A pseudospectral approach to the McWhorter and Sunada equation for two-phase flow in porous media with capillary pressure

Bjørnarå, Tore Ingvald; Mathias, Simon A.

doi:10.1007/s10596-013-9360-4

Cited by 27 publications

(28 citation statements)

References 15 publications

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“…The wetting phase volume that has imbibed into the porous medium is given by the solution of the initial and boundary value problem defined in equations . This nonlinear parabolic equation has no general solution and requires some numerical solution techniques [ Schmid et al ., ; Bjornara and Mathias , ]. However, at the onset of imbibition a distinctive front with a finite velocity propagates into the medium due to the vanishing diffusion coefficient for

\lim_{S \to S_{r}} D (S) = 0

.…”

Section: Models For Countercurrent Spontaneous Imbibitionmentioning

confidence: 99%

Accurate early‐time and late‐time modeling of countercurrent spontaneous imbibition

March

Doster

Geiger

2016

Water Resources Research

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Spontaneous countercurrent imbibition into a finite porous medium is an important physical mechanism for many applications, included but not limited to irrigation, CO 2 storage, and oil recovery. Symmetry considerations that are often valid in fractured porous media allow us to study the process in a onedimensional domain. In 1-D, for incompressible fluids and homogeneous rocks, the onset of imbibition can be captured by self-similar solutions and the imbibed volume scales with ffiffi t p . At later times, the imbibition rate decreases and the finite size of the medium has to be taken into account. This requires numerical solutions. Here we present a new approach to approximate the whole imbibition process semianalytically. The onset is captured by a semianalytical solution. We also provide an a priori estimate of the time until which the imbibed volume scales with ffiffi t p . This time is significantly longer than the time it takes until the imbibition front reaches the model boundary. The remainder of the imbibition process is obtained from a selfsimilarity solution. We test our approach against numerical solutions that employ parametrizations relevant for oil recovery and CO 2 sequestration. We show that this concept improves common first-order approaches that heavily underestimate early-time behavior and note that it can be readily included into dual-porosity models. Key Points: New physically based model for spontaneous imbibition Model captures transition from earlytime to late-time imbibition Model validated for different applications Correspondence to: R. March, rafael.march@pet.hw.ac.uk Citation: March, R., F. Doster, and S. Geiger (2016), Accurate early-time and latetime modeling of countercurrent spontaneous imbibition, Water Resour.

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\lim_{S \to S_{r}} D (S) = 0

.…”

Section: Models For Countercurrent Spontaneous Imbibitionmentioning

confidence: 99%

Accurate early‐time and late‐time modeling of countercurrent spontaneous imbibition

March

Doster

Geiger

2016

Water Resources Research

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“…At each time step, the domain deforms and we use Equations(20) and(23) as appropriate to update the grid. It is clear thatF (û),ĉ(û), and thus the right-hand side of Equation (49), are all vectors of length N .…”

mentioning

confidence: 99%

From arteries to boreholes: transient response of a poroelastic cylinder to fluid injection

Auton¹,

MacMinn²

2018

Proc. R. Soc. A.

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The radially outward flow of fluid through a porous medium occurs in many practical problems, from transport across vascular walls to the pressurization of boreholes in the subsurface. When the driving pressure is non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid can control both the steady state and the transient mechanics of the system. Very large pressures or very soft materials lead to large deformations of the solid skeleton, which introduce kinematic and constitutive nonlinearity that can have a non-trivial impact on these mechanics. Here, we study the transient response of a poroelastic cylinder to sudden fluid injection. We consider the impacts of kinematic and constitutive nonlinearity, both separately and in combination, and we highlight the central role of driving method in the evolution of the response. We show that the various facets of nonlinearity may either accelerate or decelerate the transient response relative to linear poroelasticity, depending on the boundary conditions and the initial geometry, and that an imposed fluid pressure leads to a much faster response than an imposed fluid flux.

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“…Other than the solution procedure presented by McWhorter and Sunada [1990], many algorithms have been proposed for an improved and stable performance [Fuč ık et al, 2007;Bjørnarå and Mathias, 2013]. For example, Bjørnarå and Mathias [2013] presented an approach to obtain the solutions without using the implicit integrals in equations (13), (14) and (15). This was achieved by solving the ordinary differential equation (9) using a pseudospectral differentiation matrix.…”

Section: Analytical Solutionmentioning

confidence: 99%

“…The original analytical solution for equation (9) can either be solved using the implicit integral [McWhorter and Sunada, 1990] or the pseudospectral approach [Bjørnarå and Mathias, 2013]. Both methods may require advanced programming skills and careful numerical implementation.…”

Section: Approximate Solutions To Co and Countercurrent Flowmentioning

confidence: 99%

Analytical and numerical investigations of spontaneous imbibition in porous media

Nooruddin

Blunt

2016

Water Resources Research

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We present semianalytical solutions for cocurrent displacements with some degree of countercurrent flow. The solution assumes a one-dimensional horizontal displacement of two immiscible incompressible fluids with arbitrary viscosities and saturation-dependent relative permeability and capillary pressures. We address the impact of the system length on the degree of countercurrent flow when there is no pressure drop in the nonwetting phase across the system, assuming negligible capillary back pressure at the inlet boundary of the system. It is shown that in such displacements, the fractional flow can be used to determine a critical water saturation, from which regions of both cocurrent and countercurrent flow are identified. This critical saturation changes with time as the saturation front moves into the porous medium. Furthermore, the saturation profile in the approach presented here is not necessarily a function of distance divided by the square root of time. We also present approximate solutions using a perturbative approach, which is valid for a wide range of flow conditions. This approach requires less computational power and is much easier to implement than the implicit integral solutions used in previous work. Finally, a comprehensive comparison between analytical and numerical solutions is presented. Numerical computations are performed using traditional finite-difference formulations and convergence analysis shows a generally slow convergence rate for water imbibition rates and saturation profiles. This suggests that most coarsely gridded simulations give a poor estimate of imbibition rates, while demonstrating the value of these analytical solutions as benchmarks for numerical studies, complementing Buckley-Leverett analysis.

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A pseudospectral approach to the McWhorter and Sunada equation for two-phase flow in porous media with capillary pressure

Cited by 27 publications

References 15 publications

Accurate early‐time and late‐time modeling of countercurrent spontaneous imbibition

Accurate early‐time and late‐time modeling of countercurrent spontaneous imbibition

From arteries to boreholes: transient response of a poroelastic cylinder to fluid injection

Analytical and numerical investigations of spontaneous imbibition in porous media

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