Improved near-well approximation for prediction of the gas/oil production ratio from oil-rim reservoirs

Transcription

1 Comut Geosci (1) 16:31 46 DOI 1.17/s ORIGINAL PAPER Imroved near-well aroimation for rediction of the gas/oil roduction ratio from oil-rim reservoirs Svenn Anton Halvorsen Are Mjaavatten Robert Aasheim Received: 6 January 11 / Acceted: 6 July 11 / Published online: 6 Setember 11 The Author(s) 11. This article is ublished with oen access at Sringerlink.com Abstract A model with short comutational time has reviously been develoed to redict the ratedeendent gas/oil ratio (GOR) from a horiontal well. The oil flow towards the wellbore is based on a onedimensional model by Koniecek. The model erforms remarkably well for medium-time roduction otimiation (weeks, months), while the redictions during the first days after a large change in the roduction can be oor. An imroved one-dimensional model for the flow towards the wellbore is roosed, where the oil flow is treated as a suerosition of three terms: 1) Radial flow towards the wellbore and towards a mirror well. ) Flow to correct for modified boundary conditions due to the radial flows. 3) Flow due to height variations of the gas/oil contact (GOC). The new model takes care of the current short term and near-well deficiencies: Effect of D flow close to the wellbore, gas breakthrough due to viscous gas fingering, and horiontal/vertical anisotroy. Based on analysis and reliminary testing the new model should have equally good medium and long term caabilities S. A. Halvorsen (B) Teknova AS, Kristiansand, Norway sah@teknova.no A. Mjaavatten Yara International ASA, Porsgrunn, Norway R. Aasheim Statoil ASA, Research Centre Porsgrunn, Porsgrunn, Norway and considerably imroved short term and near-well behaviour, comared to the resent imlementation. Keywords Oil-rim reservoir Horiontal well Imroved near-well model Gas coning Gas/oil roduction ratio GOR Gas breakthrough GORM One-dimensional model Production otimiation Gravitational drainage Duuit Forchheimer Porous media equation Nomenclature α Parameter in the Duuit Forchheimer (orous media) equation, m/s β Non-dimensional model arameter to be adated to the roduction data, used to define the distribution of the weighted roduction rate along the wellbore δ Non-dimensional arameter alied in GORM to define the oil reduction factor, κ δ a Asect ratio, adjusted for the effect of different ermeabilities, δ a = k h k v h W ρ Density difference between oil and gas in the reservoir, kg/m 3, ρ = ρ o ρ g γ Non-dimensional model arameter for defining the weighted (local) roduction rate, γ m /s free gas corresonds to 1 m /s local oil roduction ϕ Effective orosity, m 3 /m 3 κ Non-dimensional oil reduction factor. When free gas is roduced, the local oil roduction is reduced by the factor κ

2 3 Comut Geosci (1) 16:31 46 μ Oil viscosity, Pa s ρ o Density of oil in the reservoir, kg/m 3 ρ g Density of gas in the reservoir, kg/m 3 τ h Time constant for horiontal flow, s τ v Time constant for vertical flow, s B g Gas formation factor, m 3 /Sm 3. One standard m 3 gas has a volume of B g m 3 in the reservoir B o Oil formation factor, m 3 /Sm 3. One standard m 3 oil has a volume of B o m 3 in the reservoir c Isotroic diffusion coefficient for oil flow, m /Pa s, c = k μ c h Diffusion coefficient in the horiontal direction, m /Pa s, c h = k h μ c v Diffusion coefficient in the vertical direction, m /Pa s, c v = k v μ d w Diameter of the wellbore, m g m/s, gravity constant GOC Gas oil contact, i.e. the interface between oil and gas h Local height of the oil layer, m, h = h(, y, t) h Initial height of the oil reservoir, m h 1 Local height of the first cell in GORM, m, h 1 (y, t) h(, y, t) h Local height of the second cell in GORM, m k Isotroic ermeability, m k h Horiontal ermeability, m k v Vertical ermeability, m K(y) Non-dimensional function for the distribution of weighted roduction along the wellbore K I (Weighted) roduction integral, integral of K(y) along the wellbore, m K red oil reduction integral defined by K red = L κ (y, t) K (y) dy L Length of the wellbore, m Deviatoric ressure, i.e. ressure causing flow, Pa Non-dimensional deviatoric ressure fi Deviatoric ressure to counteract (fi) the boundary conditions due to sink,pa gravity Part of deviatoric ressure due to gravity, caused by variations of GOC, Pa h Hydrostatic ressure in the reservoir, Pa ref Reference ressure, i.e. hydrostatic (gas) ressure at the reference level, Pa sink Deviatoric ressure due to two sinks, Pa tot Total ressure in the reservoir, Pa Tyical ressure difference, Pa, = ρ gh q a Total weighted roduction rate at the reservoir, m 3 /s q g Total roduction rate of free gas at the reservoir, m 3 /s q o Total oil roduction rate at the reservoir, m 3 /s q a Weighted (local) roduction rate er meter well length, m /s, q a (y, t) def = q g(y,t) + γ q o (y, t) = u (t) K (y) q g Local roduction rate of free gas er meter well length, m /s q o Local volumetric oil roduction rate er meter well length, m /s Q g Total gas roduction, Sm 3 /s R s Solution of gas in the oil, Sm 3 /Sm 3. One standard m 3 oil from the reservoir contains R s Sm 3 dissolved gas t Time, s u(t) Time-deendent art of the weighted roduction rate er m well length, m /s, q a (y, t) = u (t) K (y) v c Stability limit for the (suerficial) vertical downwards velocity, m/s v s Tangential (suerficial) oil velocity at GOC in the direction towards the wellbore, m/s v Suerficial oil velocity comonent in the (horiontal) -direction, m/s v y Suerficial oil velocity comonent in the (horiontal) y-direction, m/s v Suerficial oil velocity comonent in the (vertical) -direction, m/s v fi_ Volume flow of oil in the -direction er m well due to fi,m /s V gravity_ Volume flow of oil in the -direction er m well due to gravity flow, m /s V sink_ Volume flow of oil in the -direction er m well due to flow towards two sinks, m /s V Horiontal oil volume flow er m normal to the wellbore, m /s = total volume flow of oil in the -direction er m well, m /s V y Horiontal oil volume flow er m arallel to the wellbore, m /s W Half width of the oil reservoir, m -coordinate (horiontal, erendicular to the wellbore), m Non-dimensional -coordinate y y-coordinate (horiontal, arallel to the wellbore), m -coordinate (vertical), m Non-dimensional -coordinate ref Reference level, i.e. vertical level for the reference ressure, m w Vertical well location (to of the wellbore), m

3 Comut Geosci (1) 16: wc wc 1 Introduction Vertical location of the well centre, m Non-dimensional vertical location of the well centre In oil-rim fields, a thin oil layer lies between an aquifer and a gas ca. Oil can be roduced from such fields by horiontal wells. The roduction will lower the local gas/oil contact (GOC) near the well in a rocess called gas coning. After some time, the GOC will come in contact with the wellbore close to the riser and the gas/oil ratio (GOR) from the well will then vary strongly with the roduction rate. The ability to redict this deendency is essential for roduction otimiation. A model for gas coning and rediction of GOR was introduced by Muskat in 1937 [5]. This early work considers the conditions for a vertical well. A model for horiontal wells in oil-rim fields is described by Koniecek [3]. His concets have been etended by Statoil to include variations of gas and oil roduction along the wellbore [4]. Statoil s GOR model (called GORM) has been develoed to redict the rate deendent GOR for eriods of several months. The model describes the essential dynamic reservoir behaviour with a simlified interaction between the well and the reservoir. Historical oil and gas roduction rates are alied to fit three adjustable model arameters. GORM has been etensively tested and adated to historical roduction data, c.f. [4]. The model has short comutational time and erforms remarkably well for medium term roduction otimiation (weeks, months). The redictions during the first days after a large change in the roduction can, however, be oor. The model equations have been analysed and the behaviour close to the wellbore has been comared to more accurate finite element simulations [1]. The analysis revealed that the model had suitable medium term roerties but that the short-term behaviour could be oor due to the chosen simlifications. Based on these results, an imroved aroimation was formulated. Current model.1 Basic assumtions The model was originally intended for short term roduction otimiation (days, ossibly a few weeks). It should be comaratively simle to enable short comutational time but sufficiently sohisticated to include the essential reservoir behaviour. A few suitable arameters should further be available to tune the model to historical roduction data. The model, GORM, alies the following assumtions [4]: The gas is assumed to behave like an inviscid fluid, comared to the viscous oil. Hence, the ressure along GOC is given by the hydrostatic gas ressure. The oil one of the well drainage region initially has the form of a rectangular aralleleied with an imermeable bottom and imermeable vertical boundaries. The vertical lane through the wellbore slits the reservoir into two symmetric arts. Vertical oil flow and flow arallel to the direction of the wellbore can be neglected and the local oil flow towards the wellbore can be described by the one-dimensional Duuit Forchheimer equation. The distribution of the total (weighted) roduction rate locally along the well is described elicitly by a linear function. Free gas is roduced at locations where GOC is in contact with the wellbore. Here, the local oil roduction is reduced roortionally to the roduction of free gas determined by the amount of GOC in contact with the well. Tyical shae of GOC from a model simulation is shown in Fig. 1.. Model equations In a Cartesian coordinate system, let the -ais be normal and the y-ais arallel to the wellbore. The -ais is ointing vertically uwards with = at the to of the aquifer. The oil reservoir is initially (before any roduction) bounded by the lanes = W, = W, y =, y = L, = and = h,wherel is the length of the well, W is the half width and h the initial height of the oil layer. Above the oil, an infinite gas ca is assumed. The hydrostatic ressure, h, within the oil layer is given by: h = ref + ρ g g ( ref h ) + ρ o g(h ) = ρgh + ref ρ o g + ρ g g ref (1) where ref is the (hydrostatic) gas ressure at some reference level, ref, ρ o and ρ g the reservoir oil and gas densities, g the gravity constant, h(, y, t) the local height of the oil layer, and ρ = ρ o ρ g.

4 34 Comut Geosci (1) 16:31 46 Fig. 1 Schematic shae of GOC as redicted by the model. (Not to scale) y Let tot be the total ressure and define the deviatoric ressure by: = tot ( ) ref ρ o g + ρ g g ref () Oil flow is caused by gradients in the deviatoric ressure. Isotroic conditions are assumed in GORM [4]. Here, we will be slightly more general and assume constant, but ossibly different ermeabilities horiontally and vertically. The deviatoric ressure will then satisfy the equation: k h + k h y + k v = (3) and the suerficial oil velocity comonents are given by: v = c h, v y = c h y, v = c v (4) where k h and k v are the horiontal and vertical ermeabilities, c h = k h μ, c v = k v where μ is the oil viscosity, μ and v, v y, v are the suerficial velocity comonents. At GOC, the oil ressure is equal to the hydrostatic gas ressure. The total ressure is then equal to the hydrostatic ressure given by Eq. 1, and the deviatoric ressure is = ρ gh at GOC. The boundary conditions along the vertical and horiontal boundaries are ero normal ressure derivatives. Mathematical analysis shows that the Duuit assumtion of vertical ressure equilibrium is a roer overall aroimation [1]. The vertical ressure variation in the oil is then given by the hydrostatic ressure and the deviatoric ressure is: (, y,, t) = ρ gh(, y, t) (5) According to Eqs. 4 and 5, the vertical oil velocity will be ero and the horiontal volume flows er m are given by: V = hv = c h ρgh h, V y = hv y = c h ρgh h y (6) The material balance, ϕ h t + (hv ) + ( ) hv y =, (7) y then gives the two-dimensional Duuit Forchheimer (orous media) equation: ( ( h t = α h h ) + ( h h )) (8) y y where α = c h ρ g = k h ρ g and ϕ is the effective orosity. ϕ μϕ When the well length is more than three times the reservoir half width, the last term in Eq. 8 can be neglected [1], and the oil drainage can be aroimated by the dominating term, as roosed by Koniecek [3]: h t = α ( h h ) (9) As boundary conditions for Eq. 9, GORM alies the volumetric local oil roduction rate at = : 1 q o (y, t) = [ [ ] hv = = c h ρ gh h ] (1) = and requires no-flow condition at = W: v = h = (11) where q o is the local volumetric oil roduction rate er meter well length.

5 Comut Geosci (1) 16: The model needs to handle two different regimes along the wellbore: h(, y, t) > w GOC is locally above the wellbore, only oil (with dissolved gas) is roduced. The local oil roduction rate can, at least in rincile, be chosen arbitrarily large. w d w h(, y, t) w GOC is in direct contact with the well at = and free gas is roduced locally in addition to oil. The oil roduction rate must now be limited to kee GOC above the bottom of the wellbore. h(, y, t) = w can be alied as an aroimation for the boundary condition at =, asd w is small comared to w. w is the vertical well location (to of the wellbore) and d w is the diameter of the wellbore. Two basic issues remain to comlete the model: How to distribute the roduction rate along the wellbore How to find the gas and oil roduction rates where GOC is in contact with the well In GORM, the distribution of the total roduction locally along the well is secified elicitly [4]. When there is no roduction of free gas, the local oil roduction can be written as: q o (y, t) = u(t) K(y) (1) where K(y) is defined elicitly, and u(t) can be regarded as a model inut for total roduction. For K(y), a simle linear function is chosen: K(y) = (β 1) y L + 1 (13) where β is a arameter to be adated to the roduction data. Where free gas is roduced, some formulation of the gas roduction rate is required. The gas roduction must be limited. Another simle linear relation has been assumed: A reduction of the oil roduction is accomanied by a roortional increase in the roduction of free gas. The roduction rates can then be written as: q o (y, t) = (1 κ) u(t) K(y) (14) q g (y, t) = γκu(t)k(y) (15) where q g is the local gas roduction rate, κ is an oil reduction factor and γ is a constant. The oil reduction factor is given by: κ(y, t) = δ (16) where the arameter δ is defined by:, h(, y, t) > w δ = w h(, y, t) (17), h(, y, t) w d w By this definition, the oil roduction is reduced smoothly from full when GOC is above the well to ero if GOC should reach the bottom of the wellbore. The central feature of the oil reduction factor is to kee h(, y, t) aroimately at w when free gas is roduced. Other ossible formulations with this roerty would only have a minor influence on the comuted results. Now define the weighted (local) roduction rate by letting γ m /s gas corresond to 1 m /s oil roduction. q a (y, t) def = q g(y, t) + q γ o (y, t) = u(t)k(y) (18) The total weighted roduction is then given by: q a (t) def = where K I L q a (y, t)dy = u(t) L K(y)dy = u(t)k I (19) (weighted) roduction integral, integral of K(y) along the wellbore The local oil and gas roduction rates can be given in terms of the weighted roduction rate: q o (1 κ(y)) u (t) K (y) (1 κ (y)) K (y) = = () q a u (t) K I K I q g γκ(y) u (t) K (y) γκ(y) K (y) = = (1) q a u (t) K I K I and the corresonding roduction rates for the whole well can be written as: q o q a = q g q a = L (1 κ (y, t)) K (y) dy K I L γκ(y, t) K (y) dy K I = K I K red K I () = γ K red K I (3) where the oil reduction integral is defined by: K red (q a, h) = L κ (y, t) K (y) dy (4) With the chosen formulation of the oil reduction factor, the integral will only deend on h(, y, t), i.e. the state

6 36 Comut Geosci (1) 16:31 46 of the reservoir at =. We will, however, assume that more general formulations might be chosen. The total gas roduction in Sm 3 /s, Q g,isthesumof the contribution from the free gas roduction and the contribution from gas dissolved in the reservoir oil. Q g = q g + q o R s = q ( a γ Kred + R ) s (K I K red ) B g B o K I B g B o where (5) B g Gas formation factor, m 3 /Sm 3 One standard m 3 gas has a volume of B g m 3 in the reservoir B o Oil formation factor, m 3 /Sm 3 One standard m 3 oil has a volume of B o m 3 in the reservoir R s solution of gas in the oil, Sm 3 /Sm 3 One standard m 3 oil from the reservoir contains R s Sm 3 dissolved gas The integral K I is given by the function K(y) and K red can be comuted elicitly when the state of the reservoir is known. The weighted roduction rate is then given by Eq. 5 and the local oil roduction is given by Eq.. Hence, when the historical and/or lanned gasroduction rate, Q g, is given, the boundary condition for Eq. 9 can be found and the artial differential equation (PDE) can be solved to redict the oil rate. In GORM s rogram code, an equation similar to Eq. 5 is alied to solve for the reservoir oil roduction rate, q o [4]. We have chosen here to derive an alternative formulation that will be alicable for more general choices for the oil reduction factor. Equation 5 also rovides, for instance, the required equation for the more comle case where κ deends on q a. GORM uses a finite volume aroach to discretie Eq. 9 to transform the PDE to a system of ordinary differential equations (ODEs) [4]. Actually, the current model version alies the D Duuit Forchheimer, Eq. 8. This modification did not in itself imrove the redictions, in accordance with a recent model analysis [1]. It has, however, been ket as the increased comutational time is minor. In addition, the simulated art of the reservoir can be etended in both ends in the y-direction along the wellbore, which is more realistic..3 Model analysis: D versus 1D flow towards the wellbore A few years ago, it was decided to investigate the nearwell roerties of the model. The 1D model in Eq. 9 should be comared to the corresonding D flow in the lane, where the deviatoric ressure and the oil velocities are given by: k h + k v = (6) v = k h μ, v = k v μ (7) The boundary conditions are secified ressure at GOC, (, h, t) = ρ gh(, t) (8) given oil velocity at the well, and no-flow across the remaining boundaries. A uniform distribution of the oil velocity was assumed at the well. The equations were imlemented in the finite element rogram COMSOL Multihysics. The D ressure Eq. 6 was solved along with an arbitrary Lagrangian Eulerian formulation to udate the element grid to follow the time evolution of GOC. A narrow oil reservoir with half width equal to m and initial height 8 m was studied (surrounding rectangle in Fig. ). The wellbore was located at m, Fig. Results from a numerical D simulation. The figure shows the deformed finite element grid. The colours illustrate the deviatoric ressure distribution ranging from 16,4 Pa at the wellbore (dark blue) to 45,5 Pa in the uer right corner (brown). GOC is slightly unstable, i.e. some oscillations can be seen close to the wellbore

7 Comut Geosci (1) 16: and the well diameter was cm. The model first showed severe convergence roblems. After the time integration arameters had been adjusted roerly, the simulations showed smooth behaviour for some time. Then, the free boundary started to oscillate above the wellbore (Fig. ). The oscillations grew and the solution rocedure diverged after some more time stes. The equations were then analysed for the isotroic case, and it was revealed that GOC would become unstable when the vertical oil velocity at GOC reached a critical value. The viscous fingering instability, c.f. for instance [], was rediscovered for the secial case with one viscous and one inviscid fluid. The simulations diverged as the model/code had not been adated for such unstable cases. The influence of the inviscid fluid is given by the boundary condition at GOC, Eq. 8, secifying the oil ressure. For isotroic conditions, the tangential velocity is then given by: v s = c s = cg ρ sin (φ) = k g ρ sin (φ) (9) μ where v s is the tangential velocity at GOC in the direction towards the wellbore, c = c h = c v is the isotroic diffusion coefficient, k = k h = k v the corresonding ermeability, and φ the inclination angle of GOC. Maimum tangential velocity along GOC is achieved when the surface is vertical. (v s ) ma = cg ρ (3) A urely gravity-driven, orous oil flow cannot achieve a higher velocity. It is, however, ossible to create higher velocities close to the well by sulying a sufficiently low well ressure. If the higher velocities etend to the surface, then the surface becomes unstable: If there is a small deression in the surface, there will be highest ressure gradient at the bottom of the di. Hence, the bottom will try to move downwards faster than the remaining art of the interface. At low velocities, the di will be filled by gravity flow, and the surface remains smooth. If the downwards velocity is higher than (v s ) ma, the gravity flow can not fill the deression, and a gas finger will raidly stretch down to the well, Fig. 3. For the GOR model, such a di can only be formed right above the well since the vertical oil velocity is highest there. For the anisotroic case with different horiontal and vertical ermeabilities, it can be shown that GOC will be unstable if the downwards velocity at GOC eceeds the critical value: v c = c v g ρ = k v μ g ρ (31) (v s ) ma (v s ) ma Stable Unstable Fig. 3 A stable di will be filled with oil, while an unstable one will grow deeer The highest downwards velocity at GOC will be right above the wellbore. From the start, this velocity will be below the critical value, for normal roduction rates. Then, it will increase gradually as GOC is lowered. Unless the roduction rate is very low, GOC will be significantly above the wellbore when the critical rate is reached. Then, there will be gas breakthrough due to gas fingering, and free gas will be sucked into the well. The roduction rate is limited by the total amount of gas roduction. Hence, after gas (fingering) breakthrough, the wellbore ressure will be reduced to control the roduction. The highest downwards oil velocity at GOC will be ket at the critical value. Higher weighted roduction will result in more free gas, while maimum oil velocity at GOC below the critical value would corresond to stable conditions with gas roduction below the target value. From a modelling oint of view, the oil roduction rate should be reduced to kee the maimum vertical velocity at GOC at the critical value, v c, after gas fingering breakthrough. This transition state should then be maintained until GOC reaches the wellbore. Then, the oil roduction should be further reduced to kee GOC aroimately in this location at =. For our test cases, the critical (suerficial) velocity was μm/s = 1. m/week. The corresonding rate of change for GOC was v c /φ = v c /. = 1 μm/s =.86 m/day = 6. m/week. Further analysis has been alied to show that the D model described by Eqs. 6 and 7 has generally four non-dimensional arameters: k h h δ a = k v, asect ratio, adjusted for the effect of W different ermeabilities

8 38 Comut Geosci (1) 16:31 46 wc = wc h = w 1 d w h, non-dimensional vertical location of the well centre dw_h = d w h, non-dimensional height of the well/nondimensional well diameter dw_w = d w W, non-dimensional width of the well These four arameters determine the qualitative behaviour of the solution. The non-dimensional width of the well is not imortant and can be neglected. If the ermeabilities are equal, the non-dimensional width can be eressed by the non-dimensional diameter and the asect ratio. In this case it is no longer a free arameter. The asect ratio determines the relative strength of horiontal versus vertical flow and also determines the sie of the near field. The non-dimensional vertical location of the well influences the flow field in the vicinity of the wellbore and hence the time evolution of GOC close to the well. The non-dimensional well diameter will basically influence the flow very close to the well, and can have a large imact on the ressure difference between the reservoir and the well. Such ressure difference is not an issue for the GOR model and this non-dimensional arameter is therefore not imortant. Similar analysis for the one-dimensional model reveals that there are no non-dimensional arameters before gas breakthrough. In addition to the non-dimensional arameters the quantities alied to scale the various variables are also (otential) free arameters. For the D equation, these arameters are: h, initial height of the oil reservoir W, half width of the oil reservoir = ρ gh, tyical ressure difference τ h = W α h = μϕ W ρ g k h h, time constant for horiontal flow, that is the time constant for draining the thin reservoir by gravitational flow. As ressure values are not of interest for the GOR model, there are only three relevant free dimensional arameters. It can be shown that the 1D model has the same three free dimensional arameters. As the time constant for horiontal flow is the same for the 1D and the D models, the 1D aroimation should be an aroriate aroimation where this constant is relevant, i.e. for medium and long term simulations [1]. For very long-term simulations significant deviations from an initially rectangular reservoir shae are likely to be imortant and roer model redictions can not be eected. The initial height of the reservoir is known from available information. GORM is then tuned to historical roduction data by adjusting the arameters W and α. This is equivalent to adjusting W and τ h,asα can be eressed by the other free arameters and therefore can be alied instead of τ h. In addition, the arameter β is tuned. A fourth ossible tuning arameter is γ, but this arameter is ket at a nominal value based on eerience [4]. Two imortant non-dimensional arameters for the D model are not resent in the current 1D aroimation. These arameters can be combined with the dimensional ones to: τ v = δa τ h = k h h W k v W α h = h k h α k v = μφ h ρ g k v, time constant for vertical flow, that is the time sent during a vertical free fall of oil from the to to the bottom of the initial reservoir height. wc = h wc, vertical location of the well centre As these arameters are imortant for the near-well flow, and hence for the short time behaviour, the current 1D model cannot be alied for accurate simulation of short-term variations..4 Model characteristics: D versus current 1D model Based on the model analysis, the current aroimation does not take account of the following characteristics found in the D model: Effect of the asect ratio, adjusted for ossible different ermeabilities Effect of the vertical location of the well centre, that is non-horiontal flow towards the well in the near-well region Ability to redict gas breakthrough due to viscous gas fingering Handling flow after (local) gas breakthrough before GOC reaches the wellbore The ratio between the horiontal and the vertical ermeabilities is a otential tuning arameter for short term time evolution, c.f. definition of δ a or τ v above..5 Flow regimes While the current model can handle two flow regimes locally along the wellbore, the analysis reveals that there will be three regimes: Stable oil flow with GOC above the wellbore the maimum vertical oil velocity at GOC is below the stability limit. Transition flow, with gas fingering the oil roduction rate tries to draw GOC downwards faster than

9 Comut Geosci (1) 16: the maimum gravity flow rate. Gas is sucked into the well due to viscous gas fingering and the oil flow is ket at the stability limit. Direct contact between the gas and the wellbore gas is sucked into the well. The oil roduction rate is limited by gravitational flow towards the wellbore. Consider the early conditions for a reservoir and assume a suitable, constant gas roduction rate (Q g = constant). Further assume that the D model is alied for the flow towards the wellbore. From the start, there will only be stable oil flow and gas coning will gradually develo. GOC will be lowest above the heel as the wellbore ressure is lowest here. After some time, the critical downwards velocity will be reached above the heel and there will be gas (fingering) breakthrough in this region. The oil roduction is then reduced as free gas is sucked into the well. Transition flow will start. First only right at the heel, but deending on how freely gas can be sucked, the transition flow region will gradually etend somewhat along the wellbore. Some model is required to describe how much free gas is roduced locally when the oil roduction is reduced, for instance by assuming a linear relation, c.f. Eqs. 14 and 15. Before GOC reaches the wellbore, the region with transition flow will gradually eand. Later, GOC will get in contact with the wellbore at the heel. The oil roduction must then be further reduced, and the roduction of free gas corresondingly increased. The higher gas roduction at the heel imlies lower gas roduction further along the well and the flow state will change from transition flow to stable at the end of the transition region. Gradually, GOC will get in contact with a larger and larger art of the wellbore, reducing the transition one. The accomanying increase in the roduction of free gas imlies that the transition one is gradually reduced also from the other end, until it disaears. From now on, the one where the gas is in direct contact with the wellbore will slowly eand towards the toe. and the flow towards the wellbore is then adequately described by: = (3) v = c = k μ, v = c = k μ (33) Oil flow in the reservoir is caused by two conditions: Lower ressure in the well and height variations of the GOC. Let the flow therefore be suerimosed by the two terms: Flow due to oil roduction, i.e. flow towards the wellbore due to low ressure in the well. The boundary condition for the corresonding deviatoric ressure is = at GOC. Flow due to gravity, i.e. flow caused by height variations of GOC, where the aroriate boundary condition at GOC is = ρ gh. The flow towards the wellbore is then aroimated by radial flow towards a sink in the vicinity of the well. A mirror sink is added to get the correct boundary condition at the bottom of the reservoir. The flow due to the two wells will, however, violate the boundary conditions at GOC and at = W. A third sinkfi flow is therefore added, to correct these boundary conditions. Figure 4 shows how the total ressure is decomosed into two and then into three terms, together with the boundary conditions. The ressure equation and boundary conditions for the gravity flow are shown in Fig. 5, while Fig. 6 shows the equation satisfied by the flow caused by the two sinks and the two boundary = = = Sink = Δρ g h OilProduction + + SinkFi GravityFlow + GravityFlow = 3 Imroved 1D aroimation 3.1 Suerosition of three flows In the following aragrahs, the modified model will be described for the case where the horiontal and vertical ermeabilities are equal, i.e. k h = k v = k and c h = c v = c. Flow in the y-direction will also be ignored = Fig. 4 Pressure as the sum of three terms W

10 4 Comut Geosci (1) 16:31 46 g h SinkFi Sink GravityFlow SinkFi SinkFi Sink W W Fig. 5 Pressure due to gravity flow, no sinks Fig. 7 Pressure to fi boundary conditions conditions satisfied by this flow. Finally, Fig. 7 shows the equation and the boundary conditions for the sinkfi flow. 3. Horiontal flows The Duuit assumtion of vertical ressure equilibrium is a roer aroimation for the gravity flow. Hence, the ressure is given by: gravity = ρ gh (34) and the volume flow of oil er m well length in the -direction is given by: V gravity_ = hv = c ρ gh h (35) The inde gravity or GravityFlow is alied for the gravity flow. The ressure for the flows towards the two D sinks can be written as: sink (, ) = + q ( o ln r w+ + ln r ) w (36) π c r r where is a reference ressure due to one of the sinks at a reference distance r from this sink. For simlicity, let = Pa and r =1m.r w+ and r w are the distances from the centre of the wellbore and the mirror wellbore, resectively: r w+ = r w = + ( wc ) (37) + ( + wc ) (38) Sink Equation 36 has two terms. Each term describes the ressure due to ure radial flow towards the resective sink. The horiontal volume flow due to the wellbore sink is equal to the radial volume flow within the aroriate sector. Hence, the horiontal volume flow er m well length is given by: W V uer_sink_ = q ( o arctan h wc + arctan ) wc π (39) Similarly, the horiontal volume flow (er m well length) due to the lower sink is: Fig. 6 Pressure from two sinks V lower_sink_ = q ( o arctan h + wc arctan ) wc π (4)

11 Comut Geosci (1) 16: The flow due to both sinks is then: V sink_ = V uer_sink_ + V lower_sink_ = q o π ( arctan h wc + arctan h + wc ) (41) The Duuit assumtion will be used for the sinkfi flow. The ressure for this flow can then be written as: fi (, t) = sink (, h (, t), t) (4) The horiontal flow is then given by: V fi_ = hc fi = hc = q o h π ( sink ( + (h wc ) h r w+ + sink ) h + + (h + wc) h r w and the total horiontal oil flow er m well length is: ) (43) V = V gravity_ + V sink_ + V fi_ (44) Observe that when the Duuit assumtion is alied, the boundary conditions at = and = W are not invoked. Due to symmetry of the ressure from the sinks, the condition at = will be satisfied while the other condition can be somewhat violated. This discreancy is easily handled by adjusting the boundary condition for the material balance. 3.3 Material balance: equation for GOC The material balance is given by: ϕ h t + V =, (45) which can be written as: h t = α ( h h ) 1 ( Vsink_ + V ) fi_ ϕ The boundary conditions are flow conditions: V () = 1 q (46) V (W) = (47) If the boundary conditions are alied for h, then: h (, t) V gravity_ (W) = = [ c ρ gh h ] =w = ( V sink_ (W) + V fi_ (W) ) (48) The flow condition at = will be satisfied by the contribution from V sink_ while the contribution from V fi_ is ero. It can be observed that the modified equation is equal to the original one, Eq. 9, lus two terms. As for GORM, the PDE can be discretied by a finite-volume aroach to derive an ODE-system to be solved by a standard ODE-solver. The comleity of the discretied modified model will be similar to the original one. 3.4 Anisotroic conditions If the horiontal and vertical ermeabilities differ, the aroriate D model is described by: k h + k v = (49) v = c h, v = c v, (5) instead of Eqs. 3 and 33. The following coordinate transformation will change the roblem to an isotroic one: = λ = k h k v (51) In the transformed roblem, all vertical arameters and the roduction rate have been changed by the same factor, λ, and the isotroic ermeability is: k = k h k v (5) The modified GOR model can then be alied on the transformed roblem. Alternatively, the aroriate horiontal flows can be derived for the anisotroic case. Hence, the modified model can also be alied for this tye of anisotroic conditions. 3.5 Analysis The aroriate non-dimensional version of Eq. 49 is: ( kh h ) k v W + = (53) Duuit s assumtion of ( vertical ressure equilibrium is valid when the factor kh h k v )is small, c.f. for instance W [1]. If the factor is less than.1, the second term in Eq. 53 will dominate. The argument can also be turned

12 4 Comut Geosci (1) 16:31 46 around: If the factor is greater than.1, ressure variation in the vertical direction, and hence vertical flow, should not be ignored. The boundary for the near-well region can then be defined as the value of W which makes the factor equal to.1: W nw = h 1 k h k v (54) where W nw is the sie of the near-well region. For isotroic conditions the half width of the near-well region is then some three times the reservoir height. For the original GOR model, two imortant nondimensional arameters are missing: the location of the well centre (before gas breakthrough) and the asect ratio (adjusted for ossible different ermeabilities). For the modified model, the location of the well centre is evidently included. The asect ratio will also be included as a non-dimensional arameter since the flow towards the two sinks is an eact solution to the original equation (but with different boundary conditions). Our model modification will only affect the flow significantly in the near-well region. Outside this region the Duuit assumtion is valid and the sinkfi flow cancels the sink flow. The sinkfi flow, as given by Eq. 43, is by definition eactly equal to minus the Duuit Forchheimer aroimation for the sink flow. Model simulations showed tyical gas fingering instability for all cases that was run for sufficiently long time: First a smooth gas coning would be develoed. Then the level of the first cell would suddenly move raidly towards the wellbore, while the remaining art of GOC would develo as before. We are convinced that the modified 1D model will have a stability limit, similar to the critical downwards velocity for the D model, but a mathematical roof remains. Some reliminary, rather technical analysis showed, however, that the instability is likely to occur as GOC aroaches the well. It remains to be studied how close the 1D instability will aroimate the D one. The modified model reserves the significant roerties of the full D roblem: The effect of D flow towards the wellbore. The near-well region, where D-flow should not be ignored, is the same as for the full D roblem. An instability similar to the viscous gas fingering instability (not comletely verified). These roerties indicate that the model will be a good aroimation, but we can not reclude that further imrovements can be found. 3.6 Oil reduction factor The oil roduction rate must be reduced when GOC reaches the wellbore. As stated above, the main function is some controller that will kee GOC aroimately at this location. For the new model GOC must be ket above the well centre. The formula in Eq. 16 can be ket: κ (y, t) = δ (55) rovided the definition of δ is roerly modified, for instance to:, h (, y, t) > ws δ = ws h (, y, t) (56), h (, y, t) ws d ws where ws is aroimately equal to w (location for to of the wellbore) and d ws is some constant such that d ws < ws wc (57) where wc is the vertical location of the well centre. The oil roduction should also be reduced, and free gas roduced, due to the gas fingering instability. It can be shown that the full roblem becomes unstable when the vertical downwards suerficial velocity eceeds the critical value given by, c.f. for instance []: v c = k v g ρ (58) μ As an aroimation in the discretied model we might limit the oil roduction when the rate of change for the level of the first cell grows too high, that is when ḣ1 v c φ = k v g ρ μφ (59) where ḣ1 is the time derivative of the innermost cell. Such deendency on the time derivative imlies that the oil reduction factor would deend on the local oil roduction rate. Hence, the oil reduction integral, K red, would deend on the weighted roduction rate and the latter can not be comuted elicitly by Eq. 5. Another aroach is to detect viscous gas fingering by monitoring the difference between the oil level in the first and the second cell. The oil reduction factor can then be eressed as: κ (y, t) = δ + δ v (6)

13 Comut Geosci (1) 16: where, h h 1 < h c δ v = (h h 1 ) h (61) c, h h 1 h c d c where h c defines the critical difference for onset of gas fingering in the model and d c is another model arameter to be selected. This choice for an additional term in the oil reduction factor will kee the difference h h 1 between h c and h c + d c during viscous gas fingering. After some time, h 1 will be at the wellbore level. Then the other term in Eq. 6 will start to grow and reduction of oil roduction due to viscous fingering will gradually diminish. Oil reduction due to gas fingering might also be controlled by the relative distance to the well centre, by defining:, h w <λ c h 1 w δ v = ( ) h w ω c λ c, h (6) w λ c h 1 w h 1 w where λ c defines the critical ratio of the distances to the wellbore and ω c is a factor for tuning the model behaviour. The last formulation for δ v may, after some time, kee h 1 suitably above the well centre. If this should be the case, the first term in Eq. 6 can be droed. The choice of formulation for the oil reduction factor is a control issue and not an essential modelling matter. The chosen formulation should: Detect viscous gas fingering Reduce local oil roduction during viscous gas fingering, while the level of the first cell aroaches the well. Reduce local oil roduction to maintain GOC close to the well centre after GOC becomes sufficiently low. The discretied model may, however, turn out to be sufficiently accurate without a controller for the viscous gas fingering instability. Our tests indicate that the resulting, discretied ODE system is numerically stable. It has only been observed that the level of the first cell suddenly moves raidly towards the wellbore, without numerical roblems. The time for this transition is comaratively small, rovided the first comutational cell (containing the well) is not too wide. The model could be further tested without a searate controller for the transition flow. It may turn out that it is sufficiently accurate for an aroriate choice of discretiation close to the wellbore. The transition flow will then only be imlicitly included and the start and the end of this eriod will not be clearly defined. During the initial art of the transition there will be no roduction of free gas, when the level of the first cell moves raidly towards the well. Then there will be gas roduction due to fingering for some time, until the levels of the neighbouring cells get close to the wellbore level. Further model testing is required to determine what is a suitable strategy to handle the viscous gas fingering. 3.7 Preliminary testing Figures 8, 9, 1, 11 show some results from reliminary model testing. The comutational region for these simulations only covers the near-well region, i.e. a narrow reservoir. The tests were focused on the stable oil flow regime, before gas breakthrough. The arameters/data in Table 1 were alied. The arameters μ, k, ρ o, ρ g,andϕ were alied to comute α; while the arameters L, B o, R s and Q g were used to comute the local oil roduction rate, q o, assuming ero roduction of free gas and uniform conditions along the wellbore. A satial discretiation of.1 m was alied for both 1D models, while the D model alied an element grid somewhat coarser than shown in Fig., but with smaller elements close to GOC above the well. The 1D models alied the routine ode15s in Matlab requiring a relative tolerance of 1 5 for the time steing. The D model was run with a similar routine and with similar tolerance in COMSOL Multihysics. Figures 8 and 9 show results for a case with comaratively high roduction rate. The original GOR model clearly deviates from the D model, while the new 1D model is reasonably close. Some differences are found above the well. Figure 9 shows that the level of the first cell moves quickly towards the wellbore after the ODE system has become unstable. Figures 1 and 11 show the result for a case with lower roduction rate. For this case, there are also clear differences between GORM and the D simulation. The modified GOR model is very close to the D results ecet for < m. The tests are romising, but insufficient to show the erformance of the imroved model. Further work should include more comarative simulations before a comlete, imroved GOR model is tested against real roduction data.

14 44 Comut Geosci (1) 16:31 46 Fig. 8 Eamle 1: redicted shae of GOC after days, D versus current and new 1D models GOC location [m] D model Current 1D model New 1D Model Horiontal distance [m] Fig. 9 Eamle 1: redicted shae of GOC after 5 days, D versus current and new 1D models GOC location [m] D model Current 1D model New 1D Model Horiontal distance [m] Fig. 1 Eamle : redicted shae of GOC after 1 days, D versus current and new 1D models GOC location [m] D model Current 1D model New 1D Model Horiontal distance [m] Fig. 11 Eamle : redicted shae of GOC after 16 days, D versus current and new 1D models GOC location [m] D model Current 1D model New 1D Model Horiontal distance [m]

15 Comut Geosci (1) 16: Table 1 Parameters/values for the simulations Parameter Value W, half width of the oil reservoir m h, initial height of the oil layer 8 m w, well location (to of wellbore) m d w, diameter of wellbore cm α, arameter in the Duuit Forchheimer m/s equation μ, oil viscosity (adated to get.967 Pa s rounded value for α) k, isotroic ermeability m ρ o, reservoir density of oil 8 kg/m 3 ρ g, reservoir density of gas kg/m 3 ϕ, effective orosity. L, length of the well 1 m B o, oil formation factor 1.17 m 3 /Sm 3 R s, solution of gas in the oil 6 Sm 3 /Sm 3 Q g, total gas roduction, Eamle 1 1, Sm 3 /day Q g, total gas roduction, Eamle 5, Sm 3 /day 4 Discussion It should be ket in mind that GORM is not suitable for reservoir management. It has been develoed for roduction otimiation and focuses on the local behaviour for each horiontal well searately. The hilosohy has been to kee the model simle, but sufficiently advanced to cature the essential reservoir behaviour associated with each well. The simlifications start by assuming a simle geometry: symmetry and a rectangular shae for the initial state of the reservoir associated with the well. These assumtions are obviously wrong, but when model arameters are aroriately fitted to roduction data, the model can redict the basic time evolution for a limited eriod of time. The assumtions are definitely not adequate for very long simulations, but the model has erformed remarkably well for medium term roduction otimiation (weeks, months). As the net simlification, only oil flow is comuted within the reservoir. The model treats the gas as an inviscid fluid comared to the viscous oil. This assumtion takes care of the flow within the oil reservoir, but some assumtion for the gas roduction is needed at the well. At any location along the wellbore only oil is roduced before local gas breakthrough. The local oil roduction will be increased whenever the ressure in the well is lowered. After gas breakthrough the oil flow is limited, either by gravitational flow towards the well or the stability limit for viscous gas fingering. If the local well ressure is lowered, more free gas is roduced while the oil roduction is reserved. The simlest assumtion for the gas roduction is a linear relation: Let the local roduction demand be. q a. After gas breakthrough the oil roduction is lower than q a, which can be eressed as: q o = (1 κ) q a (63) The remaining roduction demand imlies gas roduction, for which a linear relation gives: q g = γκq a (64) The factor γ is a constant to be determined. Since the gas is far less viscous than the oil, the value is obviously greater than one. The oil reduction factor, κ, should in rincile be determined by first comuting the oil roduction for the aroriate boundary condition at =. Forinstance, when the gas is in direct contact with the wellbore, h(, y, t) = w is a roer aroimation for the boundary condition. The local oil roduction rate can then be comuted and κ can be determined by Eq. 63. But such a formulation imlies that the equation structure deends uon the solution and the imlementation will be comaratively comle. Formulating controllers to fulfil the boundary conditions aroimately is definitely referable from a comutational oint of view. The ressure in the wellbore varies along the well with higher ressure in the toe than in the heel. Hence, the local roduction is highest at the heel. The simlest ossible assumtion is a linear function for the distribution of the roduction rate along the wellbore. A more comle descrition has been tested where a hydraulic model was included for the ie ressure. The result was much longer comutational time and no noticeable increase in simulation accuracy. The simle, linear function has therefore been ket. For the oil flow within the reservoir, the Duuit assumtion of vertical ressure equilibrium has been made. This is a standard simlification for orous flow within thin regions, i.e. when the horiontal length scales are at least a few times greater than the height of the flow region. Analysis shows that the Duuit assumtion is valid for the intermediate and far field regions, while it can be oor for the near field. It is secifically not aroriate to describe the flow close to a well. Hence, GORM can erform well for medium to long term simulations, while oor short term redictions are to be eected when changes in the near-well region are significant. The roosed, imroved model addresses the near-well deficiencies of GORM, while keeing its medium/far field roerties.

16 46 Comut Geosci (1) 16:31 46 For horiontal/vertical anisotroy, the ratio between the vertical and horiontal ermeabilities is a model arameter. This arameter can be alied to adat the model to historical short term variations. Such adatation can be vital for roer short term behaviour. Hence, horiontal/vertical anisotroy should be included in the imlementation. The imroved model is valid for flow erendicular to the wellbore. This is a good aroimation at some distance from the heel and the toe. To kee the model simle, the formulations should be alied along the comlete length of the wellbore. If the model region is somewhat etended beyond the heel and toe, the aroriate equation will be: h t ( = α ( h h ( Vsink_ ) + y ( h h y )) 1 + V fi_ (65) φ where anisotroic formulations should be imlemented for the sink and the sinkfi terms. ) Aears to be able to detect and handle gas breakthrough due to viscous gas fingering Includes an additional arameter that can be adated to short term variations Only modifies the near-well roerties of the current model Based on our analysis, the new model should behave in the same way as the current one for medium and long term redictions, while short term variations should be significantly imroved. Preliminary simulations show imroved redictions for the near-well region. Further simulations and testing are required to verify the caabilities of the new model. Oen Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which ermits any noncommercial use, distribution, and reroduction in any medium, rovided the original author(s) and source are credited. 5 Conclusions Mathematical analysis elains why the current model to redict the gas/oil ratio, GORM, can erform remarkably well for medium term redictions (weeks, months), while short-term results (hours, days) can be oor. An imroved model is roosed that will take care of the current short term deficiencies. The new model Has the same medium and far field roerties as the current model Imroves the redictions of the gas oil contact (GOC) close to the wellbore References 1. Halvorsen, S.A., Mjaavatten, A., Aasheim, R.: Analysis of an oil-rim reservoir model for rediction of the gas/oil roduction ratio. In: 6th IMA Conference on Modelling Permeable Rocks, Abstract Book, (1). Homsy, G.M.: Viscous fingering in orous media. Ann. Rev. Fluid. Mech. 9, (1987) 3. Koniecek, J.: The concet of critical rate in gas coning and its use in roduction forecasting. SPEJ-Soc. Petrol. Eng. J. AIME. (7), (199) 4. Mjaavatten, A., Aasheim, R., Saelid, S., Gronning, O.: A model for gas coning and rate-deendent gas/oil ratio in an oil-rim reservoir. SPE Reserv Eval. Eng, Soc Petrol. Eng. 11, (8) 5. Muskat, M.: The flow of homogeneous fluids through orous media. McGraw-Hill, New York, London (1937)