THEORETICAL RESERVOIR MODELS

Transcription

1 THEORETICAL RESERVOIR MODELS TIME EARLY TIME MIDDLE TIME AREA OF INTEREST NEAR WELLBORE RESERVOIR MODELS Wellbore storage and Skin Infinite conductivity vertical fracture Finite conductivity vertical fracture Partial penetrating (limited entry) well Horizontal well Homogeneous Double porosity Double permeability Radial composite Linear composite Infinite lateral extent LATE TIME RESERVOIR BOUNDARIES Single boundary Wedge (two intersecting boundaries) Channel (two parallel boundaries) Circular boundary! Sealing! Constant pressure! Sealing! Constant pressure Composite rectangle! Sealing! Constant pressure! No boundary

2 Early Time Models (1) Wellbore storage and Skin (2) Infinite conductivity vertical fracture Area of Interest: NEAR WELLBORE (3) Finite conductivity vertical fracture (4) Partial penetrating (limited entry) well (5) Horizontal well

3 (1) Wellbore storage and Skin Early Time Models Assumptions A well is generally characterized by a constant W.B.S. which governs the production due to wellbore fluid decompression/compression when the well is opened or closed in. Log - log response Both the pressure and the derivative curves follow a straight line of unit slope (n=1) until the pressure disturbance is in the wellbore (pure wellbore storage). Afterwards, the derivative passes through a hump until the wellbore effects become negligible. Parameter: C, wellbore storage constant; S, formation permeability damage (skin) In case of multiphase flow at the wellbore it is possible to have a changing WBS option The magnitude depends upon the type of completion (surface/downhole shut-in)

4 Early Time Models q surface flowrate q surface flowrate drawdown build-up sandface flowrate sandface flowrate time time log p log p' log t

5 Early Time Models (2) Infinite conductivity vertical fracture Assumptions The well intercepts a single vertical fracture plane. The flowlines pattern is orthogonal to the fracture and the transient pressure response defines a linear flow in the reservoir. The well is at the center of the fracture and there are no p losses along the fracture length. Log - log response The pressure and the derivative curves are parallel and they both follow a straight line with slope equal to n = 0.5. The derivative pressure values are half of the pressure values. Parameter: x f, fracture half length Specialized plot The linear flow has no particular shape on a semi-log plot. It is only detected on the specialized plot p -vs-( t) 0.5

6 Early Time Models no p losses along the fracture length X f log p log p' Linear flow 1/2 log t

7 Assumptions (3) Finite conductivity vertical fracture The well intercepts a single vertical fracture plane. The flowlines pattern is orthogonal to the fracture and along the fracture length. The transient pressure response defines bilinear flow in the reservoir. The well is at the center of the fracture and there are p lossesalong the fracture length. Log - log response Early Time Models The pressure and the derivative curves are parallel and they both follow a straight line with slope equal to n = Afterwards, the response starts to be linear with slope n = 0.5. Bilinear flow is a very early time feature and it is often masked by WBS effects. Parameter: x f, fracture half length ; x fx w, fracture conductivity Specialized plot The linear flow has no particular shape on a semi-log plot. It is only detected on the specialized plot p -vs-( t) 0.25

8 Early Time Models p losses along the fracture length X f log p log p' Linear flow Bilinear flow log t

9 Assumptions The well produces from a perforated interval smaller than the total producing interval. This produces spherical or hemispherical flow depending on the position of the opened interval with respect to the upper and lower boundaries. Log - log response (4) Partial penetrating well Early Time Models At very early times a first radial flow, relative to the perforated interval, may establish. This is often masked by WBS effects. Then spherical flow develops and, correspondingly, the derivative curve exhibits a n =- 0.5 slope. Eventually, later on, the radial flow in the full formation is achieved. Parameter: k z /k r, vertical to radial permeability ratio; S, permeability damage (skin) relative to the perforated interval Specialized plot The spherical flow has no particular shape on a semi-log plot. It is only detected on the specialized plot p -vs-( t) 0.5

10 Early Time Models log p log p' Spherical flow -1/2 log t

11 Early Time Models Impact of anisotropy on spherical flow log p log p' log t

12 Assumptions (5) Horizontal well The well is strictly horizontal and the vertical or slanted section is not perforated. There is no flow parallel to the horizontal well. Both the top and the bottom of the formation are sealing. Log - log response Early Time Models At first radial flow may establish in a plane orthogonal to the horizontal well with an anisotropic permeability k = (k z k r ) 0.5. When the top/bottom boundaries are reached, linear flow with a n = 0.5 slope is achieved. Later on, horizontal radial flow develops in the formation. Parameter: k z /k r, vertical to radial permeability ratio; L, producing horizontal well length; S, formation permeability damage (skin); formation k r h Specialized plot The radial flow regimes can be analyzed on a semi-log plot. The linear flow regime is only detected on the specialized plot p -vs -( t) 0.5.

13 log t Early Time Models log p log p' LINEAR FLOW (1/2 slope) RADIAL FLOW (Horizontal line) EARLY RADIAL FLOW (Horizontal line)

14 Middle Time Models (1) Homogeneous (2) Double porosity Area of Interest: RESERVOIR (3) Double permeability (4) Radial composite (5) Linear composite

15 (1) Homogeneous Middle Time Models Assumptions The reservoir is homogeneous, isotropic and has constant thickness. Log - log response At early times the pressure response is under the influence of WBS effects (n=1). When infinite acting radial flow (I.A.R.F) is established in the formation, the pressure derivative stabilizes and follows a horizontal line. Parameters: formation kh; S, formation permeability damage (skin) Specialized plot On a semi-log plot (Horner plot) the points corresponding to the horizontal trend of the derivative follow a straight line of slope m.

16 Middle Time Models log p log p' I.A.R.F. log t Pressure I.A.R.F. Horner time

17 (2) Double porosity Middle Time Models Assumptions Two distinct porous media are interacting in the reservoir: the matrix blocks, with high storativity and low permeability and the fissures system, with low storativity and high permeability. Main points: The fissures system is assumed to be uniformly distributed throughout the reservoir The matrix is not producing directly into the wellbore, but only into the fissures Only the fissure system provides the total mobility, but the matrix blocks supply most of the storage capacity.

18 Middle Time Models Parameters definition Total porosity, φ t : φ t = φ f + φ m (0.01 < φ f < 1%) Total kh : kh = (kh) f (fissure system only) Storativity ratio, ω: defines the contribution of the fissure system to the total system ω = [φvc t ] f / [(φvc t ) f + (φvc t ) m ] (0.001< ω <0.1) Interporosity flow, λ: defines the ability of the matrix to flow into the fissures λ = α r w 2 (K m / K f ) ( 10-4 < λ < 10-9 ) where α is related to the geometry of the fissure network

19 Middle Time Models A double porosity response depends upon: contrast between the parameters of the matrix and fissures (φ, k) communication degree between matrix and fissures (interface skin) Two types of flow regimes from matrix to fissures are considered: a) Restricted flow conditions (pseudo steady state regime: Skin > 0) The matrix response is slower b) Unrestricted flow conditions (transient regime: Skin = 0) The matrix response is faster

20 Middle Time Models Double porosity: restricted flow conditions (S>0) In this model, also called pseudo steady interporosity flow, it is assumed that the fissures are partially plugged and that the flow from the matrix is restricted by a skin damage at the surface of the blocks. Log - Log response Three different regimes can be observed during welltest: 1) At early times only the fissures flow into the well. The contribution of the matrix is negligible. This corresponds to the homogeneous behavior of the fissure system. 2) At intermediate times the matrix starts to produce into the fissures until the pressure tends to stabilize. This corresponds to a transition flow regime. 3) Later, the matrix pressure equalizes the pressure of the surrounding fissures. This corresponds to the homogeneous behavior of the total system (matrix and fissures).

21 log t Middle Time Models FISSURES Pressure Horner time log p log p' FEEDING MATRIX λ ω

22 Middle Time Models Wellbore storage effect on fissure flow identification log p log p' log t

23 Middle Time Models Double porosity: unrestricted flow conditions (S=0) In this model, also called transient interporosity flow, it is assumed that there is no skin damage at the surface of the matrix blocks. The matrix reacts immediately to any change in pressure in the fissure system and the first fissure homogeneous regime is often not seen. Log - Log response Only two different regimes can be observed during the welltest: 1) At early times, both the matrix and the fissure are producing, but pressure change is faster in the fissures than in the matrix.this corresponds to a transition flow regime. 2) Later, the matrix pressure equalizes the pressure of the surrounding fissures. This corresponds to the homogeneous behavior of the total system (matrix and fissures).

24 Middle Time Models log p log p' (kh) 2 = 1/2 (kh) 1 (kh) 1 slabs log t

25 Middle Time Models (3) Double permeability Assumptions Stratified reservoirs, where layers with different characteristics can be identified and grouped as two distinct porous media, are interacting with their own permeability and porosity. The double - permeability behavior is observed when crossflow establishes in the reservoir between the two porous media (main layers). Main points : In each homogeneous layer the flow is radial. In multilayer reservoirs the high k layers are grouped by convention into Layer 1 while Layer 2 describes the low k or tighter zones. The two layers can produce either simultaneously or separately into the well. Crossflow always goes from the lower K layer to the higher K layer.

26 Middle Time Models Parameters definition Total Kh : (kh) tot = (kh) 1 + (kh) 2 Mobility ratio κ : defines the contribution of the high K layer to the total Kh κ = (kh) 1 / [(kh) 1 + (kh) 2 ] if κ = 1 there is double φ Storativity ratio, ϖ: defines the contribution of the high K layer to the total storativity ω = [φhc t ] 1 /[(φhc t ) 1 + (φhc t ) 2 ] Interlayer crossflow, λ : defines the effect of vertical crossflow between layers λ = A r w 2 /[(kh) 1 + (kh) 2 ] if λ =0 there is no crossflow where A defines the vertical resistance to flow and is function of the vertical permeability, k z between layers.

27 Middle Time Models Double permeability with interlayer crossflow Anywhere in the reservoir, the interlayer crossfiow is proportional to the pressure difference between the two layers. Log - Log response Three different regimes can be observed during the welltest: 1) At early times, the layers are producing independently and the behavior corresponds to two layers without crossflow. 2) At intermediate times, when the fluid flow between the layers is activated, the pressure response follows a transition flow regime. 3) Later, the pressure equalizes in the two layers. This corresponds to the homogeneous behavior of the total system.

28 log t Middle Time Models LAYER 1 LAYER 2 (kh) 1 (kh) 2 (kh) 1 >(kh) 2 log p log p' No crossflow if λ = 0

29 (4) Radial composite (lnt( lnt/2) Middle Time Models Assumptions The well is at the center of a circular homogeneous zone of radius r i (inner region), communicating with an infinite homogeneous reservoir (outer region). The inner and the outer zones have different reservoir and/or fluid properties. There is no pressure loss at the radial interface r i. This R.C. model is characterized by a change in mobility and storativity in the radial direction. Parameters definition " Mobility Ratio, M : M = (kh/µ) 1 /(kh/µ) 2 " Storativity Ratio, D : D = (φhc t ) 1 /(φhc t ) 2 Log - log response The two reservoir regions are seen in sequence: 1) The pressure behavior describes the homogeneous regime in the inner region (kh/µ) 1 2) After a transition, a second homogeneous regime is achieved in the outer region (kh/µ) 2

30 log t Middle Time Models log p log p' (kh) 1

31 Assumptions (5) Linear composite (no lnt/2) The well is in a homogeneous infinite reservoir, but in one direction there is a change in reservoir and/or fluid properties. There is no pressure loss at the linear interface L 1 This L.C. model is characterized by a change in mobility and storativity in the linear direction. Parameters definition # Mobility Ratio, M : M = (kh/µ) 1 /(kh/µ) 2 # Storativity Ratio, D : D = (φhc t ) 1 /(φhc t ) 2 Log - log response The two reservoir regions are seen in sequence: Middle Time Models 1) The pressure behavior describes the homogeneous regime in the inner region (Kh/µ) 1 2) After a transition, a second homogeneous regime is achieved in the outer region. The average mobility of the two zones is defined as: [(kh/µ) 1 +(kh/µ) 2 ]/2

32 Middle Time Models log p log p' (kh) 1-1 log t

33 Late Time Models (1) (1) Infinite lateral extent (2) (2) Single boundary Area of Interest: RESERVOIR BOUNDARIES (3) (3) Wedge (intersecting boundaries) (4) (4) Channel (parallel boundaries) (5) (5) Circular boundary (6) (6) Composite rectangle

34 Late Time Models Assumptions One linear fault, located at some distance from the producing well, limits the reservoir extension in one direction (sealing), or provides a pressure support in one direction (water drive constant pressure). Parameters Boundary distance from the well, d Single boundary Log - log response Before the boundary is reached the reservoir response shows infinite homogeneous behavior (I.A.R.F.). Two possible cases may exist: 1) sealing fault : after the boundary is felt the reservoir behavior is equivalent to an infinite system with a permeability half of the initial response permeability. On the Horner plot, the presence of a sealing boundary is shown by the doubled straight line slope : m 2 = 2m 1 2) constant pressure : the water drive support produces a constant well pressure response. After the first radial flow regime, the derivative drops with slope n = -1.

35 Late Time Models m 2 = 2 m 1 Pressure m 2 = 0 m 1 Horner time log p log p' (kh) 2 = 1/2 (kh) 1 (kh) 1-1 log t

36 Assumptions Late Time Models Intersecting boundaries (Wedge) Two intersecting boundaries, sealing or constant pressure, located at some distance from the producing well, limit the reservoir extension in two directions. The intersection angle θ is always less then 180. The well is in any position between the two barriers. Parameters: Distances from well to boundaries, d 1 and d 2 Log - log response Intersection angle: θ = 2π π [m [ 1 / m 2 ] Before the boundaries are reached the reservoir response shows the first infinite homogeneous behavior (I.A.R.F.) with a permeability of k 1. The radial flow duration is a function of the location of the well between the two boundaries. Two cases may exist: 1) two sealing faults: when both the boundaries are reached, the reservoir behavior is equivalent to an infinite system with a permeability: k 2 = (θ /2π) k 1 2) constant pressure: If one (or both) of the boundaries is water drive, the pressure stabilizes and the derivative drops.

37 Late Time Models θ 2 d WELL CENTERED WELL OFF-CENTERED d 2 log p log p' (kh) 3 =θ/2π(kh) 1 (kh) 2 =1/2(kh) 1 (kh) 1 log t

38 Late Time Models Assumptions Parallel boundaries (Channel) Two parallel boundaries, sealing or constant pressure, located at some distance from the producing well, limit the reservoir extension in two opposite directions. In the other directions the reservoir is of infinite extent. The well is in any position between the two boundaries. Parameters: Boundary distances from the well, d 1 and d 2 Log - log response Before the boundaries are reached the reservoir response shows infinite homogeneous behavior (I.A.R.F.). The radial flow duration is a function of the location of the well in the channel. Two possible cases may exist: 1) two sealing fault: when the boundaries are reached, a linear flow regime (n = 0.5) establishes. The linear flow is detected on the specialized plot p -vs-( t) 0.5 2) constant pressure: If one (or both) of the boundaries is water drive, the pressure stabilizes and the derivative drops.

39 Late Time Models WELL CENTERED 2 WELL OFF-CENTERED log p log p' 1/2 (kh) 2 = 1/2 (kh) 1 (kh) 1 log t

40 Late Time Models Closed reservoir (composite boundaries) Assumptions The closed system behavior is characteristic of bounded reservoirs. Only the rectangular reservoir shape is here considered and each side can be either a sealing barrier, a constant pressure boundary or at infinity (i.e.: no boundary). Parameters : Boundaries distances from the well d 1, d 2, d 3, d 4 Log - log response Before the boundaries are reached the reservoir response first shows the infinite homogeneous behavior (I.A.R.F.). The radial flow duration is a function of the location of the well inside the rectangular area. Depending upon the type of the existing barriers, boundaries can be: 1) sealing faults : The effect of each sealing fault is seen according to its distance from the well. If all the sealing boundaries are reached, a closed system is then defined and pseudo steady-state conditions apply (i.e.: the flowing pressure is linearly proportional to time ).

41 Late Time Models A closed system is characterized by a loss of pressure (depletion) in the reservoir, expressed as : p = p i -p avg The pressure behavior of closed systems is totally different during drawdown and build-up periods: drawdown derivative : when all the sealing boundaries are reached both the pressure and the derivative curve follow a unit slope (n = 1) straight line. On the specialized Cartesian plot p-vs-time, the flowing pressure is a linear function of time. build-up derivative : when all the sealing boundaries are reached the reservoir pressure tends to stabilize at the average reservoir pressure p avg and, as a consequence, the derivative curve drops.

42 Late Time Models r e log p log p' log t

43 Late Time Models 2) Constant pressure : If any of the sides acts as a constant pressure boundary, due to water drive support, the loglog pressure curve tends to stabilize and the derivative drops. Only the sealing faults closer to the well may be felt but, when the effect of pressure support start to act, any other sealing boundary is masked. Because no depletion is present in this case, the pressure derivative trend is the same for both the build-up and drawdown periods.

44 Late Time Models log p log p' log t

45 Late Time Models log p log p' SEALING CONSTANT PRESSURE log t