Lumped parameters models

A key part of responsible utilization of geothermal resources is good management of the resource. Mismanagement of the geothermal resource, or field, can have serious negative consequences for the field. The most common example of this is maintaining production from a geothermal field that cannot be sustained by natural recharge, leading to an overall decrease in production efficiency. It is therefore important to have a good understanding of the resource, or field, and the field’s response to utilization. Many different models have been constructed in order to facilitate such an understanding and to aid in the management of geothermal fields.

Geothermal reservoir models can vary in scale and complexity. They can range from simple analytical models, like a simple volumetric assessment, to complex numerical models, like so called complex numerical 3d spatial reservoir models. In general, there is a trade-off between complexity and cost. The more complex the model the more information can be extracted out of it, However, to use complex models more data, and often time, is required. This means that while f.ex. complex numerical 3d spatial reservoir models are often very good to use for general management of geothermal resources they can be quite costly. This cost can be prohibitively expensive, especially for hot water systems used for district heating, where the enthalpy-to-mass ratio is lower. This means that there is a need for simpler models that can, nevertheless, be valuable for reservoir management.

Figure 1. A schematic of a simple single volume lumped model showing the tank and resistor components.

One type of such simplified models are the lumped parameter models (Grant and Bixley 2011; Axelsson 1989). Such models group different parts of the reservoir into separate, homogeneous “lumped” volumes, and then define connections between them. These lumped models simplify the reservoir by only considering a small number of volumes and connections, which reduces their complexity. There are two basic components to such lumped parameter models, shown in figure 1. The “lumped” volumes, or tanks, simulate the mass, or fluid, storage in the reservoir. They are assumed to be homogenous with respect to thermodynamic, fluid, and rock properties. The connections, or resistors, simulate fluid flow in the reservoir, and describe the flow resistance between two tanks, i.e. between two parts of the reservoir. These two components can then be used to construct a model of a reservoir, as can be seen in figure 2. The effects of utilization on the reservoir can be simulated by extracting, or inserting, mass into the lumped volumes. Lumped models have been used to reliably estimate production capacity, rates of pressure decline and effects of reinjection in many hot water geothermal reservoirs (Axelsson et al. 2005).

Figure 2. The upper figure shows a conceptual model of a geothermal system while the lower figure shows a schematic of a lumped model of the same geothermal system.

From the conservation of mass we know that any changes in the mass in a tank means that either that mass has been injected into or extracted out of the system (via production/injection) or has flowed to/from another tank. For a single tank this can be expressed as

EQUATION 1

where

m

is the mass of the fluid in the tank,

t

is the time,

Q

is the total internal mass flow into, or out of, the tank and

F

describes the external mass flow into, or out of, the tank due to production or injection. The relation between the fluid mass in tank i and the pressure in the tank can be expressed as

EQUATION 2

where

Δ p_{i}

is the pressure change in the tank,

Δ m

is the mass change in the tank and

κ_{i}

is the storage coefficient of the tank. The storage coefficients,

κ_{i}

, depend on the size of the reservoir volume the tank describes, as well as the storage mechanism for that volume. These storage mechanisms can relate to the liquid/rock compressibilites, for confined reservoirs, or to the free-surface mobility, in the case of unconfined reservoirs. The mass flow

q_{i k}

between two tanks, i and k, can be described by Darcys law

EQUATION 3

where

σ_{i k}

is the conductance between the wells,

p_{k}

is the pressure in tank k and

p_{i}

is the pressure in tank i. The conductance,

σ_{i k}

, reflect the permeability of the reservoir as well as the geometry of the flow in the reservoir. Combining equations 1, 2 and 3 an equation for the pressure change in a tank i, in a system of N tanks, can be derived

EQUATION 4

where

f_{i}

is the external mass flow (production/injection) into or out of the tank. Here, it has been assumed that the tank can be connected to an infinite volume tank at a constant pressure. Such infinite tanks can be used to describe recharge into the system. A simple analytical solution for this equation can be found if a linear lumped system is considered. Such a system is shown in figure 3.

Figure 3. Example of a simple lumped system where the pressure response can be solved analytically (Figure from Axelsson et al. 2005).

Here the reservoir is imagined as a nested collection of tanks. The first tank describes the central, or production part of the reservoir, where the production wells are located. This central part is then connected to the outer parts of the reservoir that should, in theory, be not as directly affected by production from the field. Finally, these outer parts can be connected again to even deeper and more distant parts of the field. Such linear systems have been used to successfully model several hot water systems (Axelsson et al. 2005; Thorgilsson et al. 2020). These simple linear systems work well for hot water geothermal systems where there is a small (1 – 2) number of production wells. For more complex systems, with multiple production and injection wells a more complex model is needed. One of the aims of the GeoModel project is to derive a general solution to equation 4 that allows the use of any kind of arbitrary lumped system.

References

Grant, M.A. and Bixley, P.F. (2011). Geothermal Reservoir Engineering (2^nd ed.). Academic Press.

Axelsson, G. (1989). Simulation of pressure response data from geothermal reservoirs by lumped parameter models. Proceedings, 14^th Workshop on Geothermal Reservoir Engineering, Stanford University, California.

Axelsson, G., Björnsson, G., Quijano, J.E. (2005). Reliability of Lumped Parameter Modeling of Pressure Changes in Geothermal Reservoirs., Proceedings, World Geothermal Congress 2005, Antalya, Turkey.

Thorgilsson, G., Axelsson, G., Halldorsdottir, S., Hardardottir, V., Gautason, B., Oskarsson, F. (2020). The Eskifjordur Low-Temperature Geothermal System in E-Iceland, Pressure Response Modelling and Tracer Test Analysis. Proceedings, World Geothermal Congress 2020, Reykjavík, Iceland.

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The project is financed under the Fund for Bilateral Relations through the European Economic Area Financial Mechanism (EEA FM) and the Norwegian Financial Mechanism (NFM) 2014-2021, programme “Environment, Energy and Climate Change”.

Project Operator: Ministry of Climate and Environment