We present an extension of the MUFITS reservoir simulator for modelling the ground displacement and gravity changes associated with subsurface flows in geologic porous media. Two different methods are implemented for modelling the ground displacement. The first approach is simple and fast and is based on an analytical solution for the extension source in a semi-infinite elastic medium. Its application is limited to homogeneous reservoirs with a flat Earth surface. The second, more comprehensive method involves a one-way coupling of MUFITS with geomechanical code presented for the first time in this paper. We validate the accuracy of the development by considering a benchmark study of hydrothermal activity at Campi Flegrei (Italy). We investigate the limitations of the first approach by considering domains for the geomechanical problem that are larger than those for the fluid flow. Furthermore, we present the results of more complicated simulations in a heterogeneous subsurface when the assumptions of the first approach are violated. We supplement the study with the executable of the simulator for further use by the scientific community.

Reservoir simulation remains an essential area for forecasting various parameters of subsurface exploration and natural flows. The capabilities of the reservoir simulators, i.e., the computer programs for modelling the flows in geologic porous media, are constantly improving. The modern simulators can account for additional physical phenomena and technological processes. For example, the modelling of Darcy flows is often coupled with rather sophisticated approaches for geomechanics, multiphase transport in wellbores, and other processes (Fig.

Sketch of typical processes in hydrothermal systems (left) and petroleum reservoirs (right). Magma degassing results in a plume of hot magmatic fluid. Near the surface, the fluid can mix with colder meteoric water. The observation points, where the ground displacement and gravity changes can be measured, are shown.

The numerical modelling of hydrothermal systems differs in several respects from that of petroleum reservoirs (Fig.

In the numerical modelling of hydrothermal systems, a common approach for estimating gravity changes and ground displacement assumes the application of different computer programs that are not directly compatible with each other. The fluid flow is simulated by using hydrodynamic code. Then, the gravity changes are calculated by post-processing the simulation results in separate code, i.e., by summing the contributions of every grid block to the strength of the gravitational field

MUFITS has been developed over the past decade. Initially, it was designed for modelling the subsurface storage of

As presented in Sect.

Diagram showing the software options for modelling gravity changes and ground displacement. The interface between the simulators is organised through external files.

To extend MUFITS for the modelling of gravity changes and ground displacement, a new primitive (element) of the reservoir model, namely the “observation point” (OP), is introduced. Every OP is assigned a character name and is characterised by 3 coordinates in space. For example, an OP can correspond to surface or airborne measurements (Fig.

The developed options for simulating gravity changes and ground displacement (by method A) are generally designed for 3-D simulations with domains of arbitrary complexity (Fig.

Possible symmetries that can be accounted for in the calculation of gravity changes and ground displacement by method A. The simulation domain (red) occupies a region of the subsurface reservoir (black).

We denote by

According to Newton's law of universal gravitation, the gravity change in every OP is calculated using the following equation:

Method A is restricted by the following assumptions:

The mechanical properties of the saturated porous medium are homogeneous.

The top boundary of the domain,

The domain for modelling the displacement is a semi-infinite region

Using the analytical solution of

The displacement

Method A provides a fast option for calculating ground displacement. However, this mathematical model is restricted to the case of a homogeneous distribution of thermoporoelastic moduli and is subject to other assumptions described in Sect.

To allow for the modelling of hydrothermal systems with heterogeneous mechanical properties, we have developed numerical code for calculating stresses and displacements by the finite volume method in axisymmetric domains. The solution of the elastic problem is uncoupled from the equations governing the fluid flow, i.e., deformations of the solid phase do not influence the fluid pressure or the porosity and permeability of the rock matrix. This is justified by the assumption of small deformations. In addition, uncoupling the ground displacement from the fluid flow makes it possible to implement method B in a post-processing module of the simulator, removing the necessity to re-run MUFITS simulations in order to change the mechanical properties of the rocks.

The constitutive equations for linear isotropic thermoporoelastic medium are expressed as

We seek a numerical solution of the equations of static equilibrium:

The equations of elastic equilibrium (

The advantages of the PT method in comparison to other numerical techniques that require assembling matrices are the brevity of the corresponding code and the simplicity of its development and modification. However, since the PT method involves only explicit time integration, the number of iterations required for convergence scales quadratically with the number of grid blocks due to stability conditions. This is acceptable only for 1-D problems and can be too restrictive for 2-D and 3-D problems. To overcome this limitation and accelerate convergence, we use the second-order pseudo-transient method

Average number of pseudo-transient iterations

As shown in Fig.

To validate the developed modelling options, we consider an axisymmetric study of hydrothermal activity at Campi Flegrei, a caldera located in southern Italy. An accelerated ground deformation and heating is observed in this densely populated area of Naples causing interest to understanding and predicting the hydrothermal activity. The corresponding flows in the hydrothermal system and associated observable parameters at the surface were broadly investigated by

We simulate the non-isothermal flow of a

The simulation domain and the boundary conditions for the hydrodynamic simulation. The distribution of the gas saturation (

The influx of fluid from a deep magmatic source is simulated with a point source placed at

Parameters of the benchmark study.

For modelling the non-isothermal flow associated with the formation of the plume of hot magmatic fluid, we use a standard system of governing equations. It includes the continuity equations for

We use radial non-uniform grids with the

For modelling the gravity changes and ground displacement by method A, we specify a network of equally spaced OPs along the straight line

We apply an extended grid to reduce the influence of the boundary conditions on the elastic equilibrium (Fig.

The domain and the boundary conditions for the geomechanical simulation by method B. The mesh used for the fluid flow modelling is shown in black. The distribution of total displacement is shown at

If the domains for the hydrodynamic and geomechanical modelling coincide, i.e.,

The vertical components of the gravity change and ground displacement,

Vertical components of the ground displacement (

An agreement between the

The results shown in Figs.

The results of

Radial distribution of the vertical

The distribution of

Displacements

In addition to validating the geomechanical code against the semi-analytical approach, we consider another benchmark study to demonstrate the capabilities of method B for modelling hydrothermal systems with spatial heterogeneities in elastic properties of rocks. The study is identical to that described in Sect.

The distributions of

Distributions of

The developed extension of MUFITS allows for the convenient built-in calculations of ground displacement and gravity changes. These calculations are performed automatically by the simulator without the involvement of any external post-processing utilities. As a consequence, the reservoir models developed and simulated with MUFITS can now be history matched to the observations of gravity changes and ground displacement. This software development, although quite straightforward, makes MUFITS closer to a universal package for modelling flows in hydrothermal systems. The developed modelling options, applied here in a study of hydrothermal activity, can also be utilised in other applications, including oil and gas extraction (Fig.

The simulation results also demonstrate an acceptable accuracy of the semi-empirical approach (method A) for predicting ground displacement. Given that the necessary assumptions in Sect.

The executable of the extended version of MUFITS as well as its Reference manual can be downloaded at

AA developed the modelling options for the gravity changes and ground displacement by method A. IU developed the geomechanical code (method B) and coupled it with MUFITS.

The authors declare that they have no conflict of interest.

This article is part of the special issue “European Geosciences Union General Assembly 2020, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2020, 4–8 May 2020.

The authors acknowledge funding from the Russian Science Foundation under grant # 19-71-10051. The authors would like to thank Armando Coco and an anonymous reviewer for their extensive and constructive comments during the review of this manuscript.

This research has been supported by the Russian Science Foundation (grant no. 19-71-10051).

This paper was edited by Antonio Pio Rinaldi and reviewed by Armando Coco and one anonymous referee.