The numerical prediction of land subsidence above producing reservoirs can be affected by a number of uncertainties, related for instance to the deep rock constitutive behavior, geomechanical properties, boundary and forcing conditions, etc.
The quality and the amount of the available observations can help reduce such uncertainties by constraining the numerical model outcome and providing more reliable estimates of the unknown governing parameters. In this work, we address the numerical simulation of land subsidence above a producing hydrocarbon field in the Northern Adriatic, Italy, by integrating the available monitoring data in the computational model with the aid of Data Assimilation strategies. A preliminary model diagnostic analysis, i.e. the

The numerical prediction of land subsidence above producing reservoirs can be affected by a number of uncertainties, related for instance to the definition of the constitutive law that describes the behavior of the porous medium, geomechanical properties, boundary and forcing conditions, etc.
These uncertainties involve in the lack of quality and reliability of the land subsidence prediction.
The accuracy and the amount of the available monitoring data, such as surface displacements recorded by GPS stations, can greatly help reduce such uncertainties, by constraining the computational model, solving approximately the inverse problem and providing more reliable estimates of the uncertain governing parameters.
A novel efficient workflow that combines Data Assimilation (DA) techniques and geomechanical model has been proposed in a recent work by

The aim of this methodology is to improve the prediction of land subsidence, avoiding deterministic outcomes that are not realistic.
The available measurements train the model in order to improve its accuracy and reliability as new information comes in.
This procedure has been applied to a synthetic but realistic test case, resulting in encouraging quantification and reduction of uncertainties

In this work, we address the numerical simulation of land subsidence above a real-world producing hydrocarbon field in the Northern Adriatic basin, Italy. Available data consist of GPS records for eleven years. A state-of-the-art Finite Element geomechanical model of the multi-pay layered reservoir is implemented, with a one-way coupled approach used to simulate the production program. Three ensembles of the expected land subsidence are generated by using a Modified Cam-Clay (MCC) and a visco-elasto-plastic (VEP) constitutive laws with different set of parameter values.

This methodology provides a significant reduction of the prior uncertainty on the governing geomechanical parameters, which allows to quantify and reduce as well the expected interval for the predicted ground settlement.

The paper is organized as follows. First, a brief description of the methodological approach is provided, then the producing hydrocarbon field and the outcomes of every step of the analyzed workflow are presented and discussed. A final section summarizes the conclusions.

The aim of the methodological approach originally proposed by

The starting point of this methodology is to propagate such input uncertainties to the output of the geomechanical model, i.e. the land subsidence prediction, by the generation of a group of forecast ensembles of Monte Carlo realizations that consider, for example, different possible constitutive laws to describe the deep rock behavior and variability ranges for the main mechanical parameters.
In fact, at the beginning of a land subsidence study, typically a few pieces of information only are available to characterize the porous medium behavior.
Displacement records are collected when the pressure program starts and then used with different DA techniques in order to train the model and make the predictions more accurate and reliable.
First, a diagnostic of the ensembles, i.e. the

The second step of the workflow is the RF approach, as originally proposed by

The last step consists in a smoother formulation, i.e. the ES technique.
The ES is an ensemble-based technique originally developed by

The methodology described in Sect.

Land subsidence has been predicted through a one-way coupled approach with the pore pressure values in time and space considered as a deterministic sources of strength and imposed on the reservoir and aquifer nodes.
In fact, uncertainties on the flow field are negligible if compared with those on mechanical parameters due to the large amount of available information for the history matching of the reservoir model.
Moreover, for the time and scales of interest, this simplified approach is generally acceptable

The numerical simulations last for 40 years, including a first production period of about 7 years followed by a 4-years recovery step before the forecast phase of about 29 years.

On the lateral and the bottom boundaries, homogeneous boundary conditions have been prescribed, while the top of the domain, which represents the land surface, is modeled as a traction-free boundary.

The generation of the ensembles has been done by running the geomechanical model with a set of statistical distributions of the input parameters.
Three ensembles, each composed of 50 realizations, have been created by using the MCC and the VEP constitutive laws with different set of parameter values.
In particular, we consider uncertain the modified compression index

The first two ensembles of vertical displacements

Forecast state ensemble (grey lines), i.e. relative vertical displacements

To train the model, a set of displacement measurements recorded by a GPS station is available. In particular, the GPS measurements recorded at the times when pressure records are available are used for the assimilation.

For the application of the DA techniques, the covariance matrix of measurement error has to be computed. We consider a diagonal matrix with a standard deviation equal to 5 mm that accounts for the accuracy of both the GPS station and the mathematical model used to reproduce the observed processes.

First, a diagnostic of the forecast ensembles has been carried out.
The outcome of the

Analyzed ensembles with their associated constitutive laws, parameters (

After the model diagnostic, RF is an easy and fast technique to analyze the ensemble by characterizing every realization by its own probability of occurrence.
The outcomes of RF are shown in Fig.

RF technique could be used also for a preliminary reduction of the uncertainties that affect the ensemble, by neglecting the realizations characterized by the smallest probability of occurrence that are, for example, grey lines in Fig.

Outcomes of RF for the forecast state ensemble, the third in Table

The ES technique allows to update both parameter and state variables.
The outcomes are shown in Fig.

Parameter

The ES can be a useful tool also in a predictive sense by assimilating new measurements as they are recorded.
To evaluate this capability, measurements have been assimilated by increasing steps of about 2 years.
The outcomes are shown in Fig.

Testing ES approach in a predictive sense. Variations

Outcomes of ES used in a predictive sense: parameter ensembles for increasing years of assimilated measurements.

Except for the assimilation of only 2 years of measurements, there is always a reduction of uncertainties for both parameter and state variables.
The variation

In this work, the novel methodological approach developed by

First, three ensembles have been generated by considering two non-linear constitutive laws, i.e. the MCC and the VEP behavior, and two different confidence intervals to describe the main mechanical parameter

A model diagnostic procedure carried out by the

After the diagnostic of the ensemble, every Monte Carlo realization has been characterized by its own probability of occurrence through the RF technique. This approach could be used for a preliminary reduction of the uncertainties that affect the model by neglecting the least probable realizations of the forecast ensemble.

Finally, the ES technique has been used to reduce the uncertainties without neglecting any realization, by updating the state and parameter ensembles. The model has been trained by the available measurements, resulting in an effective reduction of the uncertainties and in a more reliable subsidence prediction. The ES could be used also in a predictive sense by assimilating new observations as they become available. Adding new measurements seems to result in a more effective constraining of the model, but also a relative few years of assimilated observations appear to be enough for a first reduction of uncertainties in this case study.

On summary, the application of the proposed methodology for a real-world hydrocarbon reservoir confirms the results obtained for the synthetic test case. This workflow allows for (i) a stochastic study of land subsidence; (ii) a reduction of the uncertainties that affect the numerical model and (iii) more accurate and reliable land subsidence predictions. The approach here presented represents a way to investigate land subsidence different from the traditional methodologies. The traditional (or “deterministic”) procedures generally use the available measurements to calibrate the geomechanical model in order to define the set of parameters that allows the optimal reproduction of the observations. In this way, there could be the need to re-calibrate the model as new information become available, i.e. update the values of the geomechanical parameters. Conversely, the proposed workflow starts the geomechanical study at the very beginning of the reservoir life combining uncertainties to the numerical model by the creation of an ensemble of realization. Then, when measurements become available, such uncertainties are reduced, with the reduction depending on the observation accuracy, but not eliminated in order to avoid the ensemble collapse and keep open the possibility to stochastically reproduce further records. This approach allows to take advantages from the available measurements, in fact they are not used only to monitor the evolution of land subsidence in time, but also to train the geomechanical model.

Data non explicitly reported in this paper are property of Eni S.p.A.

All the authors have equally contributed to the development of the conceptual model. FM, GL and ZC took care of the code implementation and numerical testing. GL, FM and TP are the main contributors for paper writing and proofreading.

The authors declare that they have no conflict of interest.

This article is part of the special issue “TISOLS: the Tenth International Symposium On Land Subsidence – living with subsidence”. It is a result of the Tenth International Symposium on Land Subsidence, Delft, the Netherlands, 17–21 May 2021.