Advanced Horizontal Well Model

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SPE 127742PP

ADVANCED HORIZONTAL WELL MODEL

D.S. Kuznetsov, SPE, A.N. Cheremisin, SPE, Schlumberger; A.A. Chesnokov, S.V. Golovin, Institute of Hydrodynamics

Copyright 2010, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE North Africa Technical Conference and Exhibition held in Cairo, Egypt, 1417 February 2010.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrationsmay not b e copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

A new semi-analytical model developed allows making fast modeling of the heterogeneous reservoirs, drained by a

single horizontal well with multiple transverse hydraulic fractures. It is assumed that:

the flow in the reservoir obeys Darcy law;

the fluid contains only one phase;

the reservoir permeability tensor is a function of the variable, directed along the horizontal wellbore;

the well trajectory is not a straight line;

the wellbore has finite conductivity.

The proposed model is fast in terms of the execution time and the time required to build a new case.In the first section of this article a description of the solution method is given.The second part is dedicated to validating the proposed model. The recognized finite-difference hydrodynamic

simulator (FDS) was used as reference software, though the model and FDS are not competitors because of theirdiffering applicability. For this reason the execution times of the FDS and the Advanced Horizontal Well (AHW)model are not compared with one another. More significant is that using this model one can generate a new case infifteen minutes even when the properties of the reservoir are complex and when there are multiple fractures.

The primary focus of the AHW model is a single well productivity index analysis, and candidate selection forstimulation.

Introduction

Fracturing horizontal wells has gained wide acceptance as a viable option to maximize investment returns (i.e. increasedwell productivity). This is especially true in the case of tight gas formations. Compared to an openhole horizontalwell, this type of completion provides a greater area of oil penetration and a better flow efficiency. It also providesmore even pressure drop in the reservoir compared to a vertical well intersecting a vertical fracture, which enhancesproductivity.

The value of horizontal multiple fractured wells can be maximized when the parameters with the greatest effect

on productivity are understood. As a step towards achieving maximum value from horizontal multifractured wells,a semi-analytical Advanced Horizontal Well model (AHW) has been developed that allows the user to quickly assessthe impact of various well, fractures or reservoir properties on well performance.

The procedure described has many restrictions in use. The domain (reservoir) must be of special shape (ellipse,rectangular), the external and internal boundary conditions are usually constant values of pressure or flow rate. Theproperties of the wellbore are simplified, its deviation from the straight line is often omitted.

Taking into account all restrictions above, numerical methods for solving reservoir problems are in the vanguardnow. Nevertheless, numerical methods are powerless against high gradient solutions, which occur in singular pointsof the domain such as source (or sink), or in the neighborhood of faults. One problem caused by high gradients isaccuracy and stability of numerical methods. Another problem is time resources and machine memory needed foraccuratecalculation.

Common features of the existing analytical models are dictated by the methods of solution. These are: constant

reservoir properties with respect to spatial variables (x,y,z), type of external boundary conditions (constant pressureor no-flow). Also, the pressure drop in the wellbore is neglected.

Of course, all of this simplifies the solution, but it also simplifies the description of the fluid flow in the reservoir.


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In the review below only models with multiple transverse fractures are considered. The orientation of the fracturesto the wellbore is arbitrary.

The inner boundary condition (at the wellbore) is known flow rate q(t). This is due to fact that the majority ofresults are supposed to be used in welltest analysis.

The well is usually positioned at the geometry center of the drainage region (reservoir), and the well is treated asa straight line, no deviation.

A short overview of most relevant existing models is done below.

An analytical solution for diffusivity equation was performed[1]

. The reservoir is homogenous, external boundaryconditions are either of Dirichlet or Neumann type. Boundary condition at the wellbore is flow rateq(t). The fluid isslightly compressible. The conductivity of the wellbore is infinite. There can exist multiple transverse fractures alongthe wellbore. The conductivity of fractures is finite, non-Darcy flow is considered inside the fracture. A solution formultiple wells with fractures in an isolated reservoir is obtained.

Another semi-analytical modelof horizontal well with multiple fractures in a brick-shaped drainage volume wasdeveloped[2]. A 2D Fourier transform is used to obtain an analytical solution. At the same time pressure is calculatedusing single-phase numerical simulator. Those two values are used to calculate the effective wellbore radius. Generalsolutions are developed for horizontal wells with multiple 2D and 3D fractures. The reservoir is homogenous withconstant porosity and constant permeabilities kx, ky, kz. The reservoir has impermeable top and bottom; infiniteacting reservoir or box-shaped reservoir is considered. Multiple fractures with infinite or finite conductivity can bearbitrarily oriented to the wellbore. The inner boundary condition (at the wellbore) is total flow rate q(t).

Fully analytical modelbased on the boundary element method combined with Laplace transform is used to performa fracture flow model and a reservoir flow model[3]. Based on the solution built for vertical wells in a closed reservoir,a solution for horizontal wells intersected by multiple hydraulic fractures is developed. The fluid is supposed to beslightly compressible, the reservoir anisotropy is taken into account. The wellbore has infinite conductivity, and a flowrate q(t) of the well is known. Early- and long-time approximations are derived. It can be used in welltest analysisfor evaluation of reservoir properties.

An interesting combination of the numerical fracture model with an analytical reservoir modelcan be found[4].The reservoir is anisotropic (kx =ky =kz) and infinite acting. The solution is obtained for one well with one fracture,but the number of fractures can be extended by the method of superposition. The fracture has varying properties(e.g. conductivity), and its direction might not be straight. The boundary condition at the wellbore is known flowrate q(t). It is shown that the approximation of complex fracture geometry with pseudo-skin give non-satisfactoryerrors at the early stages of production.

The article[5] compares different methods of estimating the horizontal well productivity. These are the well-known

Joshi method, its augmented version, a new analytical solution presented in paper, and a numerical simulation. Theconstant-pressure outer boundary ellipse is used in the model. The numerical simulation showed a deviation with ananalytical one. The cause of this deviation was identified, and an augmentation to Joshis solution was done. Also anew more appropriate analytical solution provided results much closer to the simulation.

Based on the information above, and business needs, a new semi-analytical method for 3-D simulation of reservoirflow towards the horizontal well was developed. Our model is a combination of a discrete approach in one direction,and an analytical solution in the other two directions plus time.

The following requirements for the model were made:

Permeabilities are step functions along the direction of the horizontal wellbore;

Wellbore has finite conductivity;

Trajectory of the horizontal wellbore is not a straight line;

There can exist multiple transverse hydraulic fractures. Each fracture has its own size and conductivity; allfractures can be positioned at any point of the wellbore.

1 The Background

The differential equation for pressure distribution in the reservoir is a consequence of the Darcy law

w= k

p (1)

and continuity equation

()

t + div( w) = F . (2)

System (1)(2) of the differential equations for unknown velocity w and pressure p is closed if functions = (p, x),= (p, x), = (p, x),k = k(p, x) defining porosity , liquid density , dynamic liquid viscosity , and permeability


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coefficient k as functions of pressure p(t, x) and coordinates x = (x,y,z), and the right-hand side F(t,x,y,p) of theequation (2) are specified.

Assuming that

Dynamic viscosity is constant;

Permeability tensor k is diagonal and depends on x;

Liquid is almost incompressible: (p) = 0(1 + l(p p0));

Porous media is almost incompressible: (p) = 0+ p(p p0).

we arrive at the parabolic equation:

p

t= div

k

p

+ F. (3)

HereF =10 F. The horizontal wellbore is introduced by means of the additional term q(t, x)

yyw(x), zyw(x)

.

It models a well with the trajectory

W ={(x, yw(x), zw(x)) |0 x X}, (4)

and time-dependent flow rate q(t, x). The reservoir permeability is a function of the variable, directed along thewellbore: kx = kx(x), ky =ky(x),kz =kz(x). The equation (3) reduces to

p

t =

x

kx(x)

P

x

+

ky(x)

2P

y 2 +

kz(x)

2P

z 2 + q(t, x)

y yw(x), z yw(x)

+ F. (5)

Piezoconductivity equation (5) with suitable initial and boundary conditions determines the pressure distributionin the reservoir and fluid inflow to the wellbore.

2 Heterogeneous Reservoir Without Fracture

This section describes the basic algorithm to calculate the pressure distribution in the reservoir.

2.1 Formulation of the Problem The equation (5) is considered in the domain

= {(x,y,z) R3 |x (0, X), y (0, Y), z (0, H)}.

(seeFig. 1)Viscosity and storage coefficient = ct are constants. The term of equation (5) with Delta-function is

responsible for the fluid inflow q(t, x) to the wellbore.

x

z

y

Sections with

horizontal wellSections with fracture

Sections without horizontal well

Fig. 1. Schematic view of the reservoir

The bottom and the top of the reservoir are impermeable:

P

z

z=0,H

= 0. (6)

At the remaining part of the outer boundary either pressure

P

= pe(t) (7)

or zero flow condition

P

n

= 0 (8)


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is specified. Here n is the outer normal vector to the boundary . At the initial time moment the pressure distributionis uniform:

P|t=0= pe(0). (9)

Pressure along the wellbore (4) is distributed according to the Hagen-Poiseuille law:

x(x)

P

x= q(t, x), where (x) =

R4

8

(10)

FunctionsR(x),(x) are radius and empirical conductivity of the wellbore. The inner boundary condition is specifiedeither as the given pressure pw(t) at the start point of the horizontal wellbore

P =pw(t), (x= xl, y = yw(xl), z = zw(xl)) (11)

or as the given total flow rate Q(t) of the wellW

q(t, x) dx= Q(t). (12)

At the end point of the wellbore the flow rate equals zero:

(x)P

x = 0, (x= x

r, y= y

w(x

r), z = z

w(x

r)). (13)

The mathematical statement of the problem is the following: it is required to satisfy equation (5) in the domain with given

initial condition (9);

outer boundary condition (6), and either (7)or (8);

inner boundary condition either (11), or (12);

closure boundary condition (10), (13).

The reservoir pressure and either flow rate or bottomhole flowing pressure should be determined.Replacing pressurePwith the functionp(t,x,y,z) = P(t,x,y,z)pe(t) in the case of condition (7), orp(t,x,y,z) =

P(t,x,y,z) pe(0) in the case of condition (8), we arrive to the similar problem, but with zero outer boundary

conditions.2.2 Method of Solution The idea of solving the formulated problem is following:

Discretization along Ox.

The Ox-axis is splitted into N subintervals by the set of points x1, . . . , xN, where xi < xi+1 (seeFig. 2).

hi

xi-1 xi xi+1i

x =01/2 x =XN+1/2x =xN+1/2r rx =xN -1/2l l

Fig. 2. Discretization alongOx

All functions are assumed to be piecewise constant with respect to x, equal to their mean values over intervals[xi1/2, xi+1/2]. Then all equations and boundary conditions are to be rewritten using this finite-differenceapproach.

Laplace transform with respect to time.

The system of equations obtained at the previous stage is reduced using the Laplace transformation with respectto time. Thus the initial 3-dimensional evolutionary problem becomes only 2-dimensional over variables (y, z).

Fourier decomposition.

To solve the remaining problem, all functions are to be decomposed in Fourier series over sines and cosines. Thechoice of basic functions makes it possible to satisfy the outer boundary conditions: either the specified pressureor the constant inflow.

The linear algebraic system of equations for Fourier coefficients is 3-diagonal with diagonal predominance, andthe method of its solution is sweep procedure.

Closing relations for this linear system is a pressure continuity condition within the wellbore side surface.


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Inverse Fourier and Laplace transformations for solution images.

According to the workflow above, the method is semi-analytical: it uses the finite-difference approach along thewellbore, and the analytical solution in transverse directions.

3 Heterogeneous Reservoir With Multiple Hydraulic Fractures

The algorithm described in the previous section can be adopted to the case with multiple transverse hydraulic fracturesalong the horizontal wellbore. It is assumed that each layer along the Ox axis (except for the first and the last one)may contain a hydraulic fracture (HF), located at the center point x = xi of the corresponding layer (see Fig. 3). Weassume that the height of each HF equals the reservoir thickness Z. In the horizontal direction along theOy-axis eachfracture has its individual size.

xi-1 x i-1/2 xi x i+1/2 x

zy

Hydraulic fracture (HF)

wellbore

Fig. 3. Hydraulic fracture in i-th layer

Hydraulic fractures are taken into account by adding termsNr

k=Nl

k(t, y)(x xk) to the right hand side of the

main equation (5), which are responsible for the inflow to the wellbore from the fractures.

Fluid inflow to the fracture in the i-th layer is determined by the discharge intensity i(t, y). For simplicity anddue to the thickness of the reservoir in the z-direction we assume that functions i do not depend on z. In the casewhen there is no fracture in the i-th section, the corresponding function i 0.

The flow inside the fracture is supposed to be laminar, and the same approach as with the finite conductivity wellis valid here: pressure drop in HF satisfies the equation

y

CF

p

y

= (t, y). (14)

Here the empirical function CFdetermines HF conductivity. The following boundary conditions for equation (14)aretaken:

CFp

y

yil

,yir

= 0, p|yi0

= pi

W (15)

Hereyil andy

ir are the left and right limiting values ofy along HF in the i-th layer; y

i0 is the position of the wellbore;

pi

Wis the wellbore pressure in i-th layer.

The method of solution of modified equations describing the flow in the reservoir with a horizontal well andmultiple transverse hydraulic fractures remains the same as in section 2. Nevertheless, one must add closure relationsto determine unknown flow rates qi and intensities i. The following conditions will be used:

The reservoir pressure at the wellbore coincide with the wellbore pressure;

The reservoir pressure at HF coincides with the HF pressure.

At least 2 approaches can be used to determine the reservoir pressure at the HF:

Pressure at HF coincides with the average layer pressure: pf = pi.

Pressure at HF is calculated by linear approximation: pf = i1(4pi

pi1

) + i(4pi

pi+1

)3(i1+ i)

, where i is the

length of the i-th interval.


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4 Validation

A reliable finite-difference simulator FDS was used to validate this new model, which can have application not onlyfor slightly compressible fluid flow, but also for gas flows. The usual approach to switch from liquid to gas is a use ofpseudo-functions, e.g. pseudo-pressure and pseudo-time[7, 8, 9] .

Standard SPE tests for horizontal well simulators are not applicable here, because they are developed for multi-phase flow instead of 1-phase flow considered in our model [10].

Tests have started from simple reservoir models, which complexity was increased with each step. All types of testsare listed in the tree below.

Homogenous Reservoir

No fractures

Fractured

Heterogenous (along O x) reservoir

Deviated horizontal well

4.1 Models of the Reservoirs

4.1.1 Numerical Hydrodynamic Model. Productivity layer is supposed to have the form of rectangular paral-

lelepiped with dimensions 1100 100 10 m. The dimensions were chosen based on typical reservoirs for which themodel will be used and based on the widely accepted Seventh SPE comparative solution project for horizontal wellsimulation[10]. Cell dimensions of the Cartesian grid in FDS are selected according to the case tested. Several simu-lations were done on different grids to be convinced that the result is not sensitive to the grid selection. Simulationof the vertical hydraulic fracture in Oyzplane was done on a refined grid with several buffer blocks in front of andbehind the fracture (seeFig. 7for explanation).

Permeability of grid cells is chosen depending upon the case considered (type of reservoir, fractured or not).Porosity and total compressibility ct are constant values in all tests. Only the 1-phase fluid flow is considered.

A numerical aquifer is used in FDS to simulate cases with a constant outer boundary pressure condition.Fractures in FDS are modeled via a local permeability increase of the cells in the refined grid: the grid in the

Oxdirection was successively decreased from 100 m to 1 cm. Permeability of the cells containing fracture is chosenso that the equivalent fracture conductivity kfwf, where kfis the real fracture permeability, and wf is the fracturewidth, has the same value both in the model and the reservoir.

4.1.2 AHW Model. The model is determined by the following parameters:

Dimensions of the reservoir;

Fluid viscosity ;

Total compressibility ct Start and end point of the horizontal well, its deviation survey;

Number of layers of heterogeneity along the horizontal well;

Permeability of each layer in x, y and z directions;

Hydraulic fractures, its properties;

Wellbore radius and wellbore conductivity;

Boundary conditions both at the wellbore and at the outer boundary;

Time period for simulation.

4.1.3 Parameters to Compare. The AHW model makes it possible to track the following reservoir and productionparameters:

Pressure as a function of time at the starting point of the wellbore;

Flow rates from each layer as a function of time;

Total flow rate of the wellbore;

Pressure as a function ofy at any fixed time t= const, and any section x= const.

4.1.4 Validation Methods and Acceptance Criteria Seventh SPE Comparative Solution Project [10] reports com-parisons results of simulation runs performed by 14 organizations involving production from a horizontal well in a

rectangular reservoir.The results from the Seventh SPE comparison study showed that maximum deviation (among 14 simulators) in

cumulative horizontal well production for most cases is about 10%. It is impossible to find a base case here, because


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there is no exact solution for horizontal well inflow in rectangular reservoir. Therefore, SPE community reports meanvalue and standard deviation for each case. For most cases 5%, and we take this value as an acceptance criterionfor the AHW model validation. Expected results between the developed model and FDS are also 10%, but applicationof FD simulators for horizontal well inflow modeling is limited (e.g., reservoirs with extremely low permeability).Thus, the decision was made to compare the most simple cases first and then to increase complexity of the cases tovalidate all functionalities of the new model.

This approach makes it possible to identify the validity area of the model as well as to determine its weak points.

4.1.5 Stability of the ModelsBefore running the validation tests, both models in FDS and AHW were put on trialswith different grid block dimensions. FDS gives a stable result with an inessential distinction when:

the number of grid blocks 3 in both Ox, O y directions;

the number of grid blocks 1 in the O z direction.

Note that the permeability of the reservoir can be a function of only variable x (see Section2).

The AHW model showed good convergence on different grids1. Typical plots are given onFig. 4. Here the followingnotation is used: 7(5) Sections mean that the reservoir was split into 7 sections, and 5 of them in the center containa well.

The solution method of the model requires at least 5 sections in the Ox direction with mandatory 3 mid-sectionscontaining a horizontal well.

0

5

10

15

20

25

30

0 50 100 150 200

Flow

rate,m3/day

Time, days

15(13) Sections

7(5) Sections

7(3) Sections

R

a

v

Time days

14

14,1

14,2

14,3

14,4

14,5

14,6

14,7

14,8

14,9

15

25,0 25,2 25,4 25,6 25,8 26,0

Flow

rate,m3/day

Time, days

15(13) Sections

7(5) Sections

7(3) Sections

R

a

v

Time days

Fig. 4. Flow rate curves for AHW. Varying discretization along Ox (left); Zoomed plots (right).

4.2 Homogeneous Reservoirs

4.2.1 No FracturesTests start from simple homogeneous reservoirs with a straight (not deviated) horizontal wellpositioned at the symmetry axis (along O x) of the rectangular reservoir. All physical parameters of the rock and thefluid remain the same in all these runs. Conductivity of the well in our model was set manually while FDS calculatespressure drop at the wellbore due to friction according to engineering formulas used in pipeline flow calculations.

Such parameters as porosity, compressibility, and viscosity are just multiplicative constants for one-phase flows;permeability influences the duration of different flow regimes in the reservoir and the flow regime in the wellbore.Thus, the storage coefficient ct in all tests is 4.05 10

5 bar1, the permeability in all directions is 0.1 mD, the fluidviscosity= 0.88 cP.

Case #1 (see Appendix for details) shows good correspondence between FDS and AHW (Fig. 5). The overalldisagreement for cumulative production at end of simulation period is even less than 1%. Pressure and rate gradientsare always high at early times, hence, it is very hard to obtain good matching there. FDS is very time step sensitivewithin the first 1020 days of production, and it negatively affects the match between AHW and FDS. Thus, thedisagreement at early time period had no influence on validation.

1The term grid for semi-analytical AHW model means how detailed is the discretization along the Ox axis.


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200

100

150

o

w

m

FDS

AHW

0

50

0 50 100 150 200

F

Time days

m

C

m

a

v

m

Time days

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Er

ror,%

Time, days

Flow Rate

Cumulative

Fig. 5. Flow rates, Case #1 (left); Relative Error, Case #1 (right).

100

150

a

m

FDS

AHW

0

50

0 100 200 300 400 500 600

F

o

w

Time days

C

m

a

v

m

Time days

Fig. 6. Flow rates. Case #2.

Case #2 simulates a depletion drive type of production. In addition, the time of the forecast increased upto 500 days (Fig. 6). A minor disagreement between FDS and AHW cumulative production curves appeared onday 200 of the production, after some oscillations in AHW flow rates have occurred due to the transverse Laplacetransform procedure. This disappointing singularity of the semi-analytical approach can be eliminated by usinganother procedure, e.g. the Stephest algorithm instead of transformation with Legendre polynomials. Later on themismatch in flow rates (oscillations around zero) has generated a divergence of 6.5% in the cumulative production atlate time stages.

4.2.2 Multiple Transverse FracturesCase #3 (seeAppendix)is intended to make sure that AHW and FDS give similar results on the refined grid ( Fig. 7),

prepared to simulate vertical transverse fractures (Fig. 8).

fracture in one cell of dx=0.01m

5*100m

1*25m

2*10m

1*2.5m

2*1m

1*0.2

5m

2*0.1m

10*0.0

1m

2*0.1m

1*0.2

5m

2*1m

1*2.5m

2*10m

1*25m

5*100m

5*100m - 5 grid blocks with dx=100m

Fig. 7. Progressive grid refinement for fracture simulation

It is observed that both simulators perform equally on the refined grid without fractures, which shows good stabilityof the numerical methods used.


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0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

Fig. 8. Relative Error. Case #3 (left); Case #4 (right).

Hydraulic fractures with significantly different conductivities were simulated in cases #47. Case #5 showedminor oscillations of the flow rate (Fig. 9) due to very close placement of the fracture tip to the outer boundary ofthe reservoir, and it resulted in the pressure gradient of approximately 100 bars within 5 meters of porous media. To

reduce the relative error in this case (7.5% between the blue and the red lines), an increase of the number of basicfunctions is needed. Of course, the time of simulation grows much faster than the rate of convergency: the number ofbasic functions in both directions (y, z) was increased by factor 1.5, but the accuracy increased by only 2.4% (greenredlines,Fig. 9).

250

100

150

200

w

m

FDS

AHW short series

AHW extended series

0

50

0 50 100 150 200

F

o

w

Time days

3

C

m

a

v

m

Time days


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative


Cases #4, #8 with real fracture conductivity (10003000 mD m) show good agreement (approximately 2%)with the numerical simulator (Fig. 8(right),Fig. 11(left)), while other cases with extremely low fracture conductivity(Fig. 10) gave a relative error of 34% in the flow rate, and resulted in the total error of 24% in cumulative production.


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The AHW model has an option to simulate non-symmetryvertical fractures, whenthe extreme case of which iswhen only one wing of the fracture exists. Such experiment (Fig. 11,right) was launched using a grid for symmetryfractures, and the result is quite good: disagreement between AHW and FDS is small, and the diagram for pressuredistribution at the section with fracture shows how the solution is sensitive to fracture parameters (Fig. 12).

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative


100

150

200

s

T=0,6

0

50

0 10 20 30 40 50 60 70 80 90 100

P

T=3,5

T=9,1

T=21,3

T=148

Y meters

Fig. 12. Pressure distribution along the section with non-symmetry fracture. Case #9.

Before making multiple fractures, the stability of calculations was checked on the refined grid. As in Case #3, eachfracture in FDS was simulated via a local increase of permeability so that the equivalent permeability (in mD m) bothin AHW and FDS was similar. Thus,Fig. 13 contains a production forecast on the fine grid with original reservoirproperties; plots on Fig. 14 are the product of 3 fractures simulated at X0 = 550m, X0 = 650m, and X0 = 750m.Again, some oscillations in production curves occur, which has further influence on cumulative production.

200

100

150

w

m

FDS

AHW

0

50

0 50 100 150 200

F

o

w

Time days

3

e

m



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FDS and AHW coincide better (only 0.7% of relative error) when the well is moved close to the open boundary,i.e. Y= const (case #13, Fig. 16).

Such difference in two previous cases is caused by the fact that a smaller number of basic functions is used in theOzdirection in comparison with theOydirection (100 against 200). The increase in the series length in Oz directionleads to a significant increase of time for the execution with an increase in accuracy by only several fractions. Thus,the golden mean is preferable in this case: both satisfactory accuracy and running time of the program.

Case #14 is the superposition of two previous cases. The number of harmonics in the Fourier series was optimized,

and the total relative error dropped to 0.6%. This was done just to show that the adjustment between the models ispossible, but prior to forecast execution one must set an appropriate mistake according to common sense and selectthe series length. This will remain as an option of AHW, similar to FDS time step choice.

4.2.4 Cases with Extreme Permeability Values Three cases with reservoir permeability of 0.01 mD, 10 mD and10 D were studied (see table below). Extremely low permeability (case #15,Fig. 17, Appendix) leads to inefficientdisagreement between AHW and FDS at early-time flow rates. The flow rates on steady-state regime are nearly equal:0.34% of relative error.

150

100

w

m

FDS

AHW

0

50

0 50 100 150 200

F

o

w

Time days

3

C

m

a

v

m

Time days

30

15

20

25

w

m

FDS

AHW

0

5

10

0 100 200 300 400

F

o

w

Time days

3

C

m

a

v

m

Time days

Fig. 17. Flow rates. Case #14 (left); Flow rates. Case #15 (right).

An increase in reservoir permeability up to 10 mD results in higher well productivity. The flow in the wellboreswitches to the turbulent type, and the pressure gradient along the horizontal segment of the well increases. FDSautomatically tracks the type of the fluid flow in the well: as Reynolds number exceeds 4000, the friction factor iscalculated using another formula, while the AHW model still uses the Poiseuille equation ( 10). Case #16 (Fig. 18)shows this disagreement between FDS and AHW. After the conductivity of the well in AHW was manually decreased,the results of simulations nearly coincided (green curves on theFig. 18).

5000

2000

2500

3000

3500

4000

4500

w

m

FDS

AHW

AHW low conduct.

0

500

1000

1500

0 50 100 150 200

F

o

w

Time days

3

C

m

a

v

m

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

Fig. 18. Flow rates. Case #16 (left); Relative error. Case #16 (right).

Cases #17, 18 are similar except the wellbore friction option. Once increased reservoir permeability up to 10 Dresulted in a sufficiently turbulent flow in the wellbore, and the disagreement between two simulators became visibly


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SPE127742PP 13

high (Fig. 19). After the wellbore friction option was disabled, the agreement between simulators was restored(Fig. 20).

180000

80000

100000

120000

140000

160000

w

m

FDS

AHW

0

20000

40000

60000

0 50 100 150 200

F

o

w

Time days

3

C

m

a

v

m

Time days


0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

Fig. 20. Relative error. Case #18.

4.3 Heterogeneous Reservoirs

4.3.1 Constant PermeabilityThe reservoirs simulated here have constant, but different permeability in x, y , and zdirections. It means that kx, ky, kz are constant values, though they are not equal.

The results of FDS runs with different number of cells in each coordinate axis direction were analyzed beforecomparison with the AHW model. During these adjustment runs the optimal number of cells was selected: theminimal number of cells in each direction, when its further increment had a negligible effect on the result.

35

15

20

25

30

w

m

FDS

AHW

0

5

10

0 50 100 150 200

F

o

w

Time days

3

i

v

m

500

200

250

300

350

400

450

w

m

FDS

AHW

0

50

100

150

0 50 100 150 200

F

o

w

Time days

3

m

Fig. 21. Flow rates. Case #19 (left); Case #20 (right).


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14 SPE127742PP

Cases #1922 (see Appendix) are typical with respect to the variety of tested ones within this section. The overalltrend is as follows: when permeability in Oy direction dominates permeability in both Ox and Oz directions, AHWmodel production curves lay slightly below the FDS production curves (Fig. 21Fig. 22).

Then, ifkx values are dominant, the rates calculated by the AHW model exceed those obtained via FDS (Fig. 22).

The behavior of production curves described above does not depend on the number of cells in FDS or on thenumber of layers and series lengths in AHW.

200

100

150

w

m

FDS

AHW

0

50

0 50 100 150 200

F

o

w

Time days

3

C

m

a

v

m

Time days


4.3.2 Varying PermeabilityThe heterogeneity of the reservoir in the AHW model can be magnified in comparisonwith section4.3.1,meaning that the components of the permeability tensor kx, ky, and kz can be step functions ofxvariable (along the main direction of the wellbore).

New model was tested on reservoirs with the following properties (see table on page17).

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow RateCumulative

Fig. 23. Relative error. Case #23 (left); Case #24 (right).

Cases #2326 (seeAppendix) cover several extreme scenarios of the reservoir structure. It is known, that numericalalgorithms may have difficulties (instabilities) when functions in the design domain are highly discrete. To overcomesuch problems, special methods of calculations are required.

Listed cases were executed using several grids with different mesh sizes (for AHW the term mesh of the gridmeans the number of layers along Ox bounded by planes x = const). The final time of simulation T= 200 days issufficient to reach a steady-state production period.

Typical relative errors are shown on plots (Fig. 23,Fig. 24). The agreement in flow rates and cumulative productionis excellent except that AHW rates have oscillations at some time segments.

4.4 Deviated Wellbore Survey One more option is available in AHW model: the wellbore can be positioned

at any place of the reservoir, and its deviation survey can be not a straight line, but a step function determined atevery layer separately. Thus, the well becomes a broken line with infinite conductivity connections at the boundariesof layers with x= const.


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SPE127742PP 15

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative


The reservoir under consideration is still rectangular shaped with original dimensions: 1100malong Ox, 100 malongOy , and 10 malong theOz axis. The permeability is constant: 0.1 mDin all directions; the fluid and the wellbore

properties are as described in section 4.1. The well starts at x0 = 100 m and finishes at the point x1 = 1000 m. Thefluid in the wellbore flows to the left, e.g. towards the starting point x0.In case #27 (see Appendix) the well changes its direction in the Oxz plane. The vertical coordinate of the well

centerZc= const and it jumps from 0.5 m(distance to the bottom of the reservoir) toZc = 5.5 , which is approximatelythe center of the reservoir.

Next, in case #28 the wellbore lays in the Oyz plane, and switches between 16.7 and 83.3 meters stand-off theside reservoir boundary y = 0.

Case #29 is the superposition of two previous cases.

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

8,0

9,0

10,0

0 50 100 150 200

Error,%

Time, days

Flow Rate

Cumulative


From the plots above one can see how wellbore placement influences the production rates during transient andsteady-state regimes.

Relative error between AHW and FDS is less than 1% in all considered cases.

5 Area of Applicability of the Advanced Horizontal Well Model

Based on more than 50 comparative runs of AHW and FDS, one can determine the boundaries of new model appli-cability.

5.1 Good Performance

Homogenous and heterogeneous reservoirs with non-constant permeability along a horizontal well;

Reservoirs with multiple hydraulic fractures, each with its own physical and geometry properties;

Reservoirs with deviated horizontal wells, minimum recommended distance between well and outer boundary ofthe reservoir is 0.5 meters (2 feet);

Laminar flow in the wellbore.


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5.2 Poor Performance

Reservoirs with high permeability and long horizontal wells lead to turbulent flows inside the wellbore. Thenpressure distribution equation (10) needs additional adjustment via special type of function, or the modelshould be equipped with the turbulent flow regime in the wellbore (Fig. 19);

Long time simulation on depletion drive type reservoirs due to minor oscillations of the flow rate curve nearzero values may lead to unacceptable mismatch (Fig. 6). This issue, as it was mentioned on page8, is only a

problem of numerical algorithm.

6 Model Summary

The Advanced Horizontal Well Model is developed to meet the requirements listed inIntroductionon page2. Accordingto the specific of the task, we used finite-difference approximation in the Ox direction combined with Fourier seriesin both Oy and Oz directions. Also, as governing equations are linear with respect to timet, the Laplace transformtechnique was applied.

The whole reservoir was divided into sections by planes, parallel to the Oyz -plane. The model makes it possibleto describe cross-flows between these sections, and moreover, the cross-flows behind the fractures (as tips of fracturesdo not reach the boundary of the reservoir).

The pressure drop along the wellbore is taken into account, though the current realization assumes the laminartype of flow inside the wellbore only.

Brief comments on some functionalities of the model:

Varying Permeability

Components of the permeability tensor depend upon variable x, directed along the horizontal wellbore. Thisfunctionality makes it possible to simulate reservoirs with pronounced heterogeneity in one horizontal direction.

Comparison tests with FDS have showed good agreement, even when the permeability of adjacent sectionsdiffered by factor 10 (see section 4.3.2).

Hydraulic Fractures

Transverse hydraulic fractures can be placed within any section that contains a well. All fractures can havenon-symmetry wings with different conductivities.

Tests have showed that even in extreme conditions AHW gives results very close to FDS. There are no limitationsin fracture conductivity, but there is a limitation in fracture length: the tip of the fracture should not be close

to the outer boundary (see Fig. 9and the description). Finite Wellbore Conductivity

The model and the numerical algorithm used are designed to take into account finite wellbore conductivity.One of the possible solutions is to use HagenPoiseuille law (10), which is good only for laminar flows. As theReynolds number grows, the flow regime transforms from laminar to turbulent. FDS automatically tracks theflow type, while the AHW model has no such option (see Fig. 19). As a result, there may be some unacceptablediscrepancies between FDS and AHW at high flow rates.

Deviated Well

The wellbore in AHW can have curved survey. Coordinates of the well center in every section containing wellare specified independently from other sections. Thus, the wellbore becomes a broken line, the vertical segmentsof which have infinite conductivity.

The most complicated cases are when the well is positioned close to outer boundary, i.e. when the pressure

has high gradient. Cases #1214 and #2729 show that even if the wellbore is only 0.5 m stand-off from theboundary, the disagreement between AHW and FDS is only 23%.

7 Conclusion

This paper illustrates the use of a prediction tool for improved modelling of oil production from horizontal or deviatedwells with induced fractures. This was achieved by implementing original semi-analytical solutions into the model,to provide a useful diagnostic and prognostic tool for the oil industry. The model robustness and speed makes thisprogramme for well production optimization of a horizontal well with induced fractures easy to use.

The approach has been validated against industry recognized finite-difference simulator.Overall, an obvious benefit to the industry from this model is a possibility to make fast production prediction for

long horizontal wells with multiple transverse hydraulic fractures in heterogeneous reservoirs.

Acknowledgements

The authors express their gratitude to Schlumberger for the problem statement and support.


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SPE127742PP 17

Appendix. Cases

# Reservoir, m FDS Cells Boundary Cond. Fracture Conductivity,X Y Z X Y Z Wel l Outer Y(m)/N mD m

1 1100 100 10 11 3 1 BHP CP N2 1100 100 10 11 3 1 BHP NF N

3 1100 101 11 482 3 1 BHP CP N4 1100 101 11 48 3 1 BHP CP 40m 1000

5 1100 101 11 48 3 1 BHP CP 90m 10006 1100 101 11 48 3 1 BHP CP 40m 107 1100 101 11 48 3 1 BHP CP 40m 18 1100 101 11 48 3 1 BHP CP 40m 3000

9 1100 101 11 48 3 1 BHP CP 20m3 3000

10 1100 101 11 1504 3 1 BHP CP N

11 1100 101 11 150 3 1 BHP CP 3 40m5 3000

Well standoff6

Y = const Z= const

12 1100 101 11 11 101 100 BHP CP 50.5 0.513 1100 101 11 11 101 11 BHP CP 0.5 5.514 1100 101 11 11 101 100 BHP CP 0.5 0.5

Permeability, WellboremD friction (Y/N)

15 1100 100 10 11 3 1 BHP CP 0.01 Y16 1100 100 10 11 3 1 BHP CP 10 Y17 1100 100 10 11 3 1 BHP CP 10 000 Y

18 1100 100 10 11 3 1 BHP CP 10 000 N

Permeability, mDkx ky kz

19 1100 100 10 11 3 1 BHP CP 0.1 0.1 0.01 Y20 1100 100 10 11 3 10 BHP CP 1 1 0.1 Y21 1100 100 10 11 3 10 BHP CP 0.1 1 0.1 Y22 1100 100 10 11 3 10 BHP CP 1 0.1 0.1 Y

BHP: Bottomhole pressure control;CP: Constant pressure;NF: No flow.

Permeability along Ox , m in mD# 0100 100200 200300 300400 400500 500600

kx ky kz kx ky kz kx ky kz kx ky kz kx ky kz kx ky kz

23 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.06 0.0624 1 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.125 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.1 126 1 0.1 0.5 0.1 1 5 1 0.1 0.5 0.1 1 5 1 0.1 0.5 0.1 1 5# 600700 700800 800900 9001000 1000110023 0.07 0.07 0.07 0.08 0.08 0.08 0.09 0.09 0.09 0.1 0.1 0.1 0.11 0.11 0.1124 1 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.1 1 1 125 1 1 0.1 0.1 0.1 1 1 1 0.1 0.1 0.1 1 1 1 0.126 1 0.1 0.5 0.1 1 5 1 0.1 0.5 0.1 1 5 1 0.1 0.5

Wellbore Coordinates in the Layer (Yc, Zc)7, m

# 0100 100200 200300 300400 400500 500600 600700 700800Yc Zc Yc Zc Yc Zc Yc Zc Yc Zc Yc Zc Yc Zc Yc Zc

27 N N 50 0.5 50 5.5 50 0.5 50 5.5 50 0.5 50 5.5 50 0.528 N N 16.7 5.5 83.3 5.5 16.7 5.5 83.3 5.5 16.7 5.5 83.3 5.5 16.7 5.529 N N 16.7 0.5 83.3 5.5 16.7 0.5 83.3 5.5 16.7 0.5 83.3 5.5 16.7 0.5# 800900 9001000 1000-1100

Yc Zc Yc Zc Yc Zc27 50 5.5 50 0.5 N N

28 83.3 5.5 16.7 5.5 N N29 83.3 5.5 16.7 0.5 N N

N: No well in current layer.

2Progressively refined towards the layer with fractureFig. 73Non-symmetry fracture: only one wing l= 20m exist.4Progressively refined grid analogous toFig. 7 for 3 separate fractures in different Ox-sections53 fractures, each 40m of length (20m half length)6In meters from the corresponding boundary7The origin of the Cartesian coordinate system is placed at the bottom corner of the reservoir


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18 SPE127742PP

Nomenclature

= storage coefficientct = isothermal total compressibility

CF= fracture conductivity= Dirac delta-function

i = length ofi-th interval over O x axis

= porosity = discharge intensity of the fracture = wellbore conductivity = reservoir outer boundary= empirical specific wellbore conductivityk = permeability tensor

ki = diagonal elements ofk , i= x,y, z = fluid viscosityn = outer normal vector to

P, p = fluid pressurep = Laplace image ofpq= specific flow rate

Q = total flow rate = fluid density

R = wellbore radiust = time

= reservoir domainx = Cartesian coordinates, x= (x,y,z)w = fluid velocity vector

W = wellbore manifold

References

1. Brusswell, G., Banerjee, R., Thambynayagam, R.K.M., Spath, J.: Generalized Analytical Solution for ReservoirProblems With Multiple Wells and Boundary Conditions, SPE 99288

2. Wan, J., Aziz, K.Semi-Analytical Well Model of Horizontal Wells With Multiple Hydraulic Fractures// SPE81190

3. Zernar, A., Bettam, Y.Interpretation of Multiple Hydraulically Fractured Horizontal Wells in Closed Systems//SPE 84888

4. Al-Kobaisi, M., Ozkan, E., Kazemi, H.A Hybrid Numerical/Analytical Model of a Finite-Conductivity VerticalFracture Intercepted by a Horizopntal Well// SPE 92040

5. Economides, M., Deimbacher, F., Brand, C., Heinemann, Z. Comprehensive Simulation of Horizontal WellPerformance// SPE Formation Evaluation, December 1991

6. Krylov, V.I., Skoblia, N.S. Handbook on numerical inversion of the Laplace trnsformation Izdatelstvo Naukai Tekhnika, Minsk, 1968.

7. AlHussainy, Ramey, H.J., Jr., Crawford, P.B. The Flow of Real Gases Through Porous Media// JPT, May,

19668. Agarwal, R,G. Real Gas Pseudotime A New Function for Pressure Buildup Analysis of MHF Gas Wells//

SPE 8279

9. Chen, H.Y., Poston, S.W.Application of a Pseudotime Function to Permit Better Decline Curve Analysis//SPE 17051PA

10. Nghiem, L., Collins, D., Sharma, R.Seventh SPE Comparative Solution Project: Modeling of Horizontal Wellsin Reservoir Simulation // SPE 21221

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Advanced Horizontal Well Model