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_Journal of Structural Geology 32 (2010) 1827–1833_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _Buoyancy driven flow from a waning source through a porous leaky aquifer_ _Andrew W. Woods, Simon Norris_ _BP Institute, University of Cambridge, Cambridge, CB3 OEZ, United Kingdom _Nuclear Decommissioning Authority, Harwell, Oxfordshire_ _article info_ _Article history: Received 27 March 2009 _Received in revised form 21 July 2009 _Accepted 11 August 2009 _Available online 11 September 2009_ _abstract_ _We develop a series of models to describe the migration of a buoyant fluid through a layered permeable rock following release from a localized waning source. In particular, if the fluid is injected into a high permeability layer, bounded above by a layer of lower permeability, a plume migrates along the interface, with some draining into the low permeability layer if the current is sufficiently deep to overcome the capillary entry pressure. We show the motion of the fluid is controlled by a number of key factors with the dominant dimensionless numbers being the time-scale for the source to decay compared to the timescale for draining through the low permeability layer, the residual saturation of the gas and water in the formation as that phase is displaced by the other phase, and the capillary entry pressure, as measured by the critical depth of the current required for draining, as compared to the initial depth of the current. Simplified analytical models are presented to illustrate some of the key controls on and transitions in the flow, and the models are used to explore leakage and trapping prior to flow reaching a fault zone._ _© 2009 Elsevier Ltd. All rights reserved._ _1. Introduction_ _There is growing interest in the migration of gas from a localized source through a permeable rock owing to its relevance for the dispersal of CO2 sequestered in the subsurface, but more generally for the dispersal of buoyant fluids which may form in and migrate from geological waste repositories (Bickle et al., 2007; Hesse et al., 2007). In this latter situation, an interesting feature is that the rate of generation of buoyant fluid may progressively decay over a period of several hundred years. The gradual waning of the source leads to some interesting dynamical balances in the migrating buoyant plume, and has a critical impact on the dispersal pattern of the fluid; this forms the topic of the present contribution._ _Several models have been developed to describe the motion of buoyant plumes of fluid migrating through the subsurface (Barenblatt, 1996; Bear, 1972; Huppert and Woods, 1995), and recently these have been extended to include the equations for two-phase flow in a permeable rock, including the effects of the relative permeability between the two phases (Hesse et al., 2006; Nordbotten and Celia, 2006). With a localized source of buoyant fluid, these models lead to the prediction that in a confined aquifer, a buoyant plume develops adjacent to the upper boundary of the aquifer and then spreads out along the aquifer (Hesse et al., 2006; Mitchell and Woods, 2006). If it comes into contact with a fault fracture system, then some of the flow may drain upwards along the fault where it may then intersect another permeable layer, enabling part of the leakage flux to continue spreading laterally (Pritchard et al., 2001; Pritchard, 2007). If the source wanes, the continuing finite plume will then develop a trailing front. As this advances through the formation, there may be some capillary trapping of the fluid leading to a residual saturation (Barenblatt, 1996; Hesse et al., 2006; Obi and Blunt, 2006; Kharaka et al., 2006; Farcas and Woods, 2009a). As a result, the plume becomes progressively depleted as it migrates through the formation, leaving the capillary trapped zone behind._ _In the present contribution, we examine the motion of a buoyant plume supplied by a waning source which spreads through a permeable rock, bounded above by a less permeable thin layer into which the fluid may slowly drain off. We also account for the capillary retention of a fraction of the fluid at any point along the plume where the flow thickness decreases in time. This leads to predictions of the fraction of the current which may remain trapped in the original layer rather than leaking off higher into the formation._ _We note that our analysis is restricted to a two-dimensional flow, in order to identify some of the key controls on the system, although we note that three-dimensional cross-flow effects can also arise, especially far upslope of the source (cf. Vella and Huppert, 2007; Farcas and Woods, in press). However, with a long linear source, the effects of three-dimensional spreading of the flow_ _1828_ _A.W. Woods, S. Norris Journal of Structural Geology 32 (2010) 1827–1833_ _upslope of the source to points beyond the extremities of the source, may only become dominant once the flow has advanced a substantial distance upslope, and so in that case, the present modelling may provide a reasonable approximation to the flow in the near field. Also, in some situations, the flow may be structurally confined such that the two-dimensional model may provide a reasonable leading order model for the flow._ _2. The model_ _We consider the migration of a buoyant fluid of density ρ through a permeable layer of rock, saturated with fluid of density ρ − δρ, which is bounded above by a thin layer of lower permeability. We assume the injected fluid is only able to invade the low permeability layer if it is sufficiently deep, h(x,t), to overcome the capillary entry pressure, h > hc. Otherwise it will continue to run upslope through the high permeability layer, under the lower boundary of this ‘seal’ layer (Fig. 1). As the current spreads out along this layer, of inclination to the horizontal q, the alongslope motion is governed by the buoyancy forces acting on the flow, according to the relation for the transport or Darcy flux m (cf Bear, 1972; Barenblatt, 1996):_ _u = kDρg − m, _vh cos(q)_ _vx q sin(q)_ _(1)_ _where h(x,t) is the thickness of the current, x is the alongslope position, m is the viscosity and k is the effective permeability of the fluid as it migrates through the rock, where we account for the effects of relative permeability in a very simple fashion with the single permeability parameter. Most of the interest in this work is in modelling gas or supercritical fluid dispersion and we model the motion through the rock in terms of an effective permeability. This has been shown to give good leading order predictions compared to the full two-phase flow relations for such buoyancy driven flows (cf. Nordbotten and Celia, 2006; Hesse et al., 2006). The rate of loss of fluid from the current to the overlying low permeability layer, through unit length of the boundary, depends on the permeability kb, the thickness b of the seal layer, and the thickness of the current (cf. Pritchard et al., 2001) according to the relation:_ _Loss = kbgδρh − bcos(q)bm_ _(2)_ _These equations are then combined with the relation for the conservation of mass, which in the invading flow has the form:_ _f(1 − sw) + (vhR/vt) = v vx(hu/x8a) + Loss_ _for vh vt > 0_ _(3)_ _since as the invading gas advances into the formation, there is a fraction sw of the pore space which remains saturated in the original fluid, and gas then dissolves into this fluid, representing an effective additional pore volume fsw R for the injected fluid. Here R denotes the mass fraction of gas dissolved in the original fluid (which occupies the fraction sw of the pore volume) multiplied by the density of the original fluid and divided by the density of the free gas phase. Also, in this expression f denotes the porosity of the rock._ _In contrast, in any part of the flow where the depth of the current decreases with time, then as the buoyant fluid vacates the pore space and is displaced with water, there will be some residual gas trapped which occupies a fraction sg of the pore spaces. Here, for simplicity, we model this as being a constant (cf. Barenblatt, 1996; Hesse et al., 2006, 2008) and so the conservation of mass takes the form:_ _f(1 − sw) + (sgvvh/t) = v vx(hu/x8a) + Loss_ _for vh vt < 0_ _(4)_ _In order to solve for the motion of the current we require some boundary conditions. First, it follows that the nose of the current propagates at the rate:_ _dx/dt = f(1 − sw)Rsw_ _(5)_ _while we assume that the source flux, at x = 0, gradually wanes, at a rate:_ _Q(t) = Qoexp(−t/τs) _kDρgsin(q)m/(1 − cot(q)vh/0; vx/t!h/0; t!)_ _(6)_ _Fig. 1. Cartoon of the flow geometry for the present problem._ _A.W. Woods, S. Norris Journal of Structural Geology 32 (2010) 1827–1833_ _1829_ _where t is the e-folding time over which the source flux decays. It is this waning source flux, coupled with the dynamics of continuous leakage of fluid through the overlying seal rock, and the capillary retention at the tail of the current, which provides the new analytic results of this paper._ _From eq. (6), we deduce that there is no drainage if the source flux Qo is smaller than a critical value:_ _Qo < hcgDrsin(q)m_ _Qcrit_ _(7)_ _3. Approximations and analytical solutions_ _With this system of equations, we can now develop a series of solutions for the motion of the plume of gas along the inclined low permeability layer. These solutions are useful for exposing some of the key controls on the distance travelled along the layer, and also how the current partitions between that component which is retained in the original layer and that component which migrates through the low permeability partial seal layer and higher into the formation._ Ключевые слова: source region, permeable rock, conned aquifer, pressure, injected ?uid, current, td, huppert, solution, analytical solution, motion, eq, overlying, thickness, fraction, ut, time, leading edge, hc, tc, porous, trapped, critical depth, component, model, ha, shallower, form, barenblatt, spread, geology, owing layer, residual, structural, darcy, effective permeability, seal layer, layered, pore space, buoyant uid, ?uid, capillary trapping, waning, porous medium, buoyant plume, time greater, vt, permeability layer, uid leak, order, loss, journal structural, source, advection speed, fraction retained, region, case, unit length, depends, residual saturation, term, buoyant, original layer, remains trapped, xb, gravity current, buoyant ?uid, tb, constant depth, decay time, capillary retention, space, drain, rate, xc, capillary, reach, cf, leading, capillary entry, waning source, closest point, key control, seal, compared, remains, fault, point, cross-slope component, decrease, leak, draining zone, kharaka, order year, utl, xn, entry pressure, eventually, permeable, bear, depth current, journal, boundary, gas, woods, overlying layer, localized source, ?ux, entry, migrates, sw, decay, depth, injected uid, mech, leakage, boundary condition, source decrease, water, vh, injected, layer, critical, constant, result, localized region, aquifer, elsevier, original, lead, current spread, plume, formation, advance, zone, upper boundary, year, distance, drainage, fluid, sufciently deep, structural geology, relative permeability, uid, pritchard, draining, hesse, edge, gravity, ?ow, rock, alongslope, original uid, fluid mech, subsequently, doi, draining time, bs, permeability, farcas