Roger Young

Vertical Two-Phase Flow in Porous Media

abstractNew concepts are introduced to describe single-component two-phase flow under gravity. The phases can flow simultaneously in opposite directions (counterflow), but information travels either up or down, depending on the sign of the wavespeedC. Wavespeed, saturation and other quantities are defined on a two-sheeted surface over the mass-energy flow plane, the sheets overlapping in the counterflow region. A saturation shock is represented as an instantaneous displacement along a line of constant volume fluxJQ in the flow plane. Most shocks are of the wetting type, that is, they leave the environment more saturated after their passage. When flow is horizontal all shocks are wetting, but it is a feature of vertical two-phase flow that for sufficiently small mass and energy flows there also exist drying shocks associated with lower final saturations.

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock.In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived.When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established.When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.

Quasi‐steady vertical two‐phase flows in porous media

Two-phase flows in which the pressure profile is essentially constant are called quasi-steady. One-dimensional quasi-steady flows have the remarkable property that volumetric flux is essentially independent of position and is a function of time alone. Such volumetric fluxes are (to within an additive constant) inversely proportional to the spatial mean value of flowing viscosity. Consequently, the evolution of such a system is controlled by global properties of viscosity. Vertical two0phase quasi-steady flows evolving from an initial steady flow to another final steady flow are analyzed, and the corresponding theory is matched to output from a numerical simulator. Good agreement between the theory and simulator is demonstrated for simple shocks and expansion waves.

Two-phase boundary layer formation in a semi-infinite porous slab

Similarity profiles of pressure and saturation are analysed which result from one-dimensional planar withdrawal of fluid from a porous region initially containing a two phase mixture of steam and water. Approximate expressions are derived for the evolution of pressure and saturation profiles, and boundary-layer changes in saturation are identified. The existence of a similarity variable implies that the saturation conditions for the reservoir tend with time either to having both phases flowing; or to a single phase vapour. In particular, the nonlinear nature of the governing equations implies that infinitesimal changes in pressure can produce finite changes in saturation. The two mechanisms responsible for saturation changing with time involve local changes in energy storage in rock and fluid; together with spatial variations in flowing enthalpy. The latter mechanism occurred relatively slowly in the examples treated, and was responsible for boundary-layer formation when one phase was initially immobile. Dimensional analysis reveals that when a boundary layer develops, the underlying equations involve essentially only one dimensionless parameter which is typically small, being associated with the ratio of the energy density of the mobile phase relative to the total energy density.

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.

Self-similar radial two-phase flows

A general mathematical formulation is developed, appropriate to both single- and two-phase conditions in a porous medium. A new similarity solution, generalizing the well known Theis solution, is derived for radial flow to a well in a region initially containing a two-phase mixture of steam and water, in which either steam or water is immobile. This generalized Theis solution follows from a new Ricatti equation for mass flow, which includes an additional nonlinear effect resulting from quadratic pressure gradient terms. Existing results for the saturation profile are extended by inclusion of nonlinear contributions, which are shown to be necessary for accurate descriptions of the saturation profile. A boundary-layer analysis is developed for flow about the well, where both mass flux and flowing enthaply are almost constant, which enables both the pressure and saturation profiles to be determined analytically. An analysis of two-phase self-similar shocks is given, together with the associated entropy conditions constraining the existence of shocks. Finally, numerical examples are discussed showing the agreement between theory and numerical simulations.

PRESSURE TRANSIENTS IN A DOUBLE-POROSITY MEDIUM

Pressure transients associated with a constant-rate pumping test in a double-porosity (fissured or multilayered) reservoir may exhibit, in succession, several periods of characteristic log linear dependence on time. If the flow in the matrix blocks is primarily vertical then slope halving of the semilog gradient may be observed, but if the radial component of block flow becomes significant then pressure stabilization can occur. During the final period of formation flow an increase in gradient is associated with a dominant vertical component of block flow, but if the radial component is significant then a decrease in gradient will be observed.

Two-dimensional flow in a fractured medium

Transport processes within a liquid-filled fractured reservoir can be modelled using a double-diffusive mechanism in fracture and block. Then it is commonly assumed that the flow in the block is purely one-dimensional (e.g. vertical). Lateral flow within the block will, however, become significant at long times. Avdonin has given an analytic solution for the pressure response in an infinite fissure bounded by two homogeneous half-spaces, allowing vertical flow only in the blocks. We extend this solution to include horizontal flow in the blocks. There are significant qualitative differences between the two cases. In particular, we find that if fluid is injected at a constant rate into the fissure and horizontal flow in the blocks is allowed, then the long-time pressure response of the fissure/block assembly has the same character as that due to a line source in a homogeneous anisotropic porous medium.

Constant rate production of geothermal fluid from a two-phase vertical column. I: Theory

Previous theoretical results on geothermal two-phase flows in porous media are extended and applied to the case of withdrawal of fluid at a constant rate from a vertical column. Dimensional considerations show that pressure and saturation behaviour is controlled by a single parameterα which is the ratio of the withdrawal speed to buoyancy speed. For large flows (largeα) fluid withdrawal is a mining process, and a vapour dominated zone spreads out from the production level. Production enthalpies tend towards steam values. However ifα is small then gravity dominates, and buoyancy forces can lead to the formation of a steam bubble which escapes from the production boundary and rises towards the surface. Production enthalpy may then remain at the liquid value over long periods. In addition certain saturation ranges at the sink may be forbidden as a consequence of the constant rate boundary condition. Then saturation shocks will form at the production boundary and travel out from the sink. Internally generated shocks may also occur. Pressure and saturation response to a steady withdrawal of fluid is more complicated than in a two-phase gravity-free situation. Since gravity is an essential component of even ‘horizontal’ two-phase flow this suggests that two-phase studies which ignore the role of gravity may be too simplistic.

Two-phase brine mixtures in the geothermal context and the polymer flood model

Two-phase mixtures of hot brine and steam are important in geothermal reservoirs under exploitation. In a simple model, the flows are described by a parabolic equation for the pressure with a derivative coupling to a pair of wave equations for saturation and salt concentration. We show that the wave speed matrix for the hyperbolic part of the coupled system is formally identical to the corresponding matrix in the polymer flood model for oil recovery. For the class ofstrongly diffusive hot brine models, the identification is more than formal, so that the wave phenomena predicted for the polymer flood model will also be observed in geothermal reservoirs.