The unsteady flow of a weakly compressible fluid in a thin porous layer. I: Two-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a two-dimensional reservoir in an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting or extracting fluid. Numerical solution of this problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l. This is a situation which occurs frequently in the application to oil reservoir recovery. Under the assumption that epsilon=h/l<<1, we show that the pressure field varies only in the horizontal direction away from the wells (the outer region). We construct two-term asymptotic expansions in epsilon in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive analytical expressions for all significant process quantities. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the reservoir, epsilon, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighborhood of wells and away from wells.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that $\epsilon\propto h/l\ll 1$, we show that, to leading order in $\epsilon$, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

The unsteady flow of a weakly compressible fluid in a thin porous layer III: three-dimensional computations

We describe a novel method for determining the pressure and velocity fields for a weakly compressible fluid flowing in a thin three-dimensional layer composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Our approach uses the method of matched asymptotic expansions to derive expressions for all significant process quantities, the computation of which requires only the solution of linear, elliptic, two-dimensional boundary value and eigenvalue problems. In this article, we provide full implementation details and present numerical results demonstrating the efficiency and accuracy of our scheme.

Padé approximants for the acoustical properties of rigid frame porous media with pore size distributions

Expressions for the viscosity correction function, and hence bulk complex impedance, density, compressibility, and propagation constant, are obtained for a rigid frame porous medium whose pores are prismatic with fixed cross-sectional shape, but of variable pore size distribution. The lowand high-frequency behavior of the viscosity correction function is derived for the particular case of a log-normal pore size distribution, in terms of coefficients which can, in general, be computed numerically, and are given here explicitly for the particular cases of pores of equilateral triangular, circular, and slitlike cross-section. Simple approximate formulae, based on two-point Pade´ approximants for the viscosity correction function are obtained, which avoid a requirement for numerical integration or evaluation of special functions, and their accuracy is illustrated and investigated for the three pore shapes already mentioned

Consistent Dirichlet Boundary Conditions for Numerical Solution of Moving Boundary Problems

We consider the imposition of Dirichlet boundary conditions in the finite element modelling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass.

Industrial radioactive barite scale: suppression of radium uptake by introduction of competing ions

Incorporation of radioactive isotopes during the formation of barite mineral scale is a widespread phenomenon occurring within the oil, mining and process industries. In a series of experiments radioactive barite/celestite solid solutions (SSBarite-Celcstite) have been synthesized under controlled conditions by the counter diffusion of Ra-226, Ba2+, Sr24+ and SO42- ions through a porous medium (silica gel), to investigate inhibiting effects in Ra uptake associated with the introduction of a competing ion (Sr2+). From characterization studies, the particle size and the morphology of the crystals appear to be related to the initial [Sr]/[Ba] molar ratio of the starting solution. Typically, systems richer in Sr produce smaller sized crystals and clusters characterized by a lower degree of order. The activity introduced to the system is mainly incorporated in the crystals generated from the barite/celestite solid solution as suggested by the activity profiles of the hydrogel columns analysed by gamma-spectrometry. There is a relationship between the initial [Sr]/[Ba] molar ratio of the starting solution and the activity exhibited by the synthesized crystals. An effective inhibition of the Ra-226 uptake during formation of the crystals (SSBarite-Celestite) was obtained through the introduction of a competing ion (Sr2+): the higher the initial [Sr]/[Ba] molar ratio of the starting solution, the lower the intensity of the activity peak in the crystals. (C) 2003 Published by Elsevier Ltd.

Solute movement through intact columns of cryoturbated Upper Chalk

Cryoturbated Upper Chalk is a dichotomous porous medium wherein the intra-fragment porosity provides water storage and the inter-fragment porosity provides potential pathways for relatively rapid flow near saturation. Chloride tracer movement through 43 cm long and 45 cm diameter undisturbed chalk columns was studied at water application rates of 0.3, 1.0, and 1.5 cm h(-1). Microscale heterogeneity in effluent was recorded using a grid collection system consisting of 98 funnel-shaped cells each 3.5 cm in diameter. The total porosity of the columns was 0.47 +/- 0.02 m(3) m(-3), approximately 13% of pores were >15 mu m diameter, and the saturated hydraulic conductivity was 12.66 +/- 1.31 m day(-1). Although the column remained unsaturated during the leaching even at all application rates, proportionate flow through macropores increased as the application rate decreased. The number of dry cells (with 0 ml of effluent) increased as application rate decreased. Half of the leachate was collected from 15, 19 and 22 cells at 0.3, 1.0, 1.5 cm h(-1) application rates respectively. Similar breakthrough curves (BTCs) were obtained at all three application rates when plotted as a function of cumulative drainage, but they were distinctly different when plotted as a function of time. The BTCs indicate that the columns have similar drainage requirement irrespective of application rates, as the rise to the maxima (C/C-o) is almost similar. However, the time required to achieve that leaching requirement varies with application rates, and residence time was less in the case of a higher application rate. A two-region convection-dispersion model was used to describe the BTCs and fitted well (r(2) = 0.97-0-99). There was a linear relationship between dispersion coefficient and pore water velocity (correlation coefficient r = 0.95). The results demonstrate the microscale heterogeneity of hydrodynamic properties in the Upper Chalk. Copyright (C) 2007 John Wiley & Sons, Ltd.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.

Sea ice is a mushy layer

[1] Sea ice is a two-phase, two-component, reactive porous medium: an example of what is known in other contexts as a mushy layer. The fundamental conservation laws underlying the mathematical description of mushy layers provide a robust foundation for the prediction of sea-ice evolution. Here we show that the general equations describing mushy layers reduce to the model of Maykut and Untersteiner (1971) under the same approximations employed therein.

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/ porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic - viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown-Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown-Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz-clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks. (C) 2003 Elsevier B.V. All rights reserved.