Modeling and computational study of the impact of the non-linearity of theflow in the fractured porous media

ANNA KRYLOVAAdvisors: Evgeniy Chesnokov (UH), Akif Ibragimov (TTU), Eugenio Aulisa (TTU)

ABSTRACTFluid flow in fractured reservoirs is an important topic in modern

reservoir engineering. One of the most difficult aspects of the mod-eling and solution of this problem is the fact that the width of thefracture is several orders of magnitude smaller than the character-istic dimension of the reservoir and the length of the fracture itself.In this poster is presented a new model of fluid flow in a fracturedreservoir, which allows to treat the flow in the fracture as a specificboundary condition for the bulk of the porous medium.The poster will show the mathematical geometry reduction andcomparison of computational results. Applied aspects of the prob-lem will also be considered, and the value of the productivity indexfor coupled multilateral fractures connected via horizontal wells willbe reported.

INTRODUCTION AND MOTIVATION

• Hydraulic fracturing - well-stimulation technique in whichrock is fractured by hydraulically pressured liquid.The technique is commonly applied to wells for shale gas, tightoil or gas.The stimulation used to greatly increase the flow rate.• Mathematical model of the coupled flow in the porous media

comprise fracture as low dimension geometry.• Impact of the non-linearity on the rate of production in a reser-

voir with different geometries of a fracture.• Coupled model for the fluid flow to multi-fractured multi-well

system in the porous media.• Efficiency of the well performance depending on the geometry

and hydrodynamic parameters in Darcy and Forchheimer equa-tions.

Figure 1: A Horizontal well with multiple fractures. Image credit: Halliburton

MATHEMATICAL MODEL FOR A GENERAL CASE I.Ω = Ωm + Ωw + Ωf

Where Ωm: domain of the porous media, Ωf : domain of thefracture,Ωw represents the well. Γii - corresponds to the boundariesbetween ΩiEquation of the flow:

~u = −K(|∇p|)∇p (1)

Ki,forch =2

αi +√α2i + 4βi|∇p|

& αi =µ

kifor i = m, f,w (2)

Where αi = µki

and βi = Fi·ρφ1.5·k0.5i

where i = m, f,w, Fi -specificForchheimer coefficient.The system defines boundary conditions:

~u · ~n|Γm= 0 On Γm

~uf · ~nf = − ~um · ~nm ~pf = ~pm On Γf,m

~uw · ~nw = − ~um · ~nm ~pw = ~pm On Γw,m

~uf · ~nf = − ~uw · ~nw ~pw = ~pf On Γf,w

~uw · ~n|Γw = Q|Γw| On Γw

Non-Flux on "‘ends"’ of the well and fractures(completion)

(3)

Figure 2: Domain of the problem

Km << Kf << Kw

u(x) – velocityp(x) – pressureµ – viscosity

The equation of state:

ρ = ρ0 exp

(p− p0

κ

)(4)

ρ(p) – density1/κ – compressibility

MATHEMATICAL MODEL OF PSS AFTER REDUCTION II.Eq. (3,1,4) with the appropraite averaging and assumptions:

∇ ·(−Km

µ ∇pm)

= Q|Ω| . . . in Ωm

∇x ·(−Kwell

µ · rw∇xpw)

= −Km

µ ∇pm · nr+ on each Γw

∇y ·(−KForch

µ · δi∇ypf)

= −Km

µ

(∇pm · nf+ +∇pm · nf−

)on each Γf

pf = pm . . . continuity of pressure on ΓfKwell

µ · ∇p · n|r=H = 0 . . .B.C. the on well endpointKForch

µ · ∇p · n|r=L = 0 . . .B.C. on fractures endpointsqfi = qwelli

(5)here∇x = d

dx and∇y = ddy

Figure 3: Domain of the reduced problem

PRODUCTIVITY INDEX (PI) III.Productivity index according to the MBE:

PI(t) =TOTAL FLUX

PRESSURE DRAWDOWN

Boundary data ϕ0(x) and Q defines the PSS profile/regime:

p(x, t) = − Q

|Ω|t+ P (x) (6)

where P (x) is a basic profile PI for PSS regime (PI is time invariant)(Aulisa et al,2009):

PIPSS =Q

1|Ω|∫

ΩP (x)dx− 1

|Γw|∫

ΓwP (x)dx

= const (7)

IMPACT OF NON-LINEARITY ON PI IV.

Figure 4: PI for diferent cases and number of fracures

VARIATION OF PI V.

Figure 5: PI for diferent number of fractures

RESULTS VI.A. Pressure distrubution in the given domain with directions of the

flow:

Figure 6: Pressure frofile for 1 fracture. Redhighest

Figure 7: Pressure frofile for 3 fracture. Red highest

—————————-B. Dependence PI on the Production rate and length of the fracture.

For the Darcy case:

1. Increase of the production rate doesn’t change Productivity In-dex

2. PI increases if L increases

and the Forchheimer case

1. With the increase of the fracture length, PI - increases2. With the increase of production rate Q , PI -decreases? Behaivior of PI(L,Q) is the question.

Figure 8: PI(L,Q) for deifferent cases. Blue - Darcy case(linear flow), Green-Forchheimer case (light non-liearity), Red - Forchheimer case (strong non-linearity)

—————————-C. Pi of the well with 6 fractures and variation of their length. Maxi-mum fracture length is set up 120m(394ft), and minimum 20m(65ft)

Figure 9: PI for different length of each 6 fractures

CONCLUSIONS(i) The reduced model achieves less computational cost then initial formulation without loosing the physics, that allows further sofisticated calculations

(ii) Productivity index increases in both cases: Darcy and Forchheimer; however, for linear case the increase is more rapid. This leads to the overestimation of the efficiency for long fractures.(iii) Productivity index for non-linear case increases with the number of fractures. In the same time, more fractures higher the cost and expenses.(iv) The model allows to calculate the optimal relation between length of fractures and productivity index.

MAIN REFERENCES• E.Aulisa, A.Ibragimov, P.P.Valko, J.R.Walton. Mathematical

Framework of the Well Productivity Index for fast Forchheimer (Non-Darcy) Flows in Porous Media. J. Math. Models and Methods in Appl. Sciences,

Vol. 19(8), pp.1241-1275 (2009)

• E.Aulisa, L.Bloshanskaya, A.I.Ibragimov, Long-term dynamicsfor well productivity index for non-linear flows in porous media, J.

Math. Phys. 52 (2) (2011), 26 pp

ACKNOWLEDGEMENTKindly thankful to my advisors in the UH and Texas Tech for guid-

ing me through the research, Comsol Multiphysics for providing thesoftware and Texas Tech University for the accommodation and as-sistantship. Great apresiation to my family for the support.

CONTACT INFORMATIONEmail akrylova @uh.comAddress UH, SR1, rm128, 3507 Cullen Blvd. Houston, TX 77204