Semi-analytical Modelling of Water Injector Test with Fractured Channel in Tight Oil Reservoir

  • Yang Wang
  • Shiqing ChengEmail author
  • Kaidi Zhang
  • Jianchun Xu
  • Xiaoping An
  • Youwei He
  • Haiyang Yu
Original Paper


It is well known that long-term water injection may induce fractured channel(s), and that the fracture geometry would change with the decrease of bottom-hole pressure (BHP) during shut-in. This results in difficulties in modelling BHP behavior. This paper presents a pressure-transient procedure to analyze the BHP performance of water injectors by taking the dynamic behavior of the water injection fractured channel into consideration. Perturbation theory is adopted to solve the non-linear equations caused by decreasing fracture conductivity, while the finite-difference method is used to include the shrinking conductive fracture length during the shut-in period. Then, Duhamel’s principle is deployed to characterize the bi-storage effect; that is, fracture-storage caused by fracture closure and wellbore storage because of wellbore after-flow. Since the mobility outside of channel walls are much poorer than that in the channel, the walls are modeled as two parallel sealed boundaries. Therefore, the method of images is lastly applied to obtain the BHP response. The bi-storage phenomenon is characterized by two unit slopes in the pressure-derivative curve, and the variable fracture-storage is identified as a new flow regime in water injectors that is caused by fracture shrinkage. The interpreted storage coefficient will be much larger than the true value if the fracture-storage flow is mistakenly regarded as wellbore-storage flow. Because of the fractured channel walls and decreasing fracture conductivity, the pressure-derivative curve would increase in late time. Finally, two cases from the Changqing Oilfield are discussed to demonstrate the capabilities of the proposed approach.


Water injection fractured channel Shrinking fracture length Decreasing fracture conductivity Bi-storage effect Method of images Variable fracture storage 

List of Symbols


Water volume factor


Fracture-storage coefficient (m3/Pa)


Dimensionless fracture conductivity


Total compressibility (/Pa)


Wellbore-storage coefficient (m3/Pa)


Fracture shrinkage coefficient (dimensionless)


Plane-strain modulus (MPa)


Fracture height (m)


Permeability (m2)


Matrix permeability (m2)


Pressure (Pa)


Initial pressure (Pa)


Dimensionless bottom-hole pressure

\( p_{\text{wD}}^{0} \)

Dimensionless bottom-hole pressure when fracture begins to shrink


Water injection rate (m3/s)


Sand face rate (m3/s)


Fracture radius (m)


Laplace-transformation variable (dimensionless)


Fracture-face skin factor)


Choked fracture skin factor


Time (s)


Volume of fracture (m3)


Average fracture width (m)


Fracture channel width (m)

x, y

Cartesian coordinate


Waterflood-induced fracture half-length (m)


Fraction characterizes the well distance from the nearest fracture wall


Stress-sensitivity coefficient (1/Pa)


Water viscosity (mPa s)


Porosity (fraction)


Diffusivity (m2/s)


Dimensionless time variable


Dimensionless reservoir solution in different situations







Inter region


Out region




Laplace transform



We would like to thank the financial supports from National Science and Technology Major Projects (2017ZX05013002) and National Natural Science Foundation of China (11872073). This work was also supported by the China Scholarship Council for 1 year study at Pennsylvania State University (201706440082).


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of Petroleum (Beijing)BeijingChina
  2. 2.Pennsylvania State UniversityPennsylvaniaUSA
  3. 3.Lusheng Petroleum Development Company LimitedSINOPEC Shengli Oilfield CompanyDongyingChina
  4. 4.China University of Petroleum, East ChinaQingdaoChina
  5. 5.PetroChina Changqing Oilfield CompanyXi’anChina

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