Two-phase flow in a fissurized-porous media

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1 Two-phase low in a issurized-porous media P. Ø. Andersen and S. Evje Citation: AIP Con. Proc. 479, 234 (22); doi:.63/ View online: View Table o Contents: Published by the American Institute o Physics. Related Articles Hydrodynamics coalescence collision o three liquid drops in 3D with smoothed particle hydrodynamics AIP Advances 2, 426 (22) A new approach or the design o hypersonic scramjet inlets Phys. Fluids 24, 863 (22) Broadside mobility o a disk in a viscous luid near a plane wall with no-slip boundary condition. Chem. Phys. 37, 8496 (22) Predicting conditions or microscale suractant mediated tipstreaming Phys. Fluids 24, 82 (22) Insights into symmetric and asymmetric vortex mergers using the core growth model Phys. Fluids 24, 73 (22) Additional inormation on AIP Con. Proc. ournal Homepage: ournal Inormation: Top downloads: Inormation or Authors: Downloaded 3 Oct 22 to Redistribution subject to AIP license or copyright; see

2 Two-phase Flow in a Fissurized-porous Media P. Ø. Andersen and S. Evje University o Stavanger, NO-436 Stavanger, Norway. Abstract. Spontaneous imbibition can be an important drive mechanism in ractured reservoirs. We consider a low-permeable reservoir containing a high-permeable racture network. This subject is o practical interest as ractured reservoirs represent as much as a third o the reservoirs in the world. In the mathematical description a linear racture is symmetrically surrounded by porous matrix. Advective low occurs only along the racture, while capillary driven low occurs only along the axis o the matrix. For a given set o relative permeability and capillary pressure curves the behavior o the system is completely determined by the choice o 2 parameters: (i) the ratio o time scales or advective low in racture to capillary low in matrix α = τ /τ m ; (ii) the ratio o pore volumes in matrix and racture β = V m /V. A characteristic property o the low in the coupled racture-matrix medium is the linear recovery curve (beore water breakthrough) ollowed by a non-linear part where the rate is decreasing. The model can give insight to the role played by parameters like wettability, injection rate, volume o ractures relatively volume o matrix and strength o capillary orces versus injection rate. Keywords: Fractured reservoirs, spontaneous imbibition, water injection, capillary orces, porous media, multiphase low PACS: t, nb, 89.3.aj, r INTRODUCTION In this paper we discuss reservoirs that are naturally ractured: The reservoir is divided into a dense network o highly permeable tunnels that only makes a small portion o the bulk, while the rest o the rock is low permeable, but is the storage or much o the oil. As an example, the chalk reservoirs Ekoisk and Valhall in the North Sea have such properties. We construct a mathematical model to discuss the role o spontaneous imbibition as a recovery mechanism in a naturally ractured reservoir where water injection is used. More precisely, we consider the low along a single racture rom injector to producer well where the adjacent porous volume along the racture is being drained or oil. THE MODEL The model ormulation is largely inspired by models studied in [] and [2]. In [] the authors ormulated a D (racture)+d (matrix) model taking into account single-phase solute transport along racture driven by diusion whereas the combined eect o diusion and dissolution took place at the matrix domain normal to the racture. In [2] the authors considered a combined water-oil racture-matrix model where water was injected into racture and oil was produced rom matrix into racture due to countercurrent imbibition. Focus was on ormulating an appropriate boundary condition at the racture-matrix interace in a lab scale context. See also [3] or another interesting racturematrix low model in a single-phase setting, and [4] or a simulation study using the dual-porosity ormulation. Some analytical results on imbibition are ound in [5]. We reer to [2] or an excellent overview o previous research activity in this area. Consider a horizontal plane (x,y) such that the y-axis by deinition runs parallel with a linear racture o length L y and width 2b. The racture cuts the plane in hal and porous medium is located on both sides going a length o L x, respectively to the let and right side. The racture is bounded to 2b < x < and < y < L y. The injector is at y = and the producer is at y = L y such that water lows in positive y-direction along the racture. Locally we consider a system o 2 phases: water (w) and oil (o). The phase pressures p o, p w are related by a capillary pressure unction which depends on water saturation: p o p w = p c (s w ). Also, the saturations s o,s w are constrained by s o + s w =. We assume the matrix is closed at the outer boundary such that the total velocity in x-direction is. The total velocity v T in the racture is given by the injection conditions. We eliminate p o, p w,s o and express the system in terms o s(= s w ) only. φ m t s = x (λo m (s) m (s)k m x p m c (s)) ( < x < L x ;< y < L y ) () ( ) φ t s = φ v T y (s) b λo m (s) m (s)k m x p m c (s) x= + ( 2b < x < ; < y < L y ) Numerical Analysis and Applied Mathematics ICNAAM 22 AIP Con. Proc. 479, (22); doi:.63/ American Institute o Physics /$ Downloaded 3 Oct 22 to Redistribution subject to AIP license or copyright; see

3 (s) Capillary pressure m, OW m m, WW OW OW WW WW kr Relative permeabilities (s) Flow unction.2 m, OW m, WW FIGURE. Relative permeability, capillary pressure and low unctions or matrix and racture. 2 wetting states are considered or the matrix (oil-wet and water-wet). Green curves represent racture, while red and blue curves correspond to oil-wet or water-wet matrix, respectively. where the irst equation denotes low in the porous medium while the last equation denotes low in the racture channel. m/ = λ w λ w +λ o symbolizes the ractional low unction, λ i = k r,i(s) μ i phase mobility, k r,i (s) relative permeability, φ m/ porosity, K m/ absolute permeability and μ i viscosity (i = o,w). Capillary continuity between racture and matrix is given by p m c x= = pc. The system must also be speciied with initial conditions o the orm s(x,y,t = )=s (x,y). We scale the system using these dimensionless variables, parameters and unctions: μ = μ μ r, x = x L x, y = y L y, t = t τ, (2) p c = p c(s) P c,r = (s), b = b L x φ m /φ, λ i = λ iμ r where μ r = μ o, P c,r = max p c, τ = L y, τ m = φ m L v x 2 μ r K m P c,r. The mentioned reerence times τ,τ m arose rom systems T where advection or capillary orces play the dominant role respectively. It can be shown that the coupled system as given in () can be written in this scaled orm t s = α x (λo m m x m ) (3) ( t s + y = αβ λo m m x m) m x= + where α = τ τ m is the ratio between the time scales and β = V m is the ratio o the size o the pore volumes in V bφ matrix and racture. The system is solved by an operator splitting approach or low in x- or y-direction. = L xφ m COMPUTATIONAL EXPERIMENTS As input parameters we use length between injector and producer L y = m, racture width 2b =.m, porosities φ =,φ m =.2, racture spacing 2L x =.m, injection velocity along racture v T = m/d, matrix permeability K m = 5mD= 5 5 m 2, viscosities μ o = μ w = μ r = 3 Pa s and reerence capillary pressure is set to Pa. By deinition it ollows that β = 2, τ = d, τ m =.6d and α =.86 where corresponds to these base input data. As α, it ollows that both imbibition and advection will play signiicant roles. To describe the two-phase transport we require curves or capillary pressure (s) and relative permeability k r (s) or both racture and matrix. See Fig.. For the racture it has been assumed that = while k r,i = s i or the given phase i. The properties o the matrix have been considered or 2 cases, namely where the matrix is more water-wet or oil-wet. As imbibition is driven by a 234 Downloaded 3 Oct 22 to Redistribution subject to AIP license or copyright; see

4 (along rac) s(y) INIT s(y) OW s(y) NO IMB s(y) WW Capillary pressure (x,y=), OW Capillary pressure (x,y=), WW Water saturation (x,y=), OW Recovery.5.5 Water saturation (x,y=), WW Total oil recovery time (days) R(t) OW R(t) NO IMB R(t) WW Mat INIT Frac INIT Frac Mat. RV Mat.5 RV Mat. RV FIGURE 2. Water saturation along racture (let) ater injecting FV water shows ront delay due to imbibition. Matrix proiles at y = o capillary pressure and water saturation ater injecting.,.5 and. RV o water (middle). Oil-wet matrix imbibes less water than water-wet matrix. Recovery proile (right) or injecting RV=2d shows that recovery by imbibition depends strongly on wettability. gradient in capillary pressure the state where the capillary pressure is equal between racture and matrix corresponds to the water saturation that can be obtained in the matrix. In the example, as seen rom the Fig. the water-wet matrix can imbibe water until s w =.7 while imbibition in the oil-wet matrix stops when s w =.2. The matrix has been given =. at which the capillary pressure is at its highest, i. e. m (sw m,init )=. As the racture has zero capillary pressure = wesetsw,init =. since water would have been sucked into the matrix. The model was discretized using 5 cells along the matrix and 3 cells along the racture. an initial saturation s m,init w Wettability When water is injected and moves along the racture some o the water will be adsorbed into the matrix. Injecting racture volume (FV) o water is illustrated in Fig. 2. I there is no imbibition the water ront (represented by water saturation in the racture) will have reached the producer. However, as some water imbibes into the matrix the water ront along the racture is delayed. The behavior along the matrix axis at y = is shown: Due to the high capillary pressure in the matrix water will imbibe and expel oil back to the racture. The increasing matrix water saturation reduces the matrix capillary pressure and imbibition proceeds until it equals that o the racture. Because o the decreasing capillary pressure the orce drawing water into the racture is reduced making the imbibition rate decrease. O course, the local rate o imbibition is initially beore water arrives in the racture so it will increase, peak and decline. For the oil-wet rock only a little amount o water is necessary to reach equilibrium in capillary pressure compared to the water-wet rock. Thereore less water is adsorbed, allowing the water to travel with less delay o the ront movement. This is also relected in the oil recovery proile while injecting reservoir volume (RV). Due to the low potential or imbibition in oil-wet rock the recovery lies around a mere 5%, while more than 5% recovery is obtained or the water-wet rock. This illustrates the importance o wettability in evaluating spontaneous imbibition as a potential drive mechanism or ractured reservoirs. As the water-wet rock has most potential or imbibition we assume the rock has this wettability in the ollowing examples. Magnitude o capillary orces We have illustrated that the capillary orces are important in terms o the saturation at which the capillary pressure vanishes, however, the magnitude o the capillary orces are also o great importance or determining the eiciency 2342 Downloaded 3 Oct 22 to Redistribution subject to AIP license or copyright; see

5 .xα,.5 RV.xα,.5 RV xα,.5 RV xα.xα.7.6 Total oil recovery xα.xα xα.xα Recovery time (days) FIGURE 3. Impact o α. Distributions o water saturation (let) and recovery proile (right). I α is high imbibition dominates over racture low and the water eectively displaces oil. For low α most o the water lows through the racture without entering the matrix. o the imbibition process. To illustrate this we have considered a water-wet matrix and injected RV o water. The strength o capillary orces, represented by P c,r has been varied to produce dierent values o α. From Fig. 3 it is seen that i the capillary orce is weak the water will simply low through the racture while only the closest matrix region is aected. This is observed by early water breakthrough and a low rate o oil recovery. Thus, very much water would be required to reach the optimal recovery. However, i the capillary orces are strong, then the injected water is rapidly adsorbed into the matrix rather than lowing ahead along the racture. Strong imbibition implies delayed water breakthrough and a linear proile in recovery. CONCLUSIONS In a ractured reservoir important parameters or imbibition as recovery method are The volume o racture relatively volume o surrounding matrix represented by β; The wetting state o the surrounding matrix. Oil-wet matrix implies that a small amount o water will imbibe into matrix, whereas water-wet matrix implies that much water will imbibe into matrix; The strength o capillary orces versus injection rate represented by the α-parameter. It controls breakthrough time and rate o imbibition. The linear recovery period lasts longer or high α. For a ixed β: A more water-wet matrix and a high α avors recovery by imbibition. REFERENCES. M. Mainguy and F.-. Ulm, Coupled Diusion-Dissolution Around A Fracture Channel: The Solute Congestion Phenomenon, Tran. Por. Med. Vol 8, , P.S. Gautam and K.K. Mohanty, Matrix-Fracture Transer Through Countercurrent Imbibition in Presence o Fracture Fluid Flow, Tran. Por. Med. Vol 55, , G.I. Barenblatt, E.A. Ingerman, H. Shvets, and.l. Vazquez, Very Intense Pulse in the Groundwater Flow in Fissurised-porous Stratum, PNAS, 97, , H. Salimi and. Bruining, The Inluence o Heterogeneity, Wetting, and Viscosity Ratio on Oil Recovery From Vertically Fractured Reservoirs, SPE ournal Vol 6 (No 2), 4 428, D. Kashchiev and A. Firoozabadi, Analytical Solutions or D Countercurrent Imbibition in Water-wet Media, SPE ournal, Vol 8 (No 4), 4 48, Downloaded 3 Oct 22 to Redistribution subject to AIP license or copyright; see