A mathematical model for thermal single-phase flow and reactive transport in fractured porous media

Highlights

• Thermo-reactive flow.

• Fractured porous media and reduced models.

• Temporal splitting scheme to solve complex set of non-linear, non-smooth, mixed-dimensional partial differential equations.

• Guidances and solution strategies for realistic geometries.

Abstract

In this paper we present a mathematical model and a numerical workflow for the simulation of a thermal single-phase flow with reactive transport in porous media, in the presence of fractures. The latter are thin regions which might behave as high or low permeability channels depending on their physical parameters, and are thus of paramount importance in underground flow problems. Chemical reactions may alter the local properties of the porous media as well as the fracture walls, changing the flow path and possibly occluding some portions of the fractures or zones in the porous media. To solve numerically the coupled problem we propose a temporal splitting scheme so that the equations describing each physical process are solved sequentially. Numerical tests show the accuracy of the proposed model and the ability to capture complex phenomena, where one or multiple fractures are present.

Introduction

The presence of fractures has an impact on subsurface flows at all scales: flow tends to focus along highly permeable fractures, which can create shortcuts in the domain, or, in the case of cemented or low permeable fractures, they might create barriers in the domain. In the context of reactive transport fractures can be responsible for fast transport of fluid with different chemical composition with respect to the surrounding matrix: this occurs for instance in geothermal reservoirs where water with different salinity, solutes and temperature is injected in the subsurface. These differences in composition and temperature can trigger transformations such as mineral precipitation, dissolution or replacement, with an impact on porosity and fracture aperture. The effective exploitation of the geothermal system can be jeopardized by such phenomena.

Because of their thickness, or aperture, fractures are usually represented as lower dimensional objects and new equations along with interface conditions with the surrounding porous media are derived. This procedure is usually referred to as model reduction and the resulting model is named mixed-dimensional or hybrid-dimensional problem. Seminal works dealing with single-phase flow are for example [6], [5], [27], [7], [52]. During the years new models have been developed based on this idea, in particular for multi-phase flow [39], [4], transport [62], [21], and faults flow [63], [28], [35]. The geometrical complexity of the fracture networks requires to handle in an accurate way also the intersection between them, indeed the intersection may have different physical parameters than the incident fractures. In this case, new models have been derived where the intersection is part of the problem, see for example [32], [18], [61]. In the special setting of high speed circulation of the liquid in the fractures, the Darcy model may be not appropriate. Thus several authors proposed a new model based on Forchheimer or even more advanced flow model. Refer to [34], [53], [54], [50], [3]. Finally, we refer to [16] for a more detailed review on different strategies to handle the complex problem of fractured porous medium.

The numerical solution of these problems is challenging due to several aspects, in fact the fracture networks may pose severe constraints in the grid generation resulting in poor quality and too many elements. Since this work is more focused on the modeling side, we refer to the main works that dealt with different classes of numerical schemes: classical mixed finite elements [52], hybrid high-order [20], discontinuous Galerkin [8], mimetic finite differences [9], extended finite elements [32], [41], [61], [31], virtual element method [36], [37], and references therein. Important benchmark studies to validate the effectiveness of the numerical schemes are [23], [30], [38], [15]. Finally, a unified approach for numerical frameworks to solve such problems is presented in [55].

The aim of our work is to propose a model to account explicitly for the presence of fractures and their impact not only on flow, but also on temperature, and on the transport and reaction of minerals. The equations describing flow and transport are thus a coupled system of mixed-dimensional PDEs which will be approximated by means of lowest order mixed finite elements or mixed virtual elements, depending on the geometrical complexity of the computational grid. We will consider a simple model for mineral precipitation and dissolution following the model presented, among others, in [2]. On top of the usual quantities involved in the aforementioned physical processes, thus pressure, temperature and concentration, the porosity and consequently the permeability of the porous media can change, strengthening the coupling among the physical processes. Similarly in the fractures the tangential and normal effective permeability depend on the aperture, which changes with dissolution and precipitation increasing even more the complexity of the resulting model. The system we are aiming to solve results to be fully coupled, non-linear and possibly non-smooth due to the choice of the reaction model: this poses several challenges that are deal with the proposed framework. To avoid the occurrence of negative concentrations and oscillations when the amount of precipitate approaches zero we adopt an event detection/location strategy to detect the discontinuity in the ODE describing the reaction part, which is, for this reason, split from advection and diffusion by means of a first-order operator splitting. Several numerical examples will show the validity of our approach for increasing level of geometrical difficulty of the fracture network.

The paper is organized as follows. In Section 2 we introduce the mathematical model to describe fluid flow, heat transport, and solute transport with chemical reactions in porous media. The latter are particularized in Section 3. The mixed-dimensional problem to describe the physical processes in the fractures is discussed in Section 4. Section 5 presents the discretization considered to approximate the models, in particular a splitting scheme is detailed that allows for a sequential resolution of each physical process involved in the simulation. In Section 6 we run different examples to show the validity and accuracy of the proposed approach. Finally, Section 7 is devoted to the conclusions.