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Quantitative production analysis and EUR prediction from unconventional reservoirs using a data-driven drainage volume formulation

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Abstract

A novel data-driven approach was previously introduced for production analysis of unconventional reservoirs without the traditional rate transient analysis/pressure transient analysis assumptions. The approach relied on a w(τ) function, which is drainage volume geometry function to characterize the flow geometry from the transient drainage volume. It has been used to rank refracturing candidates and to determine optimal fracture spacing. In this paper, we generalize the previous study to improve the amount of quantitative reservoir information obtained during the production analysis. Our approach is based upon a transient generalization of the Matthews-Brons-Hazebroek definition of the pseudo-steady state drainage volume. It is obtained from an asymptotic solution of the diffusivity equation in heterogeneous and/or fractured media. Given field pressure and flow rate data, we can calculate the transient well drainage volume with time. The time evolution of the drainage volume can be inverted to estimate w(τ) function, which contains information of underlying flow geometries, and which is then used for quantitative analysis. The power and utility of the proposed methodology is first validated with synthetic examples and then demonstrated using a well from the Montney shale. In the examples studied, we are able to identify linear flow, the onset of fracture interference, complex nonlinear flow, and the development of the stimulated reservoir volume (SRV), leading to the quantitative calculation of matrix permeability, fracture surface area, and volume of the SRV. The proposed approach is a data-driven model-free analysis of production data without the presumption of specific flow regimes. It provides a simple and intuitive understanding of the transient drainage volume and instantaneous recovery efficiency, irrespective of the complexity of the reservoir depletion geometry. We show an improved approach for the w(τ) inversion which yields better physical resolution and which can identify more detailed characteristics of the underlying flow geometry than previous studies, e.g., complex near-fracture flow, linear flow, and fracture interference. The results of the analysis have been used for the characterization of hydraulic fracture and reservoir properties, including the prediction of fracture surface area, matrix permeability, and SRV, and extended to the calculation of estimated ultimate recovery.

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Abbreviations

A :

Cross-sectional area (ft2)

An(t):

Pressure amplitude of the n th order in the time domain (1 /h(n+ 2)/2)

\(\tilde {A_{n}}(\vec {{x}})\) :

Pressure amplitude of the n th order in the frequency domain (1 /h(n+ 2)/2)

b :

Decline curve exponent (dimensionless)

\(\bar {{B}}_{\mathrm {g}} \) :

Average gas formation volume factor (Mcf /Mscf)

c t :

Total compressibility (1 /psi)

\(\bar {{c}}_{t} \) :

Average total compressibility (1 /psi)

D :

Decline rate (1 /h)

h :

Payzone thickness (ft)

i :

Imaginary unit

J :

Well productivity (stb /psi / day)

J BDF :

Well productivity under boundary- dominated flow (stb /psi / day)

J PSS :

Well productivity under pseudo-steady state (stb /psi / day)

k :

Permeability (md)

k f :

Fracture permeability (md)

k m :

Matrix permeability (μ d)

K (τ,t):

Diffusion kernel

L res :

Reservoir length (ft)

L w :

Well lateral length (ft)

n basis :

Number of basis functions

n data :

Number of data points in the interval

n f :

Number of hydraulic fractures

n t :

Number of data points in the production history

p :

Pressure (psi)

\(\bar {{p}}\) :

Average pressure (psi)

\(\tilde {{p}}\) :

Pressure in frequency domain (psi)

p a :

Adjusted pressure for gas (psi)

p wf :

Bottomhole flowing pressure (psi)

q :

Flux (stb /day)

q sf :

Flux at sandface (stb /day)

q w :

Flux at surface (stb /day)

q w, cutoff :

Cutoff production rate at surface (Mscf /day)

Q w :

Cumulative production (ft3)

r :

Distance (ft)

r res :

Reservoir radius (ft)

r w :

Wellbore radius (ft)

RNPm,n :

RNP induced at fracture m by fracture n (stb /psi / day)

S :

Skin (dimensionless)

S w :

Water saturation (fraction)

t :

Time (h)

t a :

Adjusted time for gas (h)

t e :

Material balance time (h)

t elf :

Time of end of linear flow (h)

t h :

Production history used for w(τ) inversion (day)

t LOD :

Time to detect round-trip pressure front propagation under LOD (day)

t prod :

Production period (day)

t (a /2,dof):

Student’s t distribution with confidence level (a) and degree of freedom (dof)

T res :

Reservoir temperature (∘F)

V (t):

Generalized time-dependent volume integral (ft3)

V (τ,t):

Generalized incomplete volume integral (ft3)

Vd(t):

Drainage volume (ft3)

V det :

Reservoir pore volume detected during inversion (ft3)

Vp(τ):

Pore volume (ft3)

V res :

Reservoir volume (ft3)

w(τ):

Derivative of pore volume with respect to τ (ft3 /h1/2)

w f :

Fracture width (in)

w res :

Reservoir width (ft)

W(t):

Generalized pressure drop integral (h)

W (τ,t):

Generalized incomplete pressure drop integral (h)

\(\vec {x}\) :

Cartesian spatial coordinate vector

x f :

Fracture half length (ft)

x s :

Fracture spacing (ft)

X (t):

Generalized average pressure drop integral (ft3/h)

X (τ,t):

Generalized incomplete average pressure drop integral (ft3/h)

\(\hat {{y}}\) :

Data estimate from linear regression analysis

α :

Hydraulic diffusivity (ft2 /h)

α k :

Coefficient of the k th basis function

Δp :

Pressure drop (in time) (psi)

Δpm,n :

Pressure drop induced at fracture m by fracture n (psi)

Δp′ :

Well-test derivative (psi)

λ :

Weight of the roughness penalty term

μ :

Fluid viscosity (cp)

\(\bar {{\mu } }_{\mathrm {g}} \) :

Average gas viscosity (cp)

ξ :

Boltzmann variable (dimensionless)

ξ DOI :

Depth of investigation in terms of the Boltzmann variable (dimensionless)

τ :

Diffusive time of flight (h1/2)

τ DOI :

Depth of investigation in terms of the diffusive time of flight (h1/2)

τ fs :

Fracture spacing in terms of the diffusive time of flight (h1/2)

τ LOD :

Limit of detectability in terms of the diffusive time of flight (h1/2)

τ det :

Diffusive time of flight detected during inversion (h1/2)

τ max :

Upper limit of diffusive time of flight during inversion (h1/2)

τ w :

Diffusive time of flight at wellbore radius (h1/2)

τ wf :

Diffusive time of flight at effective radius (h1/2)

τ res :

Diffusive time of flight at reservoir boundary (h1/2)

ϕ :

Porosity (fraction)

ϕk (τ):

The k th basis function

ω :

Frequency of the asymptotic expansion (s− 1)

1-D:

one-dimensional

3-D:

three-dimensional

BDF:

boundary-dominated flow

BLF:

bounded linear flow

BU:

buildup

DOI:

depth of investigation

DTOF:

diffusive time of flight

EUR:

estimated ultimate recovery

FMM:

fast marching method

HF:

hydraulic fracture

LOD:

limit of detectability

MFSS:

modified Friedman’s super smoother

MLRA:

moving linear regression analysis

MTFW:

multiple transverse fracture well

ODE:

ordinary differential equation

PPSS:

pseudo pseudo-steady state

PSS:

pseudo-steady state

PTA:

pressure transient analysis

PV:

pore volume

RNP:

rate-normalized pressure drop

RTA:

rate transient analysis

SRV:

stimulated reservoir volume

UR:

ultimate recovery

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Acknowledgments

The authors would like to acknowledge the support of Energi Simulation (formerly Foundation CMG) through the Texas A&M chair in Robust Reduced Complexity Modeling with Dr. Eduardo Gildin and the support of the members of the MCERI joint industry project at Texas A&M University.

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Authors and Affiliations

  1. Texas A&M University, Richardson Building, Room 703, College Station, TX, 77840, USA

    Zhenzhen Wang, Andrew Malone & Michael J. King

  2. Chevron ETC, 1500 Louisiana ST, Houston, TX, 77002, USA

    Zhenzhen Wang

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  1. Zhenzhen Wang
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Correspondence to Zhenzhen Wang.

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Wang, Z., Malone, A. & King, M.J. Quantitative production analysis and EUR prediction from unconventional reservoirs using a data-driven drainage volume formulation. Comput Geosci 24, 853–870 (2020). https://doi.org/10.1007/s10596-019-09833-8

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