DFG-Graduiertenkolleg Geowissenschaftliche und Geotechnische Umweltforschung  an der TU Freiberg

 

	Dipl.-Ing. Alexander Nekrassov
Anschrift:	Institut für Bohrtechnik und Fluidbergbau   Agricolastraße 22  D-09596 Freiberg
Telefon:	(+49 - 3731) 39 3213
Fax:	(+49 - 3731) 39 2502
E-mail:	nekrass@bohr1.tbt.tu-freiberg.de

Today it is almost impossible deal effectively with fluid flow, such as ground water flow without using modeling. This holds true also for the development of oil and gas fields.

In any project dealing with fluid flow, many factors need to be considered. What will be the environmental impact of the  project and what are the consequences? What does this mean for business? These questions can be answered only by using modeling. Analytical solutions were used to meet this goal in the past. But the problems became more serious and more exactness was needed. This led to the development of numerical simulators, which take into account many physical phenomena and geological features.

A numerical simulator is based on a mathematical model of a process, a model of an object, and a numerical method. The mathematical model consists of physical laws which describe the process. The model of an object must contain the physical and geological properties. The calculations are determined by the numerical method.

There are many commercial simulators available, which are based on classical modeling principles. These programs are universal, i.e., they can be used for almost all practical tasks. Their universality has a disadvantage in that these models are not suitable for some cases. To overcome this difficulty we want to develop a simulator directed not at the solving of many different problems but at some special cases.

First of all, we want to consider the multiphase multicomponent flow, i.e., not only the flow of the main components such as methane or air in the gas phase, heavy hydrocarbons in NAPL (non aqueous liquid phase – oil phase), or water in the aqueous phase, but the flow of all components present including acid and salt. For the purposes of this work – the development of a simulator both for aquifers and for oil and gas fields – we assume as the mass transfer mechanisms together: filtration, dispersion, and diffusion. In the future, we plan to solve inverse problems on the basis of this simulator, i.e., to carry out history matching and optimization.

 
The main features of this new tool are:

– Detailed prediction of phase properties based on modified equation of state
– Accurate description of mass transfer owing to diffusion-dispersion phenomena
– Correct description of phase mobilities by the development of a new generalised model of phase properties
– New fast and accurate approach to computing advection/dispersion processes based on the Front Limitation algorithm
– Fast and accurate numerical solution method based on a special iterative solver

We assume the following mathematical model: according to the law of conservation of matter, the equation of balance of total mass and the equations of balance of mass for all excluding the first components are derived. In other words, the balance of the component masses and of the total mass in all phases is considered. The mass fluxes are described by the suitable physical laws, namely: by the Darcy law for the advective term and by the Fick’s law for the dispersion-diffusion term. The time derivatives show the mass change inside an REV (representative elementary volume). The divergence-operator means the difference between outfluxes and influxes.

(1)

(2)

The special feature of this model is that we, unlike those who hold a more traditional point of view, want to represent the phase properties, such as relative permeability and viscosity, as the dependencies not only of the pressure and the saturation, but also of the compound.

The distribution of components between the phases must naturally depend on the thermodynamic conditions, i.e., on the pressure and on the temperature. Therefore, we make an assumption about the equilibrium. Meaning that the components have equal chemical potential or fugasities.

(3)

(4)

Finally, we consider the additional conditions for the phase pressures, the phase saturations, and the phase compound:

(5)

(6)

This problem statement uses modern physics, chemistry, and thermodynamics to describe the phase behavior and, above all, the mass exchange between phases.

Instead of a vapor pressure curve between liquid and vapor states existing for the pure substances, a two-phase region arises where the components can occur in both phases. This region is bounded by the vapor pressures curves of the components, the bubble points, and the dew points curves. The location of this region depends on the components of the mixture. That is the main reason for considering the multicomponent flow of natural fluids.

The model of a geological object is represented by a grid. Every mesh must contain the following properties: permeability in three directions, porosity, compressibility, net-to-gross thickness ratio, and dispersivity.

We face three main problems in implementing our model. The first problem is which method to use. The numerical method for this goal must be stable, convergent, and fast. We suggest using the Newton method for this purpose. The Newton method consists of an iterative search for the values of state variables.

Interpolation of concentrations is another important problem. The Front Limitation algorithm will be used to interpolate the concentrations on the boundaries of elements. It lets us use the grid only for modeling the geology, i.e., we do not need to scale down the meshes.

The last problem is computing time. We suggest using red-black ordering to optimize the solver.

 
Nomenclature:

	f		– porosity
	r		– density
	s		– saturation
			– permeability
	kr		– relative phase permeability
	m		– viscosity
	p		– pressure
	q		– total flow rate
	qi		– flow rate of i-th component
	pC		– capillary pressure
	ji		– fugacity of i-th component
	T = const		– reservoir temperature
	yi,xi,o,xi,w		– phase concentrations of i-th component
	Di		– dispersion – diffusion tensors
	N		– number of components
	L		– number of water soluble components
	g,o,w		– indexes of gas, oil, water phases
	1		– index of water