniques

Computational Geosciences - A Correction to this paper has been published: https://doi.org/10.1007/s10596-021-10079-6     

A two‐invariant pseudoelastic model for reservoir compaction

Predicting the deformations of deep reservoirs due to fluid withdrawal/injection is a challenging task that could have important environmental, social, and economical impacts. Finite element models, if endowed with an appropriate constitutive law, represent a useful tool for computing the displacements, the deformations, and the stress distributions in reservoir applications. Several studies show that hypoelastic laws, based on a stress‐dependent vertical compressibility, are able to provide accurate results, confirmed by in situ and satellite measurements. On the other hand, such laws present some weaknesses related to the numerical implementation, in particular due to the nonsymmetry of the tangent operator. This paper presents a new constitutive model based on 2 invariants (the mean normal and deviatoric stresses), characterized by a variable pressure‐dependent bulk modulus K. This constitutive law allows for overcoming most shortcomings of the hypoelastic law, although preserving the same accuracy, reliability, and ease of use and calibration. This paper presents a procedure to identify the parameters of the new model, starting from the typically available data on the vertical compressibility. Numerical results show a good agreement between the 2 laws, suggesting the proposed approach as a valid alternative in reservoir applications.     

Land uplift due to subsurface fluid injection

The subsurface injection of fluid (water, gas, vapour) occurs worldwide for a variety of purposes, e.g. to enhance oil production (EOR), store gas in depleted gas/oil fields, recharge overdrafted aquifer systems (ASR), and mitigate anthropogenic land subsidence. Irrespective of the injection target, some areas have experienced an observed land uplift ranging from a few millimetres to tens of centimetres over a time period of a few months to several years depending on the quantity and spatial distribution of the fluid used, pore pressure increase, geological setting (depth, thickness, and area extent), and hydro-geomechanical properties of the injected formation. The present paper reviews the fundamental geomechanical processes that govern land upheaval due to fluid injection in the subsurface and presents a survey of some interesting examples of anthropogenic uplift measured in the past by the traditional levelling technique and in recent times with the aid of satellite technology. The examples addressed include Long Beach, Santa Clara Valley, and Santa Ana basin, California; Las Vegas Valley, Nevada; Cold Lake and other similar sites, Canada; Tokyo and Osaka, Japan; Taipei, Taiwan; Krechba, Algeria; Upper Palatinate, Germany; Chioggia and Ravenna, Italy.     

Preconditioners in computational geomechanics: A survey

The finite element (FE) solution of geomechanical problems in realistic settings raises a few numerical issues depending on the actual process addressed by the analysis. There are two basic problems where the linear solver efficiency may play a crucial role: 1. fully coupled consolidation and 2. faulted uncoupled consolidation. A class of general solvers becoming increasingly popular relies on the Krylov subspace (or Conjugate Gradient‐like) methods, provided that an efficient preconditioner is available. For both problems mentioned above, the possible preconditioners include the diagonal scaling (DS), the Incomplete LU decomposition (ILU), the mixed constraint preconditioning (MCP) and the multilevel incomplete factorization (MIF). The development and the performance of these algorithms have been the topic of several recent works. The present paper aims at providing a survey of the preconditioners available to date in computational geomechanics. In particular, a review and a critical discussion of DS, ILU, MCP and MIF are given along with some comparative numerical results. Copyright © 2010 John Wiley & Sons, Ltd.     

Scaling improves stability of preconditioned CG‐like solvers for FE consolidation equations

Preconditioned projection (or conjugate gradient like) methods are increasingly used for the accurate and efficient solution to finite element (FE) coupled consolidation equations. Theory indicates that preliminary row/column scaling does not affect the eigenspectrum of the iteration matrix controlling convergence as long as the preconditioner relies on the incomplete factorization of the FE coefficient matrix. However, computational experience with mid‐large size problems shows that the above inexpensive operation can significantly accelerate the solver convergence, and to a minor extent also improve the final accuracy, as a result of a better solver stability to the accumulation and propagation of floating point round‐off errors. This is demonstrated with the aid of the least square logarithm (LSL) scaling algorithm on FE consolidation problems of increasing size up to more than 100 000. It is shown that a major source of numerical instability rests with the sub‐matrix which couples the structural to the fluid part of the underlying mathematical model. It is concluded that for mid‐large size, possibly difficult, FE consolidation problems left/right LSL scaling is to be always recommended when the incomplete factorization is used as a preconditioning technique. Copyright © 2003 John Wiley & Sons, Ltd.     

A century of land subsidence in Ravenna, Italy

Over the last century the Ravenna area has been affected by a widespread land subsidence process of both natural and anthropogenic origin which has become a matter of increasingly great concern. Land settlement, initially of the order of a few mm/year, dramatically increased up to 110 mm/year after World War II primarily due to groundwater pumping and, subordinately, gas production from a number of deep on-shore and off-shore reservoirs. Spirit leveling was carried out in the area of the Ravenna Municipality by various authorities and agencies ever since the end of the nineteenth century using an increasingly refined network of benchmarks. The available information, spanning the time interval from 1897 to 2002, has been implemented and homogenized into a Geographic Information System (GIS). Measurements processed by kriging provide the pattern of the subsidence rates over the municipal territory during eight sub-periods. The reliability of data processing is discussed for both magnitude and areal distribution of the occurrence, showing a pronounced nonlinear behavior in time, as related to the major responsible factors. Cumulative land subsidence approaches almost 1 m over more than one third of the municipality area including the city and a significant fraction of the coastland, with peaks larger than 1.5 m over a 10 km² zone located between the historical center and the coastline. Most recent records show that at present the mainland appears to be substantially stable, whereas some coastal areas are still subsiding at a rate of about 10 mm/year with a significant threat to the environment and the infrastructures as well, although to a lesser extent than in the past decades.     

Numerical modelling of regional faults in land subsidence prediction above gas/oil reservoirs

The stress variation induced by gas/oil production may activate pre‐existing regional faults. This may enhance the expected land subsidence due to the generation of mechanically weak points close to the producing field. A class of elasto‐plastic interface elements (IE), specifically designed to address the mechanical behaviour of faults over a regional scale, is integrated into a finite element (FE) geomechanical model and used to investigate the role exerted by active faults in anthropogenic land subsidence. The importance of regional faults depends on a variety of factors including depth of the depleted reservoir, fault number, orientation and size, geomechanical properties of porous medium, pore pressure drawdown induced by fluid production, etc. With the aid of some representative examples, a useful indication is provided as to where and how fault activation may influence both magnitude and extent of the land subsidence bowl above producing gas/oil reservoirs, pointing to a generally limited impact on the ground surface. The simulation of a real faulted gas reservoir in a complex 3‐D setting shows that the proposed IE can be simply and efficiently incorporated into a FE geomechanical model, thus improving the quality of the stress and displacement prediction. Copyright © 2007 John Wiley & Sons, Ltd.     

Parallel solution to ill‐conditioned FE geomechanical problems

Parallel computers are potentially very attractive for the implementation of large size geomechanical models. One of the main difficulties of parallelization, however, relies on the efficient solution of the frequently ill‐conditioned algebraic system arising from the linearization of the discretized equilibrium equations. While very efficient preconditioners have been developed for sequential computers, not much work has been devoted to parallel solution algorithms in geomechanics. The present study investigates the state‐of‐the‐art performance of the factorized sparse approximate inverse (FSAI) as a preconditioner for the iterative solution of ill‐conditioned geomechanical problems. Pre‐and post‐filtration strategies are experimented with to increase the FSAI efficiency. Numerical results show that FSAI exhibits a promising potential for parallel geomechanical models mainly because of its almost ideal scalability. With the present formulation, however, at least 4 or 8 processors are required in the selected test cases to outperform one of the most efficient sequential algorithms available for FE geomechanics, i.e. the multilevel incomplete factorization (MIF). Further research is needed to improve the FSAI efficiency with a more effective selection of the preconditioner non‐zero pattern. Copyright © 2011 John Wiley & Sons, Ltd.     

On the uniformity of anthropogenic Venice uplift

A programme of Venice uplift because of seawater injection into a 600–800 m deep brackish aquifer underlying the lagoon has recently been advanced. As the actual spatial variability of the hydro‐geological parameters is to a large extent unknown, a controversial issue concerns the 25–30 cm heave prediction with the possible generation of differential displacements that might jeopardize the integrity of the architectural patrimony. It is shown that significant differential displacements have occurred in Venice in the past and are still presently occurring as evidenced by SAR interferometry. The results of a stochastic analysis addressing the variability of the hydraulic conductivity of the injected formation indicate that even a highly uneven aquifer expansion does not migrate to ground surface because of the smoothing effect exerted by the overburden. The predicted differential displacements are well below the values Venice is experiencing. Any a priori alarmism appears to be unfounded and unjustified.