International Journal of ISSN: 2475-5559 IPCSE

Petrochemical Science & Engineering
Review Article
Volume 1 Issue 2 - 2016
Transient Pressure Behavior for Low Permeability Oil Reservoir Based on New Darcy’s Equation
Liu Hailong* and Wang Guan
Research Institute of petroleum exploration and development, China
Received: September 17, 2016 | Published: November 08, 2016
*Corresponding author: Liu Hailong, Research Institute of petroleum exploration and development, China, Email:
Citation: Hailong L, Guan W (2016) Transient Pressure Behavior for Low Permeability Oil Reservoir Based on New Darcy’s Equation. Int J Petrochem Sci Eng 1(2): 00010. DOI: 10.15406/ipcse.2016.01.00010

Abstract

Fluid flow in low permeability porous media does not obey Darcy's law anymore, so traditional well testing interpretation methods may cause the error in estimation of low permeability oil reservoir parameters. This paper describes problems encountered when analyzing transient pressure tests in ultra-low permeability oil reservoir. Based on the new model, the theoretical well test model for dual-porosity media in low permeability oil reservoirs with complex fractures was derived, allowing the influencial analysis of non-Darcy's effect. The model was created by analytical solution and numerical approach. Besides, the curves for the pressure responses were plotted. The effects of parameters including fracture width, fracture length and the position of the fracture were fully investigated. The advantage of the solution is that it is easy to incorporate storability ratio and skin factor. Also, it can simplify the computation process and make enhance the efficiency. Sensitivity analysis shows that non-Darcy flow mainly affects the transition section between the pure wellbore storage and radial flow region, and the pressure derivative curves exhibit a gentle transition region compared with the Darcy's flow. These curves provide a better way to characterize the well testing pressure response for ultra-low permeability oil reservoir.

Introduction

At present, the oil exploration and development in China faces two severe problems--high water cut and low permeability. Plenty of studies have been done on pressure propagation in low permeability reservoirs, leading to some significant progress. Pascal [1] applied the numerical integration methods, which is a finite difference method, to solve the problem of consolidation of geotechnical engineering for the first time by considering the pressure gradient. Yan Qinglai [2] summed up the laboratory experimental results of single phase flow characteristics in low permeability reservoirs and flow curves' characteristics of distilled water, low saline water and simulated crude oil, which were made to flow through artificial and natural cores. Huang Yan zhang [3] derived mathematical equation of percolation for low permeability reservoir and studied the percolation feature and regularity of oil and water. Yao Yuedong [4] analyzed the experimental data of fluid flow in porous core with low permeability and plotted typical logarithmic curves. At the same time, three typical transient flow models for low-permeability reservoirs were built and the approximate analytical solutions in Laplace space were obtained. Zhu Shengju [5] used the steady state replacement method to study the propagation of pressure wave of unsteady flow of elastic fluid in low permeability reservoirs. Based on the percolation features of low permeability reservoirs, Yang Qingli [6] put forward the continuous model of nonlinear percolation and derived the equation of pressure distribution and moving boundary spreading for unsteady state flow. Based on the theory of pressure wave ellipse constant pressure surface spreading, Qiao Wei [7] deduced the relationship between pressure wave propagation time and elliptic mobile boundary for plane ellipse radial flow and spheroid centripetal flow of fracturing well in low permeability reservoir, and compared it to the laws of pressure wave spreading of plane parallel flow, plane radial flow and spherical centripetal flow.

The Model of Pressure Propagation

Model description

Figure 1 shows the model of a finite large homogeneous low permeability reservoir, with the thickness of h. The average permeability is k under the reservoir pressure. Due to the presence of starting pressure gradient in low permeability reservoir, the pressure continues to spread deeply to the geological formations with time going on. According to the low permeability theory, the following assumptions were made for the fluid flow in the circumstances:

Figure 1: A physical model of the plane radial diagram.

The formation of the reservoir is isopachous, bounded and heterogeneous.

  1. Before the production wells produce, the reservoir pressure is equal to the original formation pressure.
  2. The fluid of the reservoir is slightly compressible, and flows with unsteady single-phase flow instability.
  3. The effect of the permeability of the pressure-sensitive is not considered, and the fluid of porous media porosity and viscosity are regarded as constant.
  4. The reservoir fluid flow within a limited range to the wellbore. Radial flow of fluid occurs

Non-fracture model derivation

Being different from high permeability reservoirs, low permeability reservoir is not in line with the classical Darcy's law, and the most obvious feature of low permeability reservoir is the presence of starting pressure gradient, and the seepage law can be written and the following equation[8].

v= k μ (1 c 1 dp dr c 2 dp dr c 3 ) dp dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamODai abg2da9maalaaabaGaam4AaaqaaiabeY7aTbaacaGGOaGaaGymaiab gkHiTmaalaaabaGaam4yamaaBaaabaqcLbmacaaIXaaajuaGbeaaae aadaWcaaqaaiaadsgacaWGWbaabaGaamizaiaadkhaaaaaaiabgkHi TmaalaaabaGaam4yamaaBaaabaqcLbmacaaIYaaajuaGbeaaaeaada WcaaqaaiaadsgacaWGWbaabaGaamizaiaadkhaaaGaeyOeI0Iaam4y amaaBaaabaqcLbmacaaIZaaajuaGbeaaaaGaaiykamaalaaabaGaam izaiaadchaaeaacaWGKbGaamOCaaaaaaa@5624@  (1)

Where k is the reservoir effective permeability and µ is the oil viscosity at reservoir conditions.
If c 1 = c 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogalm aaBaaameaacaaIXaaabeaaliabg2da9KqbakaadogadaWgaaqaaKqz adGaaGOmaaqcfayabaGaeyypa0JaaGimaaaa@3F34@ , the equation (1) will be Darcy's model, if c 1 0, c 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogalm aaBaaameaacaaIXaaabeaajuaGcqGHGjsUcaaIWaGaaiilaiaadoga daWgaaqaaKqzadGaaGOmaaqcfayabaGaeyypa0JaaGimaaaa@4154@ , the equation (1) will be quasi start-up pressure gradient model (quasi-linear flow), if c 1 0, c 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogalm aaBaaameaacaaIXaaabeaaliabgcMi5kaaicdacaGGSaqcfaOaam4y amaaBaaabaqcLbmacaaIYaaajuaGbeaacqGHGjsUcaaIWaaaaa@4220@ , the equation (1) will be three-parameter continuous model, and if c 1 0, c 2 0, c 3 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadogalm aaBaaameaacaaIXaaabeaaliabgcMi5kaaicdacaGGSaqcfaOaam4y amaaBaaabaqcLbmacaaIYaaajuaGbeaacqGHGjsUcaaIWaGaaiilai aadogalmaaBaaameaacaaIZaaabeaajuaGcqGHGjsUcaaIWaaaaa@47BC@ , different curves can be achieved [8].
Flow velocity at any position can be written as:

v= Bq A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamODai abg2da9maalaaabaGaamOqaKqzadGaamyCaaqcfayaaiaadgeaaaaa aa@3CD5@  (2)

A=wh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai abg2da9iaadEhacaWGObaaaa@3A3A@  (3)

Where r is the drainage radius and h is the reservoir effective thickness. B is the formation volume factor of crude oil under original pressure. Q is the ground crude oil production, and A is the seepage area.

Substitute equation (2) and (3) into equation (1), by ignoring high order infinitely small quantity, (1) can be rewritten as

{ r ( dp dr ) 2 [r( c 1 + c 2 + c 3 )+ Bqμ 2πkh ] dp dr + c 3 Bqμ 2πkh =0 p| r= r w = p wf , p| r= r e = p e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaGabaabaeqabaGaam OCamaabmaabaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWGYbaa aaGaayjkaiaawMcaamaaCaaabeqaaKqzadGaaGOmaaaajuaGcqGHsi slcaGGBbGaaiOCaiaacIcacaWGJbWaaSbaaeaajugWaiaaigdaaKqb agqaaiabgUcaRiaadogalmaaBaaajuaGbaqcLbmacaaIYaaajuaGbe aacqGHRaWkcaWGJbWcdaWgaaqcfayaaKqzadGaaG4maaqcfayabaGa aiykaiabgUcaRmaalaaabaGaamOqaiaadghacqaH8oqBaeaacaaIYa GaeqiWdaNaam4AaiaadIgaaaGaaiyxamaalaaabaGaamizaiaadcha aeaacaWGKbGaamOCaaaacqGHRaWkdaWcaaqaaiaadogadaWgaaqaaK qzadGaaG4maaqcfayabaGaamOqaiaadghacqaH8oqBaeaacaaIYaGa eqiWdaNaam4AaiaadIgaaaGaeyypa0JaaGimaaqaamaaeiaabaGaam iCaaGaayjcSdWaaSbaaeaajugWaiaadkhacqGH9aqpcaWGYbWcdaWg aaqcfayaaKqzadGaam4DaaqcfayabaaabeaacqGH9aqpcaWGWbWaaS baaeaajugWaiaadEhacaWGMbaajuaGbeaacaGGSaGaaGjcVlaayIW7 caaMi8UaaGjcVlaayIW7caaMi8+aaqGaaeaacaWGWbaacaGLiWoada WgaaqaaKqzadGaamOCaiabg2da9iaadkhalmaaBaaajuaGbaqcLbma caqGLbaajuaGbeaaaeqaaiabg2da9iaadchadaWgaaqaaKqzadGaam yzaaqcfayabaGaaGjcVlaayIW7aaGaay5Eaaaaaa@96EA@  (4)

By solving equation (4), we have

dp dx = c 1 + c 2 + c 3 2 + Bqμ 4πkhr + [r( c 1 + c 2 + c 3 )+ Bqμ 2πkh ] 2 2r c 3 Bqμ πkh 2r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaqaaiaadsgaca WGWbaabaGaamizaiaadIhaaaGaeyypa0ZaaSaaaeaacaWGJbWaaSba aeaajugWaiaaigdaaKqbagqaaiabgUcaRiaadogadaWgaaqaaKqzad GaaGOmaaqcfayabaGaey4kaSIaam4yamaaBaaabaqcLbmacaaIZaaa juaGbeaaaeaacaaIYaaaaiabgUcaRmaalaaabaGaamOqaiaadghacq aH8oqBaeaacaaI0aGaeqiWdaNaam4AaiaadIgacaWGYbaaaiabgUca RmaalaaabaWaaOaaaeaacaGGBbGaaiOCaiaacIcacaWGJbWaaSbaae aacaaIXaaabeaacqGHRaWkcaWGJbWaaSbaaeaacaaIYaaabeaacqGH RaWkcaWGJbWaaSbaaeaacaaIZaaabeaacaGGPaGaey4kaSYaaSaaae aacaWGcbGaamyCaiabeY7aTbqaaiaaikdacqaHapaCcaWGRbGaamiA aaaacaGGDbWaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgkHiTmaala aabaGaaGOmaiaadkhacaWGJbWaaSbaaeaacaaIZaaabeaacaWGcbGa amyCaiabeY7aTbqaaiabec8aWjaadUgacaWGObaaaaqabaaabaGaaG Omaiaadkhaaaaaaa@7411@  (5)

Steady successive substitution method noted that in unsteady flow, pressure distribution can be approximated as a steady flow process at a certain time, and the pressure spread boundary radius is the function of time t, which is represented by r(t). By integrating equation (5), bottom whole pressure and pressure distribution at any time can be obtained.

p(r)= p wf + 1 C 1 [N C 1 f(r) N 2 C 1 ln N 2 +rN C 2 +Nf(r) 0.5r + N 2 C 2 ln N( c 1 + c 2 c 3 )+r C 1 2 + C 1 f(r) C 1 ] | r w r(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCai aacIcacaWGYbGaaiykaiabg2da9iaacchalmaaBaaajuaGbaqcLbma caWG3bGaamOzaaqcfayabaGaey4kaSYaaqGaaqaabeqaamaalaaaba GaaGymaaqaaiaadoealmaaBaaajuaGbaqcLbmacaaIXaaajuaGbeaa aaGaai4waiaad6eacaWGdbWcdaWgaaqcfayaaKqzadGaaGymaaqcfa yabaGaamOzaiaacIcacaWGYbGaaiykaiabgkHiTiaad6ealmaaCaaa juaGbeqaaKqzadGaaGOmaaaajuaGcaWGdbWcdaWgaaqcfayaaKqzad GaaGymaaqcfayabaGaaiiBaiaac6gadaWcaaqaaiaad6ealmaaCaaa juaGbeqaaKqzadGaaGOmaaaajuaGcqGHRaWkcaWGYbGaamOtaiaado ealmaaBaaajuaGbaqcLbmacaaIYaaajuaGbeaacqGHRaWkcaWGobGa amOzaiaacIcacaWGYbGaaiykaaqaaiaaicdacaGGUaGaaGynaKqzad GaamOCaaaaaKqbagaacqGHRaWkcaWGobWcdaahaaqcfayabeaajugW aiaaikdaaaqcfaOaam4qamaaBaaabaqcLbmacaaIYaaajuaGbeaaca GGSbGaaiOBamaalaaabaGaamOtaiaacIcacaWGJbWaaSbaaeaacaaI XaaabeaacqGHRaWkcaWGJbWaaSbaaeaacaaIYaaabeaacqGHsislca WGJbWaaSbaaeaacaaIZaaabeaacaGGPaGaey4kaSIaamOCaiaadoea lmaaBaaajuaGbaqcLbmacaaIXaaajuaGbeaalmaaCaaajuaGbeqaaK qzadGaaGOmaaaajuaGcqGHRaWkcaWGdbWaaSbaaeaacaaIXaaabeaa caWGMbGaaiikaiaadkhacaGGPaaabaGaam4qaSWaaSbaaKqbagaaju gWaiaaigdaaKqbagqaaaaacaGGDbaaaiaawIa7amaaDaaabaqcLbma caWGYbWcdaWgaaqcfayaaKqzadGaam4DaaqcfayabaaabaqcLbmaca WGYbGaaiikaiaadshacaGGPaaaaaaa@A0F6@  (6)

Where

f(r)= [r( c 1 + c 2 + c 3 )+N] 2 4 c 3 N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzai aacIcacaWGYbGaaiykaiabg2da9maakaaabaGaai4waiaadkhacaGG OaGaam4yamaaBaaabaqcLbmacaaIXaaajuaGbeaacqGHRaWkcaWGJb WcdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaGaey4kaSIaam4yaSWa aSbaaKqbagaajugWaiaaiodaaKqbagqaaiaacMcacqGHRaWkcaWGob GaaiyxaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaKqbakabgkHiTiaa isdacaWGJbWaaSbaaeaajugWaiaaiodaaKqbagqaaiaad6eaaeqaaa aa@5746@  (7)

C 1 = c 1 + c 2 + c 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qaS WaaSbaaKqbagaajugWaiaaigdaaKqbagqaaiabg2da9iaadogalmaa BaaajuaGbaqcLbmacaaIXaaajuaGbeaacqGHRaWkcaWGJbWaaSbaae aajugWaiaaikdaaKqbagqaaiabgUcaRiaadogadaWgaaqaaKqzadGa aG4maaqcfayabaaaaa@4863@ , C 2 = c 1 + c 2 c 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qaS WaaSbaaKqbagaajugWaiaaikdaaKqbagqaaiabg2da9iaadogalmaa BaaajuaGbaqcLbmacaaIXaaajuaGbeaacqGHRaWkcaWGJbWcdaWgaa qcfayaaKqzadGaaGOmaaqcfayabaGaeyOeI0Iaam4yaSWaaSbaaeaa jugWaiaaiodaaSqabaaaaa@4890@  (8)

N= Bqμ 2πkh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtai abg2da9maalaaabaGaamOqaiaadghacqaH8oqBaeaacaaIYaGaeqiW daNaam4AaiaadIgaaaaaaa@4036@  (9)

The situation of p(r) is complicated and it is difficult for the follow-up studies on pressure propagation, therefore, by using the method of curve fitting and approximation, the equation (5) is rewritten as the following.

dp dx = C 1 2 + N 2r + ε 1 r ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaaqaaiaadsgaca WGWbaabaGaamizaiaadIhaaaGaeyypa0ZaaSaaaeaacaWGdbWaaSba aeaajugWaiaaigdaaKqbagqaaaqaaiaaikdaaaGaey4kaSYaaSaaae aacaWGobaabaGaaGOmaKqzadGaamOCaaaajuaGcqGHRaWkcqaH1oqz lmaaBaaajuaGbaqcLbmacaaIXaaajuaGbeaacaWGYbWaaWbaaeqaba GaeqyTdu2cdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaaaaaaa@4DA7@  (10)

By integrating equation (10), we have

p(r)= p wf + C 1 2 (r r w )+ N 2 (lnrln r w )+ ε 1 ε 2 +1 ( r ε 2 +1 r w ε 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGWbGaaiikaiaadk hacaGGPaGaeyypa0JaamiCamaaBaaabaqcLbmacaWG3bGaamOzaaqc fayabaGaey4kaSYaaSaaaeaacaWGdbWaaSbaaeaajugWaiaaigdaaK qbagqaaaqaaiaaikdaaaGaaiikaiaadkhacqGHsislcaWGYbWaaSba aeaajugWaiaadEhaaKqbagqaaiaacMcacqGHRaWkdaWcaaqaaiaad6 eaaeaacaaIYaaaaiaacIcaciGGSbGaaiOBaiaadkhacqGHsislcaGG SbGaaiOBaiaadkhalmaaBaaajuaGbaqcLbmacaWG3baajuaGbeaaca GGPaGaey4kaSYaaSaaaeaacqaH1oqzdaWgaaqaaKqzadGaaGymaaqc fayabaaabaGaeqyTdu2aaSbaaeaajugWaiaaikdaaKqbagqaaKqzad Gaey4kaSIaaGymaaaajuaGcaGGOaGaamOCamaaCaaabeqaaiabew7a LnaaBaaabaqcLbmacaaIYaaajuaGbeaajugWaiabgUcaRiaaigdaaa qcfaOaeyOeI0IaamOCamaaDaaabaqcLbmacaWG3baajuaGbaWaaWba aeqabaGaeqyTdu2aaSbaaeaajugWaiaaikdaaKqbagqaaKqzadGaey 4kaSIaaGymaaaaaaqcfaOaaiykaaaa@7A1B@  (11)

p(r)= p e [ C 1 2 ( r e r)+ N 2 (ln r e lnr)+ ε 1 ε 2 +1 ( r e ε 2 +1 r ε 2 +1 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfgBPj MCPbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaac H8rrps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf ea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaa baqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGWbGaaiikaiaadk hacaGGPaGaeyypa0JaamiCaSWaaSbaaKqbagaajugWaiaadwgaaKqb agqaaiabgkHiTiaacUfadaWcaaqaaiaadoealmaaBaaajuaGbaqcLb macaaIXaaajuaGbeaaaeaacaaIYaaaaiaacIcacaWGYbWaaSbaaeaa jugWaiaadwgaaKqbagqaaiabgkHiTiaadkhacaGGPaGaey4kaSYaaS aaaeaacaWGobaabaGaaGOmaaaacaGGOaGaciiBaiaac6gacaWGYbWc daWgaaqcfayaaKqzadGaamyzaaqcfayabaGaeyOeI0IaaiiBaiaac6 gacaWGYbGaaiykaiabgUcaRmaalaaabaGaeqyTdu2cdaWgaaqcfaya aKqzadGaaGymaaqcfayabaaabaGaeqyTdu2cdaWgaaqcfayaaKqzad GaaGOmaaqcfayabaGaey4kaSIaaGymaaaacaGGOaGaamOCamaaDaaa baqcLbmacaWGLbaajuaGbaWaaWbaaeqabaGaeqyTdu2cdaWgaaqcfa yaaKqzadGaaGOmaaqcfayabaqcLbmacqGHRaWkcaaIXaaaaaaajuaG cqGHsislcaWGYbWcdaahaaqcfayabeaajugWaiabew7aLTWaaSbaaK qbagaajugWaiaaikdaaKqbagqaaKqzadGaey4kaSIaaGymaaaajuaG caGGPaGaaiyxaaaa@7E54@  (12)

When the well producing oil ata steady yield of q, after producing a given time t, the total crude oil production can be obtained.

N p =B ρ 0 qt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaabaqcLbmacaWGWbaajuaGbeaacqGH9aqpcaWGcbGaeqyWdi3a aSbaaeaajugWaiaaicdaaKqbagqaaiaadghacaWG0baaaa@423C@  (13)

Where ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi 3aaSbaaeaajugWaiaaicdaaKqbagqaaaaa@3ADC@ is the average oil density under the formation pressure, and N p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaabaqcLbmacaWGWbaajuaGbeaaaaa@3A2A@  is the total crude oil production.

According to the principle of conservation of mass, after producing a given time t, the output of crude oil without considering wellbore storage efficiency can be written as the following.

N p = r w r(t) 2πrh[ (ρϕ) e (ρϕ) r ] dr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOtam aaBaaabaqcLbmacaWGWbaajuaGbeaacqGH9aqpdaWdXaqaaiaaikda cqaHapaCcaWGYbGaamiAaiaacUfacaGGOaGaeqyWdiNaeqy1dyMaai ykaSWaaSbaaKqbagaajugWaiaadwgaaKqbagqaaiabgkHiTiaacIca cqaHbpGCcqaHvpGzcaGGPaWcdaWgaaqcfayaaKqzadGaamOCaaqcfa yabaGaaiyxaaqaaKqzadGaamOCaSWaaSbaaKqbagaajugWaiaadEha aKqbagqaaaqaaKqzadGaaiOCaiaacIcacaWG0bGaaiykaaqcfaOaey 4kIipajugWaiaadsgacaWGYbaaaa@6283@  (14)

Where (ρϕ) r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeg8aYjabew9aMjaacMcalmaaBaaajuaGbaqcLbmacaWGYbaajuaG beaaaaa@3ED3@  is the mass of oil in the unit volume of rock, which is away from the well center r under the conditions of original formation pressure. (pϕ) e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcaca WGWbGaeqy1dyMaaiykamaaBaaabaGaaiyzaaqabaaaaa@3B99@ is the mass of oil in the unit volume of rock, which locates re away from the well center as well as under the condition of boundary pressure.

From the seepage mechanics, the state equation of rock [9] can be approximated as the following.

ϕ= ϕ 0 [1+ c f (p p 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMj abg2da9iabew9aMTWaaSbaaWqaaiaaicdaaeqaaKqbakaacUfacaaI XaGaey4kaSIaam4yamaaBaaabaWaaSbaaeaajugWaiaadAgaaKqbag qaaaqabaGaaiikaiaadchacqGHsislcaWGWbWaaSbaaeaadaWgaaqa aKqzadGaaGimaaqcfayabaaabeaacaGGPaGaaiyxaaaa@4AA5@  (15)

ρ= ρ 0 [+ c 1 (p p 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYj abg2da9iabeg8aYTWaaSbaaWqaaiaaicdaaeqaaKqbakaacUfacqGH RaWkcaGGJbWaaSbaaeaajugWaiaaigdaaKqbagqaaiaacIcacaGGWb GaeyOeI0IaaiiCamaaBaaabaqcLbmacaaIWaaajuaGbeaacaGGPaGa aiyxaaaa@4965@  (16)

Multiply equation (13) and (14) multiplying and ignore high order infinitely small quantity, we have

ρϕ= ρ 0 ϕ 0 [1+ c t (p p 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYj abew9aMjabg2da9iabeg8aYnaaBaaabaqcLbmacaaIWaaajuaGbeaa cqaHvpGzdaWgaaqaaKqzadGaaGimaaqcfayabaGaai4waiaaigdacq GHRaWkcaWGJbWaaSbaaeaajugWaiaadshaaKqbagqaaiaacIcacaWG WbGaeyOeI0IaamiCaSWaaSbaaWqaaiaaicdaaeqaaKqbakaacMcaca GGDbaaaa@5088@  (17)

The mass of oil in the unit volume of rock can be obtained from the equation (17).

(ρϕ) e = ρ 0 ϕ 0 [1+ c t ( p e p 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacIcacq aHbpGCcqaHvpGzcaGGPaWcdaWgaaadbaGaaiyzaaqabaqcfaOaeyyp a0JaeqyWdi3cdaWgaaadbaWaaSbaaeaacaaIWaaabeaaaeqaaKqbak abew9aMnaaBaaabaWaaSbaaeaajugWaiaaicdaaKqbagqaaaqabaGa ai4waiaaigdacqGHRaWkcaWGJbWaaSbaaeaalmaaBaaajuaGbaqcLb macaWG0baajuaGbeaaaeqaaiaacIcacaWGWbWaaSbaaeaajugWaiaa dwgaaKqbagqaaiabgkHiTiaadchadaWgaaqaamaaBaaabaqcLbmaca aIWaaajuaGbeaaaeqaaiaacMcacaGGDbaaaa@5774@  (18)

(ρϕ) r = ρ 0 ϕ 0 [1+ c t (p p 0 )] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeg8aYjabew9aMjaacMcadaWgaaqaaSWaaSbaaKqbagaajugWaiaa dkhaaKqbagqaaaqabaGaeyypa0JaeqyWdi3aaSbaaeaalmaaBaaame aacaaIWaaabeaaaKqbagqaaiabew9aMnaaBaaabaWaaSbaaeaajugW aiaaicdaaKqbagqaaaqabaGaai4waiaaigdacqGHRaWkcaWGJbWcda WgaaadbaWaaSbaaeaacaWG0baabeaaaeqaaKqbakaacIcacaWGWbGa eyOeI0IaamiCamaaBaaabaWaaSbaaeaajugWaiaaicdaaKqbagqaaa qabaGaaiykaiaac2faaaa@54E8@  (19)

By combining equations (12)-(14) and (18)-(19), we have

t= 2πh c t Bq r(t) r e [ C 1 2 ( r e r)+ N 2 (ln r e lnr)+ ε 1 ε 2 +1 ( r e ε 2 +1 r ε 2 +1 )]rdr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9maalaaabaGaaGOmaiabec8aWjaadIgacaWGJbWaaSbaaeaa lmaaBaaameaacaWG0baabeaaaKqbagqaaaqaaiaadkeacaWGXbaaam aapedabaGaai4wamaalaaabaGaam4qamaaBaaabaqcLbmacaaIXaaa juaGbeaaaeaacaaIYaaaaiaacIcacaWGYbWcdaWgaaqcfayaaKqzad GaamyzaaqcfayabaGaeyOeI0IaamOCaiaacMcacqGHRaWkdaWcaaqa aiaad6eaaeaacaaIYaaaaiaacIcaciGGSbGaaiOBaiaadkhadaWgaa qaaKqzadGaamyzaaqcfayabaGaeyOeI0IaaiiBaiaac6gacaWGYbGa aiykaiabgUcaRmaalaaabaGaeqyTdu2cdaWgaaqcfayaaKqzadGaaG ymaaqcfayabaaabaGaeqyTdu2cdaWgaaqcfayaaKqzadGaaGOmaaqc fayabaGaey4kaSIaaGymaaaacaGGOaGaamOCamaaDaaabaqcLbmaca WGLbaajuaGbaWaaWbaaeqabaGaeqyTdu2cdaWgaaqcfayaaKqzadGa aGOmaaqcfayabaqcLbmacqGHRaWkcaaIXaaaaaaajuaGcqGHsislca WGYbWaaWbaaeqabaGaeqyTdu2cdaWgaaqcfayaaKqzadGaaGOmaaqc fayabaqcLbmacqGHRaWkcaaIXaaaaKqbakaacMcacaGGDbqcLbmaca WGYbGaamizaiaadkhaaKqbagaajugWaiaadkhacaGGOaGaamiDaiaa cMcaaKqbagaajugWaiaadkhalmaaBaaajuaGbaqcLbmacaWGLbaaju aGbeaaaiabgUIiYdaaaa@9052@ (20)

By integrating equation (20), we have

t= 2πh c t Bq { C 1 6 r 3 ε 1 ( ε 2 +1)( ε 2 +2) r ε 2 +2 +[ C 1 r e 4 + ε 1 2( ε 2 +1) r e ε 2 +1 r 2 ] }| r(t) r e + N r e 2 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9maalaaabaGaaGOmaiabec8aWjaadIgacaWGJbWcdaWgaaqc fayaaKqzadGaamiDaaqcfayabaaabaGaamOqaiaadghaaaWaaqGaae aadaGadaqaaiabgkHiTmaalaaabaGaam4qaSWaaSbaaKqbagaajugW aiaaigdaaKqbagqaaaqaaiaaiAdaaaGaamOCamaaCaaabeqaaiaaio daaaGaeyOeI0YaaSaaaeaacqaH1oqzlmaaBaaajuaGbaqcLbmacaaI XaaajuaGbeaaaeaacaGGOaGaeqyTdu2aaSbaaeaajugWaiaaikdaaK qbagqaaKqzadGaey4kaSIaaGymaKqbakaacMcacaGGOaGaeqyTdu2c daWgaaqcfayaaKqzadGaaGOmaaqcfayabaqcLbmacqGHRaWkcaaIYa qcfaOaaiykaaaacaWGYbWaaWbaaeqabaGaeqyTdu2cdaWgaaqcfaya aKqzadGaaGOmaaqcfayabaqcLbmacqGHRaWkcaaIYaaaaKqbakabgU caRiaacUfadaWcaaqaaiaadoealmaaBaaajuaGbaqcLbmacaaIXaaa juaGbeaacaWGYbWaaSbaaeaajugWaiaadwgaaKqbagqaaaqaaiaais daaaGaey4kaSYaaSaaaeaacqaH1oqzdaWgaaqaaiaaigdaaeqaaaqa aiaaikdacaGGOaGaeqyTdu2cdaWgaaqcfayaaKqzadGaaGOmaaqcfa yabaqcLbmacqGHRaWkcaaIXaqcfaOaaiykaaaacaWGYbWaa0baaeaa jugWaiaadwgaaKqbagaadaahaaqabeaacqaH1oqzlmaaBaaajuaGba qcLbmacaaIYaaajuaGbeaajugWaiabgUcaRiaaigdaaaaaaKqbakaa dkhalmaaCaaajuaGbeqaaKqzadGaaGOmaaaajuaGcaGGDbaacaGL7b GaayzFaaaacaGLiWoadaqhaaqaaKqzadGaamOCaiaacIcacaWG0bGa aiykaaqcfayaaKqzadGaamOCaSWaaSbaaKqbagaajugWaiaadwgaaK qbagqaaaaacqGHRaWkdaWcaaqaaiaad6eajugWaiaadkhalmaaDaaa juaGbaqcLbmacaWGLbaajuaGbaqcLbmacaaIYaaaaaqcfayaaiaaiI daaaaaaa@AD51@ (21)

The relationship between the pressure propagation and production time is exponential for low permeability reservoirs, which is different from high permeability reservoirs. Because in high permeability reservoir, the relationship between the pressure propagation and production time is linear. In the low permeability reservoir, the pore throat is small, and under the condition of the pressure-sensitive effects, the pore-throat diameter is reduced to 70%of the original value [10]. Starting pressure gradient exists in reservoirs with low permeability, especially when the reservoir capacity is insufficient (Pressure coefficient is too small, which is less than 1). The pressure of the pore fluid is unable to spread to the wellbore, and at this moment, the effect of starting pressure gradient becomes more apparent. The slowing down of the pressure propagation in low permeability reservoir is the principle change under the effect of starting pressure gradient and pressure sensitivity [11].

For the actual development of oil fields, in order to increase the pressure propagation velocity, we have to rely on foreign energy to supply the formation energy, such as water or gas injection. These foreign energies can help the reservoir to improve pore fluid pressure in porous media, and reduce or eliminate the effect of starting pressure gradient. In a word, all these methods are used to make the low permeability reservoir development like high permeability reservoir. Fracturing is often used to modify the low permeability reservoir formation and establish a "mobile network" in the reservoir, which can link more seepage routes in the reservoir and increase the flow area. What’s more, by coupling with external water or gas injection, the formation energy has been supplied in advance [12]. All in all, via increasing the reservoir pressure and reducing the reservoir fluid flow resistance, the low permeability reservoir pressure gradient is reduced or "disappeared."

Model validation

Nonlinear parameters measurement: In order to measure the nonlinear parameters in equation (21), laboratory experiments need to be carried out to obtain the values of the nonlinear parameters, which are c1, c2 and c3. The experimental apparatus is showed in (Figure 2). In compliance with the safe operation of the laboratory rules, the following steps were used to measure the nonlinear parameters.

Figure 2: Schematic diagram of the experimental apparatus.

Step1. Take 4 representative core samples from the actual low-permeability rectangle reservoirs, and their basic parameters are shown in Table 1.

Core Number

Length(cm)

Diameter(cm)

Porosity (%)

Permeability(mD)

a

3.724

2.5

8.01

1.789

b

3.649

2.5

7.89

2.853

c

3.801

2.5

8.11

6.127

d

3.597

2.5

7.91

7.751

Table 1: The basic parameters of the 4 cores samples.

Step2. Get the oil mixture by mixing the kerosene and crude oil in different mass ratio, and the viscosity of the oil mixture is measured at 158ËšF. Adjust the mass ratio of the kerosene and crude oil until the viscosity of the oil mixture measured at 158 ÌŠF is 1.4mPa.s

Step3. Adjust the compound ratio of NaCl, CaCl2 and MaCl2 until the viscosity of the salt water measured at 158ËšF is 1.21mPa.s. By changing the displacement pressure gradient of oil flow to drive the core sample, the quantity of the oil flow is measured at 158 ÌŠF. The relationships of pressure gradient and flow were drawn, as shown in Figure 3. The displacement pressure gradient data and the corresponding quantity of the oil flow data are put into the equation (1), and the c1, c2 and c3 can be obtained, which is showed in Table 2.

Core Number

C1(MPa. m-1)

C2(MPa. m-1)

C3(MPa.m-1)

a

1.201

-1.103

-1.409

b

1.032

-0.901

-0.911

c

0.533

-0.403

-0.201

d

0.314

-0.311

-0.089

Table 2: The results of the nonlinear parameters.

Figure 3: Non-Darcy flow curves of core simples for obtaining parameters.

E300simulation: The reservoir E300 module in Eclipse 2011 is developed for fractured heterogeneous reservoirs. E300 is used to simulate the pressure variation around a fractured vertical well in a rectangular heterogeneous reservoir. In order to meet the assumptions of equation (1), the numerical model is set as the following:

The width and the length of the rectangular heterogeneous reservoir are 1km, and there is an oil production well in the center of the reservoir, which is showed as in Figure 4a. A five-point well pattern is used to simulate the production, that is, one production well in the reservoir center and four injectors in the four corners to ensure the production of the production well. By adjusting injection volume, the oil well production under different displacement pressure can be obtained [13].

a) Reservoir well pattern     b) Meshing Schematic

Figure 4: The geometry information representation of the reservoir.

In order to describe the formation fluid heterogeneity, the triangular network of grid system is used to ensure that each crack at least has 3 grids, which is showed in Figure 4b. Therefore, the plane is divided into 20*20 meshes, and the average grid step is 50 meters. As the equation (1) describing a single-phase fluid seepage, only 1 simulation layer is divided in the vertical direction of the reservoir according to the seepage characteristics of single-phase fluid seepage. The simulation of the total number of grid 20 *20*1 = 4000.The required parameters for the numerical simulation are shown in Table 3, and the results of two methods are shown in Table 4. From Table 5, it can be seen that the relative error is under the basic control of 5%, which is in consistent with the allowable error range, suggesting that this method we offered is reliable.

Parameters

Value

Parameters

Value

Saturation pressure

25MPa

Formation temperature

158ËšF

Oil viscosity

1.21mPa.s

Water viscosity

1.6mPa.s

Oil density

0.79g/cm3

Dissolved gas and oil ratio

22.31m3/m3

Water compressibility

4.9×10-4MPa-1

Oil compressibility

8.1×10-4MPa-1

Oil volume coefficient

1.21m3/m3

Rock Compressibility

4.5×10-4MPa-1

Porosity

0.12

Permeability

1.2mD

Injection pressure

46MPa

Original formation pressure

27MPa

Effective thickness

5m

Injecting water intensity

0.044 m3/(d.MPa.m)

Table 3: The main parameters for the numerical simulation.

Production Time (t,d)

Oil Production (q,m3/d)

Pressure of this Work (p,MPa)

Pressure of E300 (p,MPa)

Relative Error (%)

30

17.492

22.984

22.249

3.201

60

17.202

22.603

21.87

3.244

90

16.951

22.274

21.539

3.298

120

16.729

21.982

21.251

3.323

150

16.532

21.723

20.993

3.363

180

16.355

21.49

20.767

3.369

210

16.194

21.279

20.558

3.39

240

16.048

21.087

20.366

3.421

270

15.913

20.91

20.196

3.412

300

15.77

20.723

19.912

3.913

330

15.627

20.538

19.731

3.929

360

15.486

20.353

19.45

4.436

Table 4: The results table of model validation.

Parameter

Value

Parameter

Value

Porosity

0.12

permeability

1.2mD

Oil viscosity

0.256mPa.s

injecting water viscosity

0.38mPa.s

Injection pressure

46MPa

original formation pressure

27MPa

Comprehensive rock coefficient

0.0034/MPa

effective thickness

5m

Pressure gradient

0.02MPa/m

injecting water intensity

0.044 m3/(d.MPa.m)

Table 5: Calculation parameters table of injection effective time.

Model Application

Predicting water effective time

Water flooding manner are often used in China in the progress of oilfield development. After water injection, the pressure waves in the formation continue to spread from the water wells to the oil wells. When the pressure wave spread to the oil well, the oil well begins to be under the effect, and this reaching time is called Water flooding effective time [14,15].

In the secondary development of oil field in China mainland, the majority of oil fields use water flooding to improve productivity. Therefore, how to predict the water flooding response time plays an important role in adjusting the water injection pressure differential and improving efficiency of water flooding. By replacing r(t) in the equation (21) with well spacing d, and oil production q with intensity of water injection, the model can be used to predict the water flooding response time, expressed as follows.

t= 2πh c t B J ws ( p inj p e ) { C 1 6 r 3 ε 1 ( ε 2 +1)( ε 2 +2) r ε 2 +2 +[ C 1 r e 4 + ε 1 2( ε 2 +1) r e ε 2 +1 r 2 ] }| d r e + Bμ r e 2 J ws ( p inj p e ) 16πkh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9maalaaabaGaaGOmaiabec8aWjaadIgacaWGJbWaaSbaaeaa lmaaBaaajuaGbaqcLbmacaWG0baajuaGbeaaaeqaaaqaaiaadkeaca WGkbWaaSbaaeaadaWgaaqaaKqzadGaam4DaiaadohaaKqbagqaaaqa baGaaiikaiaacchadaWgaaqaamaaBaaabaqcLbmacaWGPbGaamOBai aadQgaaKqbagqaaaqabaGaeyOeI0IaaiiCamaaBaaabaWcdaWgaaad baGaamyzaaqabaaajuaGbeaacaGGPaaaamaaeiaabaWaaiWaaeaacq GHsisldaWcaaqaaiaadoealmaaBaaajuaGbaqcLbmacaaIXaaajuaG beaaaeaacaaI2aaaaiaadkhalmaaCaaajuaGbeqaaKqzadGaaG4maa aajuaGcqGHsisldaWcaaqaaiabew7aLnaaBaaabaqcLbmacaaIXaaa juaGbeaaaeaacaGGOaGaeqyTdu2cdaWgaaqcfayaaKqzadGaaGOmaa qcfayabaqcLbmacqGHRaWkcaaIXaqcfaOaaiykaiaacIcacqaH1oqz lmaaBaaajuaGbaqcLbmacaaIYaaajuaGbeaajugWaiabgUcaRiaaik dajuaGcaGGPaaaaiaadkhadaahaaqabeaacqaH1oqzlmaaBaaajuaG baqcLbmacaaIYaaajuaGbeaajugWaiabgUcaRiaaikdaaaqcfaOaey 4kaSIaai4wamaalaaabaGaam4qaSWaaSbaaKqbagaajugWaiaaigda aKqbagqaaiaadkhadaWgaaqaaKqzadGaamyzaaqcfayabaaabaGaaG inaaaacqGHRaWkdaWcaaqaaiabew7aLTWaaSbaaKqbagaajugWaiaa igdaaKqbagqaaaqaaiaaikdacaGGOaGaeqyTdu2cdaWgaaqcfayaaK qzadGaaGOmaaqcfayabaqcLbmacqGHRaWkcaaIXaqcfaOaaiykaaaa caWGYbWaa0baaeaacaWGLbaabaWaaWbaaeqabaGaeqyTdu2cdaWgaa qcfayaaKqzadGaaGOmaaqcfayabaqcLbmacqGHRaWkcaaIXaaaaaaa juaGcaWGYbWcdaahaaqcgayabeaajugWaiaaikdaaaqcfaOaaiyxaa Gaay5Eaiaaw2haaaGaayjcSdWaa0baaeaajugWaiaadsgaaKqbagaa caWGYbWaaSbaaeaajugWaiaadwgaaKqbagqaaaaacqGHRaWkdaWcaa qaaiaadkeacqaH8oqBjugWaiaadkhalmaaDaaajuaGbaqcLbmacaWG LbaajuaGbaqcLbmacaaIYaaaaKqbakaadQeadaWgaaqaaKqzadGaam 4DaiaadohaaKqbagqaaiaacIcacaGGWbWcdaWgaaqcfayaaKqzadGa amyAaiaad6gacaWGQbaajuaGbeaacqGHsislcaGGWbWcdaWgaaqcfa yaaKqzadGaamyzaaqcfayabaGaaiykaaqaaiaaigdacaaI2aGaeqiW daNaam4AaiaadIgaaaaaaa@D243@  (48)

Much of research had been done about the water flooding effective time, and some of the specific models are as follows. Ge’s model [16]

t=69.44 μ c t ϕ 0 k d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9iaaiAdacaaI5aGaaiOlaiaaisdacaaI0aWaaSaaaeaacqaH 8oqBcaWGJbWaaSbaaeaalmaaBaaajuaGbaqcLbmacaWG0baajuaGbe aaaeqaaiabew9aMTWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaqa aiaadUgaaaGaamizaSWaaWbaaKqbagqabaqcLbmacaaIYaaaaaaa@4BE9@  (49)

Xiu’smodel [17]

t= π c t ϕ 0 ( dp dr ) 3 J ws ( p inj p e ) d 3 +2.89 μ w c t ϕ 0 k 4e e d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9maalaaabaGaeqiWdaNaam4yamaaBaaabaqcLbmacaWG0baa juaGbeaacqaHvpGzlmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaaca GGOaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWGYbaaaiaacMca aeaacaaIZaGaamOsamaaBaaabaqcLbmacaWG3bGaam4Caaqcfayaba GaaiikaiaacchalmaaBaaajuaGbaqcLbmacaWGPbGaamOBaiaadQga aKqbagqaaiabgkHiTiaacchalmaaBaaajuaGbaqcLbmacaWGLbaaju aGbeaacaGGPaaaaiaadsgadaahaaqabeaajugWaiaaiodaaaqcfaOa ey4kaSIaaGOmaiaac6cacaaI4aGaaGyoamaalaaabaGaeqiVd02aaS baaeaajugWaiaadEhaaKqbagqaaiaadogadaWgaaqaaiaadshaaeqa aiabew9aMTWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaqaaiaadU gaaaWaaSaaaeaacaaI0aGaeyOeI0IaamyzaaqaaiaadwgaaaGaamiz aSWaaWbaaKqbagqabaqcLbmacaaIYaaaaaaa@755E@  (50)

Li’s model [18]

t=25.128 c t ϕ 0 ( dp dr ) J ws ( p inj p e ) d 3 +69.44 μ w c t ϕ 0 k d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDai abg2da9iaaikdacaaI1aGaaiOlaiaaigdacaaIYaGaaGioamaalaaa baGaam4yaSWaaSbaaKqbagaajugWaiaadshaaKqbagqaaiabew9aMn aaBaaabaqcLbmacaaIWaaajuaGbeaacaGGOaWaaSaaaeaacaWGKbGa amiCaaqaaiaadsgacaWGYbaaaiaacMcaaeaacaWGkbWaaSbaaeaaju gWaiaadEhacaWGZbaajuaGbeaacaGGOaGaaiiCaSWaaSbaaKqbagaa jugWaiaadMgacaWGUbGaamOAaaqcfayabaGaeyOeI0IaaiiCaSWaaS baaKqbagaajugWaiaadwgaaKqbagqaaiaacMcaaaGaamizaSWaaWba aKqbagqabaqcLbmacaaIZaaaaKqbakabgUcaRiaaiAdacaaI5aGaai OlaiaaisdacaaI0aWaaSaaaeaacqaH8oqBlmaaBaaajuaGbaqcLbma caWG3baajuaGbeaacaWGJbWaaSbaaeaajugWaiaadshaaKqbagqaai abew9aMnaaBaaabaqcLbmacaaIWaaajuaGbeaaaeaacaWGRbaaaiaa dsgadaahaaqabeaajugWaiaaikdaaaaaaa@7635@ (51)

Bao Block of Bao Lang Oilfield is a low permeability reservoir [15], and the basic parameter is shown in Table 6. By using the equation (48)-(51), the effective time can be calculated. Figure 5 shows that this work has a very good match with field data, indicating that this model is effective [19].

From the perspective of oil field development and production experience of China, in the process of actual oil production, the well spacing of Bao blocks is about 250meters. It is well known that from the oil practical field experience, the generally water flooding effective time is about 5 months to 9 months, and the results of the formula (48), (50) and (51) were 212.7 days, 155.2days and 173.6 days, which are consistent with the actual water flooding effective time. The calculation results of the water flooding effective time also demonstrates that the model we offered here is feasible.

Figure 6 shows that, under the condition of pressure wave spreading the same distance, the time used in the model of this paper (this work) is much smaller than Ge’s model. Because in low permeability reservoirs, our model considers the effect of starting pressure gradient and other non-linear flow factors, however, Ge’s model ignores the effect of the starting pressure gradient and reduces the flow resistance of fluid flow in the low permeability reservoir formation. So Ge’s model overestimates the flow capability of fluid in porous media. The time used in this work is a litter bigger than Li’s model and Xiu’s model. Because although the Li’s model and Xiu’s model considered the starting pressure gradient, but they mistakenly regarded the starting pressure gradient as a constant and exaggerates the flow resistance of fluid flow in the reservoir. Whenever the pressure reaches a minimum starting pressure gradient, the fluid of the formation begins to flow in the Li’s model and Xiu’s model, however, in real situations, the fluid flow in the pressure of the formation must overcome the starting pressure. So in order to reach a certain distance, it costs more time.

Figure 5: Effective time graph of the injection wells with different models.

Figure 6: The map of pressure variation versus time in different locations.

Meanwhile, in the process of calculation, we found that in the model of predicting water flooding response time, the coefficient of the well spacing (That is the coefficient associated with the pilot pressure coefficient.) is much smaller than the coefficient of boundary radius (That is the coefficient associated with the starting pressure gradient.). If the water flooding response time is composed of two parts, namely, the time of associating with the pilot pressure and starting pressure gradient, the latter is much larger than the former, which indicates that in low permeability reservoirs, the effective water flooding response time is mainly associated with the starting pressure gradient. That is the reason why reducing the well spacing can effectively shorten the water flooding response time.

Sensitivity analysis

By setting the distance to the well shaft centerline, that is 0 meter, 50 meters, 100 meters, 150 meters and 200 meters, the pressure variations were observed with the time interval of 10 days. Taking different production time, which is 30 days, 60 days, 90 days, 120 days and 150 days, the distributions of pressure in the formation were observed, as shown in Figure 7.

Figure 8 shows that in the production process of the oil wells, before the pressure wave spreading to a certain location of the formation, the pressure remains at the value of the original reservoir pressure. After the pressure wave reaching the location, the pressure begins to decrease with time goes by. when spreading time is the same, in the pressure spreading range, the farther from the bottom, and the higher the pressure is.

In the process of actual oil development, many factors affect the pressure propagation speed in low permeability reservoir, such as formation properties, temperature and pressure systems. However, the model built in this paper shows that the productivity and lead pressure coefficient are the dominate factors affecting the pressure propagation speed. As the lead pressure coefficient is largely controlled by the properties of low permeability reservoirs considering actual production needs, production, permeability and viscosity are investigated in detail. By setting different oil production (that is Q=0.5 m3/d,1.0 m3/d,1.5 m3/d,2.0 m3/d), permeability (that is k=0.1mD,0.5mD,5.0mD,10mD) and viscosity (that is µ=0.5 mPa.s,3.0 mPa.s,10 mPa.s,30mPa.s), the relationships between the pressure and time are shown in Figure 7, 8 and 9. Figure 10 shows that: As the permeability of the reservoir increasing, the speed of the pressure propagation grows faster (the rate of the moving boundary increases quickly). Because with the increasing permeability, the capacity of the formation fluid flow enhances, and the corresponding reservoir resistance decreases, making the pressure wave spread easier and the moving boundary move faster.

Figure 9 shows that as the viscosity increases, the spread of the pressure wave become slowly (the rate of the moving boundary reduces). This is because that the larger the viscosity of the formation fluid is, the greater the fluid flow resistance is, and more time is needed to overcome fluid flow resistance. All of these factors make the rate of the pressure propagation becomes lower and moving boundary move slowly to the reservoir boundary.

Figure 10 shows that, under the condition of the same production time, with the oil production increasing, the faster the pressure wave propagates, and the larger the rate of the pressure wave is. Thus, for the same production time, we have to increase the rate of formation fluid flow to increase the oil productivity. Enlarging production pressure, increasing the pressure gradient as well as the rate of the pressure wave are the ways to increase the rate of formation fluid flow [20]. However, for the constant pressure boundary of the low permeability reservoir, there is no difference of pressure gradient in every certain location, so increasing the rate of pressure wave becomes the only way to increase oil production. On the other hand, increasing oil productivity means enlarging production differential pressure (producing pressure differential is positive with the oil production). It is necessary for the pressure wave to spread to a larger range and increase the area of pressure agitated region. All of these leads to the phenomena that with the improvement of oil well production, the pressure wave spreads faster and farther.

Figure 7: The formation pressure distribution at different time points.

Figure 8: The influence of permeability on the spread of pressure.

Figure 9: The influence of viscosity on the spread of pressure.

Figure 10: The influence of daily oil production on the spread of pressure.

Conclusion

In order to solve the defects of characterization the pressure propagation in low permeability reservoir, this paper develops mathematical models by using steady state replacement method and material balance law based on a new seepage model to study the pressure propagation phenomena in low permeability reservoirs under constant rate conditions.

For non-fracture model, in the area of pressure surge, pressure decreases with the increase of time. At the same time, within the scope of the pressure ripple, the distance from the bottom becomes farther, and the pressure becomes higher. The model of the flood-response time in the low permeability reservoir was deduced, and the relationship between the flood-response time and the injection-production well spacing is polynomial. Thus, reducing well spacing can effectively shorten the flood-response time. This model provides a theoretical basis for the water injection development and the estimation of reasonable well spacing in low permeable oil fields.

For the fracture model, a great agreement between this work and the numerical simulation results was showed in this paper, illustrating that the method in this study produces reliable transient pressure. Also, it can be seen from the sensitivity analysis that the pressure curve can be divided into four flow stream of early linear flow, mid-radial flow, late spherical flow and border control flow. The fracture scale mainly affects the early linear flow, and the fracture orientation mainly affects the late spherical flow. The method can be used to determine the optimal fracture length, fracture width and other parameters, providing theoretical guidance for reservoir engineering analysis and fracturing process design.

Nomenclature

p e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCam aaBaaabaWcdaWgaaadbaGaamyzaaqabaaajuaGbeaaaaa@3AF9@  Boundary pressure, psia

p wf MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCam aaBaaabaqcLbmacaWG3bGaamOzaaqcfayabaaaaa@3CEC@  Bottle whole pressure, psia

p inj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCam aaBaaabaWcdaWgaaqcfayaaKqzadGaamyAaiaad6gacaWGQbaajuaG beaaaeqaaaaa@3E8F@  Injection well pressure, psia

p(r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiCai aacIcacaGGYbGaaiykaaaa@3B77@  Pressure, psia

q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyCaa aa@3929@  Oil production, MSCF/D

r(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai aacIcacaGG0bGaaiykaaaa@3B7A@  Pressure propagation radius, feet

r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCaa aa@3929@  Radius to the center of the well axis, feet

r e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaabaqcLbmacaWGLbaajuaGbeaaaaa@3BF1@  Reservoir boundary radius, feet

r w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCam aaBaaabaWcdaWgaaqcfayaaKqzadGaam4Daaqcfayabaaabeaaaaa@3CBD@  Wellbore radius, feet

d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamizaa aa@391C@  Well spacing, feet

h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAaa aa@3920@  The effective thickness of the reservoir, feet

ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqy1dy gaaa@39FB@  Porosity, fraction

K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbpacaWGlb aaaa@39D3@  Effective permeability, mD

μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 gaaa@39E9@  Oil viscosity, cp

μ w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaeaalmaaBaaajuaGbaqcLbmacaWG3baajuaGbeaaaeqaaaaa @3D7C@  Water viscosity, cp

t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiDaa aa@392C@  Production time, h

ρ o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi 3aaSbaaeaalmaaBaaameaacaWGVbaabeaaaKqbagqaaaaa@3BCE@  Oil density, lbm/scf

c f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yam aaBaaabaWaaSbaaeaajugWaiaadAgaaKqbagqaaaqabaaaaa@3C04@  Liquid compressibility, psia-1

c l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yam aaBaaabaWaaSbaaeaajugWaiaadYgaaKqbagqaaaqabaaaaa@3C0A@  Fluid compressibility, psia-1

c t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4yaS WaaSbaaWqaamaaBaaabaGaamiDaaqabaaabeaaaaa@3A6D@  Total compressibility, psia-1

J ws MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0JLipgYlb91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOsam aaBaaabaWcdaWgaaadbaGaam4Daiaadohaaeqaaaqcfayabaaaaa@3BDD@  Solution gas-oil ratio

Acknowledgement

This article was supported by the Development of Continental Sedimentary Reservoir Simulation System " of Beijing (Grant Nos.Z121100004912001) and The Development of a New Generation of Reservoir Simulation Software of CNPC (2011A-1010).The authors would like to acknowledge support received from the RIPED, Petro China.

References

  1. Chu KH (2004) Improved fixed bed models for metal biosorption. Chemical Engineering Journal 97(2-3): 233-239.
  2. Shukla NB, Madras G (2012) Kinetics of adsorption of methylene blue and rhodamine 6G on acrylic acid-based superadsorbents. Journal of Applied Polymer Science 126(2): 463-472.
  3. Zawani Z, Chuah LA, Choong TSY (2009) Equilibrium, kinetics and thermodynamic studies: adsorption of Remazol Black 5 on the palm kernel shell activated carbon (PKS-AC). European Journal of Scientific Research 37(1): 67-76.
  4. JiaGuo, Ye Luo, Aik Chong Lua, Ru-an Chi, Yan-lin Chen, et al. (2007) Adsorption of hydrogen sulphide (H2S ) by activated carbons derived from oil-palm shell. Carbon 45(2): 330-336.
  5. Piar Chand, Arun Kumar Shil, Mohit Sharma, Yogesh B Pakade (2014) Improved adsorption of cadmium ions from aqueous solution using chemically modified apple pomace: Mechanism, kinetics, and thermodynamics. International Biodeterioration & Biodegradation 90: 8-16.
  6. Shuang Yang, Lingyun Li, Zhiguo Pei, Chunmei Li, Jitao Lv, et al. (2014) Adsorption kinetics, isotherms and thermodynamics of Cr(III) ongraphene oxide. Colloids and Surfaces A: Physicochem. Eng. Aspects 457(5): 100-106.
  7. HongxuGuo, Jianhua Chen, Wen Weng, Zishan Zheng, Dongfang Wang (2014) Adsorption behavior of Congo red from aqueous solution onLa2O3-doped TiO2 nanotubes. Journal of Industrial and Engineering Chemistry 20(5): 3081-3088.
  8. Jinbei Yang, Meiqiong Yu, Ting Qiu (2014) Adsorption thermodynamics and kinetics of Cr(VI) on KIP210 resin. Journal of Industrial and Engineering Chemistry 20(2): 480-486.
  9. Xuan Zhou, Honghong Yi, Xiaolong Tang, Hua Deng, Haiyan Liu (2012) Thermodynamics for the adsorption of SO2, NO and CO2 from flue gason activated carbon fiber. Chemical Engineering Journal 200-202: 399-404.
  10. Llorens J, Pera Titus M (2009) A thermodynamic analysis of gas adsorption on micro porous materials: evaluation of energy heterogeneity. Journal of Colloid and Interface Science 331(2): 302-311.
  11. Yi Peng Teoh, Moonis Ali Khan, Thomas SY Choong (2013) Kinetic and isotherm studies for lead adsorption from aqueous phase on carbon coated monolith. Chemical Engineering Journal 217: 248-255.
  12. Xue Song Wang, Li Fang Chen, Fei Yan Li, Kuan Liang Chen, Wen Ya Wan, et al. (2010) Removal of Cr (VI) with wheat-residue derived black carbon: Reaction mechanism and adsorption performance. Journal of Hazardous Materials 175(1-3): 816-822.
  13. Ahmad B Albadarin, Jiabin Mo, Yoann Glocheux, Stephen Allen, Gavin Walker, et al. (2014) Preliminary investigation of mixed adsorbents for the removal of copper and methylene blue from aqueous solutions. Chemical Engineering Journal 255: 525-534.
  14. Yuh Shan Ho (2004) Pseudo-Isotherms Using a Second Order Kinetic Expression Constant. Adsorption 10(2): 151-158.
  15. Mamdoh R Mahmoud, Gehan E Sharaf El deen, Mohamed A Soliman (2014) Surfactant-impregnated activated carbon for enhanced adsorptive removal of Ce(IV) radionuclides from aqueous solutions. Annals of Nuclear Energy 72: 134-144.
  16.  Adriana P Vieira, Sirlane AA Santana, Cícero WB Bezerra, Hildo AS Silva, José AP Chaves, et al. (2011) Removal of textile dyes from aqueous solution by babassu coconut epicarp (Orbignyaspeciosa). Chemical Engineering Journal 173(2): 334-340.
  17. Johnny Saavedra, Camilah Powell, BasuPanthi, Christopher J Pursell, Bert D Chandler (2013) CO oxidation over Au/TiO2 catalyst: Pretreatment effects, catalyst deactivation, and carbonates production. Journal of Catalysis 307: 37-47.
  18. Lu C, Su F, Hsu SC, Chen W, Bai H (2009) Thermodynamics and regeneration of CO2 adsorption on mesoporous spherical-silica particles. Fuel Processing Technology 90(12): 1543-1549.
  19. Yan G, Viraraghavan T (2001) Heavy metal removal in a biosorption column by immobilized M. rouxii biomass. Bioresource Technology 78(3): 243-249.
  20. Trgo M, Medvidovic Nv, Peric J (2011) Application of mathematical empirical models to dynamic removal of lead on natural zeolite clinoptilolite in a fixed bed column. Indian Journal of Chemical Technology 18(2): 123-131.
  21. Wang S, Peng Y (2010) Natural zeolites as effective adsorbents in water and wastewater treatment. Chemical Engineering Journal 156(1): 11-24.
  22. Handoaui O (2006) Dynamic sorption of methylene blue by cedar sawdust and crushed brick in fixed bed columns. Journal of Hazardous Materials138(2): 293-303.
  23. Sag Y, Aktay Y (2001) Application of equilibrium and mass transfer models to dynamic removal of Cr(VI) by chitin in packed column reactor. Process Biochemistry 36(12): 1187-1197.
  24. Ghorai S, Pant KK (2004). Investigations on the column performance of fluoride adsorption by activated alumina in a fixed bed. Chemical Engineering Journal 98(1-2): 165-173.
  25. C Nguyen, DD Do(2001) The Dubinin-Radushkevich equation and the underlying microscopic adsorption description. Carbon 39(9): 1327-1336.
  26. GO Wood (2001) Affinity coefficients of the Polanyi/Dubinin adsorption isotherm equations: A review with compilations and correlations. Carbon 39(3): 343-356.
  27. Ying Chu Chen, Chungsying Lu (2014) Kinetics, thermodynamics and regeneration of molybdenum adsorption in aqueous solutions with NaOCl-oxidized multiwalled carbon nanotubes. Journal of Industrial and Engineering Chemistry 20(4): 2521-2527.
  28. A Barona, A Elias, A Amurrio, I Cano R Arias (2005) Hydrogen sulphide adsorption on a waste material used in bioreactors. Biochemical Engineering Journal 24(1): 79-86.
  29. Yonghou Xiao, Shudong Wang, Diyong Wu, Quan Yuan (2008) Experimental and simulation study of hydrogen sulfide adsorption on impregnated activated carbon under anaerobic conditions. Journal of Hazardous Materials 153(3): 1193-1200.
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